Fluidity And Plasticity

246

FLUIDITY AND PLASTICITY

seems most reasonable to assume that the temperature is 'without
effect, in which case we should expect the diffusional viscosity
to vary directly as the square root of the absolute temperature.

Maxwell concluded from his experiments that the viscosity
varies directly as the first power of the absolute temperature.
Bams (1889) worked with air and with hydrogen over a very
wide range of temperature from 0 to 1,300° and found that the
viscosity increased as the two-thirds power of the absolute
temperature. Holman (1877) and (1886)) in a careful investiga¬
tion of the subj ect had found the exponent to be 0.77 for air. On
the other hand, easily condensible gases and vapors such as
mercury, carbon dioxide, ethylene, ethyl chloride and nitrogen
peroxide give values of the exponent which are nearly unity,
according to Puluj (1876) and Obermayer (1876); but E. Wiede¬
mann (1876) discovered that the value grows smaller as the tem¬
perature is elevated, which we might have anticipated since they
thus become more nearly like the permanent gases. The vis¬
cosity of many vapors increases even more rapidly than the first
power of the temperature. Schumann (1884) used the formula

n = KT 3A . ( 99 )

Sutherland (1893) believes that “the whole of the discrepancy
between theory and experiment will disappear if in the theory
account is taken of molecular force. * * Molecular attraction
has been proved to exist, and, though negligible at the average
distance apart of molecules in a gas, it is not quite negligible
when two molecules are passing quite close to one another; it
can cause two molecules to collide which in its absence might
have passed one another without collision; and the lower the
velocities of the molecules, the more effective does molecular
force become in bringing about collisions which would be avoided
in its absence.

“Molecular force alone without collisions will not carry us
far in the explanation of viscosity of gases as known to us in
nature, because in all experiments on the viscosity of gases there
is a solid body which either communicates to the gas motion
parallel to its surface or destroys such motion, so that the mole¬
cules of gas must collide with the molecules of the solid; for if the
molecules of gas and solid act on one another only as centers of

THE VIS( Y)»s*/TV OF GASES

247

f oree, then each molecule of gas when it comes out of the range
of the molecular force of the solid must have the same kinetic
Energy as when it went in, so that without collision
ttiol ecules of gas and solid there can be no communication of
motion to the gas. If, then, molecules of gas and solid collide,
txioleoules of gas must collide amongst themselves.”

I xi the theory of diffusional viscosity explained earlier it was
ionLa.de; plain that there would be viscous resistance even if the
auxolecriles failed to collide with each other entirely. But Slither-
land’s view is in accordance with the one we have developed as
oollisional viscosity” in that collisions between the molecules,
"Wlasahewer be the nature and origin of the collisions, have an effect
Upon the viscosity. Sutherland attributes the effect largely to
the attraction between the molecules, whereas the law of Bats-
cliixiskd would lead us to ascribe the effect to the volume of the
molecules. The two points of view are essentially the same.

Sutherland's theory led him to the formula

oir

1 +«C / T\^
C '\ 273 /
1+ T

( 100 )

where a is the coefficient of expansion of a gas and C is a constant.
T his formula has had the most remarkable success of any that
have been proposed, although it does not apply to vapors well.
A single example of its performance is given in the following
ha.~ble, using Holman’s (1886) data for carbon dioxide at atmos-
jptreric pressure.

Examining Sutherland’s formula, we observe that when the

constant C is small in comparison with the absolute temperature
the formula, reduces to the simple theoretical formula

i? = JET*

The discovery, (cf. Vogel (1914)), that Sutherland’s formula fails
a/fc low temperatures indicates that it does not quite correctly
take account of the deviation from the simple formula.

Quite in harmony with the above, it is found that the values

248

FLUIDITY AND PLASTICITY

Table LXVII. —Con cord an ce between Sutherland’s Formula and
Holman’s Data for Carbon Dioxide. C = 277, Vo « 0.000,138,0

Temperature, degrees
Centigrade

t\ X 10 7 observed

17 X 10 7 calculated

18.0

1,474

1,471

41.0

1,581

1,584

59.0

1,674

1,671

79.5

1,773

1,766

100.2

1,864

1,864

119.4

1,953

1,951

142.0

2,048

2,056

158.0

2,121

2,127

181.0

2,234

2,227

224.0

2,411

2,409

of C for different substances increase with the critical tempera¬
ture or boiling-point of the substance. Rankine (1910) obtained
an empirical relation between C and the absolute critical tempera¬
ture T cr

Tor = 1.12C (101)

Table LXVIII. —The Relation of the Constant C in Sutherland’s
Equation to the Boiling-point and Critical Temperature

Substance

T cr , Critical
tempera¬
ture, ab¬
solute

C

Tcr/ C

Tb, Boiling
tempera¬
ture, ab¬
solute

cm

Helium.

9.0

78.2

0.11

4.3

\ 18.3

Hydrogen.

37.0

83.0

0.45

20.4

l *i

Nitrogen.

127.0

113.0

1.12

77.5

1.45

Carbon monoxide.

133.0

100.0

1.33

83.0

1.20

Oxygen.

154.0

138.0

1.12

90.6

1.52

Nitric oxide.

179.5

167.0

1.08

120.0

1.39

Ethylene.

383.6

249.0

1.14

1.17

170.0

1.46

Carbon dioxide.

304.0

259.0

194.0

1.33

Ammonia.

423.0

352.0

1.20

240.0 !

1.47

Ethyl ether.

467.0

325.0

1.43

307.0

1.06

THE VISCOSITY OF GASES

249

which suggested to Vogel a similar relation to the absolute

boiling temperature

C = 1.477\ (102)

This formula indicates that C increases considerably more
rapidly than the temperature, and since Tb is comparatively
large for vapors, the less perfect agreement of Sutherland's
formula is partially explained. This, however, is not true of
hydrogen and helium which present curious anomalies, as shown
in Table LXVIIL

Viscosity and Chemical Composition

If the mass of a particle in a rarefied gas is increased n~fold
by changing its chemical composition, the velocity will be
times the original velocity, so that the momentum of each

Table LXIX —The Viscosities of Permanent Gases and
Vapors at 0°C

o x Molecular w , 7 ! m W1A ,

Substance weight 710 x ^ j TV , ^ X 10 7

j ■ ■ 1 1 ' "~™

Hydrogen. 2.0 850 ; 31.0 j

Helium... 4.0 1,871 5. 21

Methane. 16.0 1,033 183.

Neon. 20.2 2,081

Nitrogen. 28.0 1,678

Carbon monoxide. 28.0 1,672 j 133.

Oxygen. 32.0 1,920 j 154.

Argon. 39.9 2,102 | 155.6 1,253

Nitrous oxide. 44.0 1,362 j

Krypton. 82.9 2,334 | 210.5 1,806

Xenon. 130.2 2,107 j 288 . 2,266

Ethyl alcohol. 46.0 827 j 513.

Acetone. 58.0 725 510.

Methyl formate. 60.0 838 485.

Ethyl ether. 74.1 689 467.

Benzene. 78.0 689 561.

Methyl isobutyrate. 88.1 701 543.

Ethyl acetate. 88.1 690 523.

Ethyl propionate. 90.1 701 547.

250

FLUIDITY AND PLASTICITY

molecule will be n /z -fold that of the smaller molecule. But the
number of excursions of the molecules will be in proportion to
so that the total loss of momentum will be the same as
before, provided only that the number of particles per unit
volume remains the same.

In gases at ordinary pressure, there are considerable differences
in viscosity ranging from 0.0000689 for benzene vapor to
0.0002981 for neon, but they are inconsiderable as compared
with the vast differences we find in the liquid state and these
viscosities are measured at 0° and not under corresponding con¬
ditions. Table LXIX shows that the vapors have viscosities
which are smaller than those of the permanent gases except

Atomic Weiflht

Fig. 82.—The relation between the viscosity of the elements at their critical
temperature and their atomic weights.

hydrogen. Their viscosities are so nearly identical that it is not
certain whether the viscosity of a given class of chemical com¬
pounds such as the ethers differs from that of the esters or
ketones. It is quite impracticable with the data at hand to
assign any effect to an increase in the molecular weight within
a given class of compounds.

Since the viscosities of the permanent gases at 0° are not
simply related to each other, it is natural to seek some other
basis of comparison, and Rankine (1911) has achieved success
along this line by comparing the viscosities of the rare gases rj e
and their atomic weights M at the critical temperatures. He
finds them related together by the formula

Vc 2 = 3.93 X 10- 10 M

THE VISCOSITY OF OASES'

251

as depicted in Fig. 82. The critical constants of neon and niton

have not yet been determined. Rankine has further found that
th.o sam^ general formula applies to the halogens, but the constant
is different being 10.23 X 10 ~ 10 . He gives for chlorine •<> =
1>S<97 >C 10~ 7 and for bromine f} c = 2,874 X 10~ 7 (<*/. Fig. 82).
Were we to use the molecular weights instead of the atomic
^eights, the constant would be 5.12 X 10" 10 which is nearer that
of -the jraxe gases but still not identical with it.

The Viscosity of Gaseous Mixtures

Sinco in. a rarefied gas the viscosity is proportional to the
number of molecules in a unit volume, i.e., to the pressure,
the viscosities will be additive when gases are mixed in varying
percentages by volume; but since the viscosity of a rarefied gas
is also independent of the weight of the molecules, the law
loses its significance.

In gstseous mixtures at ordinary pressures the simple deduced

formula

v = l pTL

still applies, it being merely necessary to find the appropriate
mean -values of p, V, and L. This has been done by Maxwell
C1S68) and Puluj (1879), and one obtains the formula (cf.
Meyer’s Kinetic Theory of Gases , page 201 et seq.)

Gralnam (1846) observed that mixtures of oxygen and nitrogen
or* oxygen and carbon dioxide in all proportions have rates
of transpiration which are the arithmetical mean of the two

components. Thus for air,

0.0001678 X 0.7919 = 0.0001329
0.0001920 X 0.2081 = 0.0006399

CaLcutla-ted. viscosity of air . 0. 0001728

Observed viscosity of air . 0.0001724 Vogel (1914).

Graham and others have noticed that when hydrogen is mixed in

252

FLUIDITY AND PLASTICITY

small amounts with other gases, as carbon dioxide or methane, the
viscosity of the mixture is much greater than would be calculated
by the simple formula of additive viscosities. In these cases
Puluj (1879) and Breitenbach (1899) have found that the more
complicated formula (103) gives good agreement.

Viscosity of Gases and Diffusion and Heat Conductivity
We note that the diffusion coefficient D in a mixture of gases is
D - ~ 7 rCtfjLxfli + NiUSh)/N (104)

O

N h Li, and Oi being the number of molecules of the first kind
of gas per unit volume, the length of the mean free path, and
the mean speed respectively, etc. Also N = iVi + iW Since
the length of the mean free path can most easily be calculated
from the viscosity, it becomes possible to calculate the diffusion
coefficient from the viscosity.

In the conduction of heat the two kinds of gas become identical,
hence the above equation becomes

D = ^irQL (105)

If we neglect the small difference between Qi and 0 due to
temperature difference the conductivity of heat k becomes

& = ^ 7i QLpC„ (106)

C v being the specific heat of the gas at constant volume, and
combining this equation with the viscosity Eq. (97) we obtain

k = Ct)C v (107)

C being a constant (cjf. Eucken (1913)).

Determination of the Ultimate Electrical Charge

It is well known that Sir J. J. Thomson (1898) devised a
method for measuring the charge on the particle of a rarefied
gas e by observing the rate of fall under gravity of the particles of
an ionized fog which had been produced by sudden expansion and
then observing the rate of fall of a similar cloud when it is sub¬
jected to the action of a vertical electrical field of known intensity
superimposed upon gravity.

THE VISCOSITY OF GASES

253

If v is the velocity of a droplet of mass m, density p under the
action of gravity alone, and i’i its velocity when under the in¬
fluence of the electrical field whose strength is *Y in elatrostatic
units, then

v mg

“ = - \v~ 1 10S :

vi mg +■ Ae

Applying Stokes’ Law, Eq. (62), to the sphere whose volume
Is 4&r'J3, we obtain

A beautiful application of this method has been made by
Millikan (1909, etc.)- He has found the most probable value
for e to be 4.69 X 10~ 10 . This leads to the number of molecules
in a gram molecule N = 6.18 X 10 23 and the mass of the hydro¬
gen atom a>s 1.62 X 10” 24 g.

Chapman (1916) and Rankine (1926-1) have calculated the
diameters of the atoms of the monatomic gases from determina¬
tions of the viscosity. They regard the atoms as hard spheres
having the well-defined absolute diameters given below.

Atomic Diameters or some of the Noble Gases after Raxxint.

Viscosity

Crystal measurement

Neon... .
Ajrgon.. .

Krypton..
Xenon.. .

2.35 X 10- 1
2.87 X KT 1
3.19 X 10- 1
3.51 X 10- 1

61.30 X 10“‘
2.05 X lO" 1
2.35 X 10~ l
62 70 X 10” 1

These values agree very well with those obtained from van
der WaaTs equation hut they are somewhat greater than the
diameters of the outer electron shells of the atoms as obtained
by Bragg from his crystal measurements.

CHAPTER X

SUPERFICIAL FLUIDITY

The viscosity of a liquid may change, and it may change
in a quite extraordinary manner, as the boundary of the liquid
is approached. This must of necessity result wherever the
surface tension is such as to bring about a change in concentra¬
tion at the boundary. We should therefore naturally expect
soap and saponin solutions to show this phenomenon. Experi¬
mentally this field of study has not been much explored although,
as we shall attempt to show, the promise of reward is very great
and the need of such study in industry is pressing. However,
Stables and Wilson (1883) have proved that a saponin solution
has a viscosity at the surface which is 4,951 as compared with
3.927 for the surface of pure water. The viscosity was measured
by the oscillations of a circular nickel-plated brass disk, of 7.625
cm diameter and 0.2 cm thickness, which was suspended in the
liquid by means of a wire 119.8 cm long. As soon as the solution
was allowed to rise 0.15 cm above the disk the viscosity fell to its
normal value.

The viscosity found by Stables and Wilson indicates that
the surface layer of a supposedly dilute solution may nevertheless
have a viscosity which is over a thousand-fold that of water
at 20°C (1,260 cp) or about the viscosity of castor oil. But for
very small stresses, the viscosity may be still higher, for it
is to be particularly noted that in a saponin solution a pendulum
does not oscillate isochronously. Thus in one experiment
with vibrations of large amplitude. Stables and Wilson found the
time of vibration to be 10.52 seconds, whereas with small ampli¬
tudes the time of vibration was 9.73 sec. This would indicate
that with very small stresses the viscosity might be found to be
infinite, which would mean that we are here again dealing with,
plastic flow.

The experiments of Stables and Wilson need confirmation and

254

SUPERFICIAL FLUIDITY

extension with our more recent knowledge of the nature of flow
in mind, but whatever the surface of a given saponin solution may
be, we may profitably distinguish three typical cases: [A) where
the superficial layer is a true solution but of different concen¬
tration. from the interior and is in contact with it own vapor or
some gas; (B) where the surface is made up of a layer of immisci¬
ble liquid, which may be so thin as to be imperceptible by ordi¬
nary means; (C) where the surface is formed either by a continuous
solid or by solid particles in more or less intimate contact with
each other. It is evident that in the last two cases we are dealing
not with the superficial fluidity of the liquid but of a heterogene¬
ous mixture of liquid-liquid or solid-liquid respectively.

Soap solutions perhaps afford the best examples of the first
case and if such solutions have extraordinarily high superficial
viscosity, it serves to explain the stability of the soap bubble.
The liquid between the two highly viscous surfaces can proceed
downward very slowly in so narrow a space.

Oil films on water give frequent examples of the second sort,
and the use of oil “to calm troubled waters” is a practical appli¬
cation. of superficial viscosity in the damping of vibrations. The
simple harmonic motion of the wave causes the particles to move
in vertical circles, so that an oil film is alternately stretched and
compressed. The water underneath not being subjected to this
same tendency is pulled along by the oil film and in this viscous
flow energy is of course dissipated. A method for the measure¬
ment of viscosity by Watson (1902) depends upon the damping
of small waves in a free surface, and apparently this method is
capable of being used to measure superficial viscosity, but this
appears not to have been attempted.

The connection of superficial fluidity with emulsions must be
mentioned at this point although we cannot stop to discuss it.
We can merely refer the reader to the fascinating studies of Pla¬
teau, Quincke, and Lord Rayleigh upon the nature of contamina¬
ting films. The recent paper by Irving Langmuir (1919) on the
theory of flotation is very suggestive.

Many of the examples which we would naturally cite as exam¬
ples of the second case given above may really be examples of the
third instead.. It is certain that in most emulsions a third sub¬
stance is necessary to stabilize it and it may give rigidity. Scums

256

FLUIDITY AND PLASTICITY

are apparently examples of this class. Gurney (1908) in investi¬
gating the contamination of pure water surfaces on standing,
says “ Water surfaces become noticeably rigid in a few hours or
days: depending on the previous history of the fluid. Vigorous
stirring destroyed the rigidity of the surface.”

To prevent possible misunderstanding, it must be stated
again that rigidity in foams and emulsions arises largely from
the fact that during shear the bubbles of a foam or the
globules of an emulsion are distorted and may be disrupted,
and thus wort is done against the forces of cohesion opposing
such disruption.

Superficial viscosity has heretofore been considered at a free
surface only. Such a view is too narrow as it would leave the
most important examples out of consideration and from the
theoretical aspect the extension of our conception of superficial
fluidity involves no difficulty whatever. Having made this
extension, the phenomenon of slipping falls into the third case,
but the fluidity near the boundary is higher than that of the main
body of material. Henry Green (1920) has studied this slippage
under the microscope, using for observation paint colored with a
little ultramarine, which may be subjected to shearing stresses
in a capillary tube. With small stresses the shear takes place
exclusively in the region near the boundary, but when the stress
becomes greater than the yield value of the paint, the shearing
takes place throughout the material. Green reasons that it is
this mixture of the kinds of flow which causes the shear to fail
to be a linear function of the shearing stress, particularly when
those stresses are near the yield shearing stress. In the above
example, the layer next to the boundary was more fluid than the
main body of material, but more often the opposite is the case, the
fluid near the boundary is less fluid, and we might therefore consider
the general subject of adsorption under this head. And we would
then show that it is possible to make a fractional separation of
fluids by simply passing them through capillary tubes. Such a
separation of a mixture into its components by means of capillary
flow has actually been demonstrated, as in the case of petroleum
forced through clay by Gilpin and his co-workers (1908). 1 Since
the surface area of a capillary varies as the first power of the

i Am. Chem . J. 40, 495 (1908); 44, 251 (1910); 60, 59 (1913).

SUPERFICIAL FLUIDITY

2A7

radius whereas the volume of flow varies as the square of the
radius, Eq. (6), we may expect to find the effects of superficial
fluidity shown to the best advantage in very fine tubes.

There "are a variety of causes which may cause the fluid near
the boundary to have a different fluidity. The most important
cause results from the selective adhesion of the components of
the fluid for the solid. If one of the components of the fluid is
more strongly attracted than another, separation becomes possi¬
ble, and the magnitude of the fluidity of the mixture as measured
will theoretically be affected The adhesion between solid ami
liquid or liquid and liquid is doubtless just as specific a property
as is the better known cohesion or surface tension of liquids and
we are coming to understand the nature of adhesion better
through the efforts of Langmuir (1919) and Harkins (1920). We
have seen that it is possible to greatly affect both the friction
and the mobility of plastic substances by the addition of small
amounts of acid or alkali. Just what happens in such eases
might be subject to dispute, but it is certain that small amounts
of substances adsorbed on to the surface of a solid may cntirelv
change the character of the solid which is in contact with the
liquid. Thus Henry Green (1920) has observed that the addition
of small amounts of gum arabic to a suspension may greatly
decrease the yield value and increase the mobility, in spite of
the higlh viscosity of gum arabic solutions. This is interpreted
as being due to the decrease in adhesion between the sus¬
pended particles. The well-known work of Sckroeder (1903(
upon the effects of electrolytes on the viscosity of gelatine and of
Handowsky (1910) upon serum albumin should also be referred
to.

We have already proved on page 86 that if any cause results
in the fluid near the boundary becoming different from the re¬
mainder of the liquid, the resulting fluidity will be changed.
This theorem is therefore useful in explaining superficial fluidity.
We will now prove that the components of a mixture under these
conditions will undergo partial separation. The conditions
will be made more general by using the n on-homogen eons mixture
considered on page 86. Considering the mixture as made up
of the two components A and B, arranged in alternate plane
layers, the total quantity of A flowing in a unit of time, regardless

258

FLUIDITY AND PLASTICITY

of whether it is derived from the fluidity of A or £, is obtained
from the terms of Eq. (26) containing n, and is

(« Vi + ab<? 2 )

and similarly the rate of flow of component B is

V 2 = ^(bZw+^AAab^y

There will be separation of the two components only when the
thickness of the different layers is considerable or when the
passage through which the substances are forced is very small,
for in either case n will be small. If n = <»,

Ui a
U 2 b

and there will be no separation at all. The separation may be
calculated from the expression

Ui a na<pi + (n - 1)6^2 , 110 \

U% b (n - f- l)a<pi -f- nb<p 2

When n = 1, the component A will flow at only one-third of
the rate of B, even though the two components have the same
fluidity and are present in equal proportion; and even if the
fluidity of B is zero, it will flow twice as rapidly as A , under
the above conditions. It follows that the flow of B is greatly
increased by making the fluidity of A large, this being the layer
in contact with the stationary boundary.

An ingenious application of the principle of superficial fluidity
was made by the Southern Pacific Railroad, 1 when it was found
that the pressure required to pump certain heavy oils through
long pipe lines was inconveniently large. The problem was
to get the maximum flow of oil for a given expenditure of energy
and with a given diameter of pipe. By using a rifled pipe and
injecting about 10 per cent of water along with the oil, the
water was thrown to the outside of the pipe by the centrifrugal
action caused by the rifling, producing a high superficial fluidity;
and thus, by a seeming paradox, the water lubricated the oil so
that the delivery became from 8 to 10 times what it would have
been had the water not been added.

One may demonstrate the effect of superficial fluidity very

1 Engineering Record y 67 , 676 ( 1908 ).

SUPERFICIAL FLUIDITY

259

simply by comparing the times required by gravity to empty two
pipettes filled with a heavy oil, each of the pipettes being similar
in every respect except that one is moistened with water previous
to filling with oil.

In an experiment by the author at 25°C and a pressure
of 60 g per square centimeter a given volume of water required
33 sec. and the same volume of cottonseed oil required 1,640 sec.
A mixture was then used containing one-third oil and two-thirds
water by volume. Had the heavier water flowed completely
through the capillary ahead of the oil, the time of flow should
evidently have been 22 + 547 = 569 sec.; yet only 391 sec. were
actually required which is less than the time theoretically required
by the oil alone. The difference of 178 sec. is due to the water
forming a lubricating film for the oil as the water drained out
through the capillary t

Rate of Absorption.—It is appropriate here to show how the
rate of absorption of a fluid into a porous material depends upon
the fluidity of the medium. From Poiseuille’s Law, Eq. (8), it

follows that the rate ^ at which a liquid enters a long capillary

tube under the driving force P will be

dl = PrV
dt SI

If the capillary is very small, the surface tension y exerts a force

2y

— which must be added to the external pressure and this force

arising from the surface tension may be so great that the external
pressure is negligible in comparison, in which case

dl __ (pyr

dt “ IT

and by integration

The quantity of 0 .5<py is called the coefficient of penetrance of
the fluid and it is a measure of the tendency of a liquid to pene¬
trate a given material which it wets. ( Cf . Washburn Physical
Chemistry, 2d ed., p. 62.)

The distance that a liquid will penetrate a given porous mate¬
rial due to capillary action alone is often of practical importance.

260

FLUIDITY AND PLASTICITY

From the above equation we see that this distance is propor¬
tional to the square root of the fluidity, the surface tension, the
radius of the capillary and the time.

It is generally assumed that the material of the pore walls is
immaterial so long as the walls are wet by the liquid. Adhesion
between solid and liquid may come into play in certain cases
making such an assumption fallacious, as already pointed out.
Experiments on the impregnation of fabrics, belting, wood et
cet with oils, gums, paints et cet have shown that thorough dry¬
ing of the former materials has an extraordinary effect upon the
penetration of the latter. This may be due to increasing adhe¬
sion although it may be explained in some other way.

CHAPTER XI
LHBMCATIOM

"Wlieix & solid substance is subjected to a shearing stress,
it undergoes plastic flow if the stress is greater than the yield

value of the material. In this process of shear, lateral stresses

arise and if the material is not supported laterally by sufficient
pressure, rupture of the material will finally result. These
surfaces formed by rupture slide over each other according to the
laws of solid friction stated by Coulomb. The surfaces are
separated for the most part by a layer of fluid which may be air,
water, oil, a layer of oxide, etc. So two surfaces formed by a
rupture as, for example, two broken pieces of porcelain do not
adhere together firmly even when they seem to fit together very
nicely. So also the resistance to movement between ordinary
smooth surfaces is far less than the resistance to plastic flow.

If, however, sufficient force is brought to bear between two
sliding surfaces of similar material, there will occur, far below
the molting point of the substance, a welding together of the
surfaces into a more or less compact whole, unless there is
present, some substance which prevents such welding. Two
surfaces of glass ordinarily touch each other at very few points
and tbey do not adhere strongly, but when the two surfaces
are ground to an optical surface and cleaned, it is difficult to
separate the two surfaces without tearing them, after they have
been brought together. A motor bearing which has been care¬
fully fitted by “lapping in” may be ruined completely by a
slight turn with the hand after the surfaces have been cleaned
and again brought together. Powdered metals adhere strongly
when subjected to heavy pressures, even at temperatures con¬
siderably below the melting point. The Johannsen blocks used
in gage testing are made of hardened steel with surfaces which are
exceptionally true. When these blocks are placed one on top of
the otber, the adhesion between them is so great that a pile of

262

FLUIDITY AND PLASTICITY

them several inches high can be raised by lifting the topmost
one. In imperfect lubrication we first have excessive wear, then
scoring of the bearings and finally seizure with a more or less
complete welding together of the surfaces. Thus there is a con¬
siderable mass of evidence to prove that whenever two clean
surfaces come together they adhere and thus the conditions for
plastic flow may be reestablished. The problem of lubrication
is therefore to substitute as far as possible fluid friction for the
enormously higher resistance to shear in plastic flow.

According to the above view, “ solid friction,” as ordinarily
observed, is intermediate between true plastic flow and true
viscous flow. Under favorable conditions it approaches closely
to simple viscous flow, whereas under very unfavorable conditions
it may approach the conditions for plastic flow. It is clear
therefore that the coefficient of solid friction may vary within the
widest limits depending upon the condition of the bearing
surfaces, the temperature, speed, and character of the lubricant.

Thus at the outset we may state that it is impossible to
specify the lubricant that will be most suitable for a given
machine, provided that that machine works at variable speeds,
temperatures and loads, and where the bearings are continually
subject to wear due to defective lubrication. On the other
hand, if bearings are perfectly lubricated and run under constant
conditions, there is practically no wear, so that the problem to
find the most suitable lubricant has a definite solution. With
the steady advance of industrial development, the theory of
lubrication takes on increasing interest.

The laws of solid friction may be stated as follows: (1) When
two unlubricated smooth surfaces slide over each other, the fric¬
tional resistance P varies directly as the load W or

P = fF (111)

and the coefficient of friction f is defined as the ratio between the
friction and the load.

2. The force P 0 required to maintain an indefinitely small rate
of shear, the so-called static friction , is greater than when the
rate of shear is appreciable. The dynamic friction is independent
of the velocity.

3. The friction is independent of the area of the surfaces in

LUBRICATION 2fL1

^F>parent contact, within wide limits. The surfaces must, how-
^Ver, be large enough so that the surfaces remain intact.

Since it is impracticable to obtain a pair of smooth and entirely
^lrxlubricated surfaces, it is needless to say that these laws are very
inexact. As already intimated, well-fitting and clean surfaces of
similar material would probably seize and follow the laws of plastic
flow, which are very different from the laws given above. They
have, however, both historic interest and practical usefulness.

Just as the laws of solid friction are superficially unrelated
to the laws of plastic flow, so these laws are also in sharp contrast
to the laws of viscous flow which apply to well-lubricated surfaces.
With well-lubricated surfaces we have the relation

where S is the area of surface in contact, dv is the velocity and dr

is the thickness of the oil film. According to this relation:

1. The frictional resistance P is independent of the load.

2. The friction is directly proportional to the velocity and is

■therefore zero when the velocity is zero.

3. The friction is also directly proportional to the area of sur¬
faces in contact.

In view of the absolute antithesis between these two sets of
Jaws, it is not surprising that the results of the study of friction
as recorded in the literature are often contradictory. We may,
Jiowever, state broadly that slow-moving, poorly lubricated sur¬
faces follow approximately the laws of solid friction, whereas
arapid-moving and hence necessarily well-lubricated machinery,
such as electric dynamos and motors, follows the laws of fluid
friction. Most bearings are imperfectly lubricated and follow
neither set of laws exactly.

Pefcroff (1887) seems first to have applied the laws of fluid
friction to lubricated bearings testing out Ms views by experiment.

Most important in its relation to the development of the theory
of lubrication is the experimental work of Beauchamp Tower
(1883-4), undertaken at the instance of the Institution of
Mechanical Engineers. His experiments were conducted with
extreme care and under varied and well-chosen circumstances.
His results, as obtained under ordinary conditions of lubri-

264

FLUIDITY AND PLASTICITY

cation, “so far agree with the results of previous investigators as
to show the want of any regularity.” He perceived that this
difficulty was due to irregularity in the supply of lubricant, so
he conducted experiments in an oil bath. Not only was he thus
able to obtain a high degree of regularity but he proved that the
journal and bearing are completely and continuously separated
by a film of oil. This film is maintained by the motion of the
journal against a hydrostatic pressure in theoil, which at the crown
of the bearing was shown by actual measurement to be 625 lb.
per square inch greater than the pressure in the oil bath.

Tower demonstrated that even with an oily pad in contact
with the journal, the results were regular although the results
were different from those with the oil bath. Of lubrication less
than that afforded by the oil pad he says: “Theresults, generally
speaking, were so uncertain and irregular that they may be sum¬
med up in a few words. The friction depends on the quantity
and uniform distribution of the oil, and may be anything between
the oil bath results and seizing, according to the perfection or
imperfection of the lubrication.”

These experiments of Tower are indeed a landmark in the
development of the theory of lubrication for they stimulated
various investigators such as Osborne Reynolds, Stokes, and Lord
Rayleigh to apply the fundamental hydrodynamical equations
to the results obtained. And the labors of Reynolds, continued
by Sommerfeld (1904) and Michell (1905), have in fact enabled
us to reach a complete solution of the problem of lubrication in
certain very special cases. The mathematical integrations have
generally proved very difficult.

Reynolds 7 Theory of Lubrication

The model of viscous flow which we have considered, page 5,
does not give rise to any pressure at right angles to the direction
of flow, hence it is unable to sustain a load permanently and will
not serve for practical lubrication.

Case I. Parallel Surfaces Approaching with Tangential Motion .
Let AB in Fig. 83 represent the section of a surface which is
moving with the uniform velocity U in respect to the bearing
block CD, each being of indefinite length in the direction perpen-

LUBRICATION

2fw

dieular to the paper. As soon as a load is placed on the hearing
"block, the liquid begins to be squeezed out from bctw* »t. the
surfaces. If this space is divided originally into the equal area,*
irtdicated by the dotted lines, these lines, moving with the fluid,
will after a time occupy the positions of the curved lice-; and
the distances moved by the particles are shown by the dhtaiuv*
between the corresponding points on the two se?,> of eurvt u-
QP for the pointP, and the slopes of the curves indicate the direc¬
tions of the forces in the fluid just as if the lines were stretched

F

Fig. 83-—The simplest case of lubrication. Two parallel, plane ?urfa<’es.

elastic threads. The pressures exerted upon different points
along CD are shown in the curve of pressures CFD> the pressures
being proportional to the vertical height above the line CED.

the center of the block the pressure is a maximum and the
liquid is squeezed out to the right and left of this section. For
“this section alone, there is a uniform variation of velocity from
^AB to CD, such as would be true of all sections, if the surfaces
jAB and CD were not approaching.

Case II. Surfaces Inclined—Tangential Movement Only. —■
If now the bearing block is tilted, we have fulfilled the necessary
condition for continuous lubrication, for the bearing is able
to sustain a load without the surfaces approaching each other.

"Were we to assume that in this case the velocity varies uni¬
formly from V at AB to zero at CD, the quantity of fluid passing
any cross-section MN would be proportional to MN X U/2,
or simply to MN. But since the quantity of fluid passing every
cross-section must be the same, there must he an outflow to
the right and left of the cross-section M f N% at which the pressure
is a maximum, so the flow at any section MN is

(MN — MWO U/2

At the cross-section MN, the velocity varies uniformly from

266

FLUIDITY AND PLASTICITY

AB to CD , but the point of maximum pressure M is not at the
center of the block nor is it necessarily the point of application
of the resultant pressure exerted on the block.

If the bearing is free to move, it will move either up or down
until the pressure is just equal to the load. As the load is
increased, the surfaces approach each other, which increases
the friction and thereby the pressure so that equilibrium is
restored. But the point of application of the resultant pressure
changes with the load provided that the inclination of CD remains
the same.

Case III . Revolving Cylindrical Surface—Bearing Surface
Flat .—The curves of motion are represented in Fig. 84. To the

Fig. 84.—Simple continuous lubrication.

right of CH which is the point of nearest approach of the sur¬
faces, the curves are similar to those in Case II. At the left of
GH, the curves are quite the reverse of those on the right, being
convex toward a section M 2 N 2 on either side, just as they are
concave to a section M±N 1 on the right. The reason for this is
that with a uniformly varying velocity more fluid would be
brought in at the right of M 1 N 1 than would pass the section GH,
hence the fluid must flow outward from M x Ni, where the pressure
is a maximum in both directions. So at the left of GH more
fluid would be carried away than arrives through GH, hence an
inflow is necessary to the right and left of the section of minimum
pressure The fluid pressure acts to separate the surfaces

at the right and to draw them together at the left hence there is a
couple of forces resulting.

If the bearing is cut away at the left of GH, the negative pres¬
sures may be eliminated.

lubrication

267

If the oil supply is limited, the oil may not wet the entire
bearing but form an oil pad in. the region of GH 1 the pressures of
course reaching a zero value at the points where the oil surface
meets the bearing surface. If d is the thickness of the oil him
outside of the pad, the quantity brought up to the pad per
second will be Vd, and the quantity which passes the section
MjNx where the velocity varies uniformly is M U Y 2 V 2, and since
there is no accumulation of oil, these two values must be equal
and

MiJS i = 2 d

also MsN 2 = 2d

Case IV, Revolving Cylindrical Surface — Bearing also Cyh n-
drical. —In a very common example of lubrication we have u
cylindrical journal partly or wholly surrounded with the bearing
or “brass” CD in Fig. 85. The oil is drawn up into the space BD

Pig. 85.— The lubricated journal and bearing.

and creates a pressure which is a maximum at Q. The point of
nearest approach between journal and bearing is not at the
middle of the bearing 0 but at a point some 40° further on
at G toward the so-called “off-side” of the bearing. This is
the opposite to what happens in the unlubricated bearing, for

268

FLUIDITY AND PLASTICITY

the point of nearest approach is then on the “on-side.” Only
when the bearing is unloaded does the point of nearest approach
coincide with the.middle of the brass, 0. As the load increases
the point G moves from 0 up to a certain maximum value after
which it recedes toward 0, resulting finally in a discontinuity in
the oil just as in the case of a limited supply of oil.

We have considered only bearings of unlimited length, whereas
in practical bearings the lubricant is squeezed out at the sides,
as well as at the ends. Michell (1905) has made a study of the
changes of pressure in the oil film of bearings of various shapes.
Generally speaking the integrations necessary to define the exact
relations between load, speed and the friction have not been
effected.

The theory of lubrication is not inconsistent with the experience
that the friction in limited lubrication is proportional to the
load and independent of the velocity. Increase of load will
result in a diminution of the distance between the bearing
surfaces, a lengthening of the oil-pad, and therefore an increase
in the resistance. Increasing the velocity increases also the
resistance, but it also increases the pressure and therefore the
distance between the surfaces, provided that the load is kept
constant, and this produces a decrease in the resistance.

For further details of the development of this very important
subject the reader is referred to the original papers of Petroff,
Tower, Reynolds, Sommerfeld, Michell, Lasche to name but a few.

Lubrication and Adhesion

In the early use of lubrication, fixed oils and greases were
depended upon almost exclusively. The fixed oils, that is
the non-volatile oils of animal or vegetable origin, are expensive,
they may become gummy and rancid, which interferes with
proper lubrication and the acids developed may corrode the
machines. These oils moreover often partially solidify when only
slightly cooled. The range of viscosities obtainable is also
restricted by the small number of oils available in any quantity.
With the advent of mineral oils, these troubles were all overcome,
so the battle which was waged between the mineral and the
fixed or fatty oils was short and apparently decisive. The

lubrication

2m

Lirveyors of the fatty oik claimed that their oik po>>
renter u oiliness, n “body” or “lubricating value,"* bn! since
^se claimants seem never to have proved their ease by the

clonal measurement of “ oiliness"’ and since modern indii>-

rialism requires vastly more oil for lubrication than could pus-
iblyr be met by the available supplies of fatty oik, the concept it m
f "the property of oiliness has gradually become a sort of mill u
toe* wisp vaguely referred to in treatises on lubrication, and effee-

ively used by energetic salesmen in convincing a pro?qxx*rivf
uxyrer of the superiority of a given brand of oil over all others
Ffcte theory predicted that so long as the viscosity was sufficient to
produce the necessary pressure required to carry the loud, i*
vas of no moment what the chemical nature of the lubricant
tQ-Ight be, provided only that the quantity of lubricant was ample.
Tile practice has therefore been to use an oil which is much more
viscous than is really necessary and to accept a loss in power
in order to insure against any discontinuity in the oil film.

There are, to be sure, many instances which might be cited
wliere an experienced engineer has cooled a hot bearing by
substituting a fixed oil with which he was familiar for the mineral
oil in use. However, in comparing two oils used for practical
lubrication, there are so many factors which may affect the
comparison such as the quantity of oil, the speed, load, tem¬
perature of the oil film, the condition of the bearing surfaces,
tlaa/t instances which might be cited are easily discredited by the
skeptical. Nevertheless, there is a growing demand for lubri¬
cants which will be less wasteful of power and which will at the
same time give the maximum assurance that the bearings will not
fc»e injured in use. With the aeroplane in particular, it is neces-
sary to keep the motor going at all hazards during most of the
p>eriod of flight, and an overheated bearing may easily cause the
complete wreckage of the machine in mid-air, so the selection
of the best lubricant for severe conditions and the question of
* * oi!iness ,? becomes now vitally important. Perhaps the clearest
evidence on this point is obtained from cutting lubricants.

Cutting Lubricants.—It is the well-nigh universal testimony
of mechanicians that in certain cutting operations, fixed oils are
absolutely necessary and that mineral oils wiE not serve as a
satisfactory substitute- Voluminous correspondence with large

270

FLUIDITY AND PLASTICITY

shops all over this country, with concurring evidence from Great
Britain, establishes the fact that fixed oils, preferably lard oil,
are superior to all others. This is particularly true in operations
such as “parting off” soft steel, in threading wrought iron or
steel, in drilling deep holes in steel as in the manufacture of gun
barrels. The tool keeps its edge longer, the machine runs more
smoothly, there is less heating, a much greater speed may be
attained, the chip is less serrated and therefore longer, the cut
surface is smoother and much closer dimensions may be obtained,
when using lard oil or its equivalent.

On the other hand, there are certain operations such as planing
and reaming where a lubricant is not required. In others such
as sawing metals a liquid may be used merely to cool the work.

No lubricant is ordinarily used in cutting cast iron, brass or
aluminum. Wrought iron and “draggy” metals require a
lubricant.

Between the two extremes of those operations and materials
which absolutely require a fixed oil and those which require no
liquid at all, there are a great number of classes of work in which
mineral oils are satisfactory but where aqueous soap solutions
or oil-emulsions are widely used and found to be highly satisfac¬
tory. In these cases the oil or water serves to reduce the heating
of the work and the tool, and the soap or soda prevents the rust¬
ing of the machine. Fixed oils are often a needless extravagance
or positively disadvantageous.

Where lard oil is required it is not primarily to conduct away
the heat, for the operation may be a light surfacing operation
where the heat developed is slight as in the cutting of fine micro¬
meter screws. Its superiority does not depend on its peculiar
viscosity because a mineral oil possessing the same viscosity in
no way shares its superiority.

It is true that mineral oils increase in fluidity, when heated,
more rapidly than fatty oils, but castor oil is exceptional in this
respect resembling the mineral oils and yet it appears to be a very
useful cutting oil and lubricant.

It has also been suggested that pressure might decrease the
fluidity of the mineral oils less rapidly than that of the fixed oils,
but this explanation appears to be not even qualitatively correct
(cf. page 89, Report of the Lubricants and Lubrication Inquiry

LUBRICATION 271

Committee. Department of Srien.ee and Industrial Hese.tr ih.
(London)).

The surface tensions of mineral and of fixed oils are n< >t n u: t* -ria i!v
different. These oils are however very different in one imp. rt.ii.t
respect viz., that the fixed oils all have an active du niic.il croup
which gives them a strong adhesion for metals, so that sr.<h an
oil is mot readily squeezed out from between two metallic sur¬
faces (</• Langmuir (1919), Harkins (1920), and Bingham 1921
Lord Layleigh (1918) has shown that a layer of lubricant : f

Fig. 86. —Illustration of the necessity for high adhesion in an oil which m to
have the best lubricating quality.

monomolecular thickness possesses truly remarkable properties
in reducing the friction between solid bodies of similar material,

the contamination, probably serving to prevent the welding
together of the surfaces. According to Langmuir (1920) such
a film formed from paraffin oil can be readily removed by a gentle
stream, of running water from platinum, glass, etc,, but a film

formed by oleic acid cannot be thus removed.

Xo get a clearer idea of the action of a cutting lubricant we mill
follow the operation of an Armstrong parting tool in cutting off
disks from a rod of soft steel 1% in. in diameter, using a lathe with

a constant speed and feed and as lubricants a definite amount of
lard oil or of mineral oil of the same viscosity. Some 30 disks
were made with lard oil and at the end there was no evidence

272

FLUIDITY AND PLASTICITY

that the operation was not as satisfactory as at the beginning.
The disk shown in the left of Fig. 86 was perfectly smooth, and
there was little evidence of heating and on inspection the tool
was found to be not even slightly dulled. The chips shown below
the disk were only slightly serrated.

On substituting the mineral oil heating began at once, the
surface of the disk shown at the right of the figure was very
rough, the chips were deeply serrated, and the tool so dulled that
it failed completely on cutting the fourth disk. On examining
the cut in the bar at the time of failure, shown in the middle of the

Fig. 87.—Illustration of the forming of a chip in the cutting of metals and of
the function of the lubricant.

figure, one can plainly see two beads of metal flowing ahead of
the tool and gouging into the bottom of the cut. A burr is
being thrown up at the left.

The operation of a tool in cutting is illustrated diagrammati-
cally in Fig. 87. The metal b is being cut away by the tool c, a
chip f being formed which bears down heavily upon the tool at
a point d some distance back from the point. That this is the
actual case is proved by many facts. For example, a tool in use
is often gouged out by the shaving at some distance back from
the point, and there is sometimes found a “bead’* of metal

LUBRICATION

273

welded to the tool at this point. The tool therefore pries the
chip away rather than cuts it, and the point of the tool merely
clears up the surface, so long as the tool is well lubricated.

The surface of the chip is serrated and of about twice the
thickness of the cut. We have here evidently a case of plastic
flow* The explanation of the serrations and the thickening
is probably as follows:—As the tool moves into, the metal, the
strain gradually increases and a certain accommodation takes place
due to the elasticity of the metal and the machine. When the
shearing stress reaches the yield point, the metal flows, and the
more rapidly as the temperature rises rapidly in the region of
flow. In this process the pressure on the tool is relieved, the
stress falls again below the yield point, and the process is repeated.
If the machine is very sturdy with very little play, the cutting
will be steadier, but here comes the advantage in the use of a
good lubricant, that it is drawn into the space m, contaminates
the under side of the freshly formed surface of the chip and there¬
fore substitutes viscous flow for the energy-consuming plastic
flow to a greater or less degree depending upon the efficiency of
the lubricant ( cf . Taylor, “The Art of Cutting Metals”).

If the lubrication is not effective, the pressure on the tool must
be relieved to a greater extent by means of plastic flow of the
material. The result is greater fluctuations in pressure, the metal
flowing outward during the period of flow, producing serrations
of increased height, and possibly flowing downward into the
space m. It is this metal, flowing inward toward the work and
the point of the tool which creates the most serious condition,
for it tends to break off the edge of the tool and to gouge into the
face of the work.

With brittle substances such as cast iron, it is readily per¬
ceived why a lubricant is not necessary. The chip breaks
as it is pried off and there is comparatively little if any plastic
flow. In cutting very hard and brittle materials such as glass
and some varieties of steel, a lubricant as such is not needed, but
something which perhaps has just the opposite property of
causing the tool to adhere to the material, $.e., will cause the
tool to “take hold” or “bite.” Turpentine is used for this
purpose on steel and turpentine with or without camphor is
used on glass. It is difficult to see how these substances act

274

FLUIDITY AND PLASTICITY

unless they serve to remove the contaminating film of grease
which is already present.

These results lead one to the observation that in difficult cases
of lubrication, where seizure is always possible and is almost
certain to be very disastrous, the use of pure mineral oil may not
be the best practice. On the other hand, there is not enough of
the fixed oils to supply the imperative demands of mankind for
edible fats, soaps, leather dressing, et cet. Fortunately however
it is likely that all of the benefit of the use of lard oil as a lubricant
can be obtained very cheaply by adding to mineral oils small
amounts of certain substances possessing high adhesion, par¬
ticularly substances with unsaturated groups in their molecules,
such as are found in oleic acid, turpentine, pine oil et cet . Some
of these substances are already being used on a somewhat
extensive scale in successful substitutes for cutting oils. The
use of these substitutes opens up a field for research which
is most fascinating and in view of the approaching exhaustion
of our supplies of petroleum, the study is so practical that it
cannot long be postponed. Of its importance we can do no
better than quote from an editorial in the Chemical Trade
Journal for December 1920: u Before the war the annual expendi¬
ture on lubricants in England was £6,000,000 and it is estimated
that an annual saving of one to two millions could be effected
if a systematic investigation were undertaken and the results
made freely available to the public. Furthermore the loss
caused by improper lubrication, would represent a very large
addition to the figure given above.”

Asphalt-base Versus Paraffin-base Oils. —With lubricants in
use made from crude oils from different fields, the question has
arisen whether the paraffin-base or the asphalt-base oil is supe¬
rior, but there is a notable lack of convincing evidence in favor
of either. We offer the following evidence to prove that the
differences between them may be very considerable, and that
the chemical composition as determined by the source of the
oil is not a matter of indifference to the consumer; this is par¬
ticularly true in aeroplane lubrication where the results of faulty
lubrication are so very disastrous. 1

1 The walls of the aeroplane motor, the crankshaft et cet. are made so
light that the seizure of a single bearing will result in the wrecking of the

LUBRICATION

275

Benzene (C 6 H 6 ) represents a typical paraffin-base hydro¬
carbon, diallyl (C 6 Hi 0 ) may be taken to represent an unsaturated
nort-cyclic hydrocarbon, whereas benzene (C 6 H 6 ) and hexa-
methylene (C 6 H l2 ) represent types of cyclic hydrocarbons. All
of these compounds have the same number of carbon atoms, but

Temperature, Centigrade

3?r<a. 88.—A comparison of the fluidity-temperature curves of hydrocarbons
of different homologous series.

whereas their fluidity-temperature curves are nearly parallel,

they are widely different as shown in Fig. 88, the fluidity of
the cyclic compounds being extraordinarily low even at their
boiling points, marked by large circles in the figure. The higher

engine in mid-air, due to the sudden confining of the gas mixture within
the cylinders of the engine. Flying parts of the engine resulting from such
an explosion may also injure the steering mechanism, the supporting planes,

or even the pilot.

276

FLUIDITY AND PLASTICITY

fluidity of tlie paraffin is strikingly shown by introducing a
paraffin residue (CH 3 ) into the benzene ring, which results in
toluene (C 6 H 8 ) having a higher fluidity than benzene (C 6 H 6 ).
On the other hand, toluene has a much lower fluidity than the
purely paraffin compound heptane (C7H14) which contains the

Fig. 89.—Fluidity-vapor-pressure curves of hydrocarbons of different homolo¬
gous series. (C/. Fig. 58.)

same number of carbon atoms. It may be urged that whereas
these compounds contain the same number of carbon atoms they
do not contain the same number of hydrogen atoms. But one
should also note that diallyl contains more hydrogen atoms than
benzene and less than hexamethylene and yet has a fluidity
which is far higher than either. The cyclic compounds may owe
their low fluidity to association, but the relation of association
to the properties desired in a lubricant is not well understood.
However, the relation between fluidity and vapor-pressure,

LUBRICATION

277

already discussed (pages 155-160) is not without interest in this
connection.

Although hexane, diallyl, benzene, and hexamethylene differ
in fluidity by more than 250 absolute units at a given tempera¬
ture, they all boil within 20 degrees of each other, hence the
fluidity-vapor pressure curves for these hydrocarbons are very
distinctive, as shown in Fig. 89. If a low vapor pressure for
a given fluidity is an advantage, on the assumption that an oil
should not volatilize off from the walls of an engine cylinder or
away from an overheated bearing, then straight chain hydro¬
carbons have the apparent advantage. On the other hand, if
low vapor-pressure and high molecular weight for a given fluidity
result in a tendency toward carbonization, then cyclic com¬
pounds will be preferred.

Table LXX.— Average Fluidities and Vapor Pressures for Corre¬
sponding Temperatures

Tem-

Toluene 1

Benzene 2

Hexamethylene 3

Hexane 4

pera-

ture,

Vapor

Vapor

Vapor

Vapor

degrees

<p

pres-

*

pres-

<p

pres-

<P

pres-

C

sure

sure

sure

sure

0

110.8

25.3

252.3

45.4

10

131.5

45.2

84.7

47.0

281.8

75.0

20

154.1

75.6

101.7

76.9

312.4

120.0

30

192.3

32

178.0

120.2

121.3

121.3

344.8

185.4

40

214.6

60

203.1

183.6

141.8

181.6

378.8

276.7

50

238.6

94

228.8

271.4

164.8

269.2

414.9

400.9

60

262.7

140

256.1

390.1

188.7

385.0

452.7

566.2

70

287.8

203

284.9

547.4

214.5

540.8

494.6

787.0

80

314.5

291

313.8

751.9

|

90

342.8

405

1

100

371.4

560

110

400.0

751

1

1 Vapor pressure calculated from Kahlbaum, Zeitschr. f. physik. Chem.
26, 603 (1898). Fluidities from Bingham and Harrison, Zeitschr . /. physik.
Chem. 66, 1 (1909), and in the case of hexamethylene, hitherto unpublished
data of Bingham and van Klooster.

2 From Young, /. Chem. Soc . (London) 66, 486 (1889).

3 From Young and Fortney, J. Chem. Soc. (London) 76, 873 (1899).

4 From Thomas and Young, J. Chem. Soc . (London) 67, 1075 (1895).

278

FLUIDITY AND PLASTICITY

Anti-friction Metals. A hearing in usually made of a different
material from th(* journal, but the composition of the so-called
anti-friction metals varies within wide limits. It must be soft
enough so that the bearing may be easily scraped and quickly
run in to an exact fit. During the process of “running in” the
bearing, the particles of metal doubtless serve to wear down the
high spots of the softer metal, leaving the harder journal in a
highly polished condition. The maximum wear is naturally
where the journal and bearing are in closest proximity, hence if a
bearing has been run in with motion in a given direction, the
coefficient of friction will be altered if the direction of motion is
reversed, as observed by Tower. The bearing must be hard
enough to carry the load without flow of the metal. It seems
probable that the material of the bearing should have as small
an adhesion for the metal of the journal as practicable and in case
of necessity the material of the bearing should be capable of
acting as a lubricant. At any rate it should not tend to seize
the journal even when molten. Ice may be regarded as the
oldest anti-friction material, and from certain points of view
it is ideal. Binee it melts under pressure*, it furnishes its own
lubricant and adhesion does not occur due to pressure. A
sleigh standing on moist iee may become frozen in which is
evidence*, that adhesion is not impossible even between ice and
steel. Adhesion between unlike materials is less serious how¬
ever because of their different coeffietiemts of expansion.

Tin is a common (constituent of anti-friction metals and
there, was a serious shortage* of tin during the late war. The
lack of tin ores in the United States makes very desirable the
knowledge of alloys which do not contain tin and at the same
time are useful for bearings. Experiments indicate that lead
containing a very low percentage of metallic calcium is very
satisfactory.

Whether the alloy should have a certain motajlographic
structure, as for example, crystals of comparatively hard material
imbedded in a softer amorphous solid is a moot question.

CHAPTER XII

FURTHER APPLICATIONS OF THE VISCOMETRIC
METHOD

There are many further applications of the viscometric
methods which are destined to become of considerable importance
as soon as the theory of viscous and plastic flow is thoroughly
understood. In many cases however, our knowledge at present
is very restricted, or it is the closely guarded property of some
industry. Generally speaking however, progress has been held
back because the viscosity data at hand could not be interpreted
and because the distinction between viscous and plastic flow was
not recognized. An illuminating example of this has been
described by Mr. Gardner and Mr. Ingalls. 1 The American
Society for Testing Materials attempted to compare with all
care some 240 samples of paint, applying them to a test fence at
Arlington. The paints were all made up to have the same
“viscosity” as measured by the Stormer viscometer. Mr.
Gardner says of the tests, “The determinations were.fallacious.
What was actually done was to make some paints of a very low
and some of a very high yield value, although they all measured
up to the same viscosity. The result was that when some of the
paints were applied to the boards, they would flow and carry the
pigment particles down, leaving bare spots. Some paints
failed on this account.”

According to Batschinski’s Law the fluidity varies some
2,000 times as rapidly as the volume which is now used success¬
fully in the dilatometric method, hence the viscometric method
should be most useful in chemical control work. Dunstan,
Thole and their coworkers have led the way in solving chemical
problems by means of viscosity measurement. They have
studied, for example, the keto-enol tautomerism, the effect of
conjugate bonds, the order of chemical reactions the existence of

1 Proc. A. S. T. M., 19, Part II, (1919).

279

FLUIDITY AND PLASTICITY

2H0

racemates in solution, the location of transition points such as the
one be,tween Na^SO* and NaaBCh.lOlPiO. 1 Further work along
this line is needed to differentiate the effects of chemical composi¬
tion, constitution, and association, measuring the fluidities over a
range of temperatures. Ah in other lines of physical chemical
investigation, the importance of making determinations at more
than one temperature can hardly be overestimated because
substances must be compared under conditions which are truly
comparable.

Various colloidal solutions such as those of rubber, glue, vis¬
cose, nitrocellulose, dextrine, gluten, et cet.j offer problems of
importance which can he most appropriately solved by the vis¬
cometer. It is already known that the properties of a solution
of caoutchouc, for example, determine the character of the rubber
which can be manufactured from it. The exact relation of the
viscosity of the sol to the plasticity of the gel is practically a
closed hook. To indicate how complex the phenomena may be,
we may add that. Carl Rerquist of the Corn Products Refining
Company has found in an investigation of corn dextrines, tapioca
dextrine*, borax, gums and starches that as the percentage of
dextrine increases during the process of cam version, the* mobility
steadily rises whereas the* friction first, falls, then rises, and again
falls. 3 The quick setting of a gum seems to be associated with a
high friction. Thus the addition of .25 per cent sodium hydrox¬
ide to a 8,33 per cent Pearl starch reduced the mobility from
0.7214 to 0.3018 but increased the friction from 108 to 156 g
per square centimeter. The alkaline starch will set harder and
have 11 better body” than an add starch.

Nitrocellulose Solutions.— The fluidities of nitrocellulose solu¬
tions as calculated from the determinations of Raker (1913)
would indicate that nitrocellulose solutions never become true
solids as the percentage of nitrocellulose is increased, for the
fluidities approach the zero value asymptotically. This con¬
clusion is, however, so inherently improbable that it should be
confirmed. Since it was necessary to use a series of Ostwald
viscometers in order to get the necessary range, and each one is
calibrated from another, the possibility of error is considerable.

1 Of. Dunstan and Langion (1912).

* Privately communicated. Of. Herschel and Rergquisfc (1921).

APPLICATIONS OF THE VISCOMETRIC METHOD 281

So it may well be that nitrocellulose solutions in various non-
aqueous solvents may be brought into line with other colloidal
solutions, some of which have already been considered, page 198.
If, for a given nitrocellulose, there is a zero of fluidity which is
independent of the particular solvent, an empirical formula of
the general type

_ <Pi

where K is a constant, may be serviceable, ( cf . Duclaux and
Wollman (1920)).

Colloidal solutions of the above types which have a lattice-
work or sponge-like structure show an increase in the fluidity
when subjected to treatment which breaks up this structure.
Astonishingly small quantities of the disperse phase are necessary
to give zero fluidity or at any rate a very great viscosity.

Certain non-polar emulsion colloids, such as milk, are in some¬
what sharp contrast with the above, because fairly high percent¬
ages of the disperse phase alter comparatively little the fluidity
of the medium and the reduction of the size of the fat gluobles
by “homogenizing” decreases the fluidity.

Attempts are being made to use the plasticity method in the
study and control of butter and other fats and greases. As a
means for distinguishing between different fats and greases and
of determining the amount of the “hardening” of oils in the proc¬
ess of hydrogenation, or of oxidation in the blowing of oils, the
method offers opportunities which have not been exploited as
yet. Similarly it seems practicable to estimate the amount or
quality of gluten in samples of flour by this method.

Clay and Lime. —Suspension colloids offer a simpler set of
conditions than can be found anywhere else. Clays, plasters,
mortars, and cements, are all plastic and their plasticity is a
matter of prime importance in their respective industries.
Commenting on the influence the plasticity of plaster has on its
economic usefulness, Emley (1920) states that about 70 per cent
of the total cost of plastering a house is accounted for in the
labor required to spread the plaster. “If one plaster is more
plastic than another, it means that the plasterer can cover more
square yards in a given time with the former than with the latter,
which, of course, will reduce the cost. Furthermore the more

2K2

FLUIDITY AND PLASTICITY

plastic material entails lens physical and mental fatigue on the
part of the* plasterer, and lie* is thereby led unwittingly to produce
a better quality of work.”

Kmley points out that the method of slaking the lime has much
to do with the development of plasticity, but that quite as impor¬
tant, is the source, and by inference, the chemical composition of
the lime*. A lime high in magnesium oxide is capable of develop¬
ing a high plasticity more readily than one which is low in the
dolomitic oxide*. The growing practice of buying Ohio finishing
lime, already hydrated, even when local lime may be purchased
for about one-half the* price is a reflection of the almve facts and
is a demonstration of the industrial importance of plasticity.

In handling road-building and roofing materials, a knowledge
of the principles of plastic flow might enable m to avoid losses.
The, first principle of road building is to secure proper drainage,
which is in accord with the theoretical requirement of keeping
the yield value* tm high as practicable. The “metal” of the rail¬
road is made up of coarse crushed stone of uniform size which
gives excellent drainage and a very high yield jH>int. Where
liquid hydrocarbons are used as binder, a considerable amount of
fine material must be used in order to raise the yield point suffi¬
ciently to sustain the contemplated loads. In order to be able
to apply the material the mobility is greatly increased by raising
the temperature.

Paints and Pigments.—Paint must have a yield value high
enough ho that it will not run under the influence of gravity
but. the mobility must also be high so that the painter may spread

it without undue fatigue. Other things being equal, these ends
are, both achieved by the use of finely-divided materials, and at

the same time the covering power is augmented. Perrott (1919)
has made a study of the plasticity of “long” and “short” carbon

blacks.

Up to 1914, Austrian ozokerite was thought to be essential
in the wax used in making electrotypes. Research has shown
that a good impression can be obtained and hold with waxes
which do not contain the Austrian material

Textiles and Belting.— If a cotton window cord is run over a
free pulley a certain number of times under a load which is small
in comparison with the* tensile strength of the cord, it may fail

APPLICATIONS OF THE VISCOMETRIC METHOD 283

while another cord, apparently no better as judged by the weight,
tensile strength, method of fabrication and length of staple will
last perhaps one hundred times as long. It is evident that oxi¬
dation or decay cannot play an important part because the fail¬
ure may be brought about in a few hours. It is not due to
friction of the pulley as the pulley in all cases is running free. The
surprising thing about it is that the cord often wears out on the
side which is away from the pulley, or the center of the cord may
become completely pulverized while the outside is apparently
sound.

An analysis of what happens when a belt moves over a pulley
shows that the outside of the belt moves along a longer arc and
therefore tends to get ahead of the inside of the belt. There is
consequently a shearing stress set up within the belt. Since the
individual fibers are comparatively weak, it is of the utmost
importance that the individual fibers be protected from undue
strains. In order to obtain relief where the strains are greatest,
a lubricant between the fibers and plies should always be pro¬
vided. A rosined bow adheres to a violin string and in the pro¬
duction of sweet sound accumulates stresses advantageously,
but the workman who gets rosin on a machine belt with the idea
of gaining greater traction, may quickly bring about the destruc¬
tion of the belt. A certain amount of slipping of a belt and
particularly in the belt is necessary and desirable.

Lard and certain fixed oils are used to “ stuff ” leather, and a
good leather belt will practically never wear out if well-used and
dressed with lubricant occasionally. Window cords are often
lubricated with a soft paraffin. The paraffin has a tendency to
work out in use and since it becomes hard at low temperatures,
it then tends to make the cord stiff. Pitch and its congeners
is unsuitable for use on textile belting due to its having a high
temperature coefficient of fluidity. What is needed as a lubricant
is a substance which adheres strongly to the material, lubricates
the fibers, and has a small or negligible temperature coefficient of
fluidity. Oils which serve well with leather will not fill the
coarser pores of textile belting, hence rubber, balata, and semi¬
drying oils are often used. In ordinary fabrics a certain amount
of oil present will add to their life. Even a wire rope will last
longer if there is lubricant between the strands.

284

FLUIDITY AND PLASTICITY

Metallurgy.—The terms hardness, ductility, pliability, mallea¬
bility are terms which are probably, like the term plasticity,
complex in character and may in time come to be more precisely
defined in terms of friction and mobility. It is desirable to
know the friction and mobility of each modification of each
metal and their several alloys, and also the effect upon these
properties of changes in crystal size or shape and in the amount
of amorphous solid between the crystals. This subject merits
extended treatment. We know that annealing gives the crystals
opportunity to develop whereas cold working tends to break up
the crystal structure and thereby toughens the metal. Quench¬
ing the hot metal of course prevents crystal growth and should
decrease the yield point. There is no doubt but that polishing
and similar operations result in a plastic flow of the surface layers
of a metal.

Biology, Medicine and Pharmacy.—It would be out of place
here to treat in detail of the very numerous papers which have
been devoted to biological subjects. Beginning with Poiseuille
who was first drawn to his study of fluidity through his interest
in the circulation of the blood in the capillaries, there has been a
continued interest in the viscosities of animal liquids. The
viscosity of the blood in various individuals and species of animals,
in various pathological conditions as well as under the influence
of anaesthetics and drugs, the effect on viscosity caused by differ¬
ences in diet, age, sex, or temperature outside of the body, the
effect upon the viscosity of the blood produced by the removal of
certain organs of the body have all been subject to investigation.
The composite character of the blood has prompted inquiries in
regard to the viscosity of blood serum and defibrinated blood as
compared either with blood as it exists within the animal or as it
is freshly drawn. The other body fluids, milk, lymph, perspira¬
tion, the vitreous humor, et cet., have all been studied and carefully
reviewed by Rossi (1906).

Rossi finds that preceding the coagulation of a solution there is
an increase in viscosity which is the best measure of the progress
toward coagulation. The more viscous the original solution, the
more rapidly does the formation of the gel proceed. Fano and
Rossi (1904) found that electrolytes always first cause a drop in
the viscosity which is then followed by a rise as the concentration

applications of the VISCOMETRIC METHOD 285

is increased. All liquids in the body, whether circulating or not
have the minimum viscosity compatible with their colloidal

content.

Oxygenated blood according to Haro (1876) is much more
fluid, than blood through which carbon dioxide has been made to
bubble, the ratio between them being 5.61 to 6.08. Mere phys¬
ical solutions of small amounts of gases in liquids usually affect
the fluidity but imperceptibly, but the data on this subject needs
amplification. Ether and ethyl alcohol added to the blood
increase its fluidity, whereas chloroform has the opposite
effect.

Poiseuille compared the rates of flow of blood serum through
glass tubes and through a given vascular territory varying the
viscosity of the serum by various additions. From the correla¬
tion it has been assumed that the laws of Poiseuille apply to the
flow of blood through the capillaries of the body. Ewald (1877)
has questioned this conclusion and Huebner (1905) has noted an
incongruity in the rate of flow of solutions of known viscosity in
tbe organs of a frog. When blood flows through the capillaries
tbe corpuscles are deformed and the capillaries are more or less
elastic. The problems connected with the viscosity of the blood
are complicated by the fact that the fibrinogen of the blood in
contact with foreign substances produces coagulation which may
produce a coating on the inside of the tubes. Lewy (1897) how¬
ever has found that Poiseuille’s law holds good so long as no sedi¬
mentation takes place, hence the more viscous the blood the
longer it will take to diffuse through a given vascular territory.

Burton-Opitz (1914) found that fasting produced a pronounced
increase in the fluidity of the blood of a dog. A meat diet has
the greatest effect in lowering the fluidity, a fat diet next, and a
diet of carbohydrates least of all. The fluidity of the serum
varies in a manner similar to that of the blood in these particular
experiments.

Bleeding a dog causes the fluidity of the blood to decrease.
When a dog was kept in a bath at 43°C the fluidity of the blood
increased, and it decreased when the temperature of the bath was
lowered to 23°, the most rapid change taking place in from 5 to
15 minutes according to Huerthle (1900).

According to Huebner the red blood corpuscles account for

286

FLUIDITY AND PLASTICITY

from two-thirds to three-quarters of the viscosity of the blood.
The fluidity of the blood of cold-blooded animals is higher than
that of warm-blooded animals, but the rabbit is peculiar among
warm-blooded animals in having blood of exceptionally high
fluidity.

Milk .—The fluidity of the milk of a cow differs from day to
day as well as with different individuals and at different periods
of life. The fluidity of woman's milk is highest directly after
childbirth and falls off nearly 50 per cent during the period of
nursing. The milk of goats is less fluid than that of cows.
According to Cavazzani (1905) the addition to milk of small
amounts of NaOH or KOH produces a change in the fluidity of
the milk of a cow, goat, or horse but does not affect the fluidity of
woman's milk. According to Alexander 1 human milk contains
a protective colloid not present in cow's milk, hence coarse curds
are not formed on adding acids.

The action of ferments upon milk has been studied by Gutzeit
(1895) and Fuld (1902). The decline in the viscosity of a solu¬
tion of proteins during digestion by means of trypsin has been
the subject of study by Spriggs (1902). The greater part of the
loss in viscosity occurs considerably before the completion of
the digestion, according to Bayhss (1904). This is in accordance
with the idea that the destruction of the structure must lower the
viscosity tremendously, whereas the splitting of microscopic
particles may increase the viscosity and the splitting of amicro-
scopic particles decreases the viscosity. Spriggs (1902) and
Zanda (1911) investigated the changes in viscosity during diges¬
tion by pepsin.

Ceramics and Glass Making. —The thorough mixing of glass
melts, the removal of bubbles of gases, and the pressure necessary
to blow the glass at a given temperature all depend upon the
fluidity of the melt, hence the control of the fluidity of glass
melts is of importance.

The Seger cone method of determining temperatures suggests
the possibility of measuring high temperatures by the viscometric
method. Barns proposed to use the viscosity of a gas for this
purpose.

The manufacture of porcelain is concerned with the principles

1 J. Soc. Chem . Ind. 28, 280 (1909).

APPLICATIONS OF THE VISCOMETRIC METHOD 287

of plastic flow at every stage. Clays must have a friction high
enough so that the ware will not lose its shape while in the moist
condition and at the same time it must have a mobility which is
high enough so that the clay may be readily worked and it must
not shrink badly on drying. On heating, the more fusible parti¬
cles must soften sufficiently to weld the particles together, but
again the friction must be sufficient so that there will be no serious
loss of shape. When the glaze is added, it must fill the pores
quickly and yet not “run.” So many problems in plastic flow
seem to call for precise control of conditions in order to avoid
large losses.

It is found that considerable amounts of non-plastic clay, fine
sand, or ground porcelain (grog) may be added to a very plastic
clay without greatly lowering its plasticity. Until more data
is accumulated, this may remain something of a mystery, but
these additions are valuable and probably serve somewhat the
function of the “reinforcing” in concrete or of the colloid in
“ solidified alcohol.”

Geo-physics. —Basic lavas are notably fluid as compared with
acidic lavas which are more viscous. This has important bear¬
ings upon the character of volcanic eruptions in different parts
of the world and presumably therefore upon the past history of
the earth. For example, the Hawaiian volcanoes with a highly
basic lava tend to remain open, flow quietly, build a low-angle
cone, the lava spreading out over a large amount of territory. On
the other hand, the Mexican volcanoes with acidic lava are apt
to harden over during quiescence and then erupt violently. A
low-angle cone is impossible. In accordance with the relationship
between the fluidity of the melt and the rate of crystallization,
we should expect to find the basic lavas more coarsely crystalline
than those of a more acidic nature. The length of time required
for an obsidian to take on a cryptocrystafline, microcrystal¬
line or even macrocrystalline character will of course also de¬
pend upon the temperature and to a lesser extent upon the
pressure as well as the chemical composition, for all of these
factors influence the fluidity. Silicate melts have been studied by
Doelter (1906).

Segregations, as in the separation of iron from slag, is depend¬
ent to a certain extent upon the fluidity of the slag and of the

288

FLUIDITY AND PLASTICITY

molten metal. Feild (1918) has investigated the viscosity of
slags.

The sodium silicate used in industry contains varying hydroxyl
ion concentration. An excess of silicic acid increases the adhes¬
iveness but lowers the mobility. Excess of alkali has the opposite
effect. The alkalinity of sodium silicate is therefore obviously
an important control factor.

Conclusion.—If one plots the viscosity-concentration curves
of a colloid sol of the type of gelatine in water or of nitrocellulose
in acetone, one finds that the viscosity rapidly goes from the
very small viscosity of the pure solvent (O.Olp for water at 20°
and 0.003 for acetone at 25°) to an extremely high value which
may be regarded as infinite, in a concentration of only a few per
cent. Plotting these curves leads to unsatisfactory results,
which need not be exhibited here as they are very common in the
literature; the curves fall together at one extreme as soon as one
tries to represent more than the most dilute solutions, and where¬
as they may or may not coincide at the other extreme, we can
form no idea of what happens since that extreme is infinitely
removed from us.

If however we plot fluidities instead of viscosities the whole
problem becomes immediately simplified, for the fluidities of the
pure solvents assume their proper importance and the fluidity
goes to or, at any rate, approaches zero, which is accessible.
Moreover the concentration of zero fluidity has a definite and
important significance. If the relation turns out to be also
linear, then the problem is one of ideal simplicity.

To go over all of the data in the literature, critically examining
the data to see how far it could be used to support and further
amplify the theories set forth in this work has been a pleasant
task but far too great for a single worker. Already several
workers are in the field and in the Index and Appendix I are
bringing together a considerable number of references and tables
in order to facilitate the work.

A consideration of the following data may aid any who are
interested in the theoretical study of colloids or in their industrial
applications, since they help us to answer the very important and
novel questions: “Are fluidity-temperature curves linear in the
case of emulsoid colloids of the type of gelatine?” “Are their

APPLICATIONS of the VISCOMETRIC METHOD 289

fluidity-concentration curves linear?” and “Does the fluid ‘soP
pass into the plastic ‘gel’ at a perfectly definite concentration and
temperature ?”

ArisJZ (1915) has made a valuable study of the viscosity of a
10 per cent gelatine sol in a glycerol-water mixture of 1.175
specific gravity, with changing temperatures. Calculating the
fluidities, we obtain the linear curve

<P = 0.000227 (i t - 45.2) (112)

which represents faithfully the observed values of the fluidities
given in Table LXXI. The temperature of zero fluidity is a
little over 45°, which must therefore be regarded as the transition
point between the fluid and solid phases, i.e., the melting-point.

Table LXXI.—The Fluidities of 10 per cent Gelatine Solution in
Glycerol-water Mixture of Different Temperatures (after Arisz)

Temp erature,
degrees

Viscosity,

observed

Fluidity,

observed

Fluidity,

calculated

44

30,000

0.000003

0.0000

46

5,000 ±

0.00020+

0.0002

47

4,200-

0.00024-f

0.0004

SO

950 ±

0.00105 +

0.0011

S5

415

0.0024

0.0022

65

222

0.0045

0.0045

In considering the effect of concentration on fluidity, we cite
first the data of Lxiers and Schneider (1920) on flour-water mix¬
tures for 20°, giving the concentrations in volume percentages
(1006) and changing the viscosities to fluidities, Table LXXII.
The fluidities are again faithfully reproduced by means of a
linear formula, viz.,

<p = 100.5 - 569.65 (113)

except the last two concentrations, where the observed fluidity
is too high. This gives a concentration of zero fluidity as 17.6
per cent, which corresponds to the transition from viscous to
plastic flow. This is very close to the concentrations of zero
fluidity found by Bingham (1916) for clay suspensions, page 229.

19

290

FLUIDITY AND PLASTICITY

Table LXXII.— The Fluidities of Various Volume Concentratio
of Malt Flour-water Mixtures at 20°C (after Luers and

Schneider)

X g flour displaces 0.6766 ml of toluene

Volume, per cent

Viscosity relative,
observed

Fluidity absolute,
observed

Fluidity absolute,
calculated

0.000

1.000

100.5

100.5

0.338

1.010

99.5

98.6

0.676

1.042

96.5

96.6

1.352

1.090

92.2

92.8

2.704

1.192

84.3

85.1

5.408

1.433

70.1

69.7

8.120

1.844

54.5

54.2

10.816

2.566

39.2

38.9

13.520

3.459

29.0

23.5

16.240

4.839

20.8

8.0

There are certain cases where the fluidity-concentration
relation is not linear as in case of Baker’s data for nitrocellulose
solutions. It seems unwise to make any sweeping deductions
in regard to the meaning of these curves until they are confirmed
by further observations, for it is possible that correction terms
applied to the measurements might serve to rectify the curves.
It is perhaps needless to add that if the curvature is real, it will
have an important bearing upon the nature of colloids. I am
engaged upon a study of this whole matter at the present time.

In a few cases, of which one has been already cited, page 207,
the fluidity curve consists of two linear branches, meeting at
a sharp angle. This would seem to indicate a transition point,
but in view of the uncertainties connected with the measurements
of the viscosities of colloidal solutions, we may be pardoned for
extreme caution in making such an assumption until the real
existence of such a singular point has been thoroughly verified.
Such a singular point is found for example in the data for aqueous
solutions of sodium palmitate at 70°C as determined by Farrow
(1912). The formula

<p = 240.4 - 503.9b F (114)

represents the relation between the fluidity and the weight

APPLICATIONS OF THE VISCOMETRIC METHOD 291

concentration up to a concentration of 0.175 but this and the
remaining observations are reproduced by the formula

<p = 207.7 - m.7b w (115)

Neither formula gives the true fluidity of water at 70° (245.7)
when the weight concentration b w of palmitate is zero, and we
obtain two different concentrations corresponding to zero fluidity,
using the two formulas, of 0.477 and 0.677 respectively, which are
difficult to interpret.

Table LXXIII.— Fluidities of Aqueous Solutions of Sodium Palmitate
at 70°C (after Farrow)

Concentration by
weight

Viscosity, ob¬
served

Fluidity, observed

Fluidity, calcu¬
lated

0.026

0.00438

228 .

227'

0.064

0.00485

206

208

0.105 !

0.00530

189

188

► Formula I

0.131

0.00579

173

174

0.141

0.00592

169

169

0.175

0.00647

155

152

0.175

0.00647

155

154]

0.187

0.00665

150-

150

0.232

0.00743

135

136

0.283

0.00793

126

121

0 287

0.00866

115

120

0,293

0.00847

118

118

Formula II

0.302

0.00886

113

115

0.329

0.00949

105

107

0.380

0.01113

90

91

0.451

0.01363

73

69

0.499

0.01800

56

55.

W. L. Hyden has made a study of the plasticity of solutions
of nitrocellulose in acetone, but the results have not yet been
published. He finds that these colloidal solutions differ from
the suspensions of clay et cet. studied by Durham in that the
concentration of zero fluidity at a very low shear, i.e ., the transi¬
tion from viscous to plastic flow, occurs in an extraordinarily low
concentration of colloid, certainly less than one per cent.

Since the apparent viscosity of such colloid solutions is depen-

292

FLUIDITY AND PLASTICITY

M

dent upon the amount of the shearing force, the values of the
viscosity as such are quite illusory. For example, a 1.39 per
cent solution of nitrocellulose in acetone gave a fluidity of
52.49 using a pressure of 403.6 g per cm 2 whereas the fluidity
apparently fell to 51.31 at 214.5 g per cm 2 and to 50.76 at 62.96
g per cm 2 . These results are similar to those of Glaser (c/.
Table XVIII).

It is quite possible to measure the plasticity of materials of
this kind in the viscometer shown in Fig. 29. It is merely
necessary to measure the flow at two or more pressures and then

Fig. 90.—Friction-weight-concentration curve for colloidal dispersions of
nitrocellulose in acetone measured in dynes per square centimeter.

compare the volume of flow with the shearing stress. The
volume of flow-shearing stress curves obtained in this way by
Hyden are linear in every case. The mobility is found to in¬
crease with the temperature in a nearly linear manner and the
mobility falls off very rapidly with increasing concentration of
colloid, approaching the zero value asymptotically. Both of these
results are similar to those for clay suspensions ( cf . pages 220
and 221). The equations of these curves have not yet been
obtained.

The friction in nitrocellulose solutions increases rapidly with
increasing concentration of colloid, as shown in Fig. 90. As the
temperature is raised the friction is decreased in a linear manner,

4 " APPLICATIONS OF THE VISCOMETRIC METHOD 293

\

Fig. 91, so that at about 43 °C nitrocellulose in acetone would
appear to have the properties of a true fluid. If above 43° we
have a true solution, this temperature is a transmission point
which is analogous to the melting point of a solid. This does
not mean however that there would be any marked change in
the working properties as nitrocellulose solutions below 43° are
extremely soft solids.

0

Fig. 91.—Temperature-friction curve for a colloidal dispersion of 7.708 weight
percentage of nitrocellulose in acetone.

Solubility and Plasticity. —It is of course well-known that the
so-called “solutions” of nitrocellulose, gelatine and other colloids
are not true solutions, nevertheless the term solution as applied
to colloidal dispersions often leads to confusion. Thus acetone
is one of the best “solvents” for nitrocellulose, being superior
to let us say, amyl acetate. But what does this statement
mean? It cannot possibly mean that acetone will actually
dissolve more nitrocellulose than will a similar amount of amyl
acetate, for there is no point of saturation for either, i.e ., both
liquids will “dissolve” or better disperse an indefinite amount of
colloid; hence the term solubility has here a very special, albeit
a very definite, meaning, viz., that dispersive medium is the best
solvent which with a given amount of colloid gives an emulsion
having the maximum mobility. Here however there enters the

294

FLUIDITY AND PLASTICITY

fact, which seems from the literature not to have been sufficiently
considered, that acetone has a far greater fluidity than amyl
acetate to start with, and this must of necessity affect the
mobility of dispersions in these media. It is evident that this
must be taken into account if we are to get a true measure of the
dispersive power of different media. This work is being
continued.

APPENDIX A

PRACTICAL VISCOMETRY

The most essential part of the viscometer is shown in Fig. 29, p. 76.
To use the apparatus an appropriate amount of the liquid whose
viscosity is to be measured is pipetted into the right limb. The
liquid at the desired temperature is forced over into the left
limb until the right meniscus reaches the point N, it being noted
that there is sufficient liquid so that the surplus runs over into
the trap. The right limb is turned to air so as to prevent more
liquid from flowing into the trap. Having adjusted the working
volume, the left limb is connected with the pressure, and the time
required for the left meniscus to fall from B to D is noted. The
left limb is now turned to atmospheric pressure and the instru¬
ment is ready for an immediate duplicate determination in the
opposite direction. In this second measurement the time is
noted which is required for the left meniscus to rise from D to B.

Knowing the pressure, p in grams per square centimeter, the
time, t, in seconds, the two constants of the instrument, C and C',
and the density of the liquid, p, the viscosity rj at the given tem¬
perature is given by the formula, (c/. p. 74).

V = Cpt- C'p/t (1)

Determination of the Constants of the Instrument

The second term of the right hand member of the above equa¬
tion is the kinetic energy correction which should never exceed
5 per cent of the value of the first term. For this reason the
value of the constant C r and of the density p need be known with
an accuracy of 2 per cent only in order to allow viscosity deter¬
minations to be made with an error of only one-tenth of 1 per
cent.

C r = 0.0446F/Z (2)

where V is the volume in milliliters of the bulb C between
the marks B and Z), and l is the length of the capillary EF,
log. 0.0446 = 8.64895 - 10.

295

296

FLUIDITY AND PLASTICITY

The value of the constant C is most conveniently obtained by
filling the instrument with freshly distilled, dust-free water and
determining the time of flow for each limb, at 20°C.

C = 0 - 01005 * + C' p (3)

pt 2

for water at 20°. This constant may also be obtained by direct
measurement

C = 384.8 r*/Vl (4)

where r is the radius of the capillary in centimeters. If the
acceleration of gravitation of the locality is not 980, the value
of C must be increased 0.1 per cent for each unit in excess.

Since the bulbs C and K may differ in level, it is evident that
the pressure, p , used in calculating the viscosity is not necessarily
equal to the pressure, pi, delivered by the compressed air at the
top of the viscometer. If the bulb K is higher than the bulb C
by a distance h } then it is evident that the pressure during the left
limb determination is decreased by an amount h x p and the pres¬
sure during the right limb determination is increased by the same
amount. Hence,

Px - hxp = v ^ t f il (left limb)

Pi + hxp = v <2 (right limb)

and therefore

h x

v /I

2 Cp \ti

V* ~ P 1

2

or if the two determinations are carried out with water at
using the same pressure

0.005034 /1 1\ C'/l 1\

C \h « 2 / + 2C\t 2 *2 2 /

20 °

(5)

where t , is the time of flow from right to left and t 2 is the time of
flow from left to right. Log. 0.005034 = 7.70191 — 10.

The value of C used in calculating the hydrostatic head is an
approximate value obtained from Eq. (3) by employing the pres¬
sure, pi, uncorrected for hydrostatic head, which is legitimate
since the hydrostatic head is at the most only a small correction
term.

J. W. Temple has worked out a simpler method for calculating

APPENDIX A

297

the hydrostatic head when the flow in opposite directions is
carried out at the same manometer pressure p. Let the time of
flow in the one direction t L , under the true pressure corrected for
hydrostatic head p L = p + hip, be supposed to be less than the
time t& in the opposite direction under the pressure p R = p
— hip. Then

hi

Vl ~ Pr

2p

and substituting into this equation the values of p L and p R given
by Eq. (1), we have

1 h + C'p/t L rj + C'p/ t R \

ni ~2 P \ Ct L Ct R )

but in the kinetic energy correction, which is itself always small,
the small hydrostatic head correction is of negligible influence,
hence for our purpose we may write rj + C'p/t L = rj + C'p/t R so

hi

1 t R - k

to + C' P /t L )

2 Cp tjJ, R

but from Eq. (1) we have that rj + C'p/t L = Cp L t L
hence

, _ Vl t R - 4

hl 2p t R

( 6 )

The Trtje Average Pressure

It might inadvertently be assumed that if the two bulbs C and
K are the same in shape and volume and also at the same level,
the true pressure to be used in calculating the viscosity would
necessarily be the pressure pi delivered by the compressed air in
the viscometer because the hydrostatic head as obtained above
would then be zero. But since the hydrostatic head in the vis¬
cometer is really continually changing, the true average pressure
may not be zero under the above conditions, and it must be
obtained by integration. Bingham, Schlesinger, and Coleman
(1916) 1 have shown that for cylindrical bulbs the true average
pressure p would be

V

0.8686 hp

logi

po + hp

(7)

310 po - hp

1 For other shapes of bulbs see original paper of Bingham, Schlesinger,
and Coleman. For the possible importance of such corrections see Kendall
and Munroe (1917).

298

FLUIDITY AND PLASTICITY

where h is the height of the bulb and po is the pressure with all
other corrections made. Fortunately if the height of the bulb
BD , in Fig 29, is not more than one-thirtieth of the whole pressure,
this correction is unnecessary to attain the desired accuracy of
0.1 per cent.

In any case, however, the student should determine by experi¬
ment whether a change in manometer pressure is without effect
upon the valve of C.

The Pressure Corrections Outside the Viscometer

Let the density of the liquid within the manometer be po at a
temperature T in degrees Centigrade and the height read on the
manometer scale—corrected for scale error if necessary—be h Q ;
also let the viscometer bulbs be at a height hf above the middle
point of the manometer. The pressure delivered to the air in
the viscometer becomes

Pi == ho — K ± L for a water manometer (8)

p 1 = ikf±iVfora mercury manometer (9)

where the values of L are given in Table I and may be made
entirely negligible in the setting up of the apparatus.

Table I. —Values of L

k' in centimeters

j ho in centimeters

100

200

300

50

0.01

0.01

0.02

100

0.01

0.03

0.04

200

0.03

0.05

0.08

300

0.04

0.08

0.11

APPENDIX A

299

The values of K are given in Table II.

Table II.—Valves of K

Temperature,

Manometer

reading,

ho

degrees centi¬
grade

10

20 i

30

40

50

[

60

70

80

90

100

200

300

5

0.013

0.025

0.039

0.053

0.066

0.079

0.094

0.108

0.122

0.136

0.285

0.482

10

0.016

0.030

0.046

0.064

0.078

0.095

0.112

0.129

0.145

0.162

0.337

0.533

11

0.017

0.032

0.050

0.068

0.083

0.101

0.119

0.137

0.154

0.172

0.357

0.563

12

0.018

0.035

0.053

0.072

0.089

0.108

0.126

0.145

0.163

0.183

0.379

0.596

13

0.019

0.037

0.057

0.077

0.095

0.115

0.135

0.155

0.175

0.195

0.403

0.632

14

0.020

0.040

0.061

0.082!

0.102

0.123

0.144

0.165

0.187

0.208

0.429

0.671

15

0.022

0.043

0.065

0.088

0.110

0.131

0.154

0.177

0.199

0.222

0.457

0.713

16

0.023

0.046

0.070

0.094

0.118

0.140

0.165

0.189!

0.212

0.238

0.489

0.761

17

0.025

0.049

0.075

0.101

0.126

0.150

0.176

0.203

0.228

0.255

0.523

0.812

18

0.027

0.053

0.080

0.108

0.135

0.161

0.189 0.217

0.245

0.273

0.559

0.866

19

0.029

0.057

0.086

0.116

0.144

0.173

0.203 0.233

0.262

0.292

0.597

0.923

20

0.031

0.060

0.092

0.124

0.154

0.185

0.217 0.249

0.280

0.312

0.637

0.983

21

0.033

0.065

0.098

[0.132

0.165

0.198

0.232

0.265

0.299

0.333

0.679

1.046

22

0.035

0.069

0.105

0.141

0.176

0.211

0.247

0.282

0.319

0.355

0.723

1.113

23

0.037

0.074

0.112

0.151

'0.188!

0.225

0.264

0.301

0.341

0.379

0.770

1.184

24

0.040

0.079

0.119

0.160

[0.200

0.240

0.281

0.321

0.363

0.403

0.819

1.256

25

0.042

0.084

0.127

0.170

0.212

0.255

0.298

0.341

0.385

0.428

0.869

1.331

26

0.045

0.089

0.135

0.181

0.225

0.270

0.316

0.362

0.408

0.454

0.921

1.409

27

0.048

0.094

0.143

0.191

0.239

0.286

0.335

0.383

0.432

0.481

0.975

1.490

28

0.050

0.100

0.151

0.202

0.253

0.303

0.355

0.405

0.458

0.509

1.031

1.574

29

0.053

0.105

0.160

0.214

0.268!

0.321

0.375

0.429

0.484

0.538

1.089

1.661

30

0.056

0.111

0.169

0.226

0.283

0.339

0.396

0.453

0.511

0.568

1.149

1.751

31

0.059

0.117

0.178

0.239

0.298

0.357

0.417

0.478

0.538

0.599

1.210

1.842

32

0.062

0.124

0.188

0.251

0.314

0.376

0.439

0.503

0.567

0.630

1.273

1.937

33

0.066

0.130

0.197

0.264

0.330

0.395

0.462

0.529

0.595

0.662

1.337

2.033

34

0.069

0.137

0.207

0.277

0.346

0.415

0.485

0.555

0.625

0.695

1.403

2.132

If the pressure is read on a mercury manometer at 20°, the
heights in mercurial centimeters may be converted into grams
per square centimeter by means of Table III.

300

FLUIDITY AND PLASTICITY

Table III. —Values of M. Pressures in Grams per Square Centi¬
meter, for Heights in Mercurial Centimeters

Height, centi¬
meters of
mercury

99.3 *00.7

12.9 14.2

*02.0 *03.4
15.6 16.9

76.3 77.7

89.9 91.2
03.4 *04.8

16.9 18.3

30.5 31.8

*06.1 *07.5
19.7 21.0

44.5 45.9 47.2

58.0 59.4 60.8

71.6 72.9 74.3

85.1 86.5 87.8

98.7 *00.0. *01.4

*08.8 *10.2

22.4 23.7

*03.6 *05.0

17.2 18.5

30.7 32.1

44.3 45.6

57.8 59.2

71.4 72.7 ~~

84. 9 86. 3

98.4 99.8 -

12.0 13.3 0.

25.5 26.9 0.:

39.1 40.4 0.

0 .-

52.6 54.0 O.i

66.2 67.5 0.

79.7 81.1 0.

93.3 94.6 0.

*06.8 *08.2 0.'

20. 3 21.7 L_

33.9 35.2

47.4 48.8

61.0 62.3

74.5 75.9

J

APPENDIX A

301

Table 111 . — Continued

Height, centi¬
meters of
mercury

.0

. 1

_ 2

.3

.4

.5

.6

.7

.8

.9

70

48.1

49.5

50.8

52.2

53.5

54.9

56.2

57.6

58.9

60.3

71

61.7

63.0

64.4

65.7

67.1

68.4

69.8

71.1

72.5

73.8

72

75.2

76. 6

77.9

79.3

80.6

82.0

83.3

84.7

86.0

87.4

73

88.7

90. 1

91. 5

92.8

94.2

95.5

96.9

98.2

99.6

*00.9

74

1002.3

03. 6

05.0

06.4

07.7

09.1

10.4

11.8

13. 1

14.5

75

15.8

17.2

18. 5

19.9

21.2

22. 6

24.0

25.3

26.7

28.0

76

29.4

30.7

32. 1

33.4

34.8

36.1

37.5

38.9

40.2

41.6

77

42.9

44.3

45.6

47.0

48.3

49.7

51.0

52.4

53.8

55.1

78

56.5

57.8

59.2

60.5

61.9

63.2

64.6

65.9

67.3

68.7

79

70.0

71.4

72.7

74.1

75.4

76.8

78.1

79.5

80.8

82.2

80

83. 6

84.9

86.3

87.6

89.0

90.3

91.7

93.0

94.4

95.7

81

97. 1

98.4

99. 8

*01.2

*02.5

*03.9

<05. 2

*06.6

*07.9

*09.3

82

1110.6

12.0

13.3

14.7

16.1

17.4

18.8

20.1

21.5

22.8

—

—

83

24.2

25.5

26.9

28.2

29.6

31.0

32.3

33.7

35.0

36.4

1.4

84

37. 7

39. 1

40. 4

41. 8

43.1

44 5

45.9

47. 2

48 6

49.9

85

51.3

52. 6

54.0

55.3

56.7

58.0

59.4

60.7

62.1

63.5

0.1

0.1

86

64.8

66.2

67.5

68.9

70.2

71.6

72.9

74.3

75.6

77.0

0^2

0^3

87

78.4

79.7

81.1

82.4

83.8

85.1

86.5

87.8

89.2

90.5

0.3

0.5

88

91.9

93.3

94.6

96.0

j 97.3

98.7

*00.0

*01.4

*02. 7

*04.1

0*4

0 6

89

1205.4

06.8

08.2

09.5

10.9

12.2

13.6

14.9

16.3

17.6

0.5'

0.7

90

19.0

20.3

21.7

23.1

24.4

25.8

27.1

28.5

i 29.8

31.2

0.6

0.7

0.8

1.0

91

32.5

33.9

35.2

36.6

37.9

39.3

40.7

42.0

43.4

44.7

0.8

1.1

92

46.1

47.4

48.8

50.1

51.5

52.8

54.2

55.6

! 56.9

58.3

0.9

1.3

93

59.6

61.0

62.3

63.7

65.0

66.4

67.7

69.1

70.5

71.8

94

73.2

74.5

75. 9

77.2

78.6

79.9

81.3

82.6

84.0

85.4

95

86.7

88.1

89.4

90.8

92.1

93.5

94.8

96.2

97.5

98.9

96

1300.2

01.6

03.0

04.3

05.7

07.0

08.4

09.7

11.1

12.4

97

13.8

15.1

16.5

17.9

19.2

20.6

21.9

23.3

24.6

26.0

98

27.3

28. 7

30.0

31.4

32.8

34.1

35.5

36.8

38.2

39.5

99

40.9

42.2

43.6

44.9

46.3

47.7

49.0

50.4

51.7

53.1

100

54.4

55.8

57.1

58.5

59.8

61.2

62.5

63.9

j 65.3

66.6

200

2708.7

10.0

11.4

12.7

14.1

15.5

16.8

18.2

19.5

20.9

300

4062.8

64.2

65.5

66.9

68.2

69.6

71.0

72.3

73.7

75.0

302

FLUIDITY AND PLASTICITY

If the temperature of the mercury is other than 20° a correc¬
tion is applied using Table IV.

Table IY .— Values op N. Coerection in Pressures ( Grams per
Square Centimeter ) for Various Temperatures and Mercurial

Heights

Temperature,

Height of mercury , centimeters

degrees

Centigrade

10

20

30

40

50

60

70

80

90

100

5

0.4

0.7

1

i

1.1

1.5

1.8

2.2

2.6

2.9

3.3

3.7

0

0.3

0.7

1.0

1.4

1.7

2.1

2.4

2.7

3.1

3.4

7

0.3

0.6

1.0

1.3

1.6

1.9

2.2

2.6

2.9

3.2

8

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.6

2.9

9

0.3

0.5

0.8

1.1

1.4

1.6

1.9

2.2

2.4

2.7

10

0.2

0.5

0.7

1.0

1.2

1.5

1.7

2.0

2.2

2.4

11

0.2

0.4

0.7

0.9

1.1

1.3

1.5

1.8

2.0

2.2

12

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

13

0.2

0.3

0.5

0.7

0.9

1.0

1.2

1.4

1.5

1.7

14

0.2

0.3

0.4

0.6

0.7

0.9

1.0

1.2

1.3

1.5

15

0.1

0.2

0.4

0.5

0.6

0.7

0.9

1.0

1.1

1.2

16

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

17

0.1

0.1

0.2

0.3

0.4

0.4

0.5

0.6

0.7

0.7

18

0.1

0.1

0.2

0.2

0.2

i 0.3

0.3

0.4

0.4

0.5

19

0.0

0.0

0.1

0.1

0.1

! 0.2

0.2

0.2

0.2

0.2

20

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

21

0.0

0.0

- 0.1

- 0.1

- 0.1

- 0.2

- 0.2

- 0.2

- 0.2

- 0.2

22

- 0.1

- 0.1

- 0.1

- 0.2

- 0.2

- 0.3

- 0.3

- 0.4

- 0.4

- 0.5

23

- 0.1

- 0.1

- 0.2

- 0.3

- 0.4

- 0.4

- 0.5

- 0.6

- 0.7

- 0.7

24

- 0.1

- 0.2

- 0.3

- 0.4

- 0.5

- 0.6

- 0.7

- 0.8

- 0.9

- 1.0

25

- 0.1

- 0.2

- 0.4

- 0.5

- 0.6

- 0.7

- 0.9

- 1.0

- 1.1

- 1.2

26

- 0.2

- 0.3

- 0.4

- 0.6

- 0.7

- 0.9

- 1.0

- 1.2

- 1.3

- 1.5

27

- 0.2

- 0.3

- 0.5

- 0.7

- 0.9

- 1.0

- 1.2

- 1.4

- 1.5

- 1.7

28

- 0.2

- 0.4

- 0.6

- 0.8

- 1.0

- 1.2

- 1.4

- 1.6

- 1.8

- 2.0

29

- 0.2

- 0.4

- 0.7

- 0.9

- 1.1

- 1.3

- 1.5

- 1.8

- 2.0

- 2.2

30

- 0.2

- 0.5

- 0.7

- 1.0

- 1.2

- 1.5

- 1.7

- 2.0

- 2.2

- 2.4

31

- 0.3

- 0.5

- 0.8

- 1.1

- 1.4

- 1.6

- 1.9

- 2.2

- 2.4

- 2.7

32

- 0.3

- 0.0

- 0.9

- 1.2

- 1.5

- 1.8

- 2.1

- 2.4

- 2.0

- 2.9

33

- 0.3

- 0.0

- 1.0

- 1.3

- 1.6

- 1.9

- 2.2

- 2.6

- 2.9

- 3.2

34

- 0.3

- 0.7

- 1.0

- 1.4

- 1.7

- 2.1

- 2.4

- 2.7

- 3.1

- 3.4

35

- 0.4

- 0.7

- 1.1

- 1.5

- 1.8

- 2.2

- 2 . 6

- 2.9

- 3.3

- 3.7

APPENDIX A

303

The correction for the difference in level between the middle
of the manometer and the viscometer is made negligible in setting
up the apparatus.

Measurement of Time

We have seen that the pressure in grams per square centimeter
must always be 30 times as great as the distance between the
bulbs. On the other hand the pressure must always be kept small
enough so that the time of flow can be measured to the desired
accuracy. Thus the time should not fall below 200 sec. since one
cannot measure the time more accurately than to 0.2 sec. with a
stop-watch.

The stop-watch should be tested repeatedly against the second
hand of a good time piece. It should not gain or lose as much as
0.2 sec. in 5 min. It is well to keep the watch in the same posi¬
tion during successive measurements, as well as not to allow it to
be nearly run down during a measurement. In selecting a stop¬
watch it should be noted that watches show better performance
whose mechanism continues to run whether the split-second hand
is in use or not. The performance of the watch may be tested
at the U. S. Bureau of Standards.

Temperature

The viscometer is kept at a constant temperature by means of a
large, well-stirred bath which is regulated by hand, if a series of
temperatures are to be measured, or by a thermostat, if the bath
is to be used for a long time at a single temperature. Since at
0° the fluidity of water increases 0.1 per cent for every 0.03°
rise in temperature it is clear that the temperature regulation
must be to at least 0.03°. For more viscous substances a still
more precise regulation is necessary if the same degree of accu¬
racy is to be obtained.

A thermometer should be used which is graduaded to tenths
and calibrated through its entire length. The ice point should
be determined from time to time. If it is impracticable to have
the entire thread of mercury immersed at all times a correction
should be made for the emergent stem. The following table
may be used:

304

FLUIDITY AND PLASTICITY

Table V.— Correction of a Normal Thermometer from 0 ° to 100 °C
for Emergent Steam Graduated in Tenths of a Degree

Number of de¬
grees of mercury
exposed

Difference in temperature between mean temperature
of emergent steam and bath. Corrections in degrees to be
added to the observed temperature

30°

40°

50°

60°

o

o

O

O

00

10

0.05

0.05

0.05

0.05

0.10

0.10

20

0.10

0.15

0.15

0.15

0.20

0.20

30

0.20

0.25

0.25

0.25

0.30

0.35

40

0.30

0.30

0.35

' 0.40

0.45

0.50

50

0.35

0.40

0.45

0.50

0.55

0.60

Since measurements are always preferred for even degrees it is
a great advantage for the worker to have on the bath before him
a table showing what temperatures on the thermometer must be
employed in order to obtain a desired even temperature. The
temperatures of 0°, 10°, 20°, 40°, 60°, 80°, 100° are sufficient to
give a good curve over this range.

The Pressure Regulator

Viscosity measurements have usually been carried out without
the use of a pressure regulator, but due to the withdrawal of the
air in use and to possible small leaks in the connections and to
changes in temperature, the pressure rises and falls and is hardly
ever constant during the time of a single measurement. With
a pressure regulator the pressure will often stay constant to the
limit of the experimental error for a day or more at a time, with¬
out temperature regulation of the room, heat insulation of the
apparatus or any particular care in using the air. Not only is
this a saving of time and annoyance to the experimenter but by
using only a few pressures at the most there is a considerable
saving of time in calculation. Hence the pressure regulator
is a necessity for extended work.

The diagrammatic view of the apparatus with pressure regula¬
tor is given in Fig. 92. Air is forced in through a needle valve A
to a storage reservoir B whose pressure in pounds per square inch
is shown on the gauge C. In adjusting the pressure regulator
the air is very slowly admitted to the stabilizing reservoir F by

APPENDIX A

305

means of the needle valve D. The valve E is convenient in locat¬
ing leaks in the apparatus, etc., but is not often used. The valve
G is a direct connection to air which is also seldom used.

The pressure regulator consists of five brass tubes 6 cm. in
diameter which are filled with water let in at K> the valves O',
0" etc. being open and the valve N closed. When the water
begins to overflow at M into the drain pipe, the water is shut off

Fig. 92.—Diagram of viscometer set-up with multiple tube water stabilizer.

at K } and as soon as equilibrium is reached, the drain pipe is also
closed off at Z and the valves O', 0", etc. are closed.

By allowing air to pass very slowly through the valve D the air
will be gradually forced down the tube H ' until it bubbles out
through the water, and, if the pet-cock J' is open, into the air.
If the stream of air is very slow, say a bubble or two per second,
it is evident that the pressure will be constant. If a higher pres¬
sure is desired the pet-cock J' is closed when the pressure becomes
the sum of the pressures obtained by the two tubes separately
and so on for the five different pressures up to the maximum
capacity of the regulator. In lowering the pressure one must
be careful to turn the pet-cocks to air in the reverse order J v
J IV J 111 and J 11 J 1 in order that the air under pressure may not
cause the water to be drawn back into the system. The advan-
20

306

FLUIDITY AND PLASTICITY

tage of the drain pipe U is that of securing day by day practically
identical pressures, without the loss of time in adjustment. If
other pressures than these are desired, they may be obtained by
drawing off some of the water from one or more of the stand pipes.
The glass gage at J ', etc., aid the manipulator in adjusting the cur¬
rent of air. They may be cleaned by unscrewing the pet-cocks
above and using a small brush.

The beginner must be cautioned particularly against turning
the system to air at the viscometer since it may result in filling
the manometer, etc. with water. To prevent such an accident
and to dry the air, the reservoir P containing granular calcium
chloride is introduced. Any liquid should be drained at intervals.

The Manometer

The manometer consists of a plate glass mirror which must be
mounted vertically, on which is stretched a 2-m steel tape
graduated in millimeters. Over the tape is fixed the glass tube
of the manometer bent so that both the right and left limbs may
be read on the same tape. The manometer may be filled with
either mercury or water. If water is used for low pressures
another manometer will be desired for mercury. Since it is
possible to read the manometer to 0.01 cm one can use the mer¬
cury manometer down to 10 cm (135 g per square centimeter)
with the desired accuracy. With water one can go down to
about 50 g per square centimeter, but not much further unless a
correction is made for the true average pressure. A thermometer
near the middle of the manometer is needed to give the tem¬
perature of the manometer fluid.

The Bath

The viscometer V is mounted on a massive brass frame Fig.
93 by means of brass clips designed especially for this purpose.
The frame slides in grooves on the side of the bath so that the
viscometer may be easily kept in a vertical position. The viscom¬
eter is connected by heavy-walled rubber tubing to the pressure
by way of the three-way glass stop cocks L and R , the third
connection being to air. The temperature of the bath is raised
by means of a burner W which is connected without the use of
rubber to the gas supply. The second burner Y with stop cock
and pilot flame is used as needed to obtain the fine regulation.

308

FLUIDITY AND PLASTICITY

To assist in the regulation, cold water is admitted, when desired,
by a cock at S. A drain pipe, Q , maintains the bath at a constant
level. It may also be unscrewed to permit draining the water
from the bath. The bath is insulated on two sides.

The Density

It is not necessary to know the exact density in
order to obtain the fluidity by this method. But
the density can be measured at the same time
with accuracy with little additional labor. Since
the fluidity is very closely related to the volume,
according to the law that the fluidity is directly
proportional to the free volume, the specific volume
should usually be obtained with precision.

The instrument shown in Fig. 94 is convenient to
use and unlike the Sprengel pycnometer, it can be
used to determine the density below room tem¬
perature. It is filled to the mark with water and
weighed at every temperature at which it is to
be used. It is then cleaned, dried, weighed, and
filled with the liquid to be determined and again
weighed. The ratio of the weights of liquid cor-
Fig. 94 .— rected simply for the buoyancy of the air gives the
for Uquids^ 6 ^ correc ^ specific gravity referred to water at 4°C.

The densities of water are given in Table VI.

Table VI.—Density

and Volume

of Water in Grams

per Milliliter

Temperature

Density

Logarithm

density

Specific volume

0

0.99987

9.99994-10

1.00013

10

0.99973

9.99988-10

1.00027

20

0.99823

9.99923-10

1.00177

30

0.99568

9.99811-10

1.00435

40

0.99225

9.99662-10

1.00782

50

0.98807

9.99479-10

1.01207

60

0.98324

9.99266-10

1.01705

70

0.97781

9.99025-10

1.02270

80

0.97183

9.98759-10

1.02899

90

0.96534

9.98468-10

1.03590

100

0.95838

9.98154-10

1.04343

APPENDIX A

309

The formula to be used in obtaining the density is:

P4 e = ^ Po+ 0.0012(^^)

Wo \ w 0 /

where w' = weight of liquid at fC,
w 0 = weight of water at fC,
po = density of water at t°C.

The liquid is introduced or removed from the pycnometer by

Fig. 95.—Apparatus for cleaning and filling viscometer.

means of the capillary pipette, used also for introducing liquid
into the viscometer, shown in Fig. 95.

This rubber tubing as well as the heavy walled tubing at the
top of the viscometer should be scrupulously cleaned on the
inside to remove dust before they are used.

If the capillary stem of a 25 ml pycnometer has a bore of 0.08
cm it is capable of an accuracy of 0.01 per cent by reading the
meniscus to within 1 mm. It is well to have two pycnometers

310

FLUIDITY AND PLASTICITY

of equal size and employ the tare method in weighing.

Strictly, it is necessary to measure the density at only one
temperature by this method. The working volume of the vis¬
cometer has to be adjusted each time that the temperature of the
liquid is raised. By noting the expansion of this working volume
for each temperature interval it is readily possible to calculate
the specific volume and density. The portion of the viscometer
HGj Fig. 29, is graduated in millimeters. By filling the viscometer
with mercury from A to G, and weighing this mercury, the work¬
ing volume V f can be actually determined. And by filling a
given length of the capillary HG with mercury, the volume v f of
the capillary per centimeter is easily determined. The density
of mercury is given in Table III.

Table VII.—Density and Volume of Mercury in Grams per

Milliliter

Temperature,

degrees

Density

Logarithm

density

Specific volume

10

13.570

1.13260

0.073687

15

13.558

1.13220

0.073757

20

13.546

1.13181

0.073822

25

13.534

1.13142

0.073887

30

13.522

1.13104

0.073954

If, therefore, the specific volume of the liquid is s 0 at temperature
t 0 and on forcing the meniscus at the left just up to the trap, the
right meniscus is a distance d away from its proper level (?, then
at the new temperature t, the specific volume $ must be

s = | 7 (l+t/d) (10)

With this volumeter it must be remembered that the errors
are cumulative. On the other hand with the pycnometer method
care must be taken to wipe off drops of liquid which may adhere
to the inside of the glass, and to prevent the evaporation of
volatile substances, on account of which a stopper is added to the
pycnometer.

Assuming that a capillary is used whose radius is 0.01 cm and
that the tube HG has a radius which is ten-fold this amount, or

APPENDIX A

311

0.1 cm (c/. page 319) reading the meniscus to 0.1 mm will give
an accuracy in the specific volume of 0.01 per cent.

Cleaning and Filling the Viscometer

The viscometer is not removed from its frame during the course
of an investigation. Two hooks are screwed into a board on the
wall which will hold the viscometer frame firmly at E, Fig. 95.
Chromic acid, added with pipette, is drawn through the instru¬
ment by means of suction. The frame and viscometer are then
again placed on hooks in an inverted position D and the liquid
withdrawn by means of suction. A Woulff flask is interposed
between the rubber tubing and the suction line. The apparatus
is washed out repeatedly with dust-free water and finally with
dust-free alcohol and dust- and grease-free ether. Air which has
passed over granulated calcium chloride A and through a long
column of absorbent cotton B is then drawn through using clean
rubber tubing.

To fill the instrument an amount of liquid slightly greater than
the working volume is drawn up into the clean pipette F which is
wiped free of dust by means of chamois skin just before use. The
liquid is protected from the moisture of the air by means of the
drying tube containing calcium chloride held in position by means
of absorbent cotton.

The Viscosity Record

The data may be kept on sheets ruled somewhat as follows:
which will give a compact and systematic record of both data
and the calculations:

312

FLUIDITY AND PLASTICITY

Table VIII. —Lafayette College Viscosity Record
Page I

Substance Pure Water Remarks Calibration

Date_ Observer W. G. K. = . 0 45

Viscometer No. JL, Pycnometer No. __2_

Log C' = 8.37698-10 Log ~ =6.22122

Time

Manometer,
upper read-

Sum
differ¬
ence =

ho

Temperature

bath

Limb

Min¬

utes

Sec¬

onds

Time,

sec¬

onds

ing,

reac

j Start

lower

ling

Finish

Temper¬

ature

Weight

pyc.

20

L

5

7.0

307.0

259.46

28.12

259.46

21.2

287.58

231.34

20

R

5

8.2

308.2

259.48

28.12

259.48

21.1

287.60
231.36

p

hip

K

± hi

pK±L

Po

Cip

P

V

in cp

<p

V

Remarks

0.9982

+ .44
-.44

-.79
t- . 79

-.35

-1.23

230.99
230.13

1.76

1.75

229.23

228.38

1.0050
1.0052

99.50

99.48

water mano¬
meter

Calculation of Constants

Let us use the above data for water at 20° to show the method
of calculation of constants, etc. We record the sum of the upper
and lower manometer readings merely as a check against error
in reading, since this sum should be constant. With our instru¬
ment V = 4.0 ml, and l = 7.5 cm hence C f = 0.02377. The
value of 231.34, corrected by Table II for K gives

200 cm at 21.2° = 0.69
30 cm at 21.2° = 0.10
1.34 cm at 21.2° = 0-00
Total correction = 0.79
pi = 231.34 - 0.79 = 230.55 cm

APPENDIX A

313

The value of L is negligible. Calculating the approximate value
of C using Eq. (3) we have,

0.01005 X 307 + 0.02377 X 0.998

230.55 X 307 X 307 1 * 4 ^ 1 X 10 7

Calculating now the hydrostatic head, using this value of C, we
have from Eq. (5)

, h ± = 0.44 + 0.01 = 0.45.

Now p = 230.55 + 0.45 = 231.0 for the left limb or
= 230.57 — 0.45 = 230.1 for the right limb;

hence, on applying again Eq. (3) the true value of C becomes
_ 0.1005 X 307 + 0.02377 X 0.998 w _

°-231 X 307 X 307 ~ 1>428 X 10 *

1.428 X 10-

Example of Calculation of Viscosity and Fluidity

Suppose that we assume that we had given the constants of
the apparatus, and that we desired to calculate out the viscosity.
We have hi = + 0.45, K = —0.79, so that the corrected pressure
is 231.0. We may now apply Eq. (1) at once, but advantages
may be obtained, without extra labor, by calculating the value
of P in the equation

Cpi - C'p/t = CPT

p = p _ __ (ii)

which is evidently the pressure consumed in overcoming viscous

C f o

resistance solely. In this case = 1.76 hence P =229.2.
The fluidity <p is

1 1

<p = = - =99.5 c.g.s. units.

vi t 7]

Instead of writing the viscosity as 0.01005 we prefer to multiply
by 100 and record the datum as 1.005 centipoises (cp), which keeps
most viscosities from becoming inconveniently small fractions
and it also makes the viscosities “specific,” referred to water at
practically 20°.

In scrutinizing the data heretofore published on viscosity one
is particularly interested in the magnitude of the kinetic energy
correction and it may be subject to slight changes in the future.

314

FLUIDITY AND PLASTICITY

Publication of the temperature, time, pressures p and P, density,
viscosity, and fluidity makes the data quite complete and cor¬
rection easy.

In constructing the viscometer, the glass blower must select
a piece of capillary tubing which has not only a uniform bore
but also one which has a radius which must be selected within
rather narrow limits. This requires the measurement of the
radius, which is accomplished as follows. The capillary is filled
with mercury completely to a distance of exactly 10 cm, this
mercury is then run out on to a watch crystal and weighed. The
radius of the capillary in centimeters can be read at once from
Table IX. These measurements need not be exact, but where it
is desired to measure the average radius with exactitude, as in
absolute measurement, it is to be noted that the volume of the
mercury is calculated for 20°C and that the values are corrected
for buoyancy of the air so that there is no correction in weighing
with platinum weights. It is assumed that the mercury thread
is a true cylinder.

Having found the radius of the capillary, it becomes feasible
to cut off a length which will give a time of flow of not less than
200 sec. for the assumed maximum fluidity, e.g ., 500, with a
pressure of 50 g per square centimeter and a volume of flow of
4 ml. The lengths to be cut off for capillaries of different radii
are given in Table X (cf. also Fig. 24).

The table shows that with a maximum fluidity of 500 and a
permissible length of capillary up to 20 cm, the radius must not
be as great as 0.015 cm; and if the ratio of the length to the radius
is to be greater than 500 in order to minimize “end effects,” the
radius must be over 0.010 cm, which limits the selection within
quite narrow limits.

The viscometer with a 500 capillary will serve for quite viscous
liquids, for the pressure can be varied from 50 to 10,000 and the
time may conveniently be increased fivefold, hence one can
measure two-thousand-fold, i.e., from 500 to 0.5. Nevertheless
in an investigation in which no fluidities are to be measured above
50, it is convenient to use a viscometer with a maximum of 50,
and therefore the length of capillary will be one-tenth of that
indicated by Table X. Just what maximum to specify, as 5,000
500, 50, 5, or 0.5, may easily be judged by the use of Tables XI

APPENDIX A

315

Tables IX. —The Average Radius of a Capillary Tube in Centimeters
Corresponding to the Weight of Mercury Required to Fill a
Length of 10 Cm at 20°

Radius, centi¬
meters

Weight,

Difference

Radius,

Weight,

grams

centimeters

grams

0.0004

12

12

30

38

47

56

63

73

81

89

0.051

1.1068

0.0016

0.052

1.1506

0.0038

0.053

1.1953

0.0068

0.054

1.2408

0.0106

0.055

1.2872

0.0153

0.056

1.3344

0.0209

0.057

1.3825

0.0272

0.058

1.4315

0.0345

0.059

1.4813

0.0426

0.060

1.5319

0.0515

98

106

115

123

132

141

149

157

166

175

0.061

1.5834

0.0613

0.062

1.6357

0.0719

0.063

1.6889

0.0834

0.064

1.7430

0.0957

0.065

1.7979

0.1089

0.066

1.8536

0.1230

0.067

1.9102

0.1379

0.068

1.9677

0.1536

0.069

2.0259

0.1702

0.070

2.0851

0.1877

183

191

200

209

217

225

234

243

251

259

0.071

2.1451

0.2060

0.072

2.2059

0.2251

0.073

2.2676

0.2451

0.074

2.3302

0.2660

0.075

2.3936

0.2877

0.076

2.4579

0.3102

0.077

2.5230

0.3336

0.078

2.5889

0.3579

0.079

2.6557

0.3830

0.080

2.7234

0.4089

268

277

285

294

302

311

319

327

336

345

0.081

2.7919

0.4357

0.082

2.8613

0.4634

0.083

2.9315

0.4919

0.084

3.0026

0.5213

0.085

3.0745

0.5515

0.086

3.1472

0.5826

0.087

3.2208

0.6145

0.088

3.2953

0.6472

0.089

3.3706

0. 6808

0.090

3.4468

0.7153

353

362

370

379

387

396

404

413

421

430

0.091

3.5238

0.7506

0.092

3.6017

0.7868

0.093

3.6804

0.8238

0.094

3.7600

0.8617

0.095

3.8404

0.9004

0.096

3.9217

0.9400

0.097

4.0038

0.9804

0.098

4.0868

1.0217

0.099

4.1706

1.0638

0.100

4.2553

316

FLUIDITY AND PLASTICITY

Table X. —Lengths of Capillary for Different Radii Assuming a
Maximum Fluidity of 500, a Minimum Pressure of 50 G per
Square Centimeter, a Minimum Time of Flow of 200 Sec., and a

Volume of Flow of 4 Ml. I -

Radius in
centimeters

Length in
centimeters

Difference

Radius in
centimeters

Length in
centimeters

Difference

0.001

0.002

0.00048

0.00770

0.051

0.052

3,254

3,517

263

279

294

312

329

349

366

384

404

429

0.003

0.03896

0.053

3,796

0.004

0.12315

0.054

4,090

0.005

0.3007

0.055

4,402

0.006

0.6234

0.056

4,731

0.007

1.155

0.825

1.186

1.655

2.232

0.057

5,080

0.008

1.970

0.058

5,446

0.009

3.156

0.059

5,830

0.010

4.811

0.060

6,234

0.011

0.012

7.043

9.978

2.935
! 3.96

4.74

1 5.87

7.18

8.65

10.32

12.18

14.29

j 16.60

0.061

0.062

6,663

7,110

i 447

467

495

515

539

567

592

619

1 646

674

0.013

13.74

0.063

7,577

0.014

18.48

0.064

8,072

0.015 '

24.35

0.065

8,587

0.016

31.53

0.066

9,126

0.017

40.18

0.067

9,693

0.018

50.50

0.068

10,285

0.019

62.68

0.069

10,904

0.020

76.97

0.070

11,550

0.021

0.022

93.56

112.7

19.12

21.94

25.0

28.3

32.0

35.6

40.0

44.6

48.4

54.6

0.071

0.072

12,224

12,926

702

736

765

794

827

861

900

928

967

1,004

0.023

134.6

0.073

13,662

0.024

159.6

0.074

14,427

0.025

187.9

0.075

15,221

0.026

219.9

0.076

16,048

0.027

255.7

0.077

16,909

0.028

295.7

0.078

17,809

0.029

340.3

0.079

18,737

0.030 !

389.7

0.080

19,704

0.031

0.032

444.3

504.4

60.1
| 65.9

72.7

1 78.8

86.2

93.8

101.2

110.0

119.0

127.0

0.081

0.082

20,708

21,750

1,042

0.033

570.3

0.083

22,830

1,080

0.034

643.0

0.084

23,950

1,120

0.035

721.8

0.085

25,111

1,161

0.036

808.0

0.086

26,314

1,203
1,246
1,289
1,333
1,380
1,426

0.037

901.8

0.087

27,560

0.038

1,003.0

0.088

28,849

0.039

1,113.0

0.089

30,182

0.040

1,232.0

0.090

31,562

0.041

0.042

1,359.0

1,497.0

138.0

148.0

158.0

169.0

182.0

194.0

206.0

220.0

233.0

247.0

0.091

0.092

32,988
34,462

1,474

1,523

1,574

1,624

1,676

1,728

1,784

1,839

1,896

0.043

1,645.0

0.093

35,985

0.044

1,803.0

0.094

37,559

0.045

1,972.0

0.095

39,183

0.046

2,154.0

0.096

40,859

0.047

2,348.0

0.097

42,587

0.048

2,554.0

0.098

44,371

0.049

2,774.0

0.099

46,210

0.050

3,007.0

0.100

!

48,106

APPENDIX A

317

and XII, without any preliminary measurements. Table X
will then be used as already indicated.

Table XI.— Approximate Fluidities for Convenient Reference

Substance

Fluidity «

Castor oil at 20°.

0.1

Lard oil at 20°.

1 .

Sugar solution at 20°, 60 per cent by weight.

1.77

Water at 20°.2.

100.0

Aliphatic hydrocarbons and ethers at boiling tem¬

perature .

500.

Carbon dioxide at the critical state.

5,000.

Table XII. —Radii Limits for Different Fluidity Maxima and l/r

Jk

Ratios

Fluidity maximum

l

r

Radius limits in centi¬
meters

5,000.0

600

0.005 to 0.008

500.0

500

0.010 to 0.015

50.0

400

0.020 to 0.026

5.0

300

0.040 to 0.045

0.5

200

0.074 to 0.081

Calibrating an instrument whose maximum fluidity is below
100 offers difficulties since water at 20° is excluded. The best
suggestions at present are: a 40 per cent solution by weight of
ethyl alcohol in water at 0°, fluidity, 14.0; a 60 per cent sucrose
solution by weight at 20°, fluidity 1.77; freshly distilled aniline at
0.5°, fluidity, 9.95.

The glass-blower should make the bulb C (and K) to have the
shape of two hollow cones placed base to base, in order to secure
good drainage. Bad drainage may be detected by inequality in
the fluidity as determined by the right and left limbs with viscous
liquids at the higher rates of flow. For very viscous liquids the
time of flow must be increased so that drainage difficulties may
be obviated. Increasing the size of the bulbs is no advantage.

The bulbs must be of such size that the left meniscus will not

318

FLUIDITY AND PLASTICITY

only be at A when the right meniscus is at G , but it should be at
B when the right meniscus is at J, and at D when the right menis¬
cus is at L.

The ends of the capillary at E and F must not be contracted
in under any circumstances, and as far as practicable the ends
should not be expanded into a trumpet shaped opening, as this
may seriously affect the kinetic energy correction. The appro¬
priateness of the correction already given may be tested for each
instrument by using a given liquid at a variety of pressures.

If the liquid is to move past the marks B and D with a
velocity of not less than 0.1 cm per second when the time of
flow is 200 seconds, it is only necessary to use tubing for
that part of the instrument whose radius is not more than
0.25 cm. On the other hand, for absolute measurements,
the instrument should always be so designed that the resist¬
ance to flow outside of the capillary will be negligible.
This is ascertained as follows: Let the lengths of the tubes
B, D, L, and J, Fig. 1, be in all L and their radius be R .
Then R A /L must be greater than 1,000 r A /l. For a capillary
whose radius is 0.012 cm, R must be at least 0.07 cm if L — 1.
This value is larger than is commonly supposed. A rule which
is simple but will cover every case is to have R at least 10 times
the radius of the capillary.

APPENDIX B

PRACTICAL PLASTOMETRY

The measurement of the flow of plastic substances resembles
that of viscous substances in most respects, but since plastic
substances do not drain like liquids, it is convenient to measure
the volume (or weight) of substance extruded. For this purpose
the plastometer, shown in Fig. 30, p. 77, has been designed to
replace the viscometer. It consists of a top A , container B } and
base C with capillary D and receiver E. The top consists of a
square plate of brass, through which the pressure is admitted by
means of a copper tube F } which is enlarged at the end to make it
convenient to connect with the pressure. The rubber gaskets H
and J enable one to make the apparatus gastight, when the
thumb-screws are screwed down. A brass pin X, brazed into the
top, passes through the rubber gasket and into a hole in the body
of the container. On the opposite side of the container, a small
copper tube passes through the top, through the rubber gasket,
and into a hole which extends all of the length of the container,
and into the lower end of which a short piece of hollow copper
tubing is affixed. This tube in turn passes through the second
rubber gasket J leading into the base C, thus affording a connec¬
tion between the atmosphere and the receiver while the plas¬
tometer is immersed in the bath.

The receiver is made of glass and with a flat bottom so that it
will sit upright. It is held in position by means of the rubber
collar M . The rod G is attached to the container in order that
the plastometer may be supported by the frame shown in Fig. 89.

Through the base, there extends the capillary tube D, whose
ends have been ground off flat and whose dimensions are known.
To cement the capillary in position it is cleaned carefully with
chromic acid mixture and dried without touching the part to be
soldered. It is “tinned over” in the usual manner and soldered
in place, the space N being filled with the alloy. Two parts
bismuth, two parts lead and one part tin has been found satis-

319

320

FLUIDITY AND PLASTICITY

factory. A more certain method is to platinize the glass and
then solder in position with pure tin as solder.

To determine the effect upon the flow of changing the length
and radius of the capillary, at least four capillaries may be
required, hence it is convenient to keep all of them mounted
continuously in duplicate bases.

Since the mercury thread gives only the radius of a cylinder
which would have the same volume as the capillary, the true aver¬
age radius for flow purposes requires more elaborate estimation if
absolute measurements of flow are to be made and any consider¬
able accuracy is desired. In fact, since the flow varies as the
fourth power of the radius and it is measurable to the desired
accuracy only with difficulty, it may be said that this is the most
difficult part of absolute measurement.

Whereas one will seek to obtain a capillary tube which is a
true cylinder, it will usually be slightly elliptical. In this case
the ratio of the major to the minor axes may be obtained by
measurements of the photomicrograph of the ends, although
several other methods may be used. Prom this ratio and the
average radius obtained by means of the mercury calibration,
the actual values of the major and minor axes 2 B and 2 C may
be calculated. 1 It follows then that

2 B*C 3

*•4 - -

B 2 + C 2

If the capillary is a frustrum of a true cone,

Ri 2 ~(- R1R2 + R2 2

where R 3 and R 2 are the radii of the two ends. If the capillary
is not only conical but elliptical at the same time,
r* = 3ff 3 W . (1 - e 2 ) 3
R 3 2 -f - RaRi + R 4, 2 1 -j- e 2

jg _ Q

where # 3 and J? 4 are the average radii of the two ends, e = —,

+ o

and 2 B and 2 C are the mean major and minor axes.

The Measurement of Plasticity

Until the pressure is admitted the flow by seepage will ordi-

1 Cf. Rucker, Phil. Trans. 185A, 438 (1894) and Knibbs, J. and Proc.
Roy. Soc. New South Wales 29, 77 (1895); 30, 186 (1896).

APPENDIX B

321

narily be extremely slow. It is possible therefore to wipe off the
end of the capillary, put the weighed container in place, admit
the pressure for a known interval of time, touch off into the
container any material still adhering to the capillary and weigh.
From the weight of material, the volume of flow may be cal¬
culated from the density when desired.

There is however, another convenient method which can be
used when the material comes from the capillary in drops.
The observer turns on the pressure and simply takes the time of
formation of a convenient number of drops, making no weighing
at all. Other measurements are made at the same or other
pressures. Finally without cleaning off the end of the capillary
a certain number of drops are counted off into a weighed receiver
at the minimum pressure used and also at the maximum pressure
used. From the weight of a drop at these two pressures, one can
calculate the weight of a drop at any intermediate pressure pro¬
vided the weight is a linear function of the pressure. By this
method a large number of measurements on a given material can
be completed in a single day with an accuracy of 0.3 per cent.
According to measurements by H. D. Bruce the weight of the
drop is not always uniform at a given pressure.

The pressure pi delivered to the plastometer is calculated in
the manner already described (page 299 et seq.), correcting for the
temperature of the liquid in the manometer. The plastic
material exerts a hydrostatic head which must be corrected for
as follows.

The initial head in the container, h, may be measured by the
use of a straight, slender wire. To this is added the length of
the capillary, l , hence the pressure (h + l)p added to p 1} gives
the corrected pressure p to be used in calculating the plasticity.
The change of hydrostatic head in subsequent determinations
may be ascertained by noting the volume of plastic material
which has accumulated in the graduated receiver. In this case
it is also necessary to know how much the level of the material
in the container is lowered by the loss of 1 ml. A much better
plan is to have a graduated glass tube of just the size to fit into
the container, and open at both ends, cemented into the container.
Having cut away portions of the metal of the container, the level
of the material within may be read directly.

21

322

FLUIDITY AND PLASTICITY

In the measurements of plasticity it has been found that
high pressures give data which may be handled more simply
than the data at low pressures. But a multiple-tube stabilizer
to give two atmospheres of pressure is both complicated and
expensive, hence a mercury stabilizer seems desirable. However
a mercury stabilizer was not used at first because as soon as the
pressure became great enough to bubble through the mercury at
all, a large amount of gas suddenly came off causing a violent
fluctuation in the pressure. This intermittent flow of air is
partly due to the failure of the mercury to wet the tube allowing
a continuous air channel to be formed over a considerable
distance between the mercury and the tube. This difficulty
can be overcome by the amalgamation of the tube by means of
sodium amalgam. A further difficulty arose from the necessity
of keeping the volume of gas bubbling through the stabilizer as
small as possible while maintaining the flow continuously. This
trouble was completely overcome by placing a Davis-Bourneville
reducing valve at the point C of the apparatus shown in Fig. 92,
a flow indicator just between the needle-valve D and the pressure-
reservoir F, and another flow indicator between the valve E
and the mercury stabilizer.

The flow indicator consists of two similar vials connected by
an inverted U-tube leading to the bottom of both vials through
two-hole rubber stoppers. A little glycerol is added to one of the
vials at the start and the rate of bubbling of the gas through the
liquid serves to indicate the direction of movement of the gas
as well as its velocity.

The mercury stabilizer consists of an single iron tube of some
25 mm internal diameter into which leads the inner tube having
a diameter of 5 mm just as in the water stabilizer. The outer
tube is closed at the bottom by means of a cap but near the
bottom a side tube leads off for the attachment of a stout rubber
tube which is connected in turn with a glass receiver of about
2 liters capacity. This receiver can be raised and lowered and
hung on stout hooks provided for the purpose at frequent vertical
intervals. In order to change from one pressure to another, it is
necessary for mercury to be added to or taken from the stabilizer.
This is very easily accomplished by simply raising or lowering
the receiver. For a pressure of two atmospheres not over 10 kg

APPENDIX B

323

of mercury are required. Were a smaller tube used for the outer
tube of the stabilizer, less mercury would be required but the
manipulation might be less convenient. A photograph of the
plastometer occording to the latest design used by Mr. H. D.
Bruce is reproduced in the frontispiece.

Treatment of Plasticity Data

The data may be analyzed either algebraically or graphically.
The formula for plastic flow through a capillary tube is

where ^ is the mobility, and / the friction or yield value. The
shearing force, F — is expressed in dynes per square centi¬
meter and the pressure P is expressed in grams per square centi¬
meter. Since the kinetic energy is generally negligible this
becomes

M = (13)

where v is the volume of flow per second and if is a constant whose
R A

value is 384.8 -y. If we substitute in Eq. (13) the values

Fiy Vi and F 2 , from two observations of the flow, we find that

_ (14)

V2 - V 1

so that both ju and / are readily determined. Since however the
weight of flow w = vp, a more convenient expression for the
friction is

Fm-Fm

Wz — Wi

The friction must have a positive value for all plastic substances
and the value should be constant for a given capillary so long as
seepage, slipping, et cet ., do not intervene.

In the early stages of the development of the subject, the
graphical method of treatment is desirable from many points of
view. Plotting the weight of flow in grams per second as ordi¬
nates and the shear in dynes per square centimeter as abscissas,
the value of the intercept of the extrapolated curve gives the value
of the friction and the slope of the curve determines the mobility.
The curvature indicates to what degree seepage, et cet., enter in.

APPENDIX C

TECHNICAL VISCOMETERS

Instruments very different from those employed in scientific
work are much in vogue both in this country and abroad for
industrial purposes, particularly in the oil industry. Thus we
have the Engler Viskosimeter in Germany, the Redwood Vis¬
cometer in Great Britian, the Saybolt Viscosimeter in the United
States, the Barbey Ixometre in France and a host of others. Most
of them seem to have been devised with the idea in mind that the
time of flow of a given quantity of various liquids through an
opening is approximately proportional to the viscosity, without
much regard to the character of the opening. There is usually
a container which is filled to a certain level and a short efflux tube
opening into the air. The number of seconds required for a given
quantity of liquid to flow out under gravity is taken as an indica¬
tion of the viscosity.

As it was gradually realized that these times of flow were not
even proportional to the true viscosities, efforts have not been
wanting to reduce the times of flow to true viscosities. Since
the pressure is due to an average head of liquid h , the pressure is
hgp and the viscosity formula 1, p. 295, may be written

Having obtained the values of the constants A and B by cali¬
brating the viscometer with liquids of known viscosity it appears
possible to calculate the kinematic viscosity 77/p; but if absolute
viscosities are desired it is necessary to make a supplementary
determination of the density p. Thus elaborate tables and
charts have been devised for converting Engler “Degrees”
(c/. Ubbelohde (1907)), and Redwood (c/. Higgins (1913), Herschel
(1918) or Saybolt “Seconds”) into true viscosities.

The widespread use of the Saybolt viscometer in this country
makes desirable the inclusion here of the specifications for its use
adopted by the American Society for Testing Materials.

294 .

APPENDIX C

325

“l. Viscosity. —Viscosity shall be determined by means of the
Saybolt Standard Universal Viscosimeter.

“2. Apparatus. —(a) The Saybolt Standard Universal Viscos-

A Oil Tube Thermometer. K Stirring Paddles.

B Bath Thermometer. L Bath Vessel.

C Electric Heater. M Electric Heater Receptacle.

€ Turntable Cover. hi Outlet Cork Stopper.

E Overflow Cup. P Gas Burner.

F Turntable Handles. Q Strainer.

G Steam Inlet or Outlet. R Receiving Flask, i

H Steam U-Tube. «S Base Block.

J Standard Oil Tube. T Tube Cleaning Plunger.

Fig. 96.— The Saybolt Universal Viscometer.

imeter (see Fig. 96) is made entirely of metal. The standard oil
tube J is fitted at the top with an overflow cup E, and the tube is
surrounded by a bath L. At the bottom of the standard oil tube
is a small outlet tube through which the oil to be tested flows
into a receiving flask R } whose capacity to a mark on its neck is

326

FLUIDITY AND PLASTICITY

60 (±0.15) cc. The lower end of the outlet tube is enclosed
by a larger tube, which when stoppered by a cork N, acts as a
closed air chamber and prevents the flow of oil through the outlet
tube until the cork is removed and the test started. A looped
string is attached to the lower end of the cork as an aid to its
rapid removal. The bath is provided with two stirring paddles
K and operated by two turn-table handles F . The temperatures
in the standard oil tube and in the bath are shown by ther¬
mometers, A and B. The bath may be heated by a gas ring
burner P, steam U-tube H, or electric heater C. The standard
oil tube is cleaned by means of a tube cleaning plunger T ) and
all oil entering the standard oil tube shall be strained through a
30-mesh brass wire strainer Q. A stop watch is used for taking
the time of flow of the oil and a pipette, fitted with a rubber
suction bulb, is used for draining the overflow cup of the stand¬
ard oil tube.

“(5) The standard oil tube J should be standardized by the
U. S. Bureau of Standards, Washington, D. C., and shall conform
to the following dimensions:

Dimensions

Minimum,

centimeters

Normal,

centimeters

Maximum,

centimeters

Inside diameter of outlet tube...

0.1750

0.1765

0.1780

Length of outlet tube.

1.215

1.225

1.235

Height of overflow rim above

bottom of outlet tube.

12.40

12.50

12.60

Diameter of container of stand¬

ard oil tube.

2.955

2.975

2.995

Outer diameter of outlet tube at

lower end.

0.28

0.30

0.32

“3. Method. —Viscosity shall be determined at 100°F (37.8°C),
130°F (54.4°C), or 210°F (98.9°C). The bath shall be held
constant within 0.25°F (0.14°C) at such a temperature as will
maintain the desired temperature in the standard oil tube. For
viscosity determinations at 100 and 130°F, oil or water may be
used as the bath liquid. For viscosity determinations at 210°F,
oil shall be used as the bath liquid. The oil for the bath liquid
should be a pale engine oil of at least 350°F flash-point (open

f

l

>

i

?

t

1

APPENDIX C 327

cup). Viscosity determinations shall be made in a room free
from draughts, and from rapid changes in temperature. All oil
introduced into the standard oil tube, either for cleaning or for
test, shall first be passed through the strainer.

“To make the test, heat the oil to the necessary temperature
and clean out the standard oil tube with the plunger, using some
of the oil to be tested. Place the cork stopper into the lower
end of the air chamber at the bottom of the standard oil tube.
The stopper should be sufficiently inserted to prevent the escape
of air, but should not touch the small outlet tube of the standard
oil tube. Heat the oil to be tested, outside the viscometer, to
slightly below the temperature at which the viscosity is to be
determined and pour it into the standard oil tube until it ceases
to overflow into the overflow cup. By means of the oil tube
thermometer keep the oil in the standard oil tube well stirred and
also stir well the oil in the bath. It is extremely important that
the temperature of the oil in the oil bath be maintained constant
during the entire time consumed in making the test. When the
temperature of the oil in the bath and in the standard oil tube are
constant and the oil in the standard tube is at the desired tem¬
perature, withdraw the oil tube thermometer; quickly remove the
surplus oil from the overflow cup by means of a pipette so that
the level of the oil in the overflow cup is below the level of the oil
in the tube proper; place the 60-ml flask in position so that the
oil from the outlet tube will flow into the flask without making
bubbles; snap the cork from its position, and at the same instant
start the stop watch. Stir the liquid in the bath during the run
and carefully maintain it at the previously determined proper
temperature. Stop the watch when the bottom of the meniscus
of the oil reaches the mark on the neck of the receiving flask.

“The time in seconds for the delivery of 60 ml of oil is the
Saybolt viscosity of the oil at the temperature at which the test
was made.”

There is little to recommend any one of these instruments
except their wide use in their respective countries. They are
inaccurate and in the case of viscous oils time-consuming. With
volatile solvents they cannot be used at all due to evaporation.
The greatest source of error in the technical instruments is due
to poor temperature control. The bath around the container is

328

FLUIDITY AND PLASTICITY

small, the stirring ineffective and the end of the efflux tube is
exposed to the air. In making duplicate determinations the
liquid flows out into the air and generally cools off, so the bath is
raised to somewhat above the desired temperature in order to
bring the temperature back again to the large mass of oil in the
container. If the run is started when the temperature comes to
the proper point, it is almost impossible to prevent it going up
during the run.

Another important source of error arises from the very extra¬
ordinary kinetic energy corrections encountered. The Engler
instrument, for example, is normally calibrated with water at
20°C and the kinetic energy correction amounts to over 90 per
cent of the total energy expended. The viscosity in this case has
but little part in determining the rate of flow, and we have already
seen that the coefficient (m) of the kinetic energy correction is
subject to some uncertainty.

Closely connected with the kinetic energy correction, are the
difficulties due to end effects and possible turbulence which are
aggravated in short, wide tubes.

It is difficult to adequately clean this type of instrument or to
tell when it has been properly cleaned. The liquids readily
absorb dust, moisture and other impurities from the air and they
may thus undergo loss or chemical change. Meissner (1910) has
made a study of these sources of error. Effects of surface tension
at the end of the capillary, of the changing level of liquid in
the container, of slow drainage of oil down the side of the receiving
flask are found to be small sources of error. With the Saybolt
instrument, the flow is started by pulling out a stopper from the
hollow cylinder below the efflux tube. One must see that no
liquid accumulates in the air space above the stopper.

Instruments embodying the principles worked out by Coulomb
and Couette have been devised by Doolittle, Stormer, and Mac-
Michael. In the Stormer instrument a cylinder is rotated by the
force arising from a falling weight, suspended by a cord carried
over a pulley. The speed varies with the viscosity of the liquid
and the revolutions per minute are counted. A better plan is the
one adopted by MacMichael of using a constant speed, imparted
to an outside cup and measuring the angle of torque produced in
a disk supported in the liquid by means of a steel wire. The

APPENDIX C

329

iust.riim.ent has considerable range, for wires of differing diameters
can be used for widely differing viscosities. The readings are
instantaneous and the instrument is compact and easily manipu¬
lated. The most troublesome feature of this type of instrument
is the lack of constancy in the supporting wire. It is neces¬
sary to use these wires with considerable care and to calibrate
frequently. Since the corrections of the instrument are not
fully understood, the calibrating fluid should have nearly the
same viscosity as the viscosity to be measured (c/. Herschel
( 1020 )).

For liquids of high viscosity, the falling sphere method is used
industrially. If the containing vessel does not have a diameter
at least 10 times that of the ball, a correction must be applied
Sheppard (1917). The method is admirably adapted for abso¬
lute measurements, but usually workers have felt dependent
upon calibrating liquids, but since there is a dearth of calibrating
fluids of high viscosity liquids are often used in which the velocity
of fall is too great for the strict application of Stokes’ law and a
correction has to be made. Reproducible liquids of high viscosity
which have been accurately determined should be available for
the industrial requirements.

APPENDIX D

The measurements of Poiseuille, being somewhat inaccessible
but of great practical as well as historical interest are given
in detail in the following tables. Comparative values of the
viscosity of water by various observers with all of the known
corrections made are given in Table II. Since specific viscosities
are often used, relative to water at different temperatures, we give
the viscosity for water for every degree from 0 to 100 in Table
III, and in Tables IV and V we give the fluidities of alcohol-water
solutions and sucrose-water solutions as possible calibration fluids
where water would be too fluid. For changing viscosities to
fluidities the table of reciprocals (Table VI) is very convenient.
To get the reciprocal of a number such as 1.007, the first part
of the table is not very convenient on account of the large
differences used in enterpolation. If however one uses instead
10 X 0.10070 in the latter part of the table, fifth column, p. 343,
the number 9.93 X1G" 1 is found as the reciprocal without enter¬
polation. The part of the table from 10.0 to 15.0 may also be
used for this same purpose, in which case the reciprocal of 10.07
is found in the ninth column.

A table of four-place logarithms (Table VII) are included,
and are often sufficiently exact, since viscosities are generally not
more accurate than one part in 1,000.

APPENDIX D

331

Table I.—Measurements of Poiseuille

03

cn

u

03

Diameter of capillary in

o

03

1 ? i

3

o

centimeters

X

03

3

*

> 03

a

|

pO

•fit

o

a

"a

0 )

Open end

1 Bulb end

o

2

"3

ja

g O

x ‘7
a £

o

3

d 8

® 3

u ®

g lo

08

a

w

O oo

s*

O 03

ft

I g

8 . 8

s?

a q

o o

O 03 ®

CO

G

0 ?

.2 M

c? 'x

.2 -8

M CD

S 2 2 a

S 2 fl

O

h3

S a

s 03

2 05

S 03

t5 a

>

* S rH

Ph

H * 0

A°

iO

«5

03

8

O

10

385.870

3,505.75

Tt<

CO

rH

rH

00

o

739.114

1,830.75

d

o

o

d

CO

773.443

1,750.00

Do.

Do.

Do.

Do.

Do.

Do.

0.6

CO

774.291

2,327.75

5.0

773.400

2,025.25

10.0

773.443

1,750.00

15.0

773.597

1,528.00

20.0

775.093

1,344.50

25.0

774.886

1,195.00

30.1

775.058

1,067.50

35.1

774.451

962.25

40.1

774.354

871.50

45.0

774.827

793.25

i

51.068

20,085.0

97. 764

10,361.0

o

w

o

o

147.834

! 6,851.0

A*

00

lo

o

Do.

Do.

o

Do.

193.632

5,233.0

o

o

387.675

2,612.5

738.715

1,372.5

774.676

1,308.0

98.404

6,921.0

AH

CM

o

148.320

4,594.0

3

rH

o

o

Do.

Do.

Do.

Do.

193.421

3,515.0

lO

o

o

387.445

1,757.0

774.810

878.0

o

lO

Am

U 2

id

rH

o

Do.

Do.

Do.

Do.

387.520

880.0

*0

o

o

774.895

448.0

cm’

o

o

24.661

8,646.0

49. 591

4,355.0

A IV

*o

is

98. 233

2,194.0

«3

Do.

Do.

Do.

Do.

Do.

Do.

148.233

1,455.0

194.257

1,116.0

388.000

571.0

775.160

298.0

23. 638

5,570.0

49.185

2,699.0

A v

«3

U5

99.221

1,360.0

03

Do

Do.

Do.

Do.

Do.

Do.

148. 623

918.5

o

193.315

718.0

387. 737

381.0

774.620

207.0

332

FLUIDITY AND PLASTICITY

Table I. —( Continued )

0)

X

_d

CQ

a

Diameter of capillary in
centimeters

<D

in cc

05

.JL, >>

g 6
> <v

«4H ®

o

a

o

d

CJ

o

Open end

Bulb end

o

JL

13

X

1 1

a 3

- a

oS

a

M

XI

W)

u

O qq

Q eo

u<

O eo

i-.

2 <D

& .§
i ®

oP

S ©

2 2

3 So

o o
v o S,

s

C3

s*

.2 *S

e? '«

.2 -a

S§°o

a fi a

Q

§ «

S 03

S 88

2 *

5 a

£ «

k H H

Ph

s * 0

24.753

3,828.75

50.001

1,923.75

»o

99.343

994.00

Avi

CO

Do.

Do.

Do.

Do

Do.

Do.

148.618

682.00

d

193.010

537.75

387.887

291.50

773.790

165.75

4.783

3,926.75

6.204

3,072.00

12.129

1,685.50

24.003

974.25

A vii

£

49.040

571.75

d

Do.

Do.

Do

Do.

Do

Do.

98.832

348.75

148.475

267.00

193.501

224.00

387.972

144.00

773.717

95.00

B

id

o

id

CO

»

uo

«o

CO

o

CO

00

388.256

739.333

4,103.5

2,156.0

o

o

©

©

o

T*

777.863

2,060.0

o

o

o

©

©

d

55.286

21,430.0

o

©

97. 922

12,079.0

B 1

»o

rf*

148.275

7,981.5

o

id

©

tH

©

Do.

Do.

Do.

Do.

193.947

6 ,100.0

©

©

387.695

3,052.0

739.467

1,600.0

774.891

1,526.5

OJ

0*

99.163

7,804.0

fin

id

t>

0*

*-)

149.679

5,165.0

CO

o»

©

*-)

O

Do.

Do.

Do.

Do.

193.441

3,997.0

o

©

387.130

1,995.0

774.796

999.0

49.091

7,471.0

2 ?m

id

t>

id

CO

CO

0*

Do.

Do.

Do.

Do.

98.315

148.571

3,729.0

2,473.0

CO

o

©

193.877

1,892.0

<N

o

o

S

388.100

946.0

774.880

473.0

APPENDIX D 333

Table I. — ( Continued)

M

1

<v

d

>p

<3>

CJ

a

Diameter of capillary in
centimeters

U

05

o

o

.2

a

O £

> CD
«*-r 03

*o

a

«

a>

Open end

1 Bulb end

o

CD

-Q

11

sS

_o

tf ^

.2 §

g=j g

ca

(3

M

*a

to

c

.O m
e? ’«

t-i

O 03

.2 *M

.O 03

e?

o »

.2 *K

1 S

& a
i ®

sP

2 ©

Pressure

meters r

10°C

o o

<33 <33 ®S

S S a

Q

►3

E 03

S *

S 03

03

r° A

Eh

> *

S 53 °

24.756

5,543.0

49.857

2,762.0

Biv

o

o

CO

99.214

1,400. 0

o

05

©

tH

©

Do.

Do.

Do.

Do.

149.082

935.0

©

©

o'

193.194

728.9

387.024

375.0

774.540

199.0

24.290

2,386.0

49.578

1,193.0

J5 V

CO

99.139

621.0

©

CO

O

r-H

O

Do.

Do.

Do.

Do.

149.098

428.0

o

©

o’

193.130

34.00

387.024

189.0

773.282

110.0

0.5

774.048

2,816.75

5.0

774.047

2,422.75

6.0

773.848

2,350.50

10.0

774.030

2,093.50

CO

CO

Mean

diam.

15.2

t-

©

©

774.070

1,826.00

C

o

at 1

0°C

20.0

774.110

1,612.75

©

©

©

085

25.1

N

774.841

1,427.50

30.1

774. 503

1,280.50

35.1

774.574

1,149.50

40.1

774.676

1,042.50

45.1

774.678

949.00

iO

CO

iO

ss

©

©

o

©

©

©

385.158

4,210.00

c

CO

o

00

s

co

o

o

1

o

©

©

738.969

2,192.00

2

o

o

o’

o

o

d

774.030

2,093.50

52.257

23,135.00

98.411

12,280.00

«3

So

00

149.241

8,098.00

C*

©

8

Do.

Do.

Do.

Do.

Do.

193.314

6,250.00

>>

o'

387.562

3,118.00

738.767

1,636.00

774.757

1,560.50

99.868

7,997.00

«0

00

149.034

5,362.00

Cn

©

©

©

©

o

Do.

Do.

Do.

Do.

193.867

4,117.00

'-tf

d

©

386.915

2,065.00

774.563

1,029.00

FLUIDITY AND PLASTICITY
Table I. —( Continued )

Diameter of capillary in ( »

centimeters a> -S

- ° |

Open end Bulb end £

49.702

7,765.00

98.921

3,899.00

Do.

Do.

Do.

Do.

148.303

2,598.50

193.544

1,994.00

387.157

995.00

774.677

498.00

24.791

6,186.75

49.931

3,073.00

98. 322

1,559.75

Do.

Do.

Do.

Do.

148.795

1,029.50

194.102

788. 00

387.191

399.00

774.607

203.00

24.192
50.506
99.102
149.119

3,587.00
1,768.00
904.00
606. 50

Do.

Do.

194.

,217

470.00

387.

237

245.00

773.

,327

131.50

386.

,247

9,708.00

O

a>

738.

137

5,080.00

o

CO

773.

970

4,846.00

54.785

35,460.00

o

o

55.796

34,798.00

to

Tt<

CO

<M

0

99.508

19,517.00

o

©

8

8

8

o

Do.

149.219

13,021.00

6

o

o

o

192.907

10,071.00

386.555

5,025.00

774.617

2,506.00

Do.

Do.

Do.

Do.

5.00

Do.

774.887

2,898.50

10.00

774.617

2,506.00

15.00

773.271

2,199.00

Mean

diam.

20.00

774.119

1,928.00

0.004

40406

25.05

775.045

1,713.75

30.07

774.356

1,532.50

35.00

675.429

1,375.50

40.00

774.475

1,246.75

45.10

774.077

1,138.00

APPENDIX D

335

Table I. — (Continued)

0>

2

cy

Diameter of

capillary in

o

o

t 1'

o i

jd

fl

+■>

<u

a

centimeters

X

CD

.2

_Q

3

a

> s

60

o fl

o

a

.2

‘43

fl

©

o

Open end

Bulb

» end

*0

CD

u

fl -w

3

.ft

1 2
.s a

x ""T
fl £

a 1

a

.2

ta a

l O

« 03

a

s

.fl

M

fl

O 03

A *X

O 03

.2 *fl

u

.o OQ

a "x

O oj
fl *rj

£ 2

2 °
ft *2

a ®

2 o

a 2

p

§ !o

S 2 ©

o o

03 CO °2

s s 1

0

0j

^ 03

^ 03

W ^

S a

Fh

| *

H S rH

Ph

p 3 °

98.917

10,149.00

«3

S

-*

co

0

147.857

6,789.00

W3

8

o

o

Do.

Do.

O

Do.

193.485

5,178.00

CM

d

d

386.847

2,589.50

773.985

1,293.00

50.374

7,978.00

CO

CO

97.124

4,136.00

Dm

iO

CM

Do.

Do.

Do.

Do.

148.248

2,706.00

o>

Ol

1

O

o

192.707

2,084.75

©

o

o

387.419

1,038.00

775.866

519.00

23.884

5,479.00

50.276

2,611.75

Dnr

»0

Tti

CM

T*

97.440

1,373.00

CO

CO

8

8

Do.

Do.

Do.

Do.

147. 889

897. 50

o

o

.

o

193.459

697. 00

387. 062

349.00

1

772.117

176.00

CO

oo

o

CO

58.211

26,625.00

©

o

o

CO

os

386.218

4,020.00

B

CO

§

©

o

i

1

o

CM

737.829

2,103.00

CM

o

o

o

o

o

774.017

2,006.00

0.50

773.808

2,705.00

5.00

774.757

2,318.50

Mean

diam.

10.00

774.017

2,006.00

0.00

2938

15.00

773.709

1,756. 75

20.00

773.475

1,547.25

E

Do.

Do.

Do.

Do.

Do.

25.10

Do.

774.081

1,372.25

30.05

775.271

1,227. 25

35.07

774.563

1,102. 50

40.10

775. 329

997. 75

1

45.00

774.635

908.75

o

CO

96. 693

5,903.50

1 '

CD

147. 588

3,868.00

E*

00

1 8

1

Do.

Do.

10°

Do.

193. 100

2,955.00

o

o

o

386. 787

1,469.00

773. 880

736. 75

24. 301

5,651.00

49.994

2,751.00

Bn

o

Do.

Do

Do.

Do.

Do.

Do.

96. 123

1,426.00

cm

148. 307

925.00

°

193. 357

707. 00

386. 852

354.00

773. 223

178.00

I

)

I

I

I

I

i

APPENDIX D

337

Table I.—(i Continued )

©

-Q

0

tn

f-

OJ

•+j

OJ

a

Diameter of capillary in
centimeters

X

cd

o

C3

a

c3

g b

> as

V* 00

*o

a

_o

h3

a

OJ

o

Open, end

Bulb end

o

0)

P-c

B -p

3

x>

•2 g

S £
sa |

"5

a

•a

OJ

"S

(3

t-*

.O 03

3 *x

tH

§ .2

♦<3

'S' *X

O 03

•S ’x

ct fl

fcn 03

CD ff

ft .5

0 *-<
a ©

a?
S 2

CD m

§ .g o

S2°o

o 3
© © ®
a a

Q

S *

s 03

s s

S *

© ft

> «

P s o

74.29

114.00

83.89

130.00

162.89

63.00

j?V

»c

329.39

39.00

o

Do.

Do.

Do.

Do.

Do.

Do.

653.49

25.00

iH

1,306.69

16.00

1,985.29

13.00

2,606.37

10.75

5,146.62

7.50

10,456.65

5.00

G

o

CO

o

o

o

1.087.200

407.00

§

CO

CO

s

OS

8. 60

CD

1,586.340

281.00

<N

o

o

o

o

8.70

to

2,084.060

213.00

CO

as

6

o

o

d

8.80

1>

vH

2,602.300

170. 00

18.70

145.300

2,290.00

18. 90

269.220

1,232.00

18.90

520.240

634.00

S

18. 90

1,019.870

323.00

Gi

.

Do.

Do.

Do.

Do.

18.80

Do.

2,014.160

162.00

©

00

18. 70

3,437.360

97.00

18.80

j 6,841.370

48.75

18.80

10,191.540

33.00

<N

o

CM*

©

18.95

145. 300

1,115.00

QIl

o

o

OS

CO

O

o

o

co

OS

CO

o

CO

CO

o

19.25 •
19.30

Do.

269.220
518. 940

597.00
305.50

o

o

d

d

o

19. 50

1,019.670

155.00

19. 50

2,014.360

79.75

§

CD

to

to

CO

00

11.00

2,316. 870

9,048.00

H

©

©

§

11.00

0. 5

3,837.000

5,438.00

d

o

o

11. 10

6,117. 600

3,460.00

10. 80

3,850.160

388.00

§

o

00

o

10.90

4,610.230

319.00

I

00

o

5

11.00

1.0

5,370.130

267.00

d

<N

o

d

11.00

6,127. 360

235.00

7. 50

6,130.080

261.00

co

co

11.00

54.987

8,590.00

o

o

CO

CO

11.00

210. 129

2,250.00

K

o

©

11.00

1.0

419. 645

1,125.75

O

00

o

d

11.00

835.565

565.00

12.00

1,576.000

286.00

22

Designation of tube

338

FLUIDITY AND PLASTICITY

Table I. —( Continued)

2

3

Diameter of capillary in

C3

C3

O C3

«Q

3

03

a

centimeters

03

.2

o3

.Jh >>

> 03

^ ca

o

a

o

a

0)

o

Open end

Bulb end

O

03

u

5 45>

3

JS

Pressure in mil
meters merou
10°C

Sa

03

a

*C

03

Q

.2

.3

bo

a

,3

Major

axis

Minor

axis

Major

axis

Minor

axis

Tempera

perimen

Volume
at 10°C

Time of
ume of
onds

1

i

11.00

1.0

2,338.376

197.50

CO

CO

11.00

3,095.540 |

154.00

O

CO

CO

11.00

3,856.939

123.00

K

Tf<

o

o

11.00

4,616.534

106.25

CO

CO

©

d

11.00

5,376.534

88.25

11.00

6,136.534

77. 50

7.00

6,136.534

86.75

C3

o

05

CO

o

05

eo

o

C3

M

§

oo

s

8

8

8

10.00

8

775.000

1,240.00

d

o

o*

|

o

o

M 1

0.125

Do.

Do.

Do.

Do.

Do.

Do.

775.000

84.50

Table II. —Viscosity op Water in Centipoises as Determined by Different Observers

APPENDIX D

339

>i o3

r-

oo

It tH

O It

It

to

o

CO

XH

00

to

rH

05

to

to

to

xH

00

D T3

cm

oo

It O

to CO

O

CM

CO

0CJ

05

c D

00

to

CO

05

CO

to

CO

05

00

O r.

05

rH

o

O 05

o

CM

to

05

XH

o

CO

CO

o

IT

to

CO

rH

05

00

'cd

k) d

1>

to

CO rH

O CO

03

It

CO

to

to

to

tH

rH

xH

CO

CO

CO

00

CM

CM

0^3

rH

rH

rH rH

rH O

O

d

o

o

o

o

o

o

d

o

o

o

o

o

d

03

bJO

It

tO

rH CO

CO rH

05

to

CO

00

It

CM

rH

05

CM

xH

CO

rH

CO

rH

rH

00

to

CO o

tH tH

o

CO

to

05

It

o

to

CO

05

to

xH

tH

00

CM

00

rH

O

O 05

o

CM

to

05

XH

o

It

CO

o

IT

to

00

rH

05

00

03

It

to

CO rH

O 00

CO

It

CO

to

to

to

XH

xH

xh

CO

CO

CO

00

CM

CM

rH

rH rH

rH O

o

o

o

d

o

o

o

o

d

o

d

d

o

o

o

03

.t3

o

rH

£3 &

hH o

rH

CO

IT

rH

CO

00

CM

05

xH

00

|T

CO

CO

& a

CO

XH

O It

to rH

05

CM

to

00

05

IT

CM

CO

CO

05

to

CO

00

00

05

CM

O CO

O 05

05

CM

to

05

xH

o

It

CO

o

IT

to

CO

05

.S o3

c

It

to

CO rH

O 00

It

It

CO

to

to

to

tH

tH

xh

CO

CO

CO

00

CM

pq ^

c3 S

rH

rH rH

rH O

d

o

o

o

o

d

o

o

d

d

o

o

d

d

bJO

oo

to

CO

o

♦

3

CM

CM

O 03

CO CM

o

It

o

o

00

05

CO

CO

o

CO

to

CD

o

xH

3

05

CM

rH

O 05

o

CM

to

o

to

o

CO

CO

o

00

to

CO

rH

o

00

m

It

to

CO rH

O 00

00

It

CO

CO

to

to

tH

xH

xH

CO

CO

CO

00

CO

CM

w

rH

rH

rH rH

rH O

o

d

o

o

o

o

o

o

d

o

o

o

o

o

d

03

CO

CO

th

to O

to

o

to

05

xH

XH

CO

CO

00

CM

IT

CO

o

o

xH

ft ^

03

CO

00

rH CM

o o

CO

05

CM

to

CO

xH

It

xH

xfl

00

xH

CO

|T

f-T T

fafl

It

o

O CO

O 05

05

rH

to

05

x+l

o

CO

CO

o

IT

to

CO

05

00

5 Si?

It

to

CO rH

O 00

It

It

CO

to

to

to

XH

XH

xh

CO

CO

CO

00

CM

CM

H

rH

rH

rH rH

rH O

o

o

d

o

d

d

d

o

d

d

d

o

o

o

o

03

It

CO

CO CO

It xo

CM

CO

CO

rH

CM

05

tH

IT

It

o

CO

xfi

to

IT

rH

O

CM

rH Htl

O 05

O

CM

to

o

to

o

It

CO

o

00

to

CO

05

00

C

00

to

CO rH

O 00

00

It

CO

CO

to

to

tH

xH

xH

CO

CO

CO

00

CM

CM

CO

1 rH

1

rH

rH rH

rH O

o

o

o

o

d

o

d

o

d

o

d

o

d

d

d

bJO

05

to hH

00 IT

CO

CO

00

rH

CM

It

OO

05 CO

It

CO

rH

to

o

=

It

o

05 CO

05 05

CM

to

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to

M

It

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It

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It

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CM

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05

rH

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to

05

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CO

It

CO

to

rH

rH

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rH O

d

o

d

o

CO

00 to

CO CO

rH

o

CO

It

■*H

00 CO

CO CO

05

o

CO

Q

rH

O HU

O 05

o

to

05

Si

to

CO rH

O 00

CO

It

CO

to

rH

rH rH

rH O

d

d

d

d

§

CO

o

It

00 xH

CO tH

CO

tH

CO

o

CO

£- *0

It cO

rH

05

CM

CO

/P

O

05

rH

O TH

O 05

o

rH

to

05

Ph

i>

to

CO rH

O 00

CO

It

CD

to

rH

rH

rH rH

rH O

d

o

d

d

tO

CO

to to

oo o

00

CO

Jr

tO

O

00

CNJ O

to

to

CO

CO

It

rH

O CO

O 05

05

rH

tH

00

Ir

to

CO rH

O 00

It

It

CO

to

rH

rH

rH rH

rH O

d

d

d

o

£ £ £

o

to

O i©

O to

o

to

o

to

o

to

o

to

o

to

o

to

o

to

8

r « 5

S3

rH rH

CM CM

CO

CO

XH

to

to

CD

CO

I>

IT

00

00

05

05

H C

U *♦*

rH

340 FLUIDITY AND PLASTICITY

Table III.—Fluidity and Viscosity op Water Calculated by Formula 1
for Every Degree Between 0° and 100°C

Tem¬

pera¬

ture,

°C

Flu¬

idity

Vis¬
cosity
in cp

Tem¬

pera¬

ture,

°C

Flu¬

idity

Vis¬
cosity
in cp

Tem¬

pera¬

ture,

°C

I

Flu¬

idity

Vis¬
cosity
in cp

0

55.80

1.7921

33

132.93

0.7523

67

236.25

0.4233

1

57.76

1.7313

34

135.66

0.7371

68

239.57

0.4174

2

59.78

1.6728

35

138.40

! 0.7225

69

242.91

0.4117

3

61.76

1.6191

36

141.15

0.7085

70

246.26

0.4061

4

63.80

1.5674

37

143.95

0.6947

71

249.63

0.4006

5

65.84

1.5188

38

146.76

0.6814

72

253.02

0.3952

6

67.90

1.4728

39

149.60

0.6685

73

256.42

0.3900

7

70.01

1.4284

40

152.45

0.6560

74

259.82

0.3849

8

72.15

1.3860

41

155.30

0.6439

75

263.25

0.3799

9

! 74.28

1.3462

42

158.20

0.6321

76

266.67

0.3750

10

76.47

, 1.3077

43

161.11

0.6207

77

270.12

0.3702

11

78.66

1.2713

44

164.02

0.6097

78

273.57

0.3655

12

80.89

1.2363

45

167.00

0.5988

79

277.04

0.3610

13

83.14

1.2028

46

169.97

0.5883

80

280.53

0.3565

14

85.40

1.1709

47

172.95

0.5782

81

284.03

0.3521

15

87.69

1.1404

48

175.95

0.5683

82

287.53

0.3478

16

90.00

1.1111

49

178.95

0.5588

83

291.03

0.3436

17

92.35

1.0828

50

182.00

0.5494

84

294.54

0.3395

18

94.71

1.0559

51

185.05

0.5404

85

298.06

0.3355

19

97.10

1.0299

52

188.14

0.5315

86

301.63

0.3315

20

99.50

1.0050

53

191.23

0.5229

87

305.21

0.3276

BO. 20

100.00

1.0000

54

194.34

0.5146

88

308.78

0.3239

21

101.94

0.9810

55

197.45

0.5064

89

312.35

0.3202

22

104.40

0.9579

56

200.62

0.4985

90

315.92

0.3165

23

106.86

0.9358

57

203.78

0.4907

91

319.53

0.3130

24

109.38

0.9142

58

206.95

0.4832

92

323.13

0.3095

25

111.91

0.8937

59

210.13

0.4759

93

326.74

0.3060

26

114.45

0.8737

60

213.33

0.4688

94

330.38

0.3027

27

117.03

0.8545

61

216.54

0.4618

95

334.01

0.2994

28

119.62

0.8360

62

219.80

0.4550

96

337.65

0.2962

29

122.25

0.8180

63

223.07

0.4483

97

341.30

0.2930

30

124.89

0.8007

64

226.34

0.4418

98

344.96

0.2899

31

127.54

0.7840

65

229.64

0.4355

99

348.63

0.2868

32

130.22

0.7679

66

232.94

0.4293

100

352.30

0.2838

1 <f> = 2.1482{ (f - 8.435) + V8078.4 + (t - 8.435) 2 } - 120. Cf. p. 137.

APPENDIX D

341

Table IV. —Fluidity of Alcohol-water Mixtures 1

Weight percentage of ethyl alcohol

Tem-

0

10

20

30

39

40

45

50

60

70

80

90

100

pera-

ture

Volume percentage of ethyl alcohol at 25°C

0

12.36

24.09

35.23

44.92

45.83

50.94

j

55.93

65.56

74.80

83.59

92.01

100

0

. 55.8

30.2

18.8

14.4

13.8

14.0

14.4

15.2

17.4

21.0

27.1

36.6

56.4

5

65.8

38.8

24.6

18.9

17.8

17.9

18.2

19.0

21.6

25.6

32.0

43.3

61.6

10

76.5

45.9

31.6

24.7

22.8

22.8

23.0

23.9

26.5

30.6

36.9

47.6

68.2

15

87.7

55.8

38.2

30.7

28.4

28.3

28.5

29.1

31.8

36.1

43.3

55.5

75.1

20

99.5

65.0

45.8

36.9

34.7

34.4

34.7

34.8

37.4

42.2

49.8

62.1

83.3

25

111.9

75.6

55.1

45.9

42.5

42.5

41.9

41.7

44.6

49.1

57.2

70.2

91.2

30

124.9

86.2

64.4

53.4

50.0

49.4

49.5

49.6

51.9

56.6

65.3

78.2

99.7

35

138.4

99.4

75.1

63.3

58.6

58.3

57.7

58.0

60.1

65.4

73.8

87.2

109.4

40

152.4

110.2

86.2

73.1

67.9

67.5

66.9

66.7

69.1

74.4

83.1

96.6

119.9

45

167.0

123.2

98.5

84.1

77.9

77.6

76.5

77.3

78.7

84.1

92.5

106.5

130.8

50

182.0

136.3

110.2

95.2

89.0

88.3

87.1

86.6

88.7

94.2

103.3

117.9

142.5

55

197.4

150.9

122.9

107.6

100.7

100.2

98.4

98.0

100.3

106.0

115.3

130.8

155.2

60

213.3

164.3

135.8

119.9

113.0

112.0

110.3

109.5

110.8

116.8

126.7

142.1

168.9

65

229.6

180.5

150.1

133.0

125.3

124.7

122.6

122.3

124.1

130.6

140.7

156.0

181.5

70

246.3

194.5

164.5

146.4

138.0

137.5

135.2

135.1

137.2

143.9

153.9

169.9

198.6

75

80

263.2

280.5

210.2

232.7

178.8

198.1

160.3

176.4

151.5

167.1

150.8

166.5

148.9

164.1

148.7

163.5

150.8

165.7

157.1

166.6

183.0

212.5

Table V.—Sucrose Solutions, Bingham and Jackson

Tem¬

pera-

Percentage sucrose by-
weight

Tem¬

pera-

Percentage sucrose by
weight

ture

0

20

40

60

ture

0

20

40

60

0

55.91

26.29

6.77

0.42

55

197.16

113.12

45.06

8.57

5

65.99

31.71

8.65

0.64

60

212.72

123.79

50.47

10.17

10

76.56

37.71

10.21

0.91

65

229.41

134.81

56.24

11.99

15

87.67

44.11

13.39

1.34

70

246.18

145.97

62.17

13.98

20

99.54

51.02

16.13

1.77

75

263.57

157.56

68.41

16.12

25

111.84

58.69

19.28

2.28

80

281.21

169.53

74.96

18.51

30

124.70

66.51

22.82

2.96

85

299.31

181.80

81.92

21.14

35

138.79

75.12

26.58

3.77

90

317.87

89.06

24.07

40

153.07

83.82

30.78

4.70

95

335.46

96.41

26.85

45

50

167.84

181.92

93.42

103.07

35.13

40.05

5.82

7.14

100

354.49

104.11

29.96

1 Values given are the weighted average of those of Stephan (1802), Pagliani and Batelli
(1885), Traube (1886), Noack (1886) and Bingham and Thomas (1913).

342

FLUIDITY AND PLASTICITY

Table VI.—Reciprocals

No.

0

1

2

3

4

5

G

7

8

9

Dif.

1.0

1.0000

9901

9804

9709

9615

9524

9434

9346

9259

9174

92

1.1

0.9091

9009

8929

8850

8772

8696

8621

8547

8475

8403

1.2

8333

8264

8197

8130

8065

8000

7937

7874

7813

7752

6-5

1.3

7692

7634

7576

7519

7463

7407

7353

7299

7246

7194

1.4

7143

7092

7042

6993

6944

6897

6849

6803

6757

6711

IS

1.5

0.6667

6623

6579

6536

6494

6452

6410

6369

6329

6289

42

1.6

6250

6211

6173

6135

6098

6061

6024

5988

5952

5917

8 7

1.7

5882

5848

5814

5780

5747

5714

5682

5650

5618

558/

ra

1.8

5556

5525

5495

5464

5435

5405

5376

5348

5319

5291

29

1.9

5263

5236

5208

5181

5155

5128

5102

5076

5051

5025

2.0

0.5000

4975

4950

4926

4902

1 4878

4854

4831

4808

4785

24

2.1

4762

4739

4717

4695

4673

4651

4630

4608

4587

4566

2 2

2.2

4545

4525

4505

4484

4464

4444

4425

4405

4386

4367

2 U

2.3

4348

4329

4310

4292

4274

4255

4237

4219

4202

4184

2.4

4167

4149

4132

4115

4098

4082

4065

4049

4032

4016

2.5

0.4000

3984

3968

3953

3937

3922

3906

3891

3876

3861

15

2.6

3846

3831

3817

3802

3788

3774

3759

3745

3731

3717

2.7

3704

3690

3676

3663

3650

3636

3623

3610

3597

3584

2.8

3571

3559

3546

3534

3521

3509

3496

3484

3472

3460

2.9

3448

3436

3425

3413

3401

3390

3378

3367

3356

3344

3.0

0.3333

3322

3311

3300

3289

3279

3268

3257

3247

3236

11

3.1

3226

3215

3205

3195

3185

3175

3165

3155

3145

3135

3.2

3125

3115

3106

3096

3086

3077

3067

3058

3049

3040

9

3.3

3030

3021

3012

3003

2994

2985

2976

2967

2959

2950

9

3.4

2941

2933

2924

2915

2907

2899

2890

2882

2874

2865

3.5

0.2857

2849

2841

2833

2825

2817

2809

2801

2793

2786

8

3.6

2778

2770

2762

2755

2747

2740

2732

2725

2717

2710

3.7

2703

2695

2688

2681

2674

2667

2660

2653

2646

2639

7

3.8

2632

2625

2618

2611

2604

2597

2591

2584

2577

2571

3.9

2564

2558

2551

2545

2538

2532

2525

2519

2513

2506

4.0

0.2500

2494

2488

2481

2475

2469

2463

2457

2451

2445

6

4.1

2439

2433

2427

2421

2415

2410

2404

2398

2392

2387

4.2

2381

2375

2370

2364

2358

2353

2347

2342

2336

2331

4.3

2326

2320

2315

2309

2304

2299

2294

2288

2283

2278

4.4

2273

2268

2262

2257

2252

2247

2242

2237

2232

2227

6

4.5

0.2222

2217

2212

2208

2203

2198

2193

2188

2183

2179

4.6

2174

2169

2165

2160

2155

2151

2146

2141

2137

2132

4.7

2128

2123

2119

2114

2110

2105

2101

2096

2092

2088

4.8

; 2083

2079

2075

2070

2066

2062

2058

2053

2049

2045

4.9

2041

2037

2033

2028

2024

2020

2016

2012

2008

2004

5.0

i 0.2000

1996

1992

1988

1984

1980

1976

1972

1969

1965

4

5.1

1961

1957

1953

1949

1946

1942

1938

1934

1931

1927

5.2

! 1923

1919

1916

1912

1908

1905

1901

1898

1894

1890

5.3

i 1887

1883

1880

1876

1873

1869

1866

1862

1859

1855

5.4

1852

1848

1845

1842

1838

1835

1832

1828

1825

1821

5.5

i 0.1818

1815

1812

1808

1805

1802

1799

1795

1792

1789

8

5.6

i 1786

1783

1779

1776

1773

1770

1767

1764

1761

1757

5.7

1754

1751

1748

1745

1742

1739

1736

1733

1730

1727

5.£

! 1724

1721

1718

1715

1712

1709

1706

1704

1701

1698

5. S

i 1695

1692

1689

1686

1684

1681

1678

1675

1672

1669

APPENDIX D

343

Table VI.—( Continued )

No.

0

1

2

3

4

5

6

7

8

9

Dif.

6.0

0.16667

16639

16611

16584

16556

16529

16502

16474

16447

16420

27

6.1

16393

16367

16340

16313

16287

16260

16234

16207

16181

16155

26

6.2

16129

16103

16077

16051

16026

16000

15974

15949

15924

15898

26

6.3

15873

15848

15823

15798

15773

15748

15723

15699

15674

15649

25

6.4

15625

15601

15576

15552

15528

15504

15480

15456

15432

15408

24

6.5

0.15385

15361

15337

15314

15291

15267

15244

15221

15198

15175

28

6.6

15152

15129

15106

15083

15060

15038

15015

14992

14970

14948

28

6.7

14925

14903

14881

14859

14837

14815

14793

14771

14749

14728

22

6.8

14706

14684

14663

14641

14620

14599

14577

14556

14535

14514

21

6.9

14493

14472

14451

14430

14409

14388

14368

14347

14327

14306

21

7.0

0.14286

14265

14245

14225

14205

14184

14164

14144

14124

14104

20

7.1

14085

14065

14045

14025

14006

13986

13966.

13947

13928

13908

7.2

13889

13870

13850

13831

13812

13793

13774

13755

13736

13717

19

7.3

13699

13680

13661

13643

13624

13605

13587

13569

13550

13532

7.4

13514

13495

13477

13459

13441

13423

13405

13387

13369

13351

18

7.5

0.13333

13316

13298

13280

13263

13245

13228

13210

13193

13175

7.6

13158

13141

13123

13106

13089

13072

13055

13038

13021

13004

17

7.7

12987

12970

12953

12937

12920

12903

12887

12870

12853

12837

7.8

12821

12804

12788

12771

12755

12739

12723

12706

12690

12674

16

7.9

12658

12642

12626

12610

12594

12579

12563

12547

12531

12516

8.0

0.12500

12484

12469

12453

12438

12422

12407

12392

12376

12361

8.1

12346

12330

12315

12300

12285

12270

12255

12240

12225

12210

15

8.2

12195

12180

12165

12151

12136

12121

12107

12092

12077

12063

8.3

12048

12034

12019

12005

11990

11976

11962

11947

11933

11919

8.4

11905

11891

11876

11862

11848

11834

11820

11806

11792

11779

14

8.5

0.11765

11751

11737

11723

11710

11696

11682

11669

11655

11641

8.6

11628

11614

11601

11587

11574

11561

11547

11534

11521

11507

8.7

11494

11481

11468

11455

11442

11429

11416

11403

11390

11377

18

8.8

11364

11351

11338

11325

11312

11299

11287

11274

11261

11249

8.9

11236

11223

11211

11198

11186

11173

11161

11148

11136

11123

9.0

0.11111

11099

11086

11074

11062

11050

11038

11025

11013

11001

9.1

10989

10977

10965

10953

10941

10929

10917

10905

10893

10881

12

9.2

10870

10858

10846

10834

10823

10811

10799

10787

10776

10764

9.3

10753

10741

10730

10718

10707

10695

10684

10672

10661

10650

9.4

10638

10627

10616

10604

10593

10582

10571

10560

10549

10537

9.5

0.10526

10515

10504

10493

10482

10471

10460

10449

10438

10428

11

9.6

10417

10406

10395

10384

10373

10363

10352

10341

10331

10320

9.7

10309

10299

10288

10277

10267

10256

10246

10235

10225

10215

9.8

10204

10194

10183

10173

10163

10152

10142

10132

10121

10111

9.9

10101

10091

10081

10070

10060

10050

10040

10030

10020

10010

10.0

0.10000

9990

9980

9970

9960

9950

9940

9930

9921

9911

10

10.1

09901

9891

9881

9872

9862

9852

9843

9833

9823

9814

10.2

9804

9794

9785

9775

9766

9756

9747

9737

9728

9718

10.3

9709

9699

9690

9681

9671

9662

9653

9643

9634

9625

10.4

9615

9606

9597

9588

9579

9569

9560

9551

9542

9533

10.5

0.09524

9515

9506

9497

9488

9479

9470

9461

9452

9443

9

10.6

9434

9425

9416

9407

9398

9390

9381

9372

9363

9355

10.7

9346

9337

9328

9320

9311

9302

9294

9285

9276

9268

10.8

9259

9251

9242

9234

9225

9217

9208

9200

9191

9183

10.9

9174

9166

9158

9149

9141

9132

9124

9116

9107

9099

11.0

0.09091

9083

9074

9066

9058

9050

9042

9033

9025

9017

8

11.1

9009

9001

8993

8985

8977

8969

8961

8953

8944

8937

8

11.2

8929

8921

8913

8905

8897

8889

8881

8873

8865

8857

8

11.3

8850

8842

8834

8826

8818

8811

8803

8795

8787

8780

8

11.4

8772

8764

8757

8749

8741

8734

8726

8718

8711

8703

8

11.5

0.08696

8689

8681

8673

8666

8658

8650

8643

8636

8628

7

11.6

8621

8613

8606

8598

8591

8584

8576

8569

8562

8554

7

11.7

8547

8540

8532

8525

8518

8511

8503

8496

8489

8482

7

11.'8

8475

8467

8460

8453

8446

8439

8432

8425

8418

8410

7

11.9

8403

8396

8389

8382

8375

8368

8361

8354

8347

8340

7

344

FLUIDITY AND PLASTICITY

Table VI. —( Continued)

No.

0

1

2

3

4

5

6

7

8

9

Dif.

12.0 0.

08333

8326

8320

8312

8306

8299

8292

8285

8288

8271

12.1

8264

8258

8251

8244

8237

8230

8224 -

8217

8210

8203

12.2

8197

8190

8183

8177

8170

8163

8157

8150

8143

8137

7

12.3

8130

8124

8117

8110

8104

8097

8091

8084

8078

8071

12.4

8064

8058

8052

8045

8039

8032

8026

8019

8013

8006

12.5 0.

08000

7994

7987

7981

7974

7968

7962

7955

7949

7942

12.6

7936

7930

7924

7918

7912

7905

7899

7893

7886

7880

12.7

7874

7868

7862

7856

7849

7843

7837

7831

7825

7819 |

12.8

7812

7806

7800

7794

7788

7782

7776

7770

7764

7758

12.9

7752

7746

7740

7734

7728

7722

7716

7710

7704

7698

13.0 0.

07692

7686

7681

7675

7669

7663

7657

7651

7646

7640

6

13.1

7634

7628

7622 '

' 7616

7610

7605

7599

7593

7587

7581

13.2

7576

7570

7564

7559

7553

7547

7542

7536

7530

7524

13.3

7519

7513

7508

7502

7496

7491

7485

7480

7474

7468

13.4

7463

7457

7452

7446

7441

7435

7430

7424

7418

7413

13. 5 0.07407

7402

7396

7391

7386

7380

7375

7369

7364

7358

13.6

7353

7348

7342

7337

7332

7326

7321

7315

7310

7305

5

13. 7

7299

7294

7289

7283

7278

7273

7268

7262

7257

7252

13.8

7246

7241

7236

7231

7226

7220

7215

7210

7205

7200

13.9

7194

7189

7184

7179

7174

7169

7164

7158

7153

7148

14.0 0.

07143

7138

7133

7128

7123

7118

7113

7108

7102

7097

14.1

7092

7087

7082

7077

7072

7067

7062

7057

7052

7047

14.2

7042

7037

7032

7027

7022

7018

7013

7008

7003

6998

14.3

6993

6988

6983

6978

6974

6969

6964

6959

6954

6949

14.4

6944

6940

6935

6930

6925

6920

6916

6911

6906

6901

14.5 0.06897

6892

6887

6882

6878

6873

6868

6863

6859

6854

14.6

6849

6845

6840

6835

6931

6826

6821

6817

6812

6807

5

14.7

6803

6798

6793

6789

6784

6780

6775

6770

6766

6761

14.8

6757

6752

6748

6743

6739

6734

6729

6725

6720

6716

14.9

6711

6707

6702

6698

6693

6689

6684

6680

6676

6671

15.0 0.06667

6662

6658

6653

6649

6645

6640

6636

6631

6627

15.1

6623

6618

6614

6609

6605

6601

6596

6592

6588

6583

15.2

6579

6575

6570

6566

6562

6557

6553

6549

6545

6540

15.3

6536

6532

6527

6523

6519

6515

6510

6506

6502

6498

15.4

6494

6489

6485

6481

6477

6472

6468

6464

6460

6456

15.5 0.

.06452

6447

6443

6439

6435

6431

6427

6423

6419

6414

15.6

6410

6406

6402

6398

6394

6390

6386

6382

6378

6373

15.7

6369

6365

6361

6357

6353

6349

6345

6341

6337

6333

15.8

6329

6325

6321

6317

6313

6309

6305

6301

6297

6293

4

15.9

6289

6285

6281

6277

6274

6270

6266

6262

6258

6254

16.00

.06250

6246

6242

6238

6234

6231

6227

6223

6219

6215

16.7

6211

6207

6203

6200

6196

6192

6188

6184

6180

6177

16.2

6173

6169

6165

6161

6158

6154

6150

6146

6143

6139

16.3

6135

6131

6127

6124

6120

6116

6112

6109

6105

6101

16.4

6097

6094

6090

6086

6083

6079

6075

6072

6068

6064

16.5 0

.06061

6057

6053

6050

6046

6042

6038

6035

6031

6028

16.6

6024

6020

6017

6013

6010

6006

6002

5999

5995

5992

16.7

5988

5984

5981

5977

5973

5970

5966

5963

5959

5956

16.8

5952

5949

5945

5942

5938

5935

5931

5928

5924

5921

16.9

5917

5914

5910

5907

5903

5900

5896

5893

5889

5886

17.00

.05882

5879

5875

5872

5868

5865

5851

5858

5854

5851

17.1

5847

5844

5841

5838

5834

5861

5828

5824

5821

5817

17.2

5814

5811

5807

5804

5800

5797

5794

5790

5787

5784

3

17.3

5780

5777

5774

5770

5767

5764

5760

5757

5754

5750

17.4

5747

5744

5741

5737

5734

5731

5727

5724

5721

5718

17.50

.05714

5711

5708

5704

5701

5698

5695

5692

5688

5685

17.6

5682

5679

5675

5672

5669

5666

5663

5659

5656

5653

17.7

5650

5647

5643

5640

5637

5634

5631

5627

5624

5621

17.8

5618

5615

5612

5609

5605

5602

5599

5596

5593

5590

17.9

5587

5583

5580

5577

5574

5571

5568

5565

5562

5559

APPENDIX D

345

Table VII. —Logarithms

No.

0

1

2

3

4

5

6

7

8

9

Dif.

10

0000

0043

0086

0128

0170

0212

0253

0294

0334

0374

42

11

0414

0453

0492

0531

0569

0607

0645

0682

0719

0755

38

12

0792

0828

0864

0899

0934

0969

1004

1038

1072

1106

35

13

1139

1173

1206

1239

1271

1303

1335

1367

1399

1430

32

14

1461

1492

1523

1553

1584

1614

1644

1673

1703

1732

30

15

1761

1790

1818

1847

1875

1903

1931

1959

*1987

2014

28

16

2041

2068

2095

2122

2148

2175

2201

2227

2253

2279

26

17

2304

2330

2355

2380

2405

2430

2455

2480

2504

2529

25

18

2553

2577

2601

2625

2648

2672

2695

2718

2742

2765

28

19

2788

2810

2833

2856

2878

2900

2923

2945

2967

2989

22

20

3010

3032

3054

3075

3096

3118

3139

3160

3181

3201

21

21

3222

3243

3263

3284

3304

3324

3345

3365

3385

3404

20

22

3424

3444

3464

3483

3502

3522

3541

3560

3579

3598

19

23

3617

3636

3655

3674

3692

3711

3729

3747

3766

3784

IS

24

3802

3820

3838

3856

3874

3892

3909

3927

3945

3962

18

25

3979

3997

4014

4031

4048

4065

4082

4099

4116

4133

17

26

4150

4166

4183

4200

4216

4232

4249

4265

4281

4298

16

27

4314

4330

4346

4362

4378

4393

4409

4425

4440

4456

16

28

4472

4487

4502

4518

4533

4548

4564

4579

4594

4609

15

29

4624

4639

4654

4669

4683

4698

4713

4728

4742

4757

15

30

4771

4786

4800

4814

4829

4843

4857

4871

4886

4900

14

31

4914

4928

4942

4955

4969

4983

4997

5011

5024

5038

32

5051

5065

5079

5092

5105

5119

5132

5145

5159

5172

33

5185

5198

5211

5224

5237

5250

5263

5276

5289

5302

13

34

5315

5328

5340

5353

5366

5378

5391

5403

5416

5428

35

5441

5453

5465

5478

5490

5502

5514

5527

5539

5551

36

5563

5575

5587

5599

5611

5623

5635

5647

5658

5670

12

37

5682

5694

5705

5717

5729

5740

5752

5763

5775

5786

38

5798

5809

5821

5832

5843

5855

5866

5877

5888

5899

39

5911

5922

5933

5944

5955

5966

5977

5988

5999

6010

11

40

6021

6031

6042

6053

6064

6075

6085

6096

6107

6117

41

6128

6138

6149

6160

6170

6180

6191

6201

6212

6222

42

6232

6243

6253

6263

6274

6284

6294

6304

6314

6325

43

6335

6345

6355

6365

6375

6385

6395

6405

6415

6425

10

44

6435

6444

6454

6464

6474

6484

6493

6503

6513

6522

45

6532

6542

6551

6561

6571

6580

6590

6599

6609

6618

46

6628

6637

6646

6656

6665

6675

6684

6693

6702

6712

47

6721

6730

6739

6749

6758

6767

6776

6785

6794

6803

48

6812

6821

6830

6839

6848

6857

6866

6875

6884

6893

9

49

6902

6911

6920

6928

6937

6946

6955

6964

6972

6981

50

6990

6998

70Q7

7016

7024

7033

7042

7050

7059

7067

51

7076

7084

7093

7101

7110

7118

7126

7135

7143

7152

52

7160

7168

7177

7185

7193

7202

7210

7218

7226

7235

53

7243

7251

7259

7267

7275

7284

7292

7300

7308

7316

54

7324

7332

7340

7348

7356

7364

7372

7380

7388

7396

8

346

FLUIDITY AND PLASTICITY

Table VII.—( Continued)

No.

0

1

2

3

4

5

6

7

8

9

Dif.

55

7404

7412

7419

7427

7435

7443

7451

7459

7466

7474

56

7482

7490

7497

7505

7513

7520

7528

7536

7543

7551

57

7559

7566

7574

7582

7589

7597

7604

7612

7619

7627

58

7634

7642

7649

7657

7664

7672

7679

7686

7694

7701

59

7709

7716

7723

7731

7738

7745

7752

7760

7767

7774

60

I

7782

7789

7796

7803

7810

7818

7825

7832

7839

7846

61

7853

7860

7868

7875

7882

7889

7896

7903

7910

7917

62

7924

7931

7938

7945

7952

7959

7966

7973

7980

7987

7

63

7993

8000

8007

8014

8021

8028

8035

8041

8048

8055

64

8062

8069

8075

8082

8089

8096

8102

8109

8116

8122

65

8129

8136

8142

8149

8156

8162

8169

8176

8182

8189

66

8195

8202

8209

8215

8222

8228

8235

8241 »

8248

8254

67

8261

8267

8274

8280

8287

8293

8299

8306

8312

8319

68

8325

8331

8338

8344

8351

8357

8363

8370

8376

8382

69

8388

8395

8401

8407

8414

1 *8420 y

8426

8432

8439

8445

70

8451

8457

8463

84701

8488

8494

8500

8506

71

8513

8519

8525

8531 V

^437 ;

Is 5*49

rvs$55

18561

,v8567.

72

8573

8579

8585

8591 '

8597

8603

8609

8615

’8621

862 7*

6

73

8633

8639

8645

8651

8657

8663

8669

8675

8681

8686

74

8692

8698

8704

8710

8716

8722

8727

8733

8739

8745

75

8751

8756

8762

8768

8774 j

8779

8785

8791

8797

8802

76

8808

8814

8820

8825

8831

8837

8842

8848

8854

8859

77

8865

8871

8876

8882

8887

8893

8899

8904

8910

8915

78

8921

8927

8932

8938

8943

8949

8954

8960

8965

8971

79

8976

8982

8987

8993

8998

9004

9009

9015

9020

9025

80

9031

9036

9042

9047

9053

9058

9063

9069

9074

9079

81

9085

9090

9096

9101

9106

9112

9117

9122

9128

9133

82

9138

9143

9149

9154

9159

9165

9170

9175

9180

9186

83

9191

9196

9201

9206

9212

9217

9222

9227

9232

9238

84

9243

9248

9253

9258

9263

9269

9274

9279

9284

9289

85

9294

9299

9304

9309

9315

9320

9325

9330

9335

9340

86 1

9345

9350

9355

9360

9365

9370

9375

9380

9385

9390

87

9395

9400

9405

9410

9415

9420

9425

9430

9435

9440

5

88

9445

9450

9455

9460

9465

9469

9474

9479

9484

9489

89

9494

9499

9504

9509

9513

9518

9523

9528

9533

9538

90

9542

9547

9552

9557

9562

9566

9571

9576

9581

9586

91

9590

9595

9600

9605

9609

9614

9619

9624

9628

9633

92

9638

9643

9647

9652

9657

9661

9666

9671

9675

9680

93

9685

9689

9694

9699

9703

9708

9713

9717

9722

9727

94

9731

9736

9741

9745

9750

9754

9759

9763

9768

9773

95

9777

9782

9786

9791

9795

9800

9805

9809

9814

9818

96

9823

9827

9832

9836

9841

9845

9850

9854

9859

9863

97

9868

9872

9877

9881

9886

9890

9894

9899

9903

9908

98

9912

9917

9921

9926

9930

9934

9939

9943

9948

9952

99

9956

9961

9965

9969

9974

9978

9983

9987

9991

9996

541 - 31,1 Uiki

^ rt*k

i

J

bibliography and author index

k-faced type refers to pages in this treatise. Abbreviations are those used by Chemical
cts. The titles of articles are printed in the original language whenever convenient.

g, R. Untersuchungen fiber Diffusion in Wasserigen Salzlosungen.
3. Physik. Chem. 11, 248 (1892); 16 pp.

ms, A. Viscosity and the Sticking Strength of Binders. Textile
Vorld J. 56, 41 (1919); 5 pp.

ms, V. R., Kavanagh, J. T., and Osmond, C. H. Air Bubble Vis-
ometer. Chem. and Met. Eng. 25, 665 (1921); 2 pp.
lme, P. Viscosity and Diastatic Reactions. Hypothesis Concerning
he Nature of the Diastases. Compt. rend. 152, 1621 (1911); 4 pp.
lme, P. & Bresson, M. (1) Influence of Viscosity of the Medium
m Enzyme Reactions. Compt. rend. 152, 1328 (1911); 3 pp.; (2) Role
>f Viscosity in the Variation of the Action of Invertase According to the
Conductivity of Saccharose. Compt. rend. 152, 1420 (1911); 3 pp.

(1) Viscositat des Plasma. Berlin klin. Wochenschr. (1909); (2)
3ur Viscositat des Blutes. Z. klin. Med. 68, 177, 1009 (1910); 28 pp.
:s, F. & Nicholson, J. Experimental Investigation into the Flow of
Garble. Phil. Trans. 195, 363 (1901); 38 pp. Proc. Roy. Soc. London
►7, 228 (1900); 6 pp.

:s Prize Announcement. On the Criterion of the Stability & Insta-
>ility of the Motion of a Viscous Fluid. Phil. Mag. (5) 24, 142 (1887);
See Proc. Math. Soc. 11, 57 (-).

an, A. ficoulement de l’eau dans un tuyau cylindrique. J. physique
3) 5, 27 (1896); 2 pp.

nese, M. (1) Uber den Einfluss der Zusammensetzung der Ernah-
ungsflussigkeiten auf die Thatigkeit des Froschherzens. Arch, fur
xp. Path. u. Pharm. 32, 297 (1893); (2) Influence des propri<$t6s phy-
iques des solutions sur le coeur de grenouille. Arch. Ital. de Biol. 25,
>08 (1896); (3) Influenza degli elettroliti sulla Viscosity dei liquidi
olloidali. Arch. exp. Path. Pharm. (1908); suppl. 16, 13 pp.
nese, V. Arch. Ital. Biol. 50, 387 (1909).

siejew, U. and Cerm iataschenski, P. A. Zap. imp. russk. techn.
bschtsch. 30, pt. 6-7 (1896).

rd, L. P. Bearings and their Lubrication. McGraw-Hill Book Co.
Y. (1911); 235 pp.

n, G. Quantitative Demonstration Experiments to Show Gaseous
friction. Physik. Z. 10, 961 (1910).

n, H. 6 (1) The Motion of a Sphere in a Viscous Liquid. Phil,

dag. (5) 50, 323 (1900); 16 pp.; (2) Do., Phil. Mag. (5) 50, 519 (1900);
6 pp.; (3) Molecular Layers in Lubrication. Proc. Phys. Soc. London
I, 32, 16 (1920); 1 p.

347

348

INDEX

Amerling, K. Viscosity of the Blood in the New Born and in Infants.
Casopisl6k. cesk. (1909).

Andrade, E. N. da C. 236, The Viscous Flow in Metals, and Allied
Phenomena. Proc. Boy. Soc. London (A) 84, 1 (1911); 12 pp. Physik
Z. 11, 709 (1911); 7 pp.

Andrews, T. The Loss in Strength in Iron and Steel by Use. Nature 65,
418 (1896-97).

Anon. Vorlage der deutschen Sektion fur die Hauptversammlung. Inter-
nationalen Kommission. Wien. Jan. (1912). Exposition of Osborne
Reynold’s Theory of Lubrication. Mathematical Eng. July 30,
(1915). Colloidal Solid Lubricants. Do. 66, 169 (1920); 1 p. Lubri¬
cants and Lubrication. Chem. Trade J. 67, 769 (1921); 1 pp.;
Reduction of Viscosity of Celluloid and Acetone by Additions of Oxalic,
Tartaric, and Citric Acids. Kolloid.-Z. 27, 44 (1920). Ger. Pat.
276,661; Viscosity Determinations in Absolute Units. Eng. 106,
655 (1918). Comment on Friction at High Velocities. Proc. Inst.
Mech. Eng. 660 (1883). Comite Deutscher Verband f. die Materiol-
prufung der Technik. Jahresber. der chem. Technol. 489 (1904);

5 pp.; (1) Report of Lubricants and Lubrication Inquiry Committee.
Dept. Sci. & Ind. Research, Advisory Council, (1920); 216 pp. Cp.
Hyde; (2) Memorandum on Cutting Lubricants and Cooling Liquids
and on Skin Diseases Produced by Lubricants. Do. Bull. No. 2
(1918); 8 pp.; (3) Memorandum on Solid Lubricants. Bull. No. 4
(1920); 28 pp.; (4) Reports on Colloid Chemistry and Its General and
Industrial Applications. Cp. Hatschek and also McBain. Do. (1917-
1920). The Determination of Viscosity as Relative Viscosity. Arch.
Rubbercult. 4, 123 (1920); 14 pp. Tentative Test for the Viscosity of
Lubricants. Proc. Am. Soc. Testing Materials I, 19, 728 (1919); 4 pp.
Appell, P. Sur 1’ Equation difftsrentielle du mouvement d’un projectile
sph6rique pesant dans l’air. Arch. d. Math. u. Phys. (3) 5,177 (1903);

2 pp.

Applebey, M. P. 15, The Viscosity of Salt Solutions. J. Chem. Soc.
97, 2000 (1910); 26 pp. Proc. Chem. Soc. 26, 266 (1910). The
Determination of Viscosity. J. Chem. Soc. 103, 2167 (1913); 4 pp.
Proc. Chem. Soc. 29, 361 (1913).

Arago, Babinet, Poibert and Regnault cp. Poiseuille. Rapport
fait k l’Acad^mie des Sciences sur un M^moire de M. le docteur Poi¬
seuille, ayant pour titre: Recherches exp6rimentales sur le mouvement
des liquides dans les tubes de tr&s-petits diamdtres. Ann. de Chim. 1,
50 (3); 25 pp.

Archbutt, L. D. Tests for Curcas Oil. J. Soc. Chem. Ind. 17,1009 (1898) ;

Lubricators and Lubrication. Chem. Age (London) 4, 280 (1920); 1 p.
Archbutt, L. & Deeley, R. Lubrication and Lubricants, a Treatise on the
Theory and Practice of Lubrication and on the Nature, Properties
and Testing of Lubricants. London, Griffin, 3d Ed. (1912); 589 pp.
Cp. Deeley.

INDEX

349

Arisz, L. 289, Kolloidchem. Beihefte 7, 1 (1915); 90 pp. Sol und Gelzus-
tand von Gelatinelosungen.

Arndt, K. 6, (1) Zahigkeitmessungen bei hohen Temperaturen. Z.
Elektrochem. 13, 578 (1907); (2) Zahigkeit und Leitfahigkeit. Z.
Elektrochem. 13, 809 (1907); (3) Messung der Zahigkeit. Z. Apparat-
kunde 3, 473.

Arndt, K-U. Gessler A. Z. Elektrochem. 13, 580 (1907).

Arndt, K. and Schiff, P. Kolloidchem. Beihefte 6, 201 (1914).

Arnold, H. D. Phil. Mag. (6) 22, 755 (1911) ;-21 pp. Limitations imposed
by Slip and Inertia Terms upon Stokes’ Law for the Motion of Spheres
through Liquids.

Aronheim. Uber den Einfluss der Salze auf die Stromungsgeschwindigkeit
des Blutes. Diss. Gottingen (1868). Cf. Hoppe Seyler’s Med. Chem.
Unterssuchungen Berlin 265 (1867).

Arons, Dammer. Chem. Tech. 1, 771. Plasticity.

Arpi, R. Experimental Determination of Viscosity and Density of Certain
Molten Metals and Alloys. Inter Z. Metallog. 5, 142 (1913); 26 pp.

Arrhenius, S, 3, 7, (1) Contributions to our Knowledge of the Action of
Eluidity on the Conductivity of Electrolytes. Brit. Assoc. Rep. 344
1886); 4 pp.; (2) tJber die innere Reibung verdtinnter Wasseriger
Losungen. Z. physik. Chem. 1, 285 (1887); 14 pp.; (3) Electrolytic
Dissociation versus Hydration. Phil. Mag. (5) 28, 39 (1889); 9 pp.;
(4) Z. P. C. 9, 487 (1892); 25 pp. tJber die Anderung des elektrischen
Leitungsvermogens einer Losung durch Zusatz von kleinen Mengen
eines Nichtleiters; (5) Viscosity and Hydration of Colloid Solutions.
Medd. fram. K. Vetenskapsakad. Nobelinst. 3, (1916); (6) The Viscosity
of Pure Liquids. Do. 3, No. 20 (1912); 40 pp.; (7) The Viscosity of
Solutions. Biochem. J. 11 , 112 (1917); 22 pp.

Ashley, H. E. (1) The Colloid Matter in Clay, 65 pp. Trans. Am. Ceram.
Soc. 11, 530 (1909); 66 pp.; (2) The Colloidal Matter in Clay and its
measurement. Bull. 388, U. S. Geol. Surv. 66 pp. U. S. Bureau of
Standards. Bull. 23, (1913).

Aten, A. Electrical Conductivity in Mixtures of Metals and their Salts.
Z. physik. Chem. 66, 641 (1909); 31 pp.

Atkins & Wallace. J. Chem. Soc. 103, 1461 (1913). Molecular con¬
dition of Mixed Liquids I. Mixtures of Lower Aliphatic Alcohols
with Water.

Atterberg, A. The plasticity of clay. Intern. Mitt. Bodenkunde 1,
4 (1910); 33 pp. Cp. Tonind. Ztg. 35, 1460; (2) Barium Sulphate a
Plastic Substance. Z. angew. Chem. 24, 928 (1910); 1 p.; (3) The Plas¬
ticity of Barium Sulphates. Z. angew. Chem. 24, 2355. Cp. Ehren-
berg; (4) Internat. Mitt. Bodenk. 3, 291 (1914); 39 pp.; (5) The Plas¬
ticity and Coherence of Clays and Loams. Chem. Ztg. 34, 369; 2 pp.
379 (1910); 1 p.; (6) What constituents give to clay its plasticity and
firmness? Kgl. Landtbruks Akftd. Handlingar och Tidskrift 413
(1913); 32 pp.

350

INDEX

van Aubel, Edm. Compt. rend. 173, 384 (1921); 4 pp. Influence de la
temperature sur la viscosite des liquides normaux.

Auerbach, F. (1) Magnetische Untersuchung. Wied. Ann. 14, 308
(1881); (2) Plasticitat und Sprodigkeit. Wied. Ann. 46, 277 (1892);
15 pp.

Augenheister. Beitrage zur Kenntniss der Elasticitat der Metalle.
Diss. Berlin (1902).

Austin, L. Experimentaluntersuchungen uber die elastiche Langs-und
Torsionsnachwirkung in Metallen. Wied. Ann. 60, 659 (1893); 19
pp.; Cp. Nature 49, 239 (1894).

Axelrod, S. Gummizeitung 19, 1053 (1905); Gummizeitung 20, 105
(1905); Gummizeitung 23, 810 (1910). The Viscosity of Caoutchouc
Solutions. Cp. Gummizeitung 19, 1053 & 20, 105.

Axer, J. A Viscometer Consisting of a Vertical Cylinder Filled with the
Material into Which Another Perforated Cylinder Dips. Ger. pat.
267, 917 (1913).

Ayrton, W. & Perry J. On the Viscosity of Dielectrics. Proc. Roy. Soc.
27, 238 (1878); 7 pp.

Bachmann. (1) Die klinische Verwertung der Viscositatsbestimmung.
Deutsch. Arch. klin. Med. 94 (1908); (2) Die Viscositat des Blutes und
ihre diagnostische Bedeutung. Med. Klinik 36 (1909).

Baffani & Luciani. Physico-chemical Investigation of the Blood of the
Mother & the Fetus with Especial Reference to the Viscosity. Atti.
soc. ital. ostetricia (1908); Arch. ital. biol. 61, 246 (1900); 7 pp.
Baheux, Ch. Comparison of Viscometers. Mat. grasses 6, 3231 (1911);

3 pp. (Report on Barbey and Engler types.)

Baily, F. (1) On the Correction of a*Pendulum for the Reduction to a
Vacuum together with Remarks on some Anomalies observed in Pendu¬
lum Experiments. Phil. Trans. 122, 399 (1832); 98 pp.; (2) Do. Phil.
Mag. (3) 1, 379 (1832); 3 pp. —

Baker, F. (1) Viscosity of Ether-Alcohol Mixtures. J. Chem. Soc.
101,1409 (1912); 7 pp.; (2) The Viscosity of Cellulose Nitrate Solutions.
J. Chem. Soc. 103, 1653 (1913); 22 pp.

Baldus, A. A Viscosity Meter. Brit. 22,961, Oct. 4 (1910). (Air bubble
method.)

Bancelin, M. (1) The Viscosity of Emulsions. Compt. rend. 162, 1382
(1911); 2 pp.; (2) The Viscosity of Suspensions & the Determination
of Avogadro's Number. Z. Chem. Ind. Kolloide 9, 154 (1912); 2 pp.
Bancroft, W. D. 213, Applied Colloid Chemistry. General Theory;

345 pp. McGraw-Hill Book Co.

Barbey. 324, See Nicolardot, Baheux.

Barnes, J. (1) On the Relation of the Viscosities of Mixtures of Solutions
of Certain Salts to their State of Ionization. Proc. and Trans. Nova
Scotia Inst, of Sci. 10, 113 (1899); 26 pp.; (2) Do., Elektrochem.
Z. 7, 134 (1900); 6 pp.; (3) Do., Chem. News 86, 4, 30, 40 (1902);

ft y\t\

INDEX

351

Barnett, 31* On the Viscosity of Water as Determined by Mr. J. B.
Harm ay by means of his Microrheometer. Proc. Roy. Soc. London
66, 259 (1894); 3 pp.

BarRj Q. and Bircumshaw, L. L. The Viscosity of Some Cellulose Acetate
Solutions. Advisory Comm, for Aeronautics. Reports and Memo¬
randa, No. 663 (1919); 6 pp.

BartheleMY, H. The Measurement of Viscosity and the Cellulose Ester
Industries. Caoucthouc and Guttapercha 10, 7202 (1910); 11 pp.
Bartoli, A- Bull. Mensile Accad. Giornale di Sci. Natur. Catania 26, 4
(1892).

Bartoli, A. <& Stracciati, E. (1) Le Propriety fisiche degl' idrocarburi
O n H 2 »+ 2 dei petrolii di Pensilvania. Cim. (3) 18, 195 (1885);
24 pp*; (2) Sur les propri6t6s physiques des hydrocarbures CJELn + 2
des pdtroles d'Am<$rique. Ann. chim. phys. (6) 7, 375 (1886); 9 pp.
Barits, C. 77, 212, 235, 246, 286 (1) The Electrical and Magnetic Proper¬
ties of Iron, Bull. #14, U. S. Geol. Survey (1888); (2) Note on the Vis¬
cosity of Gases at high Temperatures and on the pyrometric Use of the
Principle of Viscosity. Am. J. Sci. (3) 35, 407 (1888); 68 pp.; (3)
Maxwell’s Theory of the Viscosity of Solids and its Physical Verification.
Phil. Mag. (5) 26, 183 (1888); (4) Die Zahigkeit der Gase bei hohen
Temperaturen. Wied. Ann. 36, 358 (1889); 41 pp.; (5) The Viscous
Effect of Strains Mechanically Applied, as Interpreted by Maxwell's
Theory. Phil. Mag. (5) 27, 155 (1889); 23 pp.; (6) On Thermo-
Electric Measurements of High Temperatures. Bull. #54, U. S. Geol.
Survey (1889); (7) The change of the Order of Absolute Viscosity
Encountered on Passing from Fluid to Solid. Phil. Mag. (5) 29,
337 (1890); 19 pp.; (8) The Viscosity of Solids. Bull. #73, U. S. Geol.
Survey (1891); 139 pp.; (9) The Thermal Variation of Viscosity and of
Electrolytic Resistance. Am. J. Sci. (3) 44, 255 (1892); 1 p.; (10)
Isothermals, Isopiestics, and Isometrics relative to Viscosity. Am. J.
Sci. (3) 46, 87 (1893); 10 pp.; (11) Note on the Dependence of Viscosity
on Pressure and Temperature. Proc. Am. Acad. N. g. 19, 13 (1893);
6 pp.; (12) Mechanism of Solid Viscosity. Bull. #94, U. S. Geol.
Survey; (13) Maxwell's Theory of Viscosity. Am. J. Sci. 36, 178
(1888).

Barits, C. Strotjhal. (1) The Viscosity of Steel and its Relation to
Temper. Am. J. Sci.; (3) 32, 444 (1886); 23 pp.; (2) Do., Concluded.
Am. J. Sci. (3) 33, 20 (1887); 16 pp.

Bary, P. The Viscosity of Colloidal Solutions. Compt. rend. 170, 1388
(1920); 2 pp.

Basch, K- Viscometric Studies of Human Milk. Wien. Klin. Wochschr.
24, 1592 (1910); 3 pp.

Bassett, A. 33. Treatise of Hydrodynamics. 2 vols. Deighton, Bell <fe
Co., Cambridge (1888); 264 + 328 pp.; (2) Stability & Instability of
Viscous Liquids. Proc. Roy. Soc. 52, 273 (1893); 3 pp.

Bateman, E. Relation between Viscosity and Penetrance of Creosote
into Wood. Chem. Met. Eng. 22, 359 (1920); 2 pp.

352

INDEX

Batschinski, A. 92, 132, 134, 142, et seq. 148, 162, 161, 166 et seq.
(1) tJber das Gesetz der Veranderlichkeit der Viskositat des Queck-
silbers mit der Temperatur. Ann. d. phys. Klasse d. Kais. Gesellsch.
v. Freunden e. Naturw. zu Moskau 10, 8 (1900); (2) Beziehung zwis-
chen der Viskositat und der chemischen Konstitution der Flussig-
keiten. Sitzungsber. d. Kaiserl. Ges. d. Naturf zu Moskau (1900);
100 pp. Cp. p. 1 (1901); p. 265 (1902); Chem. Zentr. 2, 450 (1901);
Do. 2, 180 (1902); (3) tJber die Beziehung zwischen dem Viscositats-
parameter und anderen physikalischen Konstanten. Z. physik. Chem.
37, 214 (1901); 2 pp.; (4) Untersuchungen liber dieinnereReibung der
Fltissigkeiten. Annales de la Soci^td d’encouragement des sciences
experimentales et de leurs applications du nom de Christophe Leden-
zoff. Supplement 3 (1913); 70 pp.; (5) Viscosity Law for Liquids.
Physik Z. 13, 1157 (1913); (6) Inner Friction of Liquids. Z. physik.
Chem. 84, 643 (1913); 64 pp.; (7) Association of Liquids Z. physik.
Chem. 82, 86 (1913); 6 pp.

des Baucels, Larguier. Investigations of the Physical Changes in Gela¬
tine in the Presence of Electrolytes and Non-Electrolytes. Compt.
rend. 146, 290 (1905).

Baume, G. and Vigneron. New Apparatus for Measuring Viscosities and
Fluidities. Ann. chim. anal. chim. appl. 1, 379 (1915); 5 pp.

Bayliss, W. M. 286, (1) The kinetics of tryptic action. Areh. d. Sciences
Biologiques 11, 261 (1904); 35 pp.; (2) Causes of the Rise in Electrical
Conductivity under the Action of Trypsin. J. of Physiol. 36, 221
(1907); 31 pp.; (3) The Nature of Enzyme Action. Longmans Green
& Co. (1914); 179 pp.

Bazin. Experiences nouvelles sur la distribution des vitesses dans les
tuyaux. Compt. rend. 122, 1250 0-896); 3 pp.

Beadle, C. & Stevens, H. P. The Viscosity of Rubber Solutions. India
Rubber J. 46, 1081 (1913); 1 p.

Beck, C. & Hirsch, C. (1) Die Viskositat des Blutes. Arch. f. exp. Path,
u. Pharm. 64, 54 (1905).

Beck, K. (1) Beitrage zur Bestimmung der Relativen inneren Reibung von
Flussigkeiten. Habilitationschrift, Leipzig (1904); (2) Beitrage zur
Bestimmung der Relativen inneren Reibung von Flussigkeiten, im
besondem des menschlichen Blutes. Z. physik. Chem. 48, 641 (1904);
40 pp.; (3) The Influence of the Red Corpuscles on the Internal Friction
of the Blood. Kolloid-Z. 26, 109 (1919); 1 p.

Beck, K. & Ebbinghouse, K. Beitrage zur Bestimmung der inneren
Reibung. Z. physik. Chem. 68 , 409 (1899); 16 pp.

Becker, A. tJber die innere Reibung und Dichte der Bunsenflamme.
Ann. Physik. (4) 24, 823 (1907); 39 pp.

Becker, G. F. Strain & Rupture in Rocks. Geol. Surv. 1897-1902.

Bell, J. & Cameron, F. 189, The Flow of Liquids through Capillary
Spaces. J. Phys. Chem. 10 , 658 (1906); 17 pp.

VAN Bemmelen. Z. anorg. Chem. 6 , 466; 13, 233; 18, 14, 98; 20, 185; 22,
313.

INDEX

353

B^nard. 32.

Bence, J. (1) Klinische Untersuchungen iiber die Viscositat des Blutes
bei Storungen der C0 2 Ausscheidungen. Deutsch. med. Wochschr.
#15 (1905); (2) Klinische Untersuchungen iiber die Viscositat des
Blutes. Z. klin. Med. 27, (1907); (Abt. fur innere Med.). Do. 58,
(1909).

Benton, A. F. The End Correction in the Determination of Gas Viscosity
by the Capillary-Tube Method. Phys. Rev. 14, 403 (1919); 6 pp.
Berl and Klaye. (The viscosity of oxy- and hydro-cellulose nitrates.)

Z. Schiess. Sprengstoffwesen 2, 381 (1907).

Berl and, E. and Chenvier, A. Lubrication and Viscosity of Liquids.

M6moires de la Society de sciences phys. 3, 405 (?) (1887).

Berndorf, H. The Determination of Viscosity of Transverse Waves in
the Outermost Earthcrust. Physik. Z. 13, 83 (1912); 1 p.

Bernouilli, J. Theoria nova de motu aquarum per canales qu6cunque
fluentes. Acad. St. Petersbourg (1726).

Bernstein, G. Studies on the Vulcanization of Rubber. Kolloid-Z. 11,
185 (1912).

Berson, G. & Bouasse, H. Sur l’61asticit6 de torsion d’un fil oscillant.

Compt. rend. 119, 48 (1894); 2 pp.

Bessel. Berlin Acad. (1826).

Bestelmeyer, A. (1) Die innere Reibung des Stickstoffs bei tiefen Tempera-
turen. Diss. Miinchen (1903); 59 pp.; (2) Bemerkung zu der
Abhandlung des Herrn Markowski iiber die inneren Reibung von Sauer-
stoff, Wasserstoff, chemischen und atmospharischen Stickstoff und
ihre Anderung mit der Temperatur. Ann. Physik. (4) 15, 423 (1904);

2 pp.

Biel, R. tJber den Druckhohenverlust bei der Fortleitung tropbarer und
gasformiger Fliissigkeiten. Mitt. Forschungsarbeitan Verein deutscher
Ingenieure, Heft 44. Springer Berlin (1907), abst. Zeitschr. d. Ver.
deutsch. Ing. 1035, 1065 (1908); 59 pp.

Biltz, W. & von Vegesack, A. & Steiner, H. 205, tlber den osmotischen
Druck der Kolloide. II. Der osmotische Druck einiger Farbstoff-
losungen. Section 5. tlber die Zahigkeit von Nachtblaulosungen.
Z. physik. Chem. 73, 500 (1910).

Bingham, E. C. 126, 142, 271, 289, (1) The Conductivity and Viscosity of
Solutions of Certain Salts in Mixtures of Acetone with Methyl Alcohol,
Ethyl Alcohol and Water. Diss. Johns Hopkins (1905); 78 pp. Cp.
Jones & Bingham; (2) Viscosity and Fluidity. Am. Chem. J. 36, 195
(1906); 23 pp.; (3) Viscosity and Fluidity. Am. Chem. J. 40, (1908);
4 pp.; (4) Viskositat und Fluiditat. Z. physik. Chem. 66, 238 (1909);

17 pp.; (5) Viscosity & Fluidity. Am. Chem. J. 43, 287 (1910); 23 pp.;
(6) Viscosity and Fluidity of Matter in the Three States of Aggregation,
and the Molecular Weight of Solids. Am. Chem. J. 46, 264 (1911);

18 pp.; (7) Fluidity and Vapor Pressure. Am. Chem. J. 47, 185
(1912); 12 pp.; (8) Viscosity & Fluidity. A Summary of Results. I.
Phys. Rev. 36, (1912); 26 pp.; (9) Viscosity & Fluidity. A Summary of

23

354

INDEX

Results. II. Phys. Rev. (2) 1, 96 (1913); 27 pp.; (10) A Criticism of
Some Recent Viscosity Investigations. J. Chem. Soc. 103, 959 (1913);
6 pp.; Proc. Chem. Soc. 29, 113 (1913); (11) The Viscosity of Binary
Mixtures. J. Phys. Chem. 18, 157 (1914); 8 pp.; (12) A New Vis¬
cometer for General Scientific & Technical Purposes. J. Ind. Eng.
Chem. 6, 233 (1914); 8 pp.; (13) Fluidity as a Function of Volume,
Temperature and Pressure. The Equation of State and the Two
Kinds of Viscous Resistance. The So-called a Slipping” in Gases.
J. Am. Chem. Soc. 36, 1393 (1914); 16 pp.; (14) A review of Dunstan
and Thole’s' “Viscosity of Liquids.” J. Am. Soc. 36, 1320 (1914);
2 pp.; (15) Plastic Flow. J. Wash. Acad. Sci. 6, 177 (1916); 3 pp.;
(16) An Investigation of the Laws of Plastic Flow. Bull. U. S. Bur. of
Standards 13, 309 (1916); 43 pp.; (17) The Variable Pressure Method
for the Measurement of Viscosity. Proc. Am. Soc. Testing Materials
18, Pt. II, 373 (1918); 11 pp.; (18) Cutting Fluids. Tech. Paper
204, U. S. Bur. of Standards 16, 35 (1922) 41 pp.

Bingham, E. C. and Durham, T. C. 54, 201, 208, 215, The Viscosity and
Fluidity of Suspensions of Finely-Divided Solids in Liquids. Am.
Chem. J. 46, 278 (1911); 20 pp.

Bingham, E. C. and Green, H. 220, et seq., Paint, a Plastic Material and
not a Viscous Liquid; The Measurement of Its Mobility and Yield
Value. Proc. Am. Soc. Testing Materials. II, 19, 640 (1919); 36 pp.
Cp. Green.

Bingham, E. C. & Miss Harrison, J. 121, Viskositatund Fluiditat. Z.
physik. Chem. 66, 1 (1909); 32 pp.

Bingham, E. C. and Jackson, R. F. Standard Substances for the Cali¬
bration of Viscometers. Bull. U. S. Bur. of Standards 14, No. 298,
59 (1917); 28 pp. J. Wash. Acad. Sci. 7, 53 (1917); 2 pp.

Bingham, E. C. and Sarver, L. Fluidities and Specific Volumes of Benzyl
Benzoate and Benzene. J. Am. Chem. Soc. 42, 2011 (1920); 11 pp.
Bingham, E. C., Schlesinger, H. I. and Coleman, A. B. 298, Some
Sources of Error in Viscosity Measurement. J. Am. Chem. Soc.
38, 27 (1916); 15 pp.

Bingham, E. C., Van Klooster, H. S. and Kleinspehn, W. G. The
Fluidities and Volumes of Some Nitrogenous Organic Compounds.
J. Phys. Chem. 24, 1 (1920); 21 pp.

Bingham, E. C. & White, G. F. 6, 21, 28, 97, (1) Viscosity and Fluidity
of Emulsions, Crystallin Liquids and Colloidal Solutions. J. Am. Soc.
33, 1257 (1911); 11 pp.; (2) Fluiditat und die Hydrattheorie. I. Die
Viskositat von Wasser. Z. physik. Chem. 80, 670 (1912); 17 pp.
Bingham, E. C., White, G. F., Thomas, A. & Cadwell, J. L. 142, 169,
178, Fluidity and the Hydrate Theory. II. Z. physik. Chem. 83,
641 (1913); 32 pp.

Blanchard, A. The Viscosity of Solutions in Relation to the Constitution
of Dissolved Substances. J. Am. Chem. Soc. 26, 1315 (1904); 24 pp.
(Ionic migration velocity.)

Blanchard, A. & Pushee, H. B. The -Viscosity of Solutions of the Metal

INDEX

355

Ammonia Salts. J. Am. Chem. Soc. 34, 28 (1912); 4 pp. Cp. J. Am.
Chem. Soc. 26, 1315.

Blaistch-ard, A. Stewart, M. The Viscosity of Solutions of Metallic
Salts; Its Bearing upon the Nature of the Compound between Solvent
and. Solute. Science (N. S.) 18, 98 (1903); 1 p.

Blasitxs, H. Das Ahnlichkeitsgesetz bei Reibungsvorgangen in Fliissig-
keiten. Mitt. Forschungsarbeiten, Verein deutscher Ingenieure. Julius
Springer. Heft. 131 (1913); 40 pp.

Bl.eiktin'GER, A. V. (1) The Effect of Preliminary Heating Treatment
XJpon the Drying of Clays. Technologic Paper, Bur. of Standards #1
(1911); (2) The Viscosity of Clay Slips. Trans. Am. Ceram. Soc.
lO, 389; (3) The Effect of Electrolytes upon Clay in the Plastic State.
Orig. Comm. 8th Intern. C. Appl. Chem. Cp. IJniv. Ill. Bull. 6, #25
(1909).

Bleijstxntger, A. V. & Brown, G. H. (1) Testing of Clay Refractories
with. Special Reference to their Load Carrying Capacity at Purnace
Temperatures. Tech. Paper of the IT. S. Bur. of Standards #7; (2)
Note on the V. of Clay Slips. Trans. Am. Ceram. Soc. 11, 596 (1909);
9 pp.; (3) The Effect of Prelim. Heat Treatment upon Clays. U. S.
Bur. Stand. Bui. 1. Trans. Am. Ceram. Soc. 11, 392 (1909); 15 pp.
Bleiktintger, A. “V., Clark, H. H. Note on the Viscosity of Clay Slips as
Determined by the Clark Apparatus. Trans. Am. Ceram. Soc. 12 ,
3S3 (1910); 9 pp.

BleiniisTGER, A. V. <fc Fulton, C. E. The effect of Acids and Alkalies upon
Clay in the Plastic State. Am. Ceram. Soc. 14, 827 (1912).
Bleintxntger, A. V- & Ross, D. W. (1) The Flow of Clay under Pressure.

Trans. Am. Ceram. Soc. 16, 392 (1914); 9 pp.

Bleiistiistger, A. V. and Tector, P. The Viscosity of Porcelain Bodies.
Trans. Am. Ceram. Soc. 15, 328 (1913); 9 pp. Also Tech. Paper,
XT. S. Bur. Standards #30.

Blunschy, F. Beit rage zur Lehre der Viscositat des Blutes. Diss. Zurich
(1908); 43.pp.

Bogtte, R. H. Properties and Constitution of Glues and Gelatins. II.
Chem. Met. Eng. 23, 61 (1920); 6 pp.; J. Am. Chem. Soc. 43, 1764
(1921); 10 pp. The Viscosity of Gelatin Sols.

Bolle, B. Beitrag zur Kenntniss der Viscositat des Blates, des Serums u.

des Plasmas. Diss. Berlin (1909); 29 pp.

Boltzmann, L. 237, (1) Zur Theorie der elastische Nachwirkung (2A)
Wien. Sitzungsber. 70, 275 (1875); 31 pp.; (2) Zur Theorie der elastis-
chen Nachwirkung. Pogg. Ann. Erganzungsband 7, 624 (1876);
31 pp.; (3) tTber einige Probleme der Theorie der elastischen Nach¬
wirkung und fiber eine neue Methode, Schwingungen mittels Spiegel-
ablesung zu beobachten, ohne den Schwingenden Korper mit einen
Spiegel von erheblicher Masse zu belasten. Wien. Situngsber. (2A)
76, 815 (1878); 28 pp.; (4) Zur Theorie der elastische Nachwirkung.
Wied. Ann. 6 , 430 (1878); 3 pp.; (5) Zur Theorie der Gasreibung.
Wien. Sitzungsber. (2A) 81, 117 (1880); 42 pp.; (6) Do., Part II,

356

INDEX

Wien. Sitzungsber. (2A) 84, (1881); 95 pp.; (7) Do., Part III. Wien.
Sitzungsber. (2A) 84, 1230 (1881); 34 pp.; (8) Vorlesungen uber
Gastheorie. Barth, Leipzig 2 vols. (1896); (9) Gesehichte unserer
Kenntniss der inneren Reibung und Warmeleitung in verdlinnten
Gasen. Physik. Z. 1, 213 (1900); 1 p.

Bond, W. N. The Properties of Plastic Crystals of Ammonium Nitrate.
Phil. Mag. 41, 1 (1921); 21 pp.

Borelli L. & Datta. (1) Viscometry of the Urine. Riv. crit. clin. Med.
11, 289 (1910); 7 pp.; (2) Saggi di viscosimetria clinica. La clinica
medica italiana 45, 149 (1905); (3) Saggi di viscosimetria clinica.
Nota II. Viseometria degli essudati e trasudati. Riv. crit. clin. Med.
7, 181 (1906).

Born, M. The Mobility of the Electrolytic Ions. Z. Elektrochem. 26,
401 (1920); 3 pp.

Boscovich, R. J. 1, Opera pertinentia ad opticam et Astronomician.
5 vols. (1785) Bassani vol. 5 opusculum III.

Bose, E. 96, 100, 102, 209, 210, (1) tlber die Viskositatsanomalien von
Emulsionen und von anistropen Fliissigkeiten. Physik. Z. 9, 707
(1908); 1 p.; (2) Viskositatsanomalien anistroper Fliissigkeiten in
hydraulisehen Stromungzustanden. Physik. Z. 10, 32 (1909); 5 pp.
Cp. Willers.

Bose, E. & Bose, M. The Viscosity of Liquids in the Condition of Turbu¬
lent Flow. Physik. Z. 12, 126 (1910); 10 pp.

Bose, E. & Conrat, F. The Viscosity at the Clarifying Point of So-called
Crystalline Liquids. Physik. Z. 9, 169 (1908); 5 pp.

Bose, E. & Rauert, D. 97, Experimental Study of the Viscosity of Liquids
in the Condition of Turbulent Flow. Physik. Z. 10, 406 (1909); 3 pp.

Bossut. 1 , 18, (1) Traits 616mentaire d’hydrodynamique. Paris (1775);
(2) Nouvelles experiences sur la resistance des fluides, Paris (1777).

Botazzi, F. (1) L’Orsi, giornale di chimica, formacia, ecc. Firenze 20,
253, 289 (1897). Chem. Zentr. 1, 83 (1898); (2) Recherches sur la
viscosite de quelques liquides organiques et de quelques solutions
aqueuses de substances proteiques. Arch. ital. de Biologie 29, 401
(1898); Naturw. Rundsch 14, 47 (1899); (3) Ricerche sulk attrito
intemo (viscosity) di alcune liquidi organici e di alcune soluzioni acquose
di sostanze proteiche. Principi di Fisiolgia I Chimica-Fisica Society
editrice libraria 316 (1906); (4) Some colloidal properties of hemo¬
globin. Atti. Acad. Lincei 22, II, 263 (1913).

Botazzi, F. & D’Agostino, E. Viscosity and Surface Tension of Suspensions
and Solutions of Muscular Proteins Under the Influence of Acids and
Alkalies. Atti. Accad. Lincei 22, II, 183 (1913); 9 pp.

Botazzi, F. & d’Errico, G. 207, 208, Pflugers Arch. f. Physiol. 115, 359
Biochem. Z. 7, (1908).

Botazzi, F. & Jappeli. Viscosity of blood serum of cattle. Rend. Line.
(5) 17, [2] 49 (1908), Cp. Japelli.

Botazzi, F. and Victorow, C. Surface Tension, Viscosity, and Appear-

INDEX

357

ance of Dialysed Marseilles Soap of Unknown Composition, With or
Without Addition of Alkali. Rend. R. Ac. Line. (5) 19, I, 659 (1910).

Bottomley, L. On the Secular Experiments in Glasgow on the Elasticity
of Wires. Rep. Brit. Assoc. 537 (1886); 1 p.

Bous field, W. R. 195, Z. physik. Chem. 53, 303 (1905). Ionic Size in
Relation to the Physical Properties of Silver Solutions. Phil. Trans.
A. 206, 129 (1906); 13 pp.; (2) Ionic Size in Relation to Viscosity.
Phil. Trans. (A) 206, 101 (1906); 59 pp.

Bousfield, W. & Lowry, T. 192, (1) The Influence of Temperature on the
Conductivity of Electrolytic Solutions. Proc. Roy. Soc. London 71,
42 (1902); 13 pp.; (2) The Electrical Conductivity and other Properties
of Sodium Hydroxide in Aqueous Solutions as Elucidating the Mech¬
anism of Conduction. Proc. Roy. Soc. London 74, 280 (1904); 4 pp.
Phil. Trans. (A) 204, 253 (1905); 69 pp.

Boussinesq, J. 6, 18, 50, (1) Essai sur la thdorie des eaux courantes.
Mem. prdsentds par divers savants k PAcad. des Scien. 23, (1877);
680 pp.; (2) Additions et delaircissements au mdmoire institutd: Essai
sur la thdorie des eaux courantes. Do. 24, (1877); 64 pp.; (3), Lepons
synthetiques de Mdcanique gdndrale, servant d’introduction au Cours de
Mdcanique physique de la Faculty des Sciences do Paris. Gauthier-
Villers (1889j; (4) Thdorie du regime permanent graduellement varie
qui se produit prds de Pentr6e dvasde d’un tube fin, oil les filets d’un
liquide qui s’y dcoule n’ont pas encore acquis leurs indgalitds normales
de vitesse. Compt. rend. 110, 1160 (1890); 6 pp.; (5; Thdorie de
mouvement permanent qui se produit pr&s de 1’entrde dvasee d’un
tube fin: application k la deuxi&me s6rie d’expdriences de Poiseuille.
Compt. rend. 110, 1238 (1890); 5 pp.; (6) Sur Pexplication physique
de la fluidity. Compt. rend. 112,1099 (1891); 3 pp.; (7) Sur la Mani&re
dont les vitesses, dans un tube cylindrique de section circulaire,.6vas4
k son entrde, se distribuent depuis cette entrde jus q’aux endroits on
se trouve 6tabli un rdgime uniforme. Compt. rend. 113, 9 (1891);
6 pp.; (8) Calcul de la moindre longueur que doit avoir un tube circu-
laire, 6vasd k son entrde pour qu ‘un regime sensiblement uniforme s’y
dtablisse et de la ddpense de charge qu’y entratne l’dtablissement.
Compt. rend. 113, 49 (1891); 2 pp.; (9) Thdorie analytique de la chaleur
Vol. II. Note I, 196-265. Sur la rdsistanse opposde aux petits mouve-
ments d’un fluide inddfini par un solide immergd dans ce fluide.
Gauthier-Villars, Paris, 1903; (10) Existence of Surface Viscosity in the
Thin Transition Layer Separating a Liquid from a Contiguous Fluid.
Ann. chim. phys. 29, 349 (1913); 8 pp.; (11) Application of Surface
Viscosity Formulas to the Surface of a Spheroidal drop Falling with
Uniform Velocity into a Fluid of less Specific Gravity. Ann. chim.
phys. 29, 357 (1913); 7 pp.; (12) Velocity of the Fall of a Spherical
Drop into a Viscous Fluid of Less Specific Gravity. Ann. chim. phys.
29, 364 (1913); 7 pp.; (13) Internal Friction and Turbulent Flow. Compt.
rend. 1517 (1896).

358

INDEX

Boutaric, A. Sur quelques consequences physico-chimiques des mesures
de viscosity. Rev. gen. Sci. 25, 425 (1914); 8 pp.

Bouty and Bender. 192.

Bovey, H. Some Experiments on the Resistance to Flow of Water in Pipes.

Trans. Roy. Soc. of Canada Sect. 3 (1898); 14 pp.

Boynton, W. P. Application of the Kinetic Theory to Gases, Vapors, Pure
Liquids and the Theory of Solutions. Macmillan Co. (1904);

288 pp.

Braun, W. fiber die Natur der elastischen Nachwirkung. Pogg. Am.
159, 337 (1876); 62 pp.

Braun, W. & Kurz, A. (1) fiber die Dampfung der Torsionsschwingungen
von Drahten. Carl’s Repert. Exp.-physik. 15, 561 (1879); 16 pp.;

(2) Do., II, Carl’s Repert. Exp.-physik. 17, 233 (1881); 21 pp.;

(3) fiber die elastische Nachwirkung in Drahten. III. Carl’s Repert.
Exp.-physik. 18, 665 (1882); 8 pp.

Bredig, G. 192, Beitrage zur Stochiometrie der Ionenbeweglichkeit.

Z. physik. Chem. 13, 190 (1894); 98 pp.

Breitenbach, P. 79, 252, (1) Uber die innere Reibung der Gase und deren
Anderung mit der Temperatur. Diss. Erlangen (1898); Wied. Ann.
67, 803 (1899); 25 pp; (2) Do., Ann. Physik. 5, 166 (1901); 4 pp.
Bridgman, P. W. (1) Mercury, Liquid and Solid, Under Pressure. Proc.
Amer. Acad. 47, 345 (1912); 94 pp.; (2) On the Effect of General Mech¬
anical Stress on the Temperature of Transition of Two Phases, with
a Discussion of Plasticity. Phys. Rev. (2) 7, 215 (1916); 9 pp.

Briggs, Bennett and Pierson. J. Phys. Chem. 22, 256 (1918).
Brillouin, M. 32, 38, 42, 142, (1) ThSorie yiastique de la plasticity et
la fragility des corps solides. Compt. rend. 112, 1054 (1891); 2 pp.;
(2) Viscosity of Liquids as a Function of the Temperature. Ann. chim.
phys. (7) 18, 197 (1899); 16 pp.; (3) Sur la viscosity des fluides. Compt.
rend. 144, 1151 (1907); 2 pp.; (4) Legons sur la viscosity des liquides
des gaz.; Part I. Gyndrality des viscositys des liquides. VII + 228
pp. Part II. Viscosity des gaz. Caract^res gynyraux des thdories
moiycularies. 146 pp. Gauthier-Villers. Paris (1907); (5) Diffusion
of Animated Particles in Brownian Movement. Ann. chim. phys.
27, 412 (1913); 11 pp.; (6) Thermal Conductivity and Viscosity of
Monatomic Liquids. Compt. rend. 159, 27 (1914); 3 pp.

Brinkman, C. Die innere Reibung als Htilfsmittel zur Erkennung und
Unterscheidung ahnlich konstituierter Verbindungen. Diss. Leipzig
(1903); 54 pp.

Britten, R. P. L. Hardness and Viscosity of Varnishes. Oil and Color
Chem. Assoc. Sept., 17 (1918); J. Soc. Chem. Ind. 37, 594 (1918).
Brodman, C. 30, (1) Untersuchungen fiber den Reibungscoefficienten von
Fliissigkeiten. Diss. Gottingen (1891); Wied. Ann. 45, 159 (1892);
26 pp.; (2) fiber eine zur Untersuchung sehr zaher Fliissigkeiten geeig-
nete Modification der Transpirations methode. Wied. Ann. 46, 188
(1893); 19 pp- (Glycerol).

Brown, D. F. 162, et seq. } 168, Thesis Lafayette College (1921).

INDEX

359

Brown, G. (1) The Viscosity of Some Shales at Furnace Temperatures.

Trans. Am. Ceram. Soc. 16, 571 (1914); 5 pp.

Bruce, H. D. 296, 322, Thesis Lafayette College (1922).

Bruckner, H. 6, 63, 179, Uber Reibung von Salzlosungen. Diss. Halle
(1890); Wied. Ann. 42, 287 (1891); 23 pp.

BrIjhl, J. W. Die Beziehungen zwischen der physikalischen Eigen-
schaften organiseher Korper und ihrer chemischen Constitution III.
Ber. d. deutsch chem. gesell 13, 1529 (1880); 2 pp.

Brunhes, J. & Dussy, J. Sur les variations de viscositd que prdsente le
soufre fondu. Compt. rend. 118, 1045 (1894); 2 pp.

Bubanovic F. The Influence of Fat-Soluble Materials on the Viscosity
and Surface Tension of Olive Oil. Z. Chem. Ind. Kolloide 10, 178
(1912); 3 pp.

Bucco, M. Viscometric Investigations in Relation to Blood Pressure.
Atti XXIII Congr. med. inst. Roma 373, Zentr. Biochem-Bio-phys.

16, 196.

Buchanan, J. and Malcolm, H. W. Experiments with Rotating Viscous
Liquids. Phil. Mag. (6) 9, 251 (1905); 7 pp.

Buchbock, G. (1) tJber die Geschwindigkeit hydrolytischen Zerzetzung
der Karbonylsulfids. Z. p. ch. 23, 123 (1897) 34 pp.; (2) Uber den
Einfluss des Mediums auf die Reactions. Z. physik. chem. 34, 229
(1900); 19 pp.

Buckingham, E. 223, et seq., (1) Studies on the Movement of Soil Moisture.
U. S. Bureau of Soils Bull. #3841; (2) Model Experiments and the Forms
of Empirical Equations. Trans. Am. Soc. Mech. Engineers (1915);
(3) On Plastic Flow through Capillary Tubes. Proc. Am. Soc. Testing
Materials (1921); J. Am. Chem. Soc. (1922).

Buhner, C. 209, Beitrage zur Kentniss der Kristallinischen Flussigkeiten.

Inaug. dissert., Marburg (1906); 46 pp.

Buglia, C. 211, (1) tlber einige physikalisch-chemisch Merkmale der
homogenisierten Milch. Kolloid-Z. 2, 353 (1908).

Burkhard. Diss. Zurich, Berlin (1873); Z. Rubenzuckerindustrie (1874);
99.

Burri, R. and Nussbaumer, T. Surface Tension and Viscosity Determi¬
nations on Milk by Means of the Traube Stalagmometer. Biochem.
Z. 22, 90 (19—); 13 pp.

Burton-Opitz, H. 285, (1) Uber die Veranderung der Viskositat des Blutes
unter dem Einfluss verschiedener Ernahrung und experimentaller
Eingriffe. Arch, ges Physiol. (Pfluger’s) 82, 447 (1900); 17 pp.; (2)
Vergleich der Viskositat des normalen Blutes mit der des Oxalat-
blutes, des defibrinirtes Blutes und Blutserums bei verschiedener
Temperatur. Pfluger Arch. 82, 464 (1900); 10 pp.; (3) The Viscosity
of Laked Blood. Am. J. Physiol. 36, (1914); 7 pp.; (4) The Viscosity
of Bile. Biochem. Bull. 3, 351 (1914); 1 p.; (5) J. Physiol. 32, (1904);
(6) Arch. ges. Physiol. (Pfltiger’s) 119 (1907).

Burton-Opitz, R. and Dinegar, R. The Viscosity of Urine. Am. J.
Physiol. 47, 220 (1918); 11 pp.

360

INDEX

Butcher, J. 216, On Viscous Fluids in Motion. Proc. London Math.
Soc. 8 , 103 (1876); 33 pp.

Callan, J. G. A Viscosity Indicator for Paper-Making “Beater-Stock.”
U. S. Pat. 1,331,861.

Camerer, R. Olreibung in Rohren. Zeitschr. gesamte Turbinenwesen
4, 461 (1907); 6 pp.

Campani, A. & Leopardi. Viscosity of the Blood and the Use of Alkalies.
Folia clin. 4, 91 (1912); 9 pp.

Campbell, H. The Resistance to the Blood flow. J. Physiol. 23, 301
(1898-1899).

Cantone, M. Elastic Hysteresis. Rend. Acc. Lincei 4, 437 (1895);
9 pp.

Cappenberg, H. A Viscometer for Comparison of Hot Pastes. Chem.
Ztg. 34, 218 (1909).

Carlson, O. and Thall, E. Cellulose Ester Solutions. Brit. Pat. 136,141
Sept. 9 (1919).

Carothers, S. D. Portland Experiments on the Flow of Oil in Tubes.

Proc. Roy. Soc. (London) (A) 87, 154 (1912); 9 pp.

Carpenter, C. E. Viscosity in Cylinder Oils. Power 43, 519 (1916);

Also Do., 44, 101,249, 463,631, 697,598, 791 (1916); 10 pp.

Carpini, C. 35, Veranderung der inneren Reibung magnetischen Fliissig-
keiten im magnetischen Felde. Rend. R. acc. des Line. 12, 2 Sem.
341 (1903); 13 pp. Beible 27, 1127 (1903).

Cavazzani, E. 286, (1) Viscosity degli umori dell’occhio. Arch, di farmac.
speriment. e scienze affini 4, 401 (1905); (2) Reazione viscosimetrica
del latte. Archiv. di Fisiol. 2, 513 (1905); (3) Viscositat der Milch.
Zentralblatt f. Physiol. 18, 841 (19—).

Ceconi, A. II metodo della conducibilitil elettrica nell’indagine clinica.

La clinica medica italiana 44, 635 (1905).

Cervello and Pitini. Sulle variazioni termiche della viscosity dei colloidi.

Arch, di farmacologia e terapeutica 12, 17 (1906).

Chandelon, T. Viscosity of Collodion. Bull. soc. ehim. belg. 28, 24
(1914); 8 pp.; J. Soc. Chem. Ind. 33, 222 (1913).

Chanoz, M. Dissymmetry Produced by a Direct Current at Symmetrically
Arranged Liquid Junctions; Effect of Viscosity. Compt. rend. 149,
598 (1910); 2 pp.

Chapman, S. (1) Kinetic Theory of Simple and Composite Monatomic
Gases: Viscosity, Thermal Conduction, and Diffusion. Proc. Roy.
Soc. London (A) 93,1 (1916); 20 pp.; (2) Law of Distribution of Molecu¬
lar Velocities and the Theory of Viscosity and Thermal Conduction
in a Non-Uniform Simple Monatomic Gas. Trans. Roy. Soc. London
216, 279 (1916); 68 pp.

Charbonnier, P. La thdorie du champ acoustique et le frottement int6-
rieur des Gaz. Compt. rend. 137, 378 (1903); 3 pp.

Cheboksarov, M. Influence of Iodine Compounds on the Viscosity of the

INDEX

361

Blood. Z. exp. Med. 2, 168 (1913); 6 pp.; Cp. Zentr. Biochem.
Biophys. 16, 57.

Chella, S. (1) Apparat zur absoluten Messung der Koefficienten der
inneren Reibung der Gase. Physik. Z. 7, 196 (1906); 3 pp.; (2) Mes¬
sung des inneren Reibung der Luft bei niedrigen Temperaturen.
Physik. Z. 7, 546 (1906); 2 pp.

Ch^neveatj, G. (1) Solutions of Et OH, H 2 S0 4 , HN0 3 ; Hydrates Et OH.-
3H 2 0, H 2 S0 4 . H 2 0 , HNOs. 2 H 2 0. Compt. rend. 165, 154 (1912);
1 p.; (2) The Measurement of the Viscosity of Oils. J. physique 7,
109 (1917); 6 pp.

Chevenard, P. The Viscosity of Steels at Elevated Temperatures. Compt.
rend. 169, 712 (1919); 5 pp.

Chick, PI. & Lubrzynska. (1) Viscosity of Some Protein Solutions.
Biochem. J. 8, 59 (1913); 10 pp.; (2) Pseudoglobulin and Euglobulin
(Horse). Do. 8, 261 (1914); 19 pp.

Chick, H. & Martin, C. J. The Viscosity of Casein Sols. Z. Chem. Ind.
Kolloide 11, 102 (1912); 3 pp.

Chorower, C. Behavior of Varieties of Casein (from Cow and Goat
Milk) with Reference to Causing Viscosity. Chem. Ztg. 44, 605, 613
(1920); 4 pp.

Chowne, W. Experimental Researches on the Movement of Atmospheric
Air in Tubes. Proc. Roy. Soc. London 7, 466 (1855); 13 pp.
Christiansen, C. Versuche fiber den Einfluss der Capillaritat auf die
Ausstromungsgeschwindigkeit des Fliissigkeiten. Ann. Physik (4)
6, 436 (1901); 11 pp. Cp. Wied. Am. 41, 565 (1890).

Chwolson. Reibung im Innern der Fliissigkeiten. Lehrbuch der Physik.,
Vol. I, Braunschweig (.1902).

Clamer, G. H. A Study of Alloys Suitable for Bearing Purposes. Frank.
Inst. 166, 49 (1903); 29 pp.

Clark, R. Variation of the Viscosity of Gases with the Temperature.

Trans. Roy. Soc. Canada 13, III, 177 (1919); 4 pp.

Clebsch, A. Theorie der Elasticitat fester Korper. Leipzig (1862).
Clement, L. Theory of Viscometers. Matures grasses 2, 1591 (1909); 2

pp.

Clerici, E. Viscosity of Liquids for the Mechanical Separation of Minerals.

Atti. accad. Lincei 20, I, 45 (1910); 6 pp.

Cltjlow, F. S. and Taylor, C. W. Consistency of Greases. J. Soc.
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Cmunt, E. Action of the Ingestion of Gelatin on the Viscosity of the Blood.
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Cohen, R. 138, 139, 140, Uber den Einfluss des Druckes auf die Viskosi-
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Coker, E. The critical velocity of flow of mercury in small tubes.
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Coker, E. & Clement, S. 48, An Experimental Determination of the
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362

INDEX

Phil. Trans. A 201, 45 (1903); 16 pp.; Proc. Roy. Soe. London 71,
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Coleman, J. Notes on Viscosity and other Tests for Oils. J. Soc. Chem.
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Colin, H. and Chaudun, A. Law of Action of Sucrose. Influence of
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Colson, A. Sur l’ecoulement des liquides en tubes capillaires. Compt.
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Couette, M. 6,17,22, et seq. 29,33,42, et seq .; (1) La viscosity des liquides.
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Cowley, W. L. Fluid Motion and Viscosity. Engineering 109, 101
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Crookes, W. 246, (1) On the Viscosity of Gases at High Exhaustions.
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Cunningham, E. On the Velocity of Steady Fall of Spherical Particles
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Czerny, A. Versuche fiber Bluteindickung und ihre Folgen. Arch. exp.
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D’Alembert. 1 , Traite de l’equilibre et du mouvement des fluides*
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Darcy, H. 1, 35, Reeherches exp6rimentales relatives au mouvement de
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der kaiser, Akademie der Wissenschaften 15.

INDEX

363

D'Arcy, R. Viscosity of Solutions, Phil. Mag. (5) 28, 221 (1889); 11 pp.
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Davis, N. B. The Plasticity of Clay. Trans. Am. Ceram. Soc. 16, 65
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Davis, P. B., et al. (1) Studies on Solution in its Relation to Light
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Inorganic Salts in Formamide and in Mixtures of Formamide with
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Davis, P. B., Hughes, H. & Jones, H. C. Conductivity & Viscosity of
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Davis, P. B. and Jones, H. C. Conductivity and Negative Viscosity
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Dawson, H. M. The Estimation of Mixtures of Isomers and Other
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6 pp.

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Dean, E. W. and Jackson, L. E. Effect of crystalline paraffine wax upon
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Deeley, R. M. Oiliness and Lubrication. Engineering 108, 788 (1919);
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Deeley, R. M. & Parr, P. H. 239, (1) The Viscosity of Glacier Ice.
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Deering, W. H. & Redwood, B. Report on Castor Oils from Indian
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Dennhardt, R. Ann. Phys. 67, 325 (1899).

Dewar, J. The Viscosity of Solids. Nature 50, 238 (1894); Chem. News
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Determann, H. (1) Ein einfaches, stets gebrauchfertiges Instrument zur
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and Protein Content of the Blood with Different Diets Especially with

364

INDEX

Vegetarians. Med. Klin. No. 24 (1909); Berlin, klin. Wochenschr.
664 (1909). Cp. Munich Med. Wochenschr. #23 (1907).

Determann* H. & Booking. Does the Administration of Iodine Influence
the Viscosity of the Blood? Deut. Med. Wochschr. 38, 994 (1912); lp.
Determann, H. and Weil, F. Viscosity and Gas Content of Human
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Dickenscheid, F. 209, Untersuchungen uber Dichte, Reibung, und
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Doelter, C. 287, (1) Silikatglaser und Silikaschmelzen. Chem. Tech.
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"Uber den Einfluss der Viskositat bei Silikatschmelzen. Centr.
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Doelter, C. & Sirk, H. The Determination of the Absolute Value of the
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Dolazalek, F. und Schtjlze, A. Zur Theorie der binaren Gemische und
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Dolfus, C. Bull. Soc. Ind. Mulhouse 5, 14-23.

Donlinson, H. The Influence of Stress and Strain on the Physical Prop¬
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Doolittle, O. 328, The Torsion Viscometer. J. Am. Chem. Soc. 15,
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Dorn, E. & Vollmer. Uber die Einwirkung von Salzsaure auf metallisches
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pp.

Doroshevschii, A. G. and Rozhdestornskii. 195, Electrical Conductivity
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Dowling, J. J. Steady and Turbulent Motion in Gases. Proc. Roy.

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Drapier, P. 98, 99, 100,102, 103, Viscosity of Binary Liquid Mixtures in
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Drew, E. A Determination of the Viscosity of Water. Physik. Rev.
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Drucker, K. Fluidity. I. Z. physik. Chem. 92, 287 (1917); 32 pp.;
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Drucker, K. & Kassel. 104, The Fluidity of Binary Mixtures. Z.

physik. Chem. 76, 367 (1911); 18 pp. ^

Dubrisay, R. A Method of Testing the Viscosity of Lubricating Oils.
Ann. fals, 10, 301 (1917); 4 pp.; J. Soc. Chem. Ind. 36, 1123 (1917).

INDEX

365

DtjBuat, C. 1, Principes d’hydraulique verifies par un grand nombre
d’expdriences. nouv. 6dit., Paris (1786).

Dtjcceschi, V. (1) I processi di ossidazione, di riduzione e di sintesi negli
animali stiroidati. Sperimentale 50 (1897); (2) Sui proteici del siero
sanguigno nei card animizzati e stiroidati. Morgagni Sez. biologica 1,
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Dttclaux, E. Recherches zur les lois des mouvements des liquides dans les
espaces capillaires. Ann. ehim. phys. (4) 23, 433 (1872); 69 pp.

Duclatjx, J. & Miss Wollman. 281, Recherches sur la cellulose et ses
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Dtjdetzkii. 236, “Viscosity of Asphalt. J. Russ. Phys. Chem. Soc.,
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Duff, W. 6, 34, 131, (1) The Viscosity of Polarized Dielectrics. Physik.
Rev. 4, 23 (1896); 16 pp.; (2) Empirical Formulae for Viscosity as a
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Phil. Mag. (6) 9, 685 (1905); 7 pp.

Duhem, P. (1) Sur les fluides compressibles visqueux. Compt. rend. 134,
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ordre par rapport k la vitesse au sein d ; un milieu vitreux dou4 de
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M6m. de Bordeaux (6) 3, 121 (1903); 19 pp.; (7) Recherches sur
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Dumansxii, A., Zabotinskii, E. & Evseyee, M. A Method for Deter¬
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Dttmaresq, F. #he Viscosity of Cream. Proc. Roy. Soc. Victoria (2)
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Duncan', J. & Gamuse, A. Notes of some Experiments on the Rate of

366

INDEX

Flow of Blood and some other Liquids through Tubes of Narrow
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Dunstan, A. & Jemmett, W. Preliminary Note on the Viscosity of Liquid
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Dunstan, A. E. & Langton, H. 112, Viscometric Determination of
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Dunstan, A. E. & Mussel, A. G. (1) The Application of Viscometry to
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Dunstan, A. E. & Stevens, J. E. The Viscosity of Lubricating Oils.
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Dunstan, A. Stubbs, J. The Relation between Viscosity and Chem.
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Dunstan, A. & Thole, F. B. Ill, 121,142,279, (1) The Relation Between
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INDEX

367

A., & Thole, F. B. & Hunt. Relations between Viscosity and
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•^trjsrsT-A^sr, A., Thole, P. and Benson, P. The Relation between Viscosity
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E>o- P*r.^s Denning. Cp. Pr6 Denning.

T. C. 64, 201 et seq. 208. Cp. Bingham and Durham.
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Eohsteh, H. (1) The Viscosity of Binary Liquid Mixtures. Medd. K.
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Eichwald, E. 209, Neuere XJntersuchungen iiber die fliissigen Kristalle.

Diss. Marburg (1905); 39 pp. (Phenol-water mixtures.)
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368

INDEX

Elsey, H. M. Conductivity and Viscosity of Solutions in Dimethyl-
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Emley, W. E. 281, 282, (1) Measurement of Plasticity of Mortars and
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Engler, C. Ein Apparat zur Bestimmung der sogenannten Viscositat
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Engler, C. & Kunkler, A. Viscosimeter zur Prufung von Oelen bei
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Engler. 3, 7, 324.

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Ercolini, G. 35, Attrito interno dei liquidi isolanti in campo elettrico.
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Espy, W. E. Practical Method for Calculating Viscosity of Hydrocarbon
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Eucken, A. 252, (1) Physik. Z. 12, 1101 (1901); (2) The Thermal Con¬
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Euler, H. tJber die innere Reibung elektrolytische Losungen. Z. physik.
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Eustice, J. Experiments on Stream Line Motion in Curved Pipes. Proc.
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Ewalb, C. 285, tJber die Transpiration des Blutes. Arch. f. Anat. u.

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INDEX

369

Eytelwein. 1 , Untersuchungen liber die Bewegung des Wassers.
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Fabry, C. & Perot, A. (1) Measurement of Coefficient of Viscosity of Air.
Nature 55, 383 (1896-7); Abst.; (2) Sur une nouvelle mesure du coeffi¬
cient de viscosite de l’air. Compt. rend. 124, 281 (1897); 3 pp.; Nature
55, 383 (1896-7).

Fairbanks, F. L. (1) The Lubrication of Bearings and Cylinders. Power
42, 805 (1915); 4 pp.; (2) Value of Cylinder Oils. Power 43, 701
(1916).

Falek, 0. Apparatus for Determining the Viscosity of Liquids. Chem.
App. 1, 100 (1914); 1 p.

Fano, G. and Rossi, G. 284, (1) Sulla viscosity del siero di sangue solo o
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del siero sanguigno e di soluzioni gommose per l’aggiunta de cloruro
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Faragher, W. F. A Standard Viscometer and Standard Method for Deter¬
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Farr, C. C., Coleridge, C. and Macleod, D. B. The Viscosity of Sulphur.

Proc. Roy. Soc. London A 97, 80 (1919); 18 pp.

Farrow, F. D. 290, 291, The Viscosity of Aqueous Solutions of Sodium
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Faust, O. 141, (1) Die innere Reibung von Fliissigkeitgemischen, ihre
Abhangigkeit von der Temperatur, und die Verwandschaft der inneren
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F austen, A. Versuchc zur Bestimmung einer allgemeinen Formel zum
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durch Rohren. Diss. Bonn (1906).

24

370

INDEX

Fawsitt, C. 6, (1) Studies in Viscosity. The Electrochemist and Metal¬
lurgist 3, 664 (1904); 4 pp.; Electrochemist & Metallurgist 3, 801
(1904); (2) Physico-Chemical Investigations in the Amide Group.
Proc. Roy. Soc. Edinburgh 26, 51 (1904); 9 pp.; (3) On the Deter¬
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Finkener. 17.

Fea, L. Practical Determination of the Viscosity of Oils. Proc. Int.
Assoc. Testing Materials 2 [13] xxl 2 .

Feild, A. L. (1) Slag Control in the Iron Blast Furnace by Means of
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Field, A. & Royster, P. Temperature-Viscosity Relations in the Ternary
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Fels, J. (1) Tiber die Viscositat*des Leimes, ein neues Priifungsverfahren.
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Feltz & Ritter, E. De Taction des sels biliares sur le pouls, la tension,
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Fennel, L. tJber die Bewegung eines festen Korpers in einer tropfbaren
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Ferrai, C. Ricerche viscosimetriche sul sangue asfittico. Arch, di
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Arch, fisiol. 1, #4.

Ferrero, E. SulT attrito intemo nelle soluzioni di alume di cromo.
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Fick, A. Tiber den Druck in den Blutcapillaren. Pfltigers Arch. 42, 482
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Filippi, E. Relation between Changes in Viscosity and Surface Tension
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Findlay, A. 172, The Viscosity of Liquid Mixtures' at the Temperatures
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Fischer, E. & Schneidauer, E. Tiber das Aufsteigen von Salzlosungen in
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Fischer, F. Tiber Untersuchungen von Schmierolen. Dingler’s polytech.
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Fischer, R. A New Viscometer. Chem. Ztg. 44, 622 (1920); Z. angew.
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INDEX

371

Fished 50, 58, 59, (1) Temperature coefficients of Gas Viscosity.

XV- Apparent Relation between Viscosity and Specific Heat. Pliys.
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73 (1909); 33 pp.; (5) The Kinetic Pressure-drop Correction in the

Transfer Method for Gas Viscosity. Phys. Rev. 32, 216 (1911); 3 pp.

Fischer, J. Molecular Conductivity and Inner Friction of Mixtures of
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Fitzgeji^-A-^ 0 ) F. (1) The Viscosity of Liquid Ammonia, Sulfur Dioxide,
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Fitzgeji^A 1 ^ G. L. Turbulent Motion in Viscous Liquids. Trans. Roy.
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Flowers, A. E. Viscosity Measurement and a New Viscometer. Proc.
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Fol, J- -A. (1) Relation between the Amount of Resins and the Viscosity of

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Forbes. Viscosity Theory of Glacial Motion.

Forcbc, O. Oberflachenspannung und Reibungskoefficient fliissiger Luft.
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Forchheimer, P. L. Hydraulik. Teubner, Berlin (1914); 566 pp.

Fouguet be Nassandres. Viscosity des sirops. Bui. Ass. Chimistes 22,
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Foussereau, G. 6, 183, 193, 194, (1) Recherches expdrimentales sur la
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Frank, JT. Physical Properties of Colloidal Solutions. Kolloidchem. Bei-
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Franke nsruEiM, M. Krystallization und Amorphie. J. Prakt. Chem. 54,
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Frexjni> lich, H. The difference of potential acting in electroosmosis and
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Freundeich, H. <& Ishizake, C. (1) Die Koagulationgeschwindigkeit von

372

INDEX

A1 (OH) 3 -Solen, gemessen an der Aenderung ihrer Zahigkeit. Trans.
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B pp.

Frey, V. The Viscosity of the Blood. Transvaal Medical J. April (1908).

Friedlander, J. 7, 94, 102, fiber merkwurdige Erscheinungen in der
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Friedlander, J. & Tammann, G. Ueber die Krystallisationsgeschwindig-
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Frumpp. The Viscosity, the Haemoglobin and Protein Content of Chil¬
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Fueth & Kroenig. Centralblatt f. Gynakologie 701 (1901).

Fuld, E. 286, fiber die Milchgerinnung durch Lab. Hofmeister’s Beitr.
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Galdi, F. (1) Contribution to the Study of the Relation between Viscosity,
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Galileo. 1 , Works 2, 537 (1718).

Gardner, H. A. and Holdt, P. C. The Measurement of the Consistency
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Garnier, G. Sur la fluidity du nickel fondu. Compt. rend. 124, 1447
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Garrett, H. 212, 213, The Viscosity and Composition of some Colloidal
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Gartenmeister, R. 2, 7, 17, Die Zahigkeit fliissiger Kohlenstoffverbind-
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INDEX

373

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Gazetti, C. The Influence of Alkali Salts on the Viscosity of Proteins.
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Gerstner. 1, 6 , 127, Versuche iiber die Fliissigkiet des Wassers bei
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Getman, P. 186, (1) J. chim. phys. 4, 386 (1906); 20 pp.; (2) J. chim.
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Gibson, A. 170, 171, (1) An Investigation of the Resistance to the Flow of
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Gibson, W. The Influence of Volume Change on the Fluidity of Mixtures
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Gibson, W. H. and Jacobs, L. M. The Falling-Sphere Viscometer. J.
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Gibson, W. McCall, R. The Viscosity of Solutions of Nitrocellulose
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Gilchrist, L. 242, An Absolute Determination of the Viscosity of the Air.
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Gill, A. H. (1) Drugs, Oils & Paints 26, #56-7; (2) Oil Analysis, Lippin-
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Girard. 1 , 6, 30, Sur le mouvement des fluides dans les tubes capillaires et
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374

INDEX

Op. Extract of same Ann. Chim. phys. (2) 4, 146 (1817); 19 pp. Also
Am. Chim. phys. (2) 1, 436 (1816); 8 pp. Sur F6coulement de lather
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Gokum. (1) tlber die Beeinflussung der Viskositat der Kolloide dureh
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Goldschmidt, F. Viscosity of Soap Solutions. Seifensieder Ztg. 41,
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Goldschmidt, F. & Weisman, L. Aqueous Solutions of Ammonia Soaps.
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Goodwin, H. M. and Kalmus, H. T. On the Conductance and Fluidity
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Goodwin, H. and Mailey, R. (1) On the Density, Electrical Conductivity,
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Gorke, H. 179, et seq., tlber Losungen stark dissocierter Electrolyte.
Diss. Leipzig (1915); 49 pp. (Densities, conductivities, and viscosities
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Gostunin, M. P. & LeDantu, P. A. Contributions to the Study of
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Graetz, L. 128,129,134, (1) Schlomilchs Zeitsch. f. Math. 25, 316 (1879);
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Graham, M. A Study of the Change from Violet to Green in Solutions
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Graham, T. 2, 79, 106, 198, 213, 241, 251, (1) On the motion of Gases.

INDEX

375

Phil. Trans. Lond. 136, 573 (1846); 59 pp.; (2) On the Motion of Gases
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Grawitz, E. (1) Klinisch-experimentelle Blutuntersuchungen. Z. f. klin.

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Gray, A. On the Relation between Temperature and Internal Viscosities
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Gray, A., Blyth, V. and Dunlop, J. On the Effects of Changes of Tempera¬
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Gray, A. and Wood, A. On the Effect of a Longitudinal Magnetic Field
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Gray, T. T. A comparison of the Engler and Saybolt Viscosities of Mixed
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pp.

Green, Henry. 231, 266, Further Development of the Plastometer and
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Green, W. H. 196, Studies on the Viscosity and Conductivities of some
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Greiner, E. tTber die Abhangigkeit der Viscositat in Silikatschmelze
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57 pp.

Griffiths. 197, On the Viscosity and Electrolytic Resistance of a Gelatine
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Grinahovskii, K. P. Cause of the Abnormal Linear Velocity of Crystal¬
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Grindley, J. H. and Gibson, A. H. 61, 242, On the Frictional Resistance

376

INDEX

to the Flow of Air through a Pipe. Proc. Roy. Soc. London 80A, 114
(1908); 26 pp. Cp. Gibson.

Gronau. tJber die Bewegung schwingender Korper im widerstehenden
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Grossmann, L. 192, (1) Theorie der numerischen Berechnung der Con-
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pp.

Grotian, O. 6, 127,192, (1) Die Reibungskonstanten einiger Salzlosungen
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Gulbring, A. Significance of Leucocytes in Connection with Viscosity

INDEX

377

of the Blood. Inaug. Biss. Stockholm (1913); 140 pp., Hygiea 75;
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Gumbel. Der Widerstand geschmierter Flachen. Monatsblatter des
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Gunzburg, I. The Influence of Uranium and Potassium on the Viscosity
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Gurney, H. P. A Method of Measuring Absolute Viscosity. J. Am.

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Gurney, L. 256, (1) The Viscosity of Water at Very Low Rates of Share.
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Guthe, K. Some Cases of Excessive Damping of Torsional Vibrations.
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Guthrie, F. On the Influence of Temperature on the Passage of Air
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Gutzeit. 286, Uber Xnderungen in der physikalischen Beschaffenheit der
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Guy, J. S. and Jones, H. C. Conductivity and Viscosity in Mixed Solvents
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Guye, C. E. (1) Internal Friction of Solids; Its V ariation with the Tempera¬
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Guye, C. E. and Einhorn-Bodzechowski. The internal Friction of
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Guye, C. E. and Mintz, S. 1, (1) fitude sur la viscosite de quelques
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Guye, C. E. and Morein, A. The Interior Friction of Quartz Filaments
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Guye, C. and Vassileff, S. (1) Internal Friction of Glasses as a Function

378

INDEX

of the Temperature. Arch. sci. phys. nat. 37, 214 (1914); 11 pp.;
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de Haas. 130, Measurements on the Influence of Temperature on the
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Hagenbach, E. 2, 13, 17, 18, (1) Uber die Bestimmung der Zahigkeit
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INDEX

379

Haro. 285, (1) Sur l’dcoulement du sang par dcs tubas de petit calibre.
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Harrison, W. J. (1) The Hydrodynamical Theory of Lubrication with
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Hartley, H., Thomas, N. and Applebey. M. 196, Some Physico-Chemical
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Hasselblate, M. Ubor die lineare Kristallisationsgcschwindigkeit isom-
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Hauser, L. 140, tJber den Einfluss dcs Druckes auf die Viscositat dcs
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380

INDEX

(3) 16, 195 (1888) 21 pp.; Phil. Mag. (5) 28, 220 (1889); (6) Throne
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Helmholtz, H. 2,14,29, (1) t)ber Reibung und Warmeleitung verdtinnter
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Henry, et Calugareanu. 213, Diffusion des mati&res colorantes dans la
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Heraeus, W. C. Zeitchr. f. angewandte chem. 18, 49 (1905).

Herschel, W. H. 324, 329, (1) Determination of Absolute Viscosity by the
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(1918); 25 pp.; (3) A Viscometer for Gasoline. Proc. Am. Soc. Testing

INDEX

381

Materials IT, 19, 676 (1919); 11 pp.; U. S. Bur. of Standards Tech,
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Higgins, W. F. 324, (1) Methods and Apparatus Used in Petroleum
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382

INDEX

Higgins, E. F. and Pitman, E. C. Measurement of the Viscosity of
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Holland, R. Uber die Anderung der electrischen Leitfahigkeit einer
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INDEX

383

Holman, S. 246, 247, (1) On the Effect of Temperature on the Viscosity
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Holmgren, I. The Influence of White Blood Corpuscles upon the Viscosity
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Honda, K. and Konno, S. On the Determination of the Coefficient of
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Horton, F. 237, The Effects of Changes of Temperature on the Modulus
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Hosking, R. 6, (1) Viscosity of Solutions. Phil. Mag. (5) 49, 274 (1900);
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Howel and Cooke. Action of the Inorganic Salts, of Serum, Milk, Gastric
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384

INDEX

Hyde, J. H. 141, (1) The Construction of an Apparatus for the Determi¬
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Ishizaka, N. fiber die Beziehung zwischen Kolloidfallung und Adsorption
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Israel, H. Theorie der Ausfliisszeiten einer Fliissigkeit. Diss. Rostock
(1905); 65 pp.

Izar, G. Lowering of Viscosity by Gelatine Antiserum. Z. Immunitat
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Jablczynski, K. The Velocity of the Formation of Precipitates. Z.
physik. Chem. 82, 115 (1913); 6 pp.

Jackson, H. Glass and Some of its Properties. J. Roy. Art 68, 134
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Jacobson, H. 2, 14, 17, 32, (1) Beitrage zur Haemodynamik. Arch. f.
Anat. und Phys. 80 (1860); 33 pp.; (2) Zur Einleitung in die Haemo¬
dynamik. Arch. f. Anat. und Phys. 304 (1861); 25 pp. Cp. Ber. d.
Naturf. Vers, in Konigsberg (1862) (1867).

Jager, G. 131, (1) tlber die kinetische Theorie der inneren Reibung der
Fliissigkeiten. Wien. Sitzungsber. (2A) 102, 253 (1893); 12 pp.;
(2) fiber die innere Reibung der Losungen. Wien. Sitzungsber. 261
(1894); 15 pp.; (3) fiber den Einfluss des Molecularvolumens auf die
innere Reibung der Gase. Wien. Sitzungsber. (2A) 108, 447 (1899);
9 pp.; (4) Do. Wien. Sitzungsber. (2A) 109, 74 (1900); 7 pp. Cp.
Wien. Anzeiger Kaiserl. Akad. Wissens. math.-naturw. Kl. 11 (1900);

1 p.; (5) Der innere Druck, die innere Reibung. die Grosse der Molekeln
und deren mittlere Weglange bei Fliissigkeiten. Wien. Sitzungsber.
(2A) 111, 697 (1902); 10 pp.; (6) Kinetische Theorie der Gasen. Hand-

INDEX

385

buch der Physik. Winklemann. 2 ed. 3, 734 (1906); 13 pp.; (7) The
Kinetic Theory of the Internal Friction of Gases. Sitz. Akad. Wiss.
Wien. (IIA) 127, 849 (1918); 22 pp.

Jappeli, G. Contributo alio studio dell’influenza della aumentata viscosity
del sangue sulla meccanica cardio-vascolare. Arch, di Fisiologia 4,
101 (1907). Cp. Botazzi.

Jeans. The Dynamical Theory of Gases. Camb. Univ. Press (1904);
347 pp.

Jeaucard & Satie. Tension superficielle et viscosity de quelques huiles
essentielles. Bull. Soc. chim. (3) 25, 519 (1901); 5 pp.

Jevons, W. S. On the Movement of Microscopic Particles Suspended in
Liquids. Quarterly J. of Science, London (1878); 22 pp.

Job. Nouvelle m^thode experimentale pour T<$tude de la transpira¬
tion des gaz. Soc. franc, d. phys. 167, 2 (1901). Cp. Fortsch. Physik.
67, 280 (1901).

Johnson & Blake. On Kaolinite and Pholerite. Amer. J. Sci. (2) 93,
351 (1867); 11 pp.

Johnston, J. 196,196, (1) The Change of the Equivalent Conductance of
Ions with the Temperature. J. Am. Chem. Soc. 31,1010 (1909); 11 pp.;
(2) A Correlation of the Elastic Behavior of Metals with Certain of
their Physical Constants. J. Am. Chem. Soc. 34, 788 (1912); 15 pp.

Johnston, J. & Adams, L. H. On the Effect of High Pressures on the
Physical and Chemical Behavior of Solids. (Effect of Pressure on
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Jones, G. C. Ann. Reports on the Progress of Chem. 9, 195 (1913); 1 p.

Jones, H. C. & Collaborators. The Freezing-Point Lowering, Conduc¬
tivity and Viscosity of Solutions of certain Electrolytes in Water.
Methyl Alcohol, Ethyl Alcohol, Acetone and Glycerol, Carnegie Publ.
180. Cp. Davis.

Jones, H. C. <fc Bingham, E. C. The Conductivity and Viscosity of
Solutions of Certain Salts in Mixtures of Acetone with Methyl Alcohol,
with Ethyl Alcohol, and Water. Am. Chem. J. 34, 481 (1905);
Cp. Bingham.

Jones, H. & Bingham, E. & McMaster, L. Cp. Jones and Bingham; and
Jones and McMaster. Z. physik. Chem. 57, 193 (1906); 115 pp.

Jones, H. <fc Carroll, C. A Study of the Conductivities of Certain Elec¬
trolytes in Water, Methyl and Ethyl Alcohola, and Mixtures of these
Solvents—Relations between Conductivity and Viscosity. Am. Chem.
J. 32, 521 (1904); 63 pp.; Z. physik. Chem. 57, 257 (1906); 63 pp.

Jones, H., Lindsay, C., Carroll, C., Bassett, H., Bingham, E., Rotjiller,
C., McMaster, L. & Veazey, W. Conductivity and Viscosity of
Mixed Solvents. Carnegie Institution of Washington, Publ. 80
(1907), 227 pp.

Jones, H. <fc Mahin, E. Conductivity and Viscosity of Dilute Solutions
of Certain Salts in Water, Methyl Alcohol, Ethyl Alcohol, Acetone,
and Binary Mixtures of these Solvents. Am. Chem. J. 36, 325 (1906);
85 pp.; Conductivity and Viscosity of Dilute Solutions of Lithium

25

386

INDEX

Nitrate and Cadmium Iodide in Binary and Ternary Mixtures of
Acetone with Methyl Alcohol, Ethyl Alcohol, and Water. Z. Physik.
Chem. 69, 389 (1909); 30 pp. Cp. Guy.

Jones, H. <fc Veazey, W. 183, Possible Explanation of the Increase in
Viscosity when Alcohols are Mixed with Water and of the Negative
Viscosity Coefficients of Certain Salts when Dissolved in Water. Am.
Chem. J. 37, 405 (1907); 5 pp.

Jones, O. G. 6, The Viscosity of Liquids. Phil. Mag. (5) 37, 451 (1894);

12 pp.

Johns. Studien zur Viscositat des menschlichen Blutes beim Gesunden
und Kranken. Med. Klinik #28 (1909).

Kanitz, L. F. Einige Bermerkungen iiber Coulomb’s Verfahren die Cohe¬
sion der Fliissigkeiten zu bestimmen. Pogg. Ann. 70, 74 (1847); 4 pp.
Cp. Moritz.

Kaess. Untersuehungen iiber die Viscositat des Blutes bei Morbus
Bosedowi (1913); 17 pp.

Kagan, G. Zur Technik der Viskositatsbestimmung. Inaug. Diss. Bern
(1911); 24 pp.

Kahlbaum, W. tJber die Durchgangsgeschwindigkeit verdunnter Luft
durch Glasrohren verschiedenen Durchmessers. Verhandl. d. Ges.
deutscher Naturforscher, Niirnberg 1893, 56 (1894).

Kahlbaum, G. & Raber, S. Die Konstante der inneren Reibung des
Ricinusols und das Gesetz ihrer Abhang von der Temperatur. Nova
Acta, Abhand. Kaiserl. Leop. Carol, deutsch. Akad. Naturf. 84, 203
(1905). Cp. Raber.

KAhrs, F. Viscometer. U. S. Pat. 1,062,159, May 20 (1913).

Kalmus, H. Electrical Conductivity and Viscosity of Electrolytes.
(1906); 54 pp.; Electrical Conductivity and Viscosity of Some Fused
Electrolytes. Diss. Zurich (1906); 54 pp. Cp. Lorenz and Kalmus,
and Goodwin and Kalmus.

Kammerer. Mitteilung iiber Forschungsarbeiten auf dem Gebiete des
Ingenieurwesens. Ver. d. Ingenieure. Heft 132, Berlin Julius
Springer (1913).

Kanitz, A. 179, tiber die innere Reibung von Salzlosungen und ihren
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Kann, L. Uber die innere Reibung des Broms und dessen Abhangigkeit
von der Temperatur. Wien. Sitzungsber. (2A) 106, 431 (1897); 5 pp.
Kapff. Die Reibung von Schmierolen bei hoheren Warmegraden. Kraft,
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Kaplan, V. The Laws of Flow with regard to Fluidity and Friction.
Z. Ver. deut. Ing., Sept. 28 (1912).

Kariya, S. Influence of Adrenaline on Viscosity of the Blood in Acute
Beriberi. Mitt. Med. Ges. Tokio, 25, #15 Zentr. Biochem. Biophys.
12, 372.

von Karmian, T. The Viscosity of Liquids in the State of Turbulent Flow.
Physik. Z. 12, 283 (1911); 2 pp.

INDEX

387

Karss, W. Untersuchungen liber die Viscositat des Blutes bei Morbus
Basedowi. Diss. Heidelberg (1912); 17 pp.

Kassel, R. Viskositat binarer Fliissigkeitgemischen. Cp. Drucker &
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Katzenelsohn, N. Diss. Berlin (1867); Wied. Beibl. 12, 307 (1888).
Kawalki, W. (1) Untersuchungen liber die Diffusionsfahigkeit einiger
Electrolyte in Alcohol. Ein Beitrag zur Lehre von der Constitution
der Losungen. Wied. Ann. 62, 166 and 300 (1894); 25 pp., 28 pp.;
(2) Die Abhangigkeit der Diffusionsfahigkeit von der Anfangsconcen-
tration bei verdiinnten Losungen. Wied. Am. 69, 637 (1896); 15 pp.
Kawamura, S. Measurements of Viscosity Particularly Fitted for the
Study of the Coagulation Phenomena of Al(OH) a . J. Coll. Science
Imp. Univ. of Tokyo, Japan 26, #8 (1908); 29 pp.

Lord Kelvin (Thomson, Sir. W.). 218, 238, (1) On the Elasticity and

Viscosity of Metals. Phil. Mag. (4) 30, 63 (1865); 9 pp.; Do., Proc.
Hoy. Soc. London 14, 289 (1865); 9 pp.; (2) Stability of Fluid Motion.
Rectilinear M otion of a Viscous Fluid between two Parallel Planes. Phil.
Mag. (5) 24, 188 (1887); 9 pp.; (3) Stability of Motion. Broad River
flowing down on Inclined Plane Bed. Do. (5) 24, 272 (1887); 6 pp.;

(4) On the Propagation of Laminar Motion through a Turbulently
Moving Inviscid Liquid. Phil. Mag. (5) 24, 342 (1887); 12 pp.; (5)
On the Stability of Steady and of Periodic Fluid Motion. Phil. Mag.

(5) 23, 459 (1887); 6 pp.; (6) On the Stability of Steady and of
Periodic Fluid Motion. Phil. Mag. (5) 23, 529 (1887); 11 pp.

Kendall, J. 92, 104, 106, (1) The Viscosity of Binary Mixtures. Medd.
K. Vetenskapsakad. Nobelinst. 2, #25, 1 (1913); 16 pp.; (2) The Exten¬
sion of the Dilution Law to Concentrated Solutions. J. Am. Chem.
Soc. 36, 1069 (1914); 20 pp.

Kendall, J. and Brakeley. Compound Formation and Viscosity in
Solutions of the Types Acid: Ester, Acid: Ketone and Acid: Acid.
J. Am. Chem. Soc. 43, 1826 (1921); 9 pp.

Kendall, J. and Monroe, K. P. (1) Viscosity of Liquids. Viscosity-
Composition Curves for Ideal Liquid Mixtures. J. Am. Chem. Soc.
39, 1787 (1917); 15 pp.; (2) Ideal Solutions of Solids in Liquids. Do.
39, 1802 (1917); 4 pp.; (3) Ideality of the System: Benzene-Benzyl
Benzoate and the Validity of the Bingham Fluidity Formula. Do.
43, 115 (1921); 11pp.

Kendall, J. and Wright, A. H. 168, Ideal Mixtures of the Types Ether-
Ether and Ester-Ether. J. Am. Chem. Soc. 42, 1776 (1920); 9 pp.
Keppeler, G. & Spangenberg, A. Increasing the Plasticity and Binding
Power of Clay, Kaolin, etc. U. S. Pat. 1,013,603.

Kernot, G. and Pomilio, U. Cryoscopic and Viscometric Behavior of Some
Quinoline Solutions. Rend. Acad. Sci. fis. mat. Napoli 17, 358
(191-); 15 pp.

Kharichkov, K. V. tlber den Einfluss des Wassers auf den Entflammungs-
punkt und Viskositat von Mineralschmierol und Naphtaruckstanden.
Chem. Ztg. Rep. 376 (1907).

38.8

INDEX

Killing, C. (1) Eine einfache Methode zur Untersuchung von Butter
auf fremde Fette. Z. angew. Chem. 642 (1894); 3 pp.; (2) Do. Chem.
Ztg. 22, 78, 100 (1898); (3) Do. Chem. Rev. Fettind. 9, 202 (1902).
King, L. V. Turbulent Flow in Pipes and Channels. Phil. Mag. 31, 322
(1916).

Kingsbury, A. (1) Trans. Am. Soc. M. E. 17, 116 (1895); (2) Experiments
with an Air Lubricated Journal. J. Am. Soe. Nav. Eng. 9, #2 (1897);
29 pp.

Kinnison, C. S. A Study of the Atterberg Plasticity Method. Trans.
Am. Ceram. Soc. 16, 472 (1914); 13 pp.; Tech. Paper, U. S. Bur. of
Standards #46 (1915).

Kirchof, F. Math. Physik. 26 Vorelesung. Mechanik 2 AufL. 370 (1877).
Kirchof, F. The Influence of the Solvent on the Viscosity of India-Rubber
Solutions. Kolloid-Z. 15, 30 (1914).

Kirkpatrick, F. A. and Orange, W. B. Tests of Clays and Limes by the
Bureau of Standards Plasticimeter. J. Am. Ceram. Soc. 1, 170 (1918);
15 pp.

Koch, K. R. 35, External Friction in Liquids. Ann. Phys. 35, 613 (1911);
4 pp.

Koch, S. (1) tlber die Abhangigkeit der Reibungskonstanten des Queck-
silbers von der Temperatur. Wied. Ann. 14, 1 (1881); (2) liber die
Reibungskonstante des Quecksilberdampfes und deren Abhangigkeit
von der Temperatur. Wied. Ann. 19, 857 (1883); 15 pp.

Klaudy, J. Werth eines Schmiermittels. Jahresber. der chem. Tech¬
nology. 45, 1088 (1899); 2 pp.

Kleinhans, K. The Dependence of the Plasticity of Rock Salt on the
^Surrounding Medium. Physic. Z. 15, 362 (1914); 1 p.

Kleint, F. Innere Reibung binarer Mischungen zwischen Wasserstoff,
Sauerstoff, und Stickstoff. Verh. D. physik. Gesell. 7, 146 (1905).
Diss. Halle (1904).

Klemencic, I. (1) Beobachtungen iiber die elastische Naehwirkung am
Glase. Carl’s Repert. Exp.-physik. 15, 409 (1879); 18 pp.; (2) Beitrag
zur Kenntniss der inneren Reibung im Eisen. Carl’s Repert. Exp.-
physik. 15, 593 (1879); 7 pp.; (3) Ober die Dampfung der Schwingungen
fester Korper in Flussigkeiten. Wien. Sitzungsber. (2A), 84, 146

(1881); 22 pp.

Kling. La viscosity dans ses rapports avec la constitution chimique.

Rev. gen. sci. Paris 17, 271 (1906); 6 pp.
van Klooster, W. S. Normal and Abnormal Cases of Specific Volume of
Binary Liquid Mixtures. J. Am. Chem. Soc. 35, 145 (1913); 5 pp.
Kntbbs, G. 18, 19, 20, 25, 26, 34, 58, 127, (1) The History, Theory, and
Determination of the Viscosity of Water by the Efflux Method. J.
Proc. Roy. Soc. N. S. W. 29, 77 (1895); 70 pp.; (2) Note on Recent
Determinations of the Viscosity of Water by the Efflux Method. J.
Proc. Roy. Soc. N. S. W. 30, 186 (1896); 8 pp.

Knietsch, R. Tiber die Schwefelsaure und ihre Fabrication nach der
Contactverfahren, Ber. 34, 4113 (1901); 2 pp.

INDEX

&89

Knudsen, M. The Law of Molecular Flow and Viscosity of Gases Moving
* through Tubes. Ann. Physik. (4) 28, 75 (1909); 55 pp.; Polemical.
Physic. Rev. 31, 586 (1910;; 2 pp. Cp. Fisher.

Knudsen, M. & Weber, S. Luftwiderstand gegen die langsame Bewegung
kleinen Kugeln. Ann. Phys. 36, 981 (1911); 14 pp.

Kobler, B. Untersuchungen liber Viskositat und Oberflaschenspannung
der Milch. Diss. Zurich (1908); 72 pp. ^

Koch, K. R. Uber die aiissere Reibung tropf barer MpiSgkeiten. Ann.
Phys. 35, 613 (1911); 4 pp.

Koch, S. 128, (1) tTber die Abhangigkeit der Reibungskonstanten des
Quecksilbers von der Temperatur. Wied. Ann. 14, 1 (1881); (2)
Uber die Reibungskonstante des Quecksilberdampfes und deren
Abhangigkeit von der Temperatur. Do., 19, 857 (1883).

Konig, W. 6, 34, (1) Bestimmung einiger Reibungscoefficienten und
Versuche liber den Einfluss der Magnetisirung und Elektrisirung auf
die Reibung der Fllissigkeit. Wied. Ann. 26, 618 (1885); 8 pp.; (2)
fiber die Bestimmung von Reibungscoefficienten tropbarer Fliissig-
keiten mittelst drehender Schwingungen. Wied. Ann. 32, 193 (1887).

Koller, H. Uber den elektrischen Widerstand von Isolatoren. Wien.
Sitzungsber. (2A) 98, 894 (1889;; 15 pp.

Kohl. Viscometer nach Engler mit konstanter Niveau. Z. f. Chem.
App.-kunde 3, 342 (1908).

Kohlrausch, F. 237, (1) fiber die elastische Nachwirkung bei der Torsion.
Pogg. Ann. 119, 337 (1863); 32 pp.; (2) Beitrage zur Kenntniss der
elastischen Nachwirkung. Pogg. Ann. 128, 1 (1866); 20 pp.; (3)
Do. Pogg. Ann. 128, 207 (1866); 21 pp.; (4) Do., Pogg. Ann. 128,
339 (1866); 21 pp.; (5) Bemerkungeji zu Hrn. Neesen’s Beobach*iij|§en
liber die elastische Nachwirkung. Pogg* Ann. 155, 579 (1875); 9
pp.; (6) Experimental-Untersuchungen iiber die elastische Nach¬
wirkung bei der Torsion, Ausdehnung, und Biegung. Pogg. Ann.
158, 337 (1876); 39 pp.; (7) Beitrag zu Boltzmann’s Theorie der elas¬
tischen Nachwirkung. Pogg. Ann. 160, 225 (1877); 14 pp.; (8; tJber
den Temperatureinfluss auf das elektrische Leitvermogen von Losungen,
in besondere die Beweglichkeit der einzelnen Ionen im Wasser. Sig-
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Ions and the Mechanical Resistance of the Solvent. Proc. Roy. Soc.
London 71, 338 (1903); 13 pp.; (10) Weitere Untersuchungen liber
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Kooper, W. Determination of the Viscosity of Milk to Detect Added
Water. Milchwirtsch 43, 169, 201 (1914); 17 pp.

Kopsch, W. Die innere Reibung von Wasserstoff und Argon bei niederen
Temperaturen. Diss. Halle (1909); 44 pp.

von Koranyi and Bence. Viscosity of Blood. Arch. ges. Physiol.
(Pfluger’s) 110.

Korn, A. (1) Viscosity and Elastic Impact. Sitz. math.-physik. Klasse
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Reibung in kontinuierlichen Massensystemen. Berlin Dummler
(1901); 219 pp.; (3) Allgemeine Losung des Problems kleiner, station-

390

INDEX

arer Bewegungen in reibenden Fliissigkeiten. Rend. Circ. mat.
Palermo 25, 253 (1908); 18 pp.

Kotlmann. Zeitschr. f. klin. Med. 69 (1910).

Kraus, C. A. The Relation between the Conductance and the Viscosity
of Electrolytic Solutions and its Bearing upon the Theory of these
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Krehl. Pathol. Physiologie II Aufl. (1898).

Kreichgauer. Quoted by Arrhenius as Wied. Ann. 26, 161 (1885).
Kreidl, A. & Leak, E. Comparative Studies on the Viscosity of Milk
by the Determination of its Capillary Rise. Wien. klin. Wochschr.
24, 1667 (1913); 2 pp.

Kremann, R. & Ehrlich, R. Uber die Fortexistenz von Molekulver-
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Monatshefte 28, 831 (1907); 62 pp.

Krotkov, S. The Viscosity of the Pleural Exudate. Russki. Vratch 12,
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Kruger, F. Viscosity of Anistropic Liquids. Physik. Z. 14, 651 (1913);
4 pp.

Krummel, O. & Ruppin, E. Innere Reibung des Seewassers. Wiss.

Meeres, unters, Kiel. Abt. Kiel (N. F.) 9, 27 (1906); 9 pp.

Krupsay. Zur Frage der Bildsamkeit der Tone. Tonindustrie-Ztg.
32, 1289 (1908); 1 p.

Krusche, A. Die Anderung des Coefficienten der inneren Reibung von
Machinenolen mit der Temperatur. Zurich. Lohbauer (1904); 67 pp.
K undig, H. fiber die Viskositat des menchlichen Blutes bei Schwitz-
proceduren. Diss. Jena (1903); 25 pp.

Kuenen, J. The Diffusion-Coefficients of Gases and the Viscosity Coeffi¬
cients of Gas Mixtures. Verslag. Akad. Wetenschappen 22, 1158
(1913); 4 pp.

Kuenen, J. P. & Visser, S. W. A Viscometer for Volatile Liquids. Ver¬
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Kullgren, C. Om sambandet mellan inra friktion och den kemiska
konstitutionen. Oefversigt Vet. Acad. Forh. (Stockholm) 53, 647
(1896); 4 pp.

Kundt. Vorl. Exper. Physik. Lect. 36, 6 pp.; Do. 84, 7 pp.

Kundt, A. & Warburg, E. 244, (1) tlber Reibung und Warmeleitung
verdtinnter Gase. Pogg. Ann. 155, 337 (1875); 29 pp.; (2) Do., Pogg.
Ann. 155, 525 (1875); 26 pp.; (3) Do., Pogg. Ann. 166, 17? (1875);
35 pp. Cp. Warburg.

Kunkler, A. (1) Zur Kenntniss der Mineralmachinenole. Dingier's
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Zeitschr. d. Verein d. Ingen. 36, 633 (1892); 5 pp.; (5) Do., 74 (1893);
Dingier's Polyt. J. 290, 942 (1893); (6) Werthbestimmung der Schmier-
mittel. Jahresber. d. Chem. Technologie 45, 1088 (1895); 1 p.
Kupffer. 237, 238, (1) Recherches experimentales sur Y elasticity des
m6taux faites a l'observatoire physique central de Russie. (1860);

INDEX

391

(2) Uber den Einfluss der Warme auf die elastische Kraft der festen
Korper und insbesondere der Metalle. Memoires de FAcad£mie.
St. Pdtersbourg., Sixi&me S6rie, Sciences MathǤmatiques et physiques
5, 233 (1853); & 6, 397 (1857); 97 pp.

Kurnakov, N. S. & Efrenor, N. M. Inner friction of the Systems:
Chloral Water and Chloral-Alcohol. Z. physik. Chem. 85, 401 (1914);
17 pp.; Do., J. Russ. Phys. Chem. Soc. 45, 329 (1913); 19 pp.

Kurnakov, N. S., Perlmutter, S. I. and Kanov, F. P. Viscosity of
Binary Systems Containing Stannic or Antimony Chloride. J. Russ.
Phys. Chem. Soc. 48, 1658 (1916); 5 pp.

Kurnakov, N. & Zhemchuznui, S. (1) Die innere Reibung der binaren
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Chem. 83, 481 (1913); 26 pp.; (2) Flow, Pressure & Hardness of Plastic
Substances. J. Russ. Phys. Chem. Soc. 45, 1004 (1913); 72 pp.;

(3) Inner Friction of Binary Systems. Characteristics of definite
compounds. J. Russ. Chem. Soc. 44, 1964 (1912); 27 pp.; Z. physik.
Chem. 83, 481 (1913); 25 pp.; (4) The Viscosity and Hardness of
Plastic Substances. Jahrb. Radioakt. Electronik 11, 1 (1914); 66 pp.

Kurz, A. Die Reibungsconstante des Wassers. Ein Vorlesungsversuch.

Exner’s Repert. Exp. physik. 23, 567 (1887); 4 pp.

Kurzmann, I. The New Multi-Viscometer. Chem. Ztg. 37, 234 (1914);
Kolloidchem. Beiheft 5, 427 (1914); Diss. Karlsruhe (1914).

Labatut. Influence of Hysteresis and Viscosity of Flexure on the Reading
of Metallic Barometers. Bulletin de la soci<§t6 de statisque, des
Sciences naturelles et des arts industriels du department de FIs&re.
28, 109 (1895).

Laby, T. H. & Carse, G. A. On a Relation Between the Velocity and the
Volume of the Ions of Certain Organic Acids and Bases. Proc. Camb.
Phil. Soc. 13, 287 (1906); 8 pp.

Lachmann, R. (1) Plasticity of Salt Rocks. Centr. Min. 46 (1912); 2 pp.;

(2) The Plasticity Question. Do., 745 (1912); 12 pp.

Ladd, G. E. Clays of Georgia. Ga. Geol. Surv. Bull. 6a, 29 (1898); 7 pp.
Ladenburg, R. 6, 33, 34, (1) TJber die innere Reibung zaher Fliissigkeiten
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Physic. (4) 22, 287 (1907); 23 pp.; (2) Uber den Einfluss der Reibung
auf die Schwingungen einer mit Fllissigkeit gefullten Kugel. Ann.
Physik. (4) 27, 157 (1908); 29 pp.; (3) tTber den Einfluss von Wandein
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Physik. (4) 23, 447 (1907); 11 pp.

Lamansky, S. (1) Untersuchungen liber Schmierole. Dingler’s Poly tech.
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ler’s Polytech. J. 256, 176 (1885); 13 pp.

Lamb, H. Hydrodynamics, A Treatise on the Mathematical Theory of the
Motion of Fluids. Univ. Press. Cambridge 597 (1879); 223 pp.
Lambert. 1 , Sur les fluides consid6r6s relativement & P hydrodynamique.
M&noires de FAcad. de Berlin (1784).

392

INDEX

Lampe, C. Programm des stadtischen Gymnasiums in Danzig (1866).

Schriften der naturforschenden Gesell. zu Danzig (N. F.) 3, (1872).
Lampel, A. tjber Drehschwingungen einer Kugel mit Luftwiderstand.

Wien. Ber. (2A) 93, 291 (1886); 23 pp.
von Lang, V. (1) Versuche uber Einstromung von Gasen. Wien. Sitzungs-
ber. (2A) 63, 604 (1871); 15 pp.; (2) Zur dynamischen Theorie der Gase.
Wien. Sitzungsber. (2A) 64, 485 (1871); 4 pp.; (3) Do., Wien. Sit-
zungsber. (2A) 66, 415 (1872); 4 pp.; (4) Experiments on the Friction
between Water and Air. Phil. Trans. London 166B, 589 (1876); 17 pp.
Lansa, L. & Vergano, R. Iodine Preparations and Viscosity of the
Blood. Pensiero med. No. 22-23. Zentr. Biochem. Biophys. 14,
602 (1912).

Laquertjr, E. & Sacktjr, 0. Hofmeisters Beitr. 3, 193 (1903).

Lasche, O. 268, Die Reibungsverhaltnisse in Lagern mit hoher Unfangs-
geschwindigkeit. Traction & Transmission, Jan., 1903); p. 80; Z.
d. Ver. d. Ingen. 46, 1932, 1961 (1902); Do. Mitt, liber Forschungs-
arbeiten. Springer. Heft 9 (1903); 27 pp.

Lauenstein, C. Untersuchungen liber die innere Reibung wasseriger
Natronsalzlosungen organiseher Sauren. Z. physik. Chem. 9, 417
(1892); 18 pp.

Lauer, L. & Tammann, G. Tiber Verschiebungselastizitat bei Fliissig-
keiten. Z. physik. Chem. 63, 141 (1908); 10 pp. Cp. Tammann.
Lawaczeck, F. Viscosity and Its Measurement. Z. Ver. deut. Ing.
63, 677 (1919); 6 pp.

LeBas, G. The Theory of Molecular Volumes I. The Existence of
Additive Relations in Molecular Volumes. Phil. Mag. 27, 344 (1914);

12 pp.

Le Chatelier, H. and F. Mechanical Properties of Plastic Substances
(steel, glass). Compt. rend. 171, 695 (1920); 4 pp.

Lechner, G. Turbulence in the Flowing of Water and Mercury through a
Spirally Wound Capillary Tube. Ann. Phys. 42, 614 (1914); 28 pp.
Lees, C. 82, On the Viscosities of Mixtures of Liquids and Solutions.
Proc. Phys. Soc. 17, 460 (1900); 20 pp.; Phil. Mag. (6) 1, 128 (1901);

20 pp.

Lehmann, O. 96, Liquid crystals. Physik. Z. 7, 578 (1906); 1 p.

Lemcke. Elektrische Leitvermogen und innere Reibung. J. Russ. Phys.
Chem. Soc. 37, 1134 (1906).

Lemstrom, S. Uber das Verhalten der Flussigkeiten in Capillarrohren
unter Einfluss eines elektrischen Luftstromes. Ann. Physik. (4) 6,
729 (1901); 27 pp.

Leppla, G. Die Bildsamkeit (Plastizitat) des Thones. Baumaterialen-
kunde 9, 124 (1904); 2 pp.

Levi. 213, Contributo alio studio della dissociazione in soluzioni colloidoli.
Gaz. chim. italiana 30, 64 (1900).

Levites. (1) Sur la friction intdrieure des solutions colloidales. J. Russ.
Phys. Chem. Soc. 36, 253 (1903); 10 pp.; Do. 36, 401 (1904); 16 pp.;
(2) Beitrage zur Kenntnis der Gelatinierungsvorganges. Uber die
innere Reibung kolloiden Losungen. Kolloid.-Z. 2, 210 (1907); 8 pp.

INDEX

393

Lewy, B. 285, (1) Die Reibung des Blutes. Pflugers Arch. 65, 447 (1897);
26 pp.; (2) tlber die Reibung des Blutes in engen Rohren und ihren
Einfluss auf das Gefalle im Gefasssystem. Arch. f. Physiol. 147
(1879); 3 pp.

Leysieffer, G. (1) Viscosity of Cellulosenitrate Solutions. Diss.
Berlin (1917); Kolloidch. Beihefte 10, #5 (1918); (2) Relation between
Viscosity of Cellulose Nitrate Solutions and the Nitration Process.
Koll. Chem. Beihefte 10,145 (1918); 33 pp.

Lichtwitz, L. & Renner, A. Variation of the “Gold Value” and the
Viscosity of Colloidal Solutions with the Temperature. Z. Physiol.
Chem. 92, 113 (1915); 5 pp.

Lidstrom, F. M. (1) A Mercurial Viscometer. J. Ind. Eng. Chem. 36,
270, 317 (1917); 3 pp.; (2) A Modified Mercurial Viscometer for
Determining the Viscosity of Volatile Liquids. J. Ind. Eng. Chem.
37, 148 (1918).

Liebermann, L. V. Apparatus for the Determination of Viscosity Particu¬
larly that of Serum and other Animal Liquids. Biochem. Z. 33,
218 (1911); 4 pp.

Lindmann, K. Klinische und experimentelle Beitrage zur pharmakolog-.

ischen Beeinflussung der Blutviscositat. Diss. Marburg (1908); 24 pp.
Linebarger, C. 82, 90, 92, On the Viscosity of Mixtures of Liquids. Am.
J. Sci. (4) 2, 331 (1896); 10 pp.

Lisbonne, M. & Margarot, J. Viscosity of Blood. Arch. Malad. coeur
6, 279, 330 (1912;.

Loeb, J. (1) Volumetric Analysis of Ion-Protein Compounds. Proc. Soc.
Exp. Biol. Med. 16, 39 (1918); (2) Amphoetric Colloids. IV. The
Influence of the Valency of Cations upon the Physical Properties of
Gelatine. J. Gen. Physiology 1, 483 (1919); 22 pp.; (3) Do. V. The
Influence of Anions upon the Physical Properties of Gelatine. Do.
1, 559 (1919). (4) Do., 3, 247, 391, 547, 557, 667, 691, 827 (1920);

142 pp. (5) Do., 4, 73, 97, 187 (1921); 66 pp.

Lowenstein, A. Viscosity of the Eye Fluids under Normal and Patho¬
logical Conditions. Arch. Augenheilk. 70, 27 (1913). Zentr. Biochem.
Biophys. 12, 535.

Lowenthal, J. (1) Die Transpiration der Flussigkeiten als Hulfsmittel
fur die Wissenschaft und Technik. Z. anal. Chem. 10, 298 (1871); 8
pp.; Do. Z. anal. Chem. 11, 43 (1872); 3 pp.

Loewinson & Lessing. Eine mogliche Beziehung zwischen Viskositats-
kurven und Molekularvolumena bei Silikaten. Centralblat Min. 289
(1906); 2 pp.

Lohr, E. Ein einfacher Zusammenhang zwischen Brechungsexponent,
Zahigkeit, und Dichte bei Gasen. Wien. Ber. (2A) 116, 1281 (1907);
7 pp.

Lommel, F. Uber die Viskositat des menschlichen Blutes bei Schwitz-
proceduren. Deutch. Arch. f. klin. med. 80, 308 (19—).

Lorentz. Verlagen der Akademie Van Wittenschappen te Amsterdam
6 , 28 (1897).

Lorenz, R. & Kalmus, H. Die Bestimmung der inneren Reibung einiger

394

INDEX

geschmolzene Salze. Z. physik. Chem. 59, 217 (1907); 8 pp. Cp.
Goodwin & Kalmus, and Kalmus.

Loria, G. Viscosity of the Blood under the Action of Various Diuretics.
Riv. crit. chim. med. 12, #5, 6, 7; Zentr. Biochem. Biophys. 13, 174.

Love. Enzyklopodie der Mathematischen. Wissenshaft. 4, II, 80; 4, III,
64 (1901-1908).

Lucius, F. (1) tJber Farbstoffabsorption. I. Kryoskopie und Viskositat
der Milch. II. Diss. Leipzig. (1906); 54 pp.

Ludwig, C. <fc Stephan, J. tJber den Druck, den das fliessende Wasser
senkrecht zu seiner Stromrichtung ausiibt. Wien. Sitzungsber. (2A)
32, 25 (1858); 18 pp. Cp. Stephan.

Ludwik, P. tJber die Anderung der inneren Reibung der Metalle mit der
Temperatur. Zeitschr. physik. Chem. 91, 232 (1916); 15 pp.

Ludeking, C. 197, 213, Leitungsfahigkeit gelatinehaltiger Zinkvitriol-
losungen. Wied. Ann. 37, 172 (1889); 5 pp.

Luers, H. and Ostwald, Wo. Colloidal Chemistry of Bread. II. Visco-
metry of Flour. Kolloid-Z. 25, 82, 116 (1919); 29 pp. Cp. Ostwald.

Luers, H. and Schneider, M. 289, The Viscosity-Concentration Function
of Polydispersed Systems. Kolloid-Z. 27, 273 (1920); 5 pp.;

Lunge, G. (1) Zur Untersuchung der Zahflussigkeit von Schmiermate-
rialien und dgl. Z. angew. Chem. 189 (1895); 3 pp.; (2) Examen de la
quality dans les gommes adragantes au moyen d’un viscosim&tre tr&s
simple. Bull. soc. ind. Mulhouse 66, 64 (1896); 9 pp.

Lunge, G. & Zilchert, P. Untersuchung der Zahflussigkeit von Gummi
und Traganthlosungen mittels des Lung4schen Viscosimeters. Z.
angew. Chem. 437 (1895); 3 pp.

Lussana, F. Sulla viscosity del latte. Bologna (1905).

Lussanna, S. & Cinelli, M. L’attrito interno e Tattrito elettrolitico nelle
soluzioni. Atti d. R. Academia dei Fisiocritici Siena (4) 9, 49 (1897);
17 pp.

Lust, F. The Viscosity of the Blood in Healthy and Sick Infants. Arch.
Kinderheilk. 54, 260 (1911); 19 pp.

Luthe, W. Ballistische Messungen der Magnetischen Viskositat an Ringen
aus Elektrolyleisen, Kobalt, und Nickel. Diss. Halle (1912); 54 pp.;
Ber. physik. Ges. 458 (1913); 28 pp.

Lyle, T. & Hosking, R. The Temperature Variation of the Specific
Molecular Conductivity and of the Fluidity of Sodium Chloride
Solutions. Phil. Mag. (6) 3, 487 (1902); 11 pp. Cp. Hosking.

Mabery, C. F. & Matthews, J. H. On Viscosity and Lubrication. J.
Am. Chem. Soc. 30, 992 (1908); 10 pp.

MacGill, A. Proc. & Trans. Roy. Soc. Canada (2) 1, III, 97 (1895).

MacGaskey. The Viscosity of the Blood. J. Am. Med. Assoc. #20
(1908). Cp. Berlin klin. Wochenschr. (1908).

MacGregor, J. On the Relation of the Physical Properties of Aqueous
Solutions to their State of Ionization. Trans. N. S. Inst. Sci. 9 , 219
(1896-7); 27 pp. Cp. Barnes.

INDEX

395

MacInnes, D. A. The Ion Mobilities, Ion Conductances, and the Effect
of Viscosity on the Conductances of Certain Salts. J. Am. Chem.
Soc. 43, 1217 (1921); 10 pp.

MacMichael, R. F. 328.

MacNider, G. M. (1) A Method for Determining the Value of Com¬
mercial Starches for Use in Cotton Mills. J. Ind. Eng. Chem. 4, 417
(1912); 12 pp.; (2) A Practical Method for Determining the Viscosity
of Starch for Mill Purposes. J. Ind. Eng. Chem. 9, 597 (1917); 2 pp.
Madella. Sopra alcune determinazioni d’attrito interno del latte. Le
Stazioni Sperimentali agrarie italiane 37, 383.

Magnus, G. (1) tlber die Bewegung der Fliissigkeiten. Pogg. Ann.
80, 1 (1850); 36 pp.; (2) Hydraulische Untersuchungen. Pogg. Ann.
96, 1 (1855); 59 pp.

Mahin, E. Cp. Jones and Mahin. Diss. Johns Hopkins (1908); 42 pp.
Mahr, H. W. Determination of the Melting Point of Greases by Means of
the New York Testing Laboratory Viscometer. J. Ind. Eng. Chem.
6, 674 (1913); 1 p.

Main, J. Note on some Experiments on the Viscosity of Ice. Proc. Roy.
Soc. London 42, 329 (1886); 2 pp.

Mair, J. G. Experiments on the Discharge of Water of Different Tempera¬
tures. Proc. Inst, of Civil Engineering 84, II, 426 (1886); 12 pp.
Malcolm. Phil. Mag. (6) 12, 508 (1906).

Mallock, A. 6, (1) Determination of the Viscosity of Water. Proc.
Roy. Soc. London 45, 126 (1888); 7 pp.; (2) Experiments on Fluid
Viscosity. Proc. Roy. Soc. London 69, 38 (1896); 2 pp.; Phil. Trans.
(A) 187, 41 (1896); 16 pp.

Malus, C. (1) Etude de la viscositd du soufre aux temperatures sup6-
rierures & la temperature du maximum de viscosity. Compt. rend.
130, 1708 (1900); 3 pp.; (2) Recherches sur la viscosity du soufre.
Ann. chim. phys. (7) 24, 491 (1901).

Marcusson, J. (1) Mitt. a. d. Konigl. Materialprufungsamt 29, 50 (1911);

4 pp.; (2) Chem. Rev. 45 (1909). (The use of blown oils in lubrication).
Marey. Changements de direction et de vitesse d’un courant d’air qui
rencontre des corps de formes diverses. Compt. rend. 132, 1291
(1901); 5 pp.

Margules, M. 29, tlber die Bestimmung des Reibungs- und Gleitungs-
coefficienten aus ebenen Bewegung einer Eliissigkeit. Wien. Sitz-
ungsber. (2A) 83, 588 (1881); (2) Wien. Sitzungsber. (2A) 84, 491
(1881).

Marie, C. Surtension et viscosit6. Compt. rend. 147, 1400 (1908);

2 pp.

Mariost. A Liquid Passing through Another without Mixing. M6m.

soc. sci. phys. nat. Bordeau 2, 51 (1886).

Mariotte. 1 , Traitd du mouvement des eaux. Paris (1700).

Markowski, H. Die innere Reibung von Sauerstoff, Wasserstoff, chemis-
chem und atmospharischem Stickstoff und ihre Anderung mit der
Temperatur. Ann. Physik. (5) 14, 742 (1904); 13 pp.; Diss. Halle;
41 pp. Cp. Bestelmeyer.

396

INDEX

Mark well, E. Coefficient of Viscosity of Air by the Capillary Tube
Method. Phys. Rev. 8 , 479 (1916); 5 pp.

Martens, A. tJber die Bestimmung des Fliissigkeitsgrades von Schmierol.

Mitt, der techn. Versuchsanst 8 , 143 (1890); 8 pp.

Martici, A. 210, Contributo alia conoscenza delle emulsioni. Arch. d.
Fisiol. 4, 133 (1907).

Martin, H. M. Lubrication. Proc. Phys. Soc. London, II, 32, 11 (1919);
4 pp.

Martins, F. Die Ershopfung und Ernaherung des Froschherzens. Arch,
f. (Anat.) u. Phys. 543 (1882).

Masi, N. Le nuove vedute nelle ricerche teoriche ed esperimentali sull’
attrito und esperenze d’attrito. Zanichelli, Bologna (1897).

Masson. A Preliminary Note on the Effect of Viscosity on the Conduc¬
tivity of Solutions. Austr. Assoc. Adv. Sci. 3 (1901).

Masson, I. & McCall, R. Viscosity of Solutions of Nitrocellulose in
Mixtures of Acetone and Water. J. Chem. Soc. 117, 118, 819 (1920).
Massoulier, P. 197, (1) Relations entre la conductibilit<§ 61ectrolytique
et le frottement interne dans les solutions salines. Compt. rend.
130, 773 (1900); 2 pp.; (2) Relation qui existe entre le resistance <§lec-
trique et la viscosite des solutions electrolytiques. Compt. rend.
143, 218 (1906); 2 pp.

Mastrobttono. Viscosity of the Aqueous Humor. Arch. Ottalm.-Centr.

Augenheilk. Erganzungsheft (1908).

MATHiEtr, £). 14, Sur le mouvement des liquides dans les tubes de tr&s-

petit diam&tre. Compt. rend. 57, 320 (1863); 5 pp.

Matthews, Brander. 7.

Matzdorff, O. A New Viscometer for the Comparison of Hot, Pasty
Substances. Z. Spiritusind. 33, 420 (1910).

Maxwell, J. 2, 5, 6, 128, 152, 215, 241, 243, 246, 251, Constitution of
Bodies. Encyclopedia Brittanica. Cp. Theory of Heat; (2) On the
Dynamical Theory of Gases. Report Brit. Assoc. (Pt. 2) 9 (1859); 1
p.; (3) Illustrations of the Dynamical Theory of Gases. Part I. On
the Motions and Collisions of Perfectly Elastic Spheres. Phil. Mag.
(4) 19, 9 (1860); (4) Do., Part II. On the Process of Diffusion of Two
or More Kinds of Moving Particles among One Another. Part III.
On the Collisions of Perfectly Elastic Bodies of any Form. Phil. Mag.
(4) 20 , 21 (1860); (5) On the Internal Friction of Air and Other Gases.
Phil. Trans. London 166, 249 (1886); 20 pp.; (6) On the Dynamical
Theory of Gases. Phil. Mag. (4) 35, 129 (1868); (7) Do., Phil.
Mag. (4) 35, 185 (1868); (8) Cp. Collected papers.

Mayer, A. (1) Role de viscosity dans les ph&iom&nes osmotiques et dans
les ^changes organiques. Compt. rend. Soc. Biol. 53, 1138 (1901);
(2) fitudes viseosimStriques sur la coagulation des albumino'fdes du
plasma sanguin par la chaleur. Compt. rend. Soc. Biol. 54, 367
(1902); (3) Coefficients de viscosity du serum et du plasma sanguins
normaux. Compt. rend. Soc. Biol. 64, 365 (1902).

Mayer, A., Schaeffer, G. & Terroine, E. Viscosity of soap solutions.
Compt. rend. 146, 484 (1908).

INDEX

397

Mayesima, J. Clinical and Experimental Researches on the Viscosity of
the Blood. Mitt. Grenz. Med. Chir. 24, 413 (1912); 25 pp.

McBain, J. W. Colloid Chemistry of Soap. Dept. Sci. Ind. Research,
Brit. Assoc. Adv. Sci. Third Report on Colloid Chem. (1920); 31 pp.
McBain, Cornish, and Bowden. Trans. Chem. Soc. 101, 2042 (1912).
McConnel, J. 239, (1) On the Plasticity of Glacier and other Ice. Proc.
Roy. Soc. London 44, 331 (1888); 36 pp.; (2) On the Plasticity of an
Ice Crystal. Proc. Roy. Soc. London 44, 259 (1890); 1 p.; Proc. Roy.
Soc. London 49, 323 (1891); 21 pp.

McGill, A. Viscosity in Liquids and Instruments for its Measurement.
Trans. Roy. Soc. Canada (2) 1, 97 Sect. III. (1895); 7 pp.; Canadian
Record of Science 6, 155 (1896).

McIntosh, D. & Steele, B. 1, Viscosity and Viscosity Temperature
Coefficients of Liquids, Hydrochloric, Hydrobromic, Hydriodic,
Hydrosulphuric Acids and Phosphine. Phil. Trans. (A) 206, 99 (1906);
68 pp.; Proc. Roy. Soc. London 73, 450 (1904).

McKeehan, L. W. The terminal velocity of fall of some spheres in air at
reduced pressures. Phys. Rev. 33, 153 (1911); 16 pp.

McMaster, L. Cp. Jones and McMaster. Diss. Johns Hopkins (1906).
Meggitt. A New Viscometer. J. Soc. Chem. Ind. 21, 106 (1902).
Meissner, W. 328, (1) The Influence of Errors in the Dimensions of
Engler’s Viscometer. Chem. Rev. Fett-Harz-Ind. 17, 202 (1909); 8 pp.;

(2) Chem. Revue iiber die Fett-Harz-Industrie 17, 202 (1910); 7 pp.;

(3) Vergleichende Untersuchungen iiber den Englischen, Redwood’
schen, u. Sayboltschen Zahigkeitsmesser. Chem. Rev. Fett-Harz-
Ind. 19, 9 (1912); 9 pp.; Book, Vienna (1912); (4) Comparison of the
Engler, Redwood, and Saybolt Viscometers. Chem. Rev. Fett-Harz-
Ind. 19, 30, 44 (1912); 10 pp.; Petroleum 7, 405; (5) Comparative
Examination of Viscometers. Chem. Rev. Fett-Harz-Ind. 21, 28
(1913); 4 pp.; (6) Viscosity of Nitrocellulose. Moniteur Scientifique
79 (1915).

Melis-Schirru, B. Changes in the Viscometric Coefficient of Human
Blood Serum after Blood-Letting. Biochem. terap. sper. 4,49 (1914);
8 pp.; Zentr. Biochem. Biophys. 16, 596 (1914).

Mellor, J. W. Clay & Pottery Industries. Griffin & Co., London (1914).
Menneret, M. Oscillatory and Uniform Motion of Liquids in Cylindrical
Tubes. J. physique (5) 1, 753 (1912); 13 pp.; Do. 1, 797 (1912);
7 pp.

Mercanton, P. L. Simple Lecture Exps. Physik. Z. 13, 85 (1912); 1 p.
Merczyng, H. H. (1) J. Russ. Phys. Chem. Soc. 21, 29 (1889); (2) Uber
die Bewegung von Fliissigkeiten, Wasser und Petroleum in weiten
Rohren. Wied. Ann. 39, 312 (1890); 7 pp.; (3) Sur le mouvement des
liquides k grande vitesse par conduits tr&s larges. Compt. rend. 144,
70 (1907); 2 pp.

Merrill, G. P. Non-metallic Minerals. Wiley & Sons, p. 221 (1904).
Merry, E. & Turner, W. The Viscosities of Some Binary Liquid Mix¬
tures containing Formamide. J. Chem. Soc. 106, 748 (1914); 1 p.
Cp. English.

39.8

INDEX

Merton, T. R. The Viscosity and Density of Caesium Nitrate Solutions.

J. Chem. Soc. London 97, 2460 (1910); 10 pp.

Merveau, M. J. Reeherehes sur le Viscosity Lons le Saunier. 64 (1910);

8 pp.

de Metz, G. Rigidity des liquides. Compt. rend. 136, 604 (1903); 3 pp.
Meyer, J. & Mylius, B. Viscosity of Binary Liquid Mixtures. Z.
Physik. Chem. 96, 349 (1920); 29 pp.

Meyer, L. 245, (1) Uber Transpiration von Dampfen. Part I. Wied.
Ann. 7, 497 (1879); 39 pp.; for Part II Cp. Meyer and Schumann;
for Part III Cp. Steudel; (2) Do. Part IV. Wied. Ann. 16, 394 (1882);
5 pp.

Meyer, L. & Schumann, O. Uber Transpiration von Dampfen. Part

II. Wied. Ann. 13, 1 (1881); 19 pp.

Meyer, O. E. 2, 6, 29, 60, 79, 127, 242, 243, 251, (1) Uber die Reibung
der Flussigkeiten. Crelle’s J. rein. Angew. Math. 59, 229 (1861); 75
pp.; (2) Do., Pogg. Ann. 113, 55 (1861); 32 pp.; (3) Do., Pogg. Ann.
113, 193 (1861); 46 pp.; (4) Do., Pogg. Ann. 113, 383 (1861); 42 pp.;
(5) tlber die innere Reibung der Gase. Part I. Pogg. Ann. 125, 177
(1865); 33 pp.; (6) Do., Pogg. Ann. 125, 401 (1865); 20 pp.; (7)
Do., Pogg. Ann. 125, 564 (1865); 36 pp.; (8) tlber die Reibung der
Gase. Pogg. Ann. 127, 253 (1866); 29 pp.; (9) Do., Pogg. Ann.
127, 353 (1868); 30 pp.; (10) tlber die innere Reibung der Gase. Part

III. Pogg. Ann. 143, 14 (1871); 12 pp.; (11) Do., Part IV. Pogg. Ann.
148, 1 (1873); 44 pp.; (12) Do., Part V. Pogg. Ann. 148, 203 (1873);
33 pp.; (13) Pendelbeobachtungen. Pogg. Ann. 142, 481 (1871);
43 pp.; (14) Theorie der elastische Nachwirkung. Pogg. Ann. 161,
108 (1874); 11 pp.; (15) Hvdraulische Untersuchungen. Pogg.
Ann. Jubelb. 1 (1874); (6) Bemerkung zu der Abhandlung von Dr.
Streintz uber die Dampfung der Torsionsschwingungen von Drahten.
Pogg. Ann. 154, 354 (1875); 7 pp.; (17) Beobachtungen von A. Rosen-
cranz liber den Einfluss der Temperatur auf der innere Reibung von
Flussigkeiten. Wied. Ann. 2 , 387 (1877); 20 pp.; (18) Uber die elastis¬
che Wirkung. Wied. Ann. 4, 249 (1878); 19 pp.; (19) Uber die Bestim-
mung der inneren Reibung nach Coulomb’s Verfahren. Wied. Ann.
32, 642 (1887); 7 pp.; Sigzungsber. Bayr. Acad. 17, 343 (1887); 21
pp.; (20) Ein Verfahren zur Bestimmung der inneren Reibung von
Flussigkeiten. Wied. Ann. 43, 1 (1891J; 14 pp.; (21) De Gasorum
Theoria. Diss. Uratislaviae (1866); 15 pp.; (22) Uber die Bestimmung
der Luftreibung aus Schwingungsbeobachtungen. Carl’s Repert f.
Exper. Physik. 18, 1 (1882); 8 pp.; (23) The Kinetic Theory of Gases.
Longman & Co. (1899); 466 pp.; (24) Uber die pendelnde Bewegung
einer Kugel unter dem Einflusse der inneren Reibung des ungebenden
Mediums. J. f. du reine und ungewandte Math. 73, 1 (1870); 40 pp.;
(25) Uber die Bewegung einer Pendelkugel in der Luft. Do., 336
(1872); 12 pp.

Meyer, O. & Rosencranz, A. 6, 93, 127, 134, Cp. Meyer, Wied. Ann.
2, 387 (1877).

INDEX

399

Meyer, O. & Springmuhl, F. fiber die innere Reibung der Gase. VI.
Pogg. Ann. 148, 526 (1873); 30 pp,

Meyer, P. ‘‘Apparatus for determining the viscosity of liquids. Ger.
Pat. 244,098, June 16 (1911).

Michaelis, G. fiber die Theorie der elastisehen Nachwirkung. Wied.
Ann. 17, ‘726 (1882); 11 pp.; The Viscosity of Protein Sols. Biochem.
Z. 28, 354 (1911).

Michaelis, L. & Mostynski, B. Viscosity of Protein Solutions. Bio¬
chem. Z. 26, 401 (1910); 11 pp.

Michell, A. G. M. 264, 268, The Lubrication of Plane Surfaces. Zs.

f. Math. u. Phys. 62, 123 (1905); 15 pp.

Mie, G. Remarks upon the Work of U. Sorkau upon Turbulence Viscosity.
Physik. Z. 14, 93 (1913); 3 pp.

Mifka, V. The Internal Friction of Colloidal Metal Solutions. Chem.
Ztg. 35, 842 (1912).

Milch, L. The Increase of Plasticity of Crystals with Rise of Temp.

Neu. Jahrb. Min. Geol. Pol. 1, 60 (1909); 12 pp.

Millikan, R. A. 188, 242, 263, (1) A New Modification of the Cloud
Method of Determining the Elementary Electrical Change and the
most Probable Value of that Charge. Phil. Mag. (6) 19, 215 (1910);
20 pp.; (2) Most Probable Value of the Coefficient of Viscosity of the
Air. Ann. Physik. 41, 759-66 (1913).

Miloradov, A. A. & Tolmachev. Viscosity of Asphalt. J. Russ. Phys.

Chem. Soc. Phys. Pt. 44, 505 (1913); 8 pp.

Minnemann, J. Note on Restoration of Plasticity to Pottery Scrap Clay.

Trans. Am. Ceram. Soc. 16, 96 (1914).
v. Mises, R. Elemente der Technical Hydrodynamik. Phys. Z. 12,
812 (1911).

Moles, E., Marquina, M. & Santos, G. Viscosidad y conductibilidad
el6ctrica en soluciones concentradas de FeCls. Anales soc. espafi.
fis. y. quim. 11, Pt. I, 192 (1913).

Molin, E. (1) Calculation of Degree of Viscosity of Mineral Oil Mixtures.
Chem. Ztg. 38, 857 (1914); 2 pp.; (2) Examination of Searle’s Method
for Determining the Viscosity of very Viscous Liquids. Proc. Cambridge
Soc. I, 20 , 23 (1920); 12 pp.

Monroe. 105, Cp. Kendall and Monroe.

Monstrov, S. (1) Study of Substances Having Large Coefficients of
Viscosity. VI. Determination of some Mechanical Properties of
Asphalt. J. Russ. Phys. Chem. Soc. Phys. Pt. 44, 492 (1913); 10 pp.;
(2) VII. Supplement to the Article by S. I. Monstrov. Do., 44, 503,
B. P. Veinberg (1913); 1 p.; (3) VIII. Viscosity of Asphalt. Do., 44,
505 (1913); 8 pp.

Montemartini, C. The Relations between the Water of Crystallization
of certain Salts and the Viscosity of their Solutions. Atti. A. Acc.
delle Scienze Torino 28, 378 (1892-3); 6 pp
Monti, V. Atti. R. Acad. Sci. Torino 28, 476 (1893).

Moore, B. On the Viscosity of Certain Salt Solutions. Phyic. Rev. 3,
321 (1896); 14 pp.

400

INDEX

Moore, H. Valuation of Motor Fuels. Automobile Eng. 245 (1918);

3 pp.; J. Soc. Chem. Ind. 37, 681A (1918).

Morgan, J. D. Lubricants and Lubrication. Power 43, 317 (1916); 1 p.
Morin. 18, Hydraulique, 45.

Moritz, A. 2, 6, Einige Bemerkungen uber Coulomb’s Verfahrendie
Cohasion der Fliissigkeiten zu bestimmen. Pogg. Ann. 70, 74 (1847).
Morrell, R. S. Varnishes, Paints, and Pigments. Dept. Sci. Ind.
Research, Brit. Assoc. Adv. Sci., Third Report on Colloid Chem.
(1920); 12 pp.

Morris-Airey, H. 218, On the Rigidity of Gelatine. Mem. & Proc.
Manchester 49, #4 (1905).

Morton, W. B. The Displacements of Particles and Their Paths in some
Cases of Two Dimensional Motion of a Frictional Liquid. Proc. Roy.
Soc. London (A) 89, 106 (1913); 19 pp.

Moruzzi, G. The Effect of Area on the Viscosity and Conductivity of
Protein Solutions. Biochem. Z. 28, 97 (1911); 9 pp.; Do., 22, 232
(1909).

Moseley, H. (1) On the Uniform Flow of a Liquid. Phil. Mag. (4) 41,
394 (1871); 3 pp.; (2) On the Steady Flow of a Liquid. Phil. Mag.
(4) 42, 184 (1871); 14 pp.; (3) Do., Phil. Mag. (4) 42, 349 (1871);
13 pp.; (4) Do., Phil. Mag. (4) 44, 30 (1872); 2? pp.

Muchin, G. Fluidity Measurements of Solutions. Z. Elektrochem. 19
819 (1914); 2 pp.

Mueller. Romberg Deutsch Med. Woch. 48 (1904).

Munzer, E. & Bloch, F. (1) Die Bestimmung der Viskositat des Blutes
mittels der Apparate von Determann und Hess nebst Beschreibung
eines eigenen Viskosimeters. Z. exp. Path. Ther. 11, 294 (1913);
Med. Klinik, #9, #10, #11 (1909); (2) Experimentelle Beitrage zur
Kritik der Viskositatsbestimmungsmethoden. Z. exp. Path. Ther. 7,
(1909).

Mussel, A. G., Thole, F. B. and Dunstan, A. E. The Viscosity of
Compounds Containing Tervalent Nitrogen. Proc. Chem. Soc. 28,
70 (1912); J. Chem. Soc. 101, 1008 (1912); 8 pp. Cp. Dunstan.
Mugge, O. 239, Plasticity of Ice. Nach. G. Wiss. Gottingen. 173 (1895);
3 pp.

Muhlenbein, J. tjber die innere Reibung von Nichtelectrolyten. Diss.

Teipzig (1901); p. Schettler’s Erben (1901). Cp. Wagner.

Muller, A. Tiber Suspensionen in Medien von hoherer innerer Reibung.
Ber. 37, 11 (1904).

Muller. Studien zur Viscositat des Blutes bei chirurgischen Erkrank-
ungen. Berlin, klin. Wochschr. 2276 (1909).

. Mutzel, K. 3, 6, 179, Tiber innere Reibung von Fliissigkeiten. Wied.
Ann. 43, 15 (1891); 28 pp.

Napiersky. Versuche iiber die Elasticitat der Metalle. Pogg. Ann.
Ergsbd. 3, 351 (1853).

Natanson, L. (1) Tiber die Gesetze der inneren Reibung. Z. physik.

INDEX

401

Chem. 38, 690 (1901); 15 pp.; Cp. Phil. Mag. (6) 2, 342 (1901); (2)
Uber die temporare Doppelbrechung des Lichtes in bewegten reibenden
Fliissigkeiten. Z. physik. Chem. 39, 355 (1902); 9 pp.; (3) Ober die
Fortpflanzung einer kleinen Bewegung in einer Flussigkeit mit innerer
Reibung. Z. physik. Chem. 40, 581 (1902); 16 pp.; (4) tJber die Dissi-
pationsfunction einer zahen Flussigkeit. Z. physik. Chem. 43, 179
(1903); 6 pp.; (5) Uber die Deformation einer plastisch-viskosen
Scheibe. Z. physik. Chem. 43, 185 (1903); 18 pp.; (6) tlber einige
von Herrn B. Weinstein zu meiner Theorie der inneren Reibung
gemachte Bemerkungen. Physik. Z. 4, 541 (1903); 2 pp.; Cp. Bull.
Int. Acad. Scienc. de Cracovie 95, 161 (1901); Do. 19, 488, 494 (1902);
Do., 268, 283 (1903); Also Krakauer Anz., 95 (1901); Do., 488 (1902);
Do., 268, 283 (1903).

Navier. 1 , 29, (1) Mdmoire sur les lois du mouvement des fluides. M6m.
de FAcad. roy. des Sciences de V inst. de France 6, 389 (1823); 52 pp.;
(2) M6moire sur lAcoulement des fluides 61astiques dans les vases et
les tuyaux de conduit. M6m. de PAcad. roy. des Sciences de l’lnst.
de France 9, 311 (1830); 68 pp.

Naylor, R. B. Testing Device for Determining the Viscosity of Rubber.
U. S. Pat. 1,327,838, Jan. 13 (1920).

Neesen, F. (1) Beitrag zur Kenntniss der elastischen Nachwirkung bei
Torsion. Pogg. Ann. 153, 498 (1874); 27 pp.; (2) Uber elastische
Nachwirkung. Pogg. Ann. 167, 579 (1876); 17 pp.; (3) Monatsber. d.
Kgl. Preuss. Acad d. Wissens. (1874); Feb.

Nensbrugghe, G. van der. Superficial Viscosity of Films of Solutions of
Saponine. Bull. sci. acad. roy. belg. 29, 368 (1870).

Nettel, R. Eine neue Viscositatsbestimmung fur helle Mineralole.
Chem. Ztg. 29, 385 (1905); 2 pp.

Netjfeld, M. W. Influence of a Magnetic Field on the Velocity of Flow of
Anistropic Liquids from Capillaries. Diss. Danzig. (1913); Physik.
Z. 14, 646 (1912); 4 pp. Cp. Kruger.

Neumann, F. 2, 14, 17, (1) Vorlesungen liber die Theorie der Elasticitat
der Festen Korper und des Lichtathers. Leipzig. Teubner (1885);
374 pp.; (2) Einleitung in die theoretische Physik. Herausgegeben von
C. Pape. Leipsig. Teubner (1883); 291 pp.

Nevitt, H. G. Chart of Viscosities in Different Systems. Chem. Met.
Eng. 22 , 1171 (1920).

Newton, I. 1, The Mathematical Principals of Natural Philosophy
(1729). Of the Motion of Bodies. Vol. 2. Of the Motions of Fluids
and the Resistance Made to Projected Bodies. Section VII.

Newton, J. F. and Williams, F. N. Testing Illuminating Oils. Petro¬
leum Age 6, 81 (1919); 3 pp.

Nicolardot, P. & Baum£, G. A Contribution to the Study of the Viscosity
of Lubricating Oils. Chimie & Industrie 1, 265 (1918); 6 pp.
Nicolardot, P. and Masson, P. J. Dubrisay's Method of Examining
lubricating Oils. Ann. fals. II, 77 (1918); 2 pp.; Analyst 43, 276 (1918);
Nicolls, W. Haemodynamics. J. of Physiology 20, 407 (1896).

26

402

INDEX

Nishida, H. (1) Viscosity of Solutions of Nitrocellulose in Alcoholic
Solutions of Camphor. Le Caoutchouc et le Gutta-Percha 121, 8103
(1914); (2) Viscosity of Nitrocellulose Solutions. Kunststoffe 4, 81,
105 (1914); 4 pp.

Nissen. Inaug. Diss. Bonn (1880).

Noack, K. 127, (1) tlber den Einfluss der Temperatur und Konzentration
auf die Fluiditat von Fltissigkeitgemischen. Wied. Ann. 27, 289
(1886); 12 pp.; (2) tlber die Fluiditat der absoluten und der verdunnter
Essigsauxe. Wied. Ann. 28, 666 (1886); 19 pp.

Nordltjnd, I. The Validity of Stokes’ Law for the Motion of Liquid Drops
in other Liquids. Ark. Mat. Astron. Fysik 9, #13, 18 pp.

Noyer, G. Viscosity of the Acetates of Cellulose. Caoutchouc & Gutta
percha 10, 7009 (1913); 2 pp.

Noyes, A. & Al. Conductance and Ionization of Salts, Acids, and Bases
at High Temperatures. Carnegie Institution (1908).

Noyes, A. A. & Folk, K. G. The Properties of Salt Solutions in Relation
to the Ionic Theory. III. Electrical Conductance. J. Am. Chem.
Soc. 34, 454 (1912); 31 pp.

Noyes, A. & Goodwin, H. The Viscosity of Mercury Vapor. Physic.
Rev. 4,207 (1896); 10 pp.; Z. physik. Chem. 21, 671 (1896); 9 pp.

Nutting, P. G. A New General Law of Deformation. J. Franklin Inst.
191, 679 (1921); 8 pp.; Proc. Am. Soc. Testing Materials (1921).

Oberbeck, A. 6, tlber die Reibung in freien Flussighkeitsoberflachen.
Wied. Ann. 11, 634 (1880); 19 pp.

v. Obermayer, A. 246, (1) tlber die Abhangigkeit desReibungscoeffieienten
der atmospharisehen Luft von der Temperatur. Wien. Sitzungsber.
(2A) 71, 281 (1875); 28 pp.; Carl’s Repert. Exp.-physik. 12, 13 (1876);
26 pp.; (2) tlber die Abhangigkeit des Coefficienten der inneren Reibung
der Gase von der Temperatur. Wien. Sitzungsber. (2A) 73, 433
(1876); 42 pp.; Carl’s Repert. Exp.-physik. 12, 465 (1876); 1 p.; (3)
Ein Beitrag zur Kenntniss der zahflussigen Korper. Wien. Sitz¬
ungsber. (2A) 76, 665 (1977); 14 pp.; (4) Das absolute Maas fur die
Zahigkeit der Flussigkeiten. Carl's Repert. Exp.-physik. 16, 682
(1879); 5 pp.; (5) tlber die innere Reibung der Gase. Carl’s Repert.
Exp.-physik. 13, 130 (1877); 29 pp. Cp. Wien. Anz. 90 (1877).

Od&n, S. 203, Physikalisch-chemische Eigenschaften der Schwefel-
hydrosole. Z. phj^sik. Chem. 80, 709 (1912); 38 pp.

Oehm. Einige Versuche Tiber Gummilosung als Nahrfliissigkeit fur Frosch-
herz. Arch. exp. Path. Pharm. 34, 29 (1904).

Oeholm, L. W. 189, (1) Free Diffusion of Non-Electrolytes—the Hydro-
Diffusion of some Organic Substances. Medd. K. Vetenskapakad.
Nobel. Inst. 2 , No. 23, 52 pp.; (2) Investigation of the Diffusion of
some Organic Substances in Ethyl Alcohol. Do. No. 24, 34 pp.; (3)
The Dependence of the Diffusion on the Viscosity of the Solvent. Do.
No. 26, 21 pp.

Oelschlager, E. The Viscosity of Lubricating Oils. Z. Ver. deut. Ing.
62, 422 (1918); 6 pp.

INDEX

403

Oertel, E. tlber die Viscositat der Milch. Diss. Leipzig. (1908); 47 pp.
Oertel, F. Eine Abiinderung der Poiseuilleschen Methode zur TJnter-
suchung der inneren Reibung in stark verdunnten wasserigen Salz-
losungen. Diss. Breslau (1903); 48 pp.

Oholm, L. W. (1) Innere Reibung von wasserigen Losungen einiger
Nichtelektrolyten liber die Reinigung des hierbei angewandte Wassers.
Oversight. Finska Vetenskaps soc., Forhandlingar 47, 1 (1904); 18 pp.;

(2) The Influence of Non-Electrolytes on the Diffusion of Electrolytes
and on the Electric Conductivity, also a Study of the Viscosities of Solu¬
tions of those Substances. Oversigt. Finska Vetenskaps. soc., For-
handlingar 55, afd. A., #5 (1913); 99 pp. Cp. Akad. afh. Helsing¬
fors (1902).

Offermann. Viscosity Determinations of Small Quantities of Oil in
Engler’s Viscometer. Chem. Rev. Fett.-Hartz-Ind. 18, 272 (1911);
3 pp.

Onfray <fc Baladoine. Viscosity of the Blood and Hemorrhage of the
Eye. Soc. ophth. Paris, Dec. 5 (1911); Klin. Monatsbl. Augenheilk.

13, 242.

Onnes, Kamerling. 131, (1) The Coefficients of Viscosity for Fluids in
Corresponding States. Communications from the Laboratory of
Physics at the Univ. of Leyden #2; (2) Arch. Nerrl. 30, 134 (1897); also
Enzykl. d. mathem. Wisenschaften V. 10, 699.

Onnes, K., Dorsman & Weber, S. The Internal friction of gases at low k
temps. I. Hydrogen, II. Helium. Verslag. K. Akad. Wetenschappen
1375 (1913); 10 pp.

Orr. Proc. Roy. Irish Acad. 27A, §2 & 3 (1907).

Orth, P. Viscosity of Saccharin Solutions. Bull, assoc, chim. sue. dist.
29, 137 (1911); 11 pp.

Ortloff, W. tJber die Reibungskoeffieienten der drei Gasen Aethan
(C 2 Hg), Aethylen (C 2 H 4 ), Acetylen (C 2 H 2 ). Diss. Jena (1895).

Orton, E. (1) Keram. Rundschaw (1901); (2) The plasticity of clay.
Brick 14,216 (1901); 4 pp.

Oseen, C. (1) Zur Theorie der Bewegung einer reibenden Fliissigkeit.
Arkiv. for Mat. Astron. och Fys. 3, 84 (1907); (2) Do., Arkiv. Mat.
Astron. Fysik. 4, 1 (1908); #9, 23 pp.

Osmond, F. Sprodigkeit & Plastizitat. Paris (1893); 8 pp.

Ost, H. The Viscosity of Cellulose Solutions. Z. angew. Chem. 24, 1892
(1911); 4 pp.; also J. Soc. Chem. Ind. 30, 1247 (1911).

Ostwald, Wm. 3, 7, 76, 76, Physikalisch-chemische Messungen.

Ostwald, Wm. & Stebtjtt. 94, Lehrb. der. allgem. Chem. 2 auf. 22 pp.;
684 etc. (1897).

Ostwald, Wo. 206, (1) Zoolog. Jahrbucher f. Systemat. Geogr. u. Biol.
18 (1903); (2) Uber das Zeitgesetz des Capillarenaufsteige von Flussig-
keiten und die Beziehung derselben zur chemischen Konstitution
usw. Z. Chem. Ind. Kolloide 2 , Suppl.-heft 2 , 20 (1908); 19 pp.;

(3) Importance of Viscosity for Study of Colloidal State. Trans.
Faraday Soc. 9, 34 (1913); 12 pp. Z. f. Chem. u. Ind der Koll.
12, 213 (1913); 9 pp.; (4) Zur Theorie der kritischen Triibungen.

404

INDEX

Ann. Phys. 36, 848 (1911); 7 pp.; (5) Pfluger’s Arch. 108, 563 (1905);
(6) Do., 109, 277 (1905); (7) Do., Ill, 581 (1906); (8) Grundriss der
Kolloidchemie. Theodor Steinkopff. Dresden. 179 (1911); 44 pp.;
(9) Zur Systematik der Kolloide. Kolloid.-Z. 1,333 (1907); 11 pp.; (10)
Cp. Liiers.

Ostwald, Wo. and Mundler, K. The Osmosis and Swelling of Disperse
Systems. Kolloid-Z. 24, 7 (1919); 20 pp.

Ostwald, Wo. tj. Genthe, A. Viscosity of Gases Dissolved in Liquids.
Zoolog. Jahrb. Abt. f. Biol. 18, 12 (1903).

Ott, I. tlber die Bildung von Serumalbumin in Nagen tiber die Fahig-
keit der Milch das Froschherzleistungs fahig zu halten. Arch. (Anat.),
Physiol. 1, (1883).

Pacher, G. Anomalia delkattrito interno dell’acqua in prossimith ai
4 gradi. Atti. R. Inst. Veneto Scienze 68 , 785 (1898); 30 pp.; Cimento
(4) 10, 435 (1899).

Packer, G. & Finazzi, L. 36, (1) Sulk attrito interno dei liquidi isolanti in un
campo elettrico constante. Atti. R. 1st. Veneto di Scienze, II, 69,
389 (1900); 14 pp.; (2) Sulk attrito interno dell’acqua distillata intorno
alia temperatura del massimo di density. Atti. R. 1st. Veneto di
Scienze 68 , II, 785 (1899). (3) Anomalia delk attrito interno delle
soluzioni acquose in vicinanza alia temperatura del loro massimo di
densith. Atti. R. 1st Veneto di Scienze II. 59, 1053 (1901); 15 pp.

Pagliani, S. (1) Ingegnere Civile e le arti industr. 13, 16 (1887); (2)
Suppl. annuale alia Enciclopedia di Chimica 5, (1888-9).

Pagliani, S. & Batelli, A. SuirAttrito Interno Nei Liquidi. Atti.
d. R. Acc. di Torino 20, 607, 845 (1885); 28 pp.

Pagliani, S. & Oddone, E. Sulk Attrito Interno Nei Liquidi. Atti.
d. B. Acc. di Torino 22, 314 (1887); 9 pp.

Painlev£, P. Legons sur le frottement. Ill pp.

Parish, W. High Viscosity Oklahoma Oil Versus Low Viscosity of Pennsyl¬
vania Oil. Nat. Petroleum News 6, #9, 36 (1913); 7 pp.

Parnell, J. and Higgins. National Physical Lab. Collected Researches
13 (1916).

Parturier, M. and Dons-Kaufmann, M. (1) Influence of Digitalis on
the Viscosity of the Blood in Cardiac Asystole. Compt. rend. soc.
biol. 80, 407 (1917); 4 pp.; (2) Influence of Potassium Iodide on the
Viscosity of the Blood. Compt. rend. soc. biol. 80, 456 (1917); 3 pp.

Patterson, W. A Constant Pressure Viscometer. Proc. Chem. Soc.
29, 172 (1913).

Paxjli, W. 188,213, (1; Viskositat und Elektrochemie der Eiweisslosungen.
Z. Chem. Ind. Kolloid 12, 222 (1913); 8 pp.; Cp. Trans. Faraday Soc.
9, 54 (1913); 12 pp.

Pauli, W. & Wagner, B. The Viscosity of Protein Sols. Biochem. Z.
27, 296 (1910); 8 pp.

Pauschmann. Diss. Erlangen (1910).

Peddie, W. On the Torsional Oscillations of Wires. Phil. Mag. (5)
38, 36 (1894); 19 pp.

INDEX

405

Pederson, F. The Influence of Molecular Structure upon the Internal
Friction of Certain Isomeric Ether Gases. Phys. Rev. 25, 225 (1907).
Pellet, M. Relation between the Fluidity (Barbey) and the Viscosity
(Engler) of Lubricating Oils. Bull. Assoc. Chem. Suer. dist. 29, 622
(1913;; 2 pp.

Perrin. 190.

Perrott, G. St. J. and Thiessen, R. 218, 282, Carbon Black—Its
Properties and Uses. J. Ind. Eng. Chem. 12, 324 (1920); 8 pp.

Perry, J., Graham, J. & Heath, C. 6, 131, Liquid Friction. Phil. Mag.

(5) 35, 441 (1893;; 18 pp.

Petroff, N. 263, 268, (1) Neue Theorie der Reibung. J. des Ing&nieurs
Nos. 1, 2, 3 (1883); Imp. Russ. Acad, of Sciences, St. Petersburg;
(2) Frottement dans les machines. Mdmoires de TAcaddmie de St.
Petersburg (8) 10 , No. 4 (1890); (3) Experimentale Untersuchungen
fiber die Reibung der Fliissigkeiten. Petersburg (1886); (4) Neue
Theorie der Reibung. Trans, by Wurzel. Voss. Leipzig (1887);
pp.; (5) Uber ein physikalisches Verfahren zur Bestimmung der Eigen-
schaften eines Schmiermittels. Baumaterialienkunde, 269 (1899);

(6) Prosddd de determination des qualites d'un liquide lubrifiant.
Congr&s international des methodes d'essai des matdriaux de con¬
struction 2, 1 (1901); 6 pp.; Paris. Vve. Ch. Dunod.; (7) Reibung der
Fliissigkeiten und Machinen (russ). Ber. d. St. Peterburger tech¬
nolog. Instituts, 1885-6; J.. f. Ingenieur (russ.), (1883). Nos. 1, 2, 3, 4;
J. d. russ. Phys.-chem. Ges 16, 14 (1884); Bull. d. kaiserl. Akad. d.
Wissen. zu. St. Petersburg 5, 365 (1896).

Pfaff, F. 239, Versuche liber die Plasticitat des Eises. Pogg. Ann.
165, 169 (1875); 6 pp.; Sitzungsber. der phys. med. Soc. zu Erlangen,
72 (1875).

Philip, A. Reducing the Viscosity of Petroleum Oils. U. S. Pat. 1,286,091.
Phillips, P. 143, 146, 245, Viscosity of Carbon Dioxide. Proc. Roy. Soc.
London 87, 48 (1912); 13 pp.

Picciati, G. Sul moto di un cilindro indefinite in un liquido viscoso.

Line. Rend. (5) 16, (2) 174 (1907); 10 pp.

Piccinini, G. M. Viscometric and Cryoscopic Variations in the Blood after
Administration of Antipyrine, Phenacetin and Antifebrin. Arch,
farm. sper. 12, 193 (1911); 17 pp.

Pick, H. Viscosity of Fluid Crystallin Mixtures of p-Azoxyanisole and
p-Azoxyphenetole. Z. physik. Chem. 77, 577 (1911); 10 pp.
Pickering, S. U. Uber Emulsionen. Roll. Zeitschr. 7, 11 (1910).

Pierce, C. On the Influence of the Internal Friction upon the Correction
of the Length of the Seconds Pendulum for the Flexibility of the
Support. Proc. Am. Acad. 13, 396 (1878); 6 pp.

Piest, C. (1) The Viscosity of Nitrocellulose Solutions. Z. Ges. Schiess-
Sprengstoffw. 6, 409 (1911); 5pp.;Z. angen. Chem.24,968 (1911); 4 pp.
Pisarshewski, L. and Karp, E. The Relation between Diffusion Constant,
the Viscosity and the Electrical Conductivity. Z. physik. Chem. 63,
257 (1908); 12 pp.; Cp. J. Russ. Phys. Chem. Soc. Phys. Pt. 40, 599.

406

INDEX

Pisarshewski, L. & Lempke. 196, ElectrocomductibilitE et frottement
intErieur. J. Russ. Phys. Chem. Soc. 37, 492 (1905); 10 pp.

Pisati, G. 6, 237, (1) Su la dilatazione, la capillarity e la viscosity del
solfo fuso. Atti. R. Acc. Lined 74, 150 (1877); 5 pp.; (2) Sulla elas¬
ticity dei metalli a diverse temperature. Gazz. chim. ital. 6 (1876);
Cp. Do. 7 (1877); (3) Beitrage zur Kenntniss der elastischen Nach-
wirkung. Wien. Sitzungsber. (2A) 80, 427 (1879); 12 pp.

Pivnikievicz, H. A Simple Apparatus for the Absolute Determination of
the Viscosity Coefficient and the Demonstration of Maxwell’s Law.
Physik. Z. 14, 305 (1913); 3 pp.

Plateau, J. Statique experimentale et theorique des liquides soumis aux
seules forces molEculaires. Paris, Ganthier-Villars (1873).

Pleissner, M. Untersuchungen uber die relative innere Reibung von
Speisefetten und fetten Olen. Arch. Pharm. 242, 24 (1904); 7 pp.

Plessi, A. and Vandine, D. Viscometry of the Blood in Various Diseases.
Riv. crit. clin. med. 12, 609 (1911); 5 pp.

Pochettino, A. 235, Su le propriety dei corpi plastici. (Viscosity of
Pitch). Nuovo cimento 8, 77 (1914); 31 pp.

Poincares, L. Recherches sur les Electrolytes fondus. Ann. chim. phys.
(6) 21,289 (315) (1890); 2 pp.

Poiseuille. 1, 6, 8 et seq . 17, 56, 62, 68, 70, 72, 127, 134, 138, 178, 179,

284, 285, (1) Recherches experimentales sur le mouvement des liquides
dans les tubes de trEs-petits diamEtres. .Compt. rend. 15, 1167 (1842);
20 pp.; Cp. Ann. chim. phys. (3) 7, 50 (1843); 24 pp.; Pogg. Ann.
58, 424 (1843); 24 pp.; MEm. prEs. par divers savants y Tacad. Roy.
des. Scienc. de l’inst. de France 9, 433 (1846); 111 pp.; Ann. 64, 129
(1848); (2) Recherches sur les causes du mouvement du sang dans les
vaisseaux capillaires. Ann. chim. u. Pharm. 64, 129; (3) Recherches
expErimentales sur le mouvement des liquides de nature diffErente dans
les tubes de tres-petits diametres. Ann. chim. phys. (3) 21, 76 (1847);
34 pp.

Poisson. 1 , Calcul des pressions dans les fluides en mouvement. Equa¬
tions differentielles de ce mouvement. J. de l’Ecole polytechn. 13,
139 (1831); 36 pp.

Pollock, J. A. Relation between the Thermal Conductivity and the
Viscosity of Gases. J. Proc. N. S. Wales 53,116 (1919); 3 pp.

Pontio, M. The Viscosity of Rubber Solutions. Caoutchouc et Gutta¬
percha 8, 5108 (1911); J. Soc. Chem. Ind. 30, 699 (19—).

Poole, C. P. Cylinder Oil Viscosity. (Use of Aluminium Oleate). Power
43, 885 (1916).

Porst, C. E. G. and Moskowitz, M. Comparison of the Various Corn
Products Starches as Shown by the Bingham-Green Plastometer.
J. Ind. Eng. Chem. 14, 49 (1922); 4 pp.

Porter, A. W. (1) Notes on the Viscosity of Liquids. Phil. Mag. 23,
458 (1912); 4 pp.; (2) Vapor Pressure. Phil. Mag. 23, 4912);8 (15
5 pp.

Post. Chem.-tech. Analyse 1, Heft 2, 318.

INDEX

407

Pound. Physical Properties of Mixtures of Ether and Sulphuric Acid.
J. Chem. Soc. 99, 708 (1911); 15 pp.

Powell, C. (1) The Viscosity of Sugar Solutions. J. Chem. Soc. 106,
1 (1914); 23 pp.; (2) Determination of the Viscosity of Refined Syrups
and Molasses. J. Soc. Chem. Ind. 33, 238 (1914); 2 pp.

Poynting, J. Change of State: Solid-Liquid. Phil. Mag. (5) 12, 32
(1887); 17 pp.

Peandtl, L. (1) Handworterbuch der Naturwissenschaften. Jena (1913)
vide Flussigkeitsbewegung; (2) Neue Untersuchungen uber die stro-
mende Bewegung der Gase und Dampfe. Physik. Z. 8, 23 (1907); 9 pp.
Pr^Denning. Viscositat und magnetische Doppelbrechung des colloid-
alen Eisenoxyhydrates. Diss. Heidelberg (1904); 40 pp. Rep.
British Assoc. Cambridge 476 (1904); 1 p.; Proc. Roy. Soc. (B) 78,
328 (1906); 30 pp.

Peeston, A. C. (1) Plow of Oil in Pipes. Chem. Met. Eng. 23, 607 (1920);

6 pp.; (2) Do., do. 23, 685 (1920); 4 pp.

P&ibeam. Uber die Beziehungen zwischen inneren Reibung und der
chemischen Zussammansetzung fitissiger Substanzen. Lehrbuch der
Chem. Graham Otto, Vol. 1, Pt. Ill, Chap. 3.

Pribram, R. & Handl, A. 2, 6, 62, 63, 75, 106, (1) Uber die spezifische
Zahigkeit der Fliissigkeiten und ihre Beziehung zur chemischen Kon-
stitution. Part I. Wien Sitzungsber. (2A) 78, 113 (1878); 52 pp.;
Carl’s Repert. Exp.-Physik. 16, 465 (1879); (2) Do., Part II. Wien.
Sitzungsber. (2A) 80, 17 (1879); 41 pp.; (3) Do., Part III. Wien.
Sitzungsber. (2A) 84, 717 (1881); 73 pp.; (4) Do., Z. physik. Chem.
9,529 (1892); 11pp.

Prichaed, H. S. The Effects of Straining Structural Steel and Wrought
Iron. Proc. Am. Soc. Civ. Eng. 42, 69 (1916); 45 pp. Discussion in
March and April Nos.

Prinsen. Die Viscositat der Rohrzuckersirupen (der Einfliiss von gelosten
Salzen und Andern. Holland. Arch. Java Suiker 11, 3 (1903); 18 pp.
Prony. 1, Recherches physico-mathdmatiques sur la thdorie des eaux
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Proudman, J. Notes on the Motion of Viscous Liquids in Channels. Phil.
Mag. (6) 28, 30 (1914); 7 pp.

Puccianti, L. (1) Misure di viscosity sopra i cristalli fluidi del Lehmann.

Atti accad. Lincei (5) 16,1, 754 (1907); 3 pp.

Pullen & Finley. Mech. Engineering 24, 493. Inst. Mech. Engineerg,
Gt. Brit. 2, 43 (1909).

Puluj, J. 79, 246, 261, 252, (1) Uber die Reibungsconstante der Luft
als Function der Temperature. Wien. Sitzungsber. (2A) 69, 287
(1874); 35 pp.; (2) Do. Wiener Sitzungsber. (2A) 70, 243 (1875;;
25 pp.; (3) Uber die Abhangigkeit der Reibung der Gase von der
Temperatur. Wien. Sitzungsber. (2A) 73, 589 (1876); 40 pp.; Cp.
Carl's Repert. Exp.-physik. 13, 293 (1877); 15 pp.; (4) Uber die
Reibung der Dampfe. Wien. Sitzungsber (2A) 78, 279 (1878); 33 pp.;
Cp. Carl’s Repert. Exp.-physik. 16, 427 (1879); 31 pp.; (5) Uber die

408

INDEX

innere Reibung in einem Gemische von Kohlensaure und Wasserstoff.
Wien. Sitzungsber. (2A) 79, 97 (1879); 17 pp.; (6) Do., Part II., Wien
Sitzungsber. (2A) 79, 745 (1879); 12 pp.

Purdy, R. C. Discussion of paper by Ashley on Colloid Matter of Clay and
its Measurement. Trans. Am. Ser. Soc. 11, 555 (1909); 40 pp.; Ill.
Geol. Surv. 9 (1908).

Pyhala, C. The Viscosity of Technical Naphthenic Acids. Petroleum
9, 1373 (1914); 1 p.; Do., 217 (1911).

Pytnkhov, S. J. Russ. Phys. Chem. Soc. 41, 665 (1909); 2 pp.

Quartaroli. Ricerche sperimentali sull’attrito interno delle soluzioni
colloidali vicino al punto di massima density. Forli (Medri e C.)
(1901); 14 pp.

Quincke, C. 34, Die Klebrigkeit isolirender Fliissigkeiten im constanten
elektrischen Felde. Wied. Ann. 62, 1 (1897). Cp. Schaufelberger.

Raber, S. Die Konstante der inneren Reibung des Ricinusols, und das
Gesetz ihrer Abhangigkeit von der Temperatur. Cp. Kahlbaum &
Raber. Diss. Halle (1904); 107 pp.

Ragosine. liber das Viscosimeter Engler-Ragosine. Chem. Ztg. 25,
628 (1901).

Rakkuk, F. 237, Die elastische Nachwirkung bei Silber, Glas, Kupfer,
Gold, Platin, und Zink, inbesondere deren Abhangigkeit von der
Temperatur. Wied. Ann. 35, 476 (1888); 20 pp.

Rakusin. Untersuchung der Erdole und seiner Produkte. Braunschweig
(1906); 271 pp.

Ramsay and Shields. 121, 123.

Ranken, C. & Taylor, W. 179, 186, Viscosity of Solutions. Trans. Roy.
Soc. Edinburgh 45, 397 (1906); 9 pp. Cp. Taylor & Ranken.

Rankine, A. O. 7, 250, 251, 253, (1) Method of Determining the Viscosity
of Gases, Especially Those Available only in Small Quantities. Proc.
Roy. Soc. London A 83, 265 (1910); 11 pp.; (2) Viscosity of the Gases of
the Argon Group. Physik. Z. 11, 497 (1910); 5 pp.; (3) The Variation
with Temperature of the Viscosity of the Gases of the Argon group.
Proc. Roy. Soc. London (A) 84, 181 (1910); 12 p.; Physik. Z. 11, 745
(1910); 8 pp.; (4) Relation between Viscosity and Atomic Weight
for the Inert Gases with its Application to the Case of Radium Eman¬
ation. Phil. Mag. (6) 2, 145 (1911); 9 pp.; (5) Viscosities of Gaseous
Chlorine and Bromine. Proc. Roy. Soc. London (A) 86 , 162 (1912);
6 pp.; (6) Method of Measuring the Viscosity of the Vapors of Volatile
Liquids with the Application to Bromine. Proc. Roy. Soc. London (A)
88, 575 (1913); 13 pp.; (7) Phil. Mag. 40, 516 (1920); (8) On the prox¬
imity of atoms in gaseous molecules. Proc. Roy. Soc. A 98, 360 (1921);

10 pp.

Rankine, W. J. M. (1) Thermodynamic Acceleration and Retardation
of Streams. Phil. Mag. 40, 288 (1870); (2) Mathematical Theory of
Combined Streams. Proc. Roy. Soc. London 19, 90 (1871).

INDEX

409

Rapp, I. M. 242, The Plow of Air through Capillary Tubes. Phys. Rev.
2, 363 (1914); 19 pp.

Rappenecker, K. Viscosity Coefficient of Vapors and their Variation
with the Temperature. Diss. Freiburg i B. (1909); Z. physik. Chem.
72, 695 (1910); 27 pp.

Rassow, B. & Dorhle, O. Kolloid.-Z. 12, Feb. (1913).

.Rayleigh, Lord. 264, (1) On the Motion of Solid Bodies through a Vis¬
cous Liquid. Phil. Mag. (6) 21, 697 (1911); 15 pp.; (2) Notes on the
Theory of Lubrication. Phil. Mag. (6) 35, 1 (1918); 12 pp.; (3) On the
Lubricating and other Properties of Thin Oily Films. Phil. Mag.
(6) 35, 157 (1918); 6 pp.; (4) On Flow of Viscous Liquids, especially
in Two Dimensions. Sci. papers IV p. 78; (5) Viscosity of Argon and
Helium. Sci. papers, IV., p. 222; (6) Viscosity of Hydrogen as Affected
by Moisture. Sci. papers, III, p. 375; Proc. Roy. Soc. 62, 112 (1897);
5 pp.; (7) On the Viscosity of Argon as affected by the Temperature.
Sci. papers, IV., 452; Proc. Roy. Soc. 66, 68 (1900); 7 pp.; (8) On the
Viscosity of Gases as affected by Temperature. Do., p. 484; Proc.
Roy. Soc. 67, 137 (1900); 3 pp.; (9) Foam. Discussion of Superficial
Viscosity. Do., p. 351; (10) On the Superficial Viscosity of Water.
Do., p. 363; (11) Friction and Heat Conduction. Theory of Sound 2,
312 Chap. 19, and Appendix A; (12) Further Remarks on the Stability
of Viscous Fluid Motion. Phil. Mag. (6) 28, 609 (1914); 11 pp.;
(13) On the Stability of the Flow of Fluids. Phil. Mag. (5) 34, 59
(1892); (14) Scientific Papers. 3, 351, 363; 4, 78, 222, 336, 375, 452
and 481.

Redwood, B. 324, (1) On Viscometry. J. Soc. Chem. Ind. 5, 121 (1886);
11 pp.; (2) Action of Oils on Metals. J. Soc. Chem. Ind. 362 (1886);
2 pp.; (3) Petroleum. London, Griffin & Co. 2 (1896); 873 pp.; (4)
Petroleum and Its Products (1906).

R^g^czy-Nagy, E. Stromung von Fliissigkeiten in Capillarrohren.

Math. Naturw. Ber. aus Ungarn. 1, 232 (1882); 1 p.

Reiger, R. 58, 239, (1) Inncre Reibung plastischer und fester Korper.
Physik. Z. 2, 213 (1901). Diss. Erlangen (1901); 5 pp.; (2) Uber die
Guitigkeit des Poiseuilleschen Gesetz bei zahfltissigen und festen
Korpern. Ann. Physik. (4) 19, 985 (1906); 22 pp.; (3) t)ber die station-
are Stromung einer Substanz mit innerer Reibung und den Einfluss der
Elastizitat der Wand. Erlanger Ber. 38, 203 (1906); 15 pp. Cp. Glaser;
(4) Propagation of Shearing Deformations in Liquids. Ann. Physik.
(4) 31, 51 (1909); 41 pp.; Erlanger Ber. 40, 160 (1908); 15 pp.
Reinganum, M. (1) Uber die Theorie der Zustandsgleichung und der
inneren Reibung der Gase. Physik. Z. 2, 241 (1901); 5 pp.; (2) Varia¬
tion with Temperature of the Viscosity of Gases of the Argon Group.
Physik. Z. 12, 779 (1911).

Rellstab, L. 2, 6, 106, Uber die Transpiration homologer Fliissigkeiten.
Inaug. Diss. Bonn (1868).

Repin, C. Experiences du lavage mdcanique du sang. Compt. rend. 141,
221 (1905;.

410

INDEX

Reyher, R. 3, tlber die innere Reibung von Losungen. Z. physik. Chem.
2, 744 (1888); 14 pp.

Reynolds, F. (1) The Viscosity Coefficient of Air and an Inquiry into the
Effect of Roentgen Rays thereon. Physic. Rev. 18, 419 (1904); 22 pp.;
(2) Do., Physic. Rev. 19, 37 (1904); 10 pp.

Reynolds, O. 18, 21, 29, et seq. 130, 264, et seq. (1) An Investigation of the
Circumstances which determine whether the Motion of Water shall be
Direct or Sinuous, and of the Law of Resistance in Parallel Channels.
Phil. Trans. London 174, 935 (1883); 48 pp.; Cp. Roy. Inst, of Gt.
Brit. 9, 44 (1884); (2) On the Theory of Lubrication and its Application
to Mr. B. Tower’s Experiments, including an Experimental Deter¬
mination of the Viscosity of Olive Oil. Phil. Trans. London 177 A, 157
(1886); 78 pp.; (3) On the Dynamical Theory of Incompressible Viscous
Fluids and the Determination of the Criterion. Phil. Trans. London
186 A, 123 (1895); 42 pp.; Cp. Brit. Assoc. Rep. Martical (1884); Phil.
Trans. 160 (1886); 83 pp.; Collected Papers 2 , 228 (1886).

Ribatjcour, A. Hydrodynamic Phenomena in Mixed Turbid and Clean
Water. Compt. rend. 252 (1885).

Richards, T. W. and Palitzsch, S. Compressibility of Aqueous Solutions,
especially of Urethan, and the Polymerization of Water. J. Am.
Chem. Soc. 41, 59 (1919); 10 pp.

Richardson, Clifford. 204, 205, (1) The Modern Asphalt Pavement;
(2) The Colloidal State of Matter in Its Relation to the Asphalt
Industry. Dept. Sci. Ind. Research. Brit. Assoc. Adv. Sci., Third
Report on Colloid Chem. (1920); 5 pp.

Riddell, M. The Fluidity of Molten Cast Iron. Foundry 46, 408 (1918);
3 pp.; Foundry Trade J. 20, 364 (1918); 3 pp.; J. Am. Soc. Mech. Eng.
40, 860 (1918); 1 p.

Rieke, R. Plasticity of Clay. Ceramique 16, 87 (191-).

Rigg, G. and Carpenter, J. L. Stormer Viscometer and the Value of
Viscosity Determinations by its Use. J. Ind. Eng. Chem. 4, 901
(1913); 2 pp.

Ringer, W. E. Protein Acid Combinations and Viscosity. Van Bem-
melen-Festschrift, 243 (1911); 18 pp.

Rivett, A. C. D. & Sidgwick, N. V. The Rate of Hydration of Acetic
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Roberts, H. T. Method of Investigating the Transpiration of Gases
through Tubes. Phil. Mag. 23, 250 (1912); 5 pp.

Roberts-Atjsten, W. C. Properties Common to Fluid and Solid Metals.

Proc. Royal Institution 11 , 395 (1887).

Robertson, T. B. Notiz liber emige Faktoren, welche die Bestandteile
von Ol-Wasser Emulsionen betimmen. Kolloid-Z. 7, 7 (1910); 3 pp.
Rontgen, W. 138, (1) tiber einen Vorlesungsapparat zur Demonstration
des Poiseuille’schen Gesetz. Wied. Ann. 20, 268 (1883); 4 pp.; (2)
tiber den Einfluss des Druckes auf die Viscositat der Fliissigkeiten,
speciell des Wassers. Wied. Ann. 22 , 510 (1884); 9 pp.; (3) Kurze
Mitteilung von Versuchen liber den Einfluss des Druckes auf einige

INDEX

411

physikalische Erscheinungen. Ann. Phys. Chem. (3) 46, 98 (1892);
10 pp. (Viscosity and conductivity.)

Rogers, A. & Sabin, A. H. Consistency of Paints by the Stormer Visco¬
meter. J. Ind. Eng. Chem. 3, 737 (1911); 1 p.

Rohland, P. (1) tJber Plasticitat der Thone. Z. anorg. Chem. 31, 158
(1902); 3 pp.; (2) The Means for Altering the Amount of Plasticity of
Clays. Spreck. 39, 1371 (1906); (3) Explanation of Plasticity, Bonding
Power, Shrinkage, and Adsorption Properties of Clay. (Due to col¬
loidal content); (4) Sprechsaal 47, 129 (1914); 1 p.; (4) The Causes of
Plasticity and the Allied Properties of Clays and Kaolin. Silikat
Ztg. 2, 30 (1911); 3 pp.

Roiti, A. La viscosity e F elasticity susseguente nei liquidi. Cimento
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Ronceray, P. 22, 36, Flow in Capillary Tubes. Am. chim. phys. 22,
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Rosencranz, A. Cp. O. E. Meyer.

Rosenow, M. The Plasticity of Clay. Konigl. Tech. Hochsch. Erlangen
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Rossander, G. Om Gasers Utstromning Genom Kapillarror vid Laga
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Rossem, V. On the “Tackiness” of India Rubber. Kolloid-Z. 12, 78 (1913).

van Rossem, A. The Viscosity of Crude Rubber Solutions. Kolloidchem.
Beihefte 10, 83 (1918); 46 pp.

Rossi, G. 284, (1) Sulla viscosity degli idrosole in generale c sulla funzione
di essa negli csseri viventi. Arch, di Fisiol. 3, 507 (1906); 25 p.; (2)
Sulla viscosity di alcuni colloidi inorganici. Arch, di Fisiol. 2, 246
(1905); (3) La viscosity e la resistanza elettrica del siero di sangue a
temperature diverse e prossimo a quella del Torganismo. Arch, di Fisiol.
1, 500 (1904); 5 pp.; (4) La viscosity e l’azione denaturante del calore
in soluzioni di sieroalbumina. Arch, di Fisiol. 2, 272 (1905); (5)
Sulla temperatura e sul tempo di coagulazione delle proteinc del siero
di sangue in rapporto con la viscosity di questo. Arch, di Fisiol. 2, 599
(1905). Cp. Fano.

Rothlin, E. (1) The Technic of the Estimation of the Viscosity of Organic
Colloids. Biochem. Z. 98, 34 (1919); 59 pp.; (2) Critical Studies of
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Rothmund, V. 95, Studien liber die kritische Trubung. Z. physik. Chem.
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Rotinjanz, L. Die Zahigkeitsanderung des flussigen Schwefels. Z. physik.
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Roux, J. Stokes’ Law and the Charge of the Electron. Compt. rend.
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Rubinstein, H. Untersuchungen liber Beziehungen zwischen Schwer-
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Ruches, W. 51, Untersuchung liber den Ausfluss komprimierter Luft aus

412

INDEX

Kapillaren und die dabei auftretenden Turbulenzerscheinungen.

Phys. 25, 983 (1908); 39 p.

Rudorf, G. (1) Zur Kenntniss der Leitfahigkeit und inneren Reibung
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Rudski, P. Note on the Flow of Water in a Straight Pipe. Phil. Mu#

(5) 35, 439 (1893); 2 pp.

Rucker. 69.

Sacc. Sur Fessai des gommes employ6es pour dpaissir les couleurs. Bull'
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Sachanov, A. & Ryachovskii, N. Viscosity of Liquid Mixtures. '/>*
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Sacher, J. Determination of Viscosity of oils. Farben Ztg. 18, 247* r *
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Sachs, J. tJber den Einfluss der Dichtigkeit auf die Viscositat tropfbarrr
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Sahlborn, N. Roll. Chem. Beih. 2, Heft 3 (1910).

Salomon, L. Description of the Barbet “Ixom&tre.” Revue G6n<$rab #
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Samele, E. Viscosity of the Blood in Glucosuria. La. clin. med. itn 1
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Sammet, C. F. Relative Viscosity of Oils at Room Temperature. J..

Ind. Eng. Chem. 10, 632 (1918); 1 p.

Sanders, H. Motion of a Viscous Liquid under a Rotating Plate. Ber.

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Saph & Schroder. Proc. Am. Soc. Civ. Eng. 51, 253 (1903).

Saybolt. 324.

Scarpa, O. 95, 100, (1) Determinazione della viscosity del fenolo alio
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the Viscosity of Liquids and Lubricants. Atti. ist. incorag. Napoli

(6) 7, Gazz. chim. itaL 40, II, 261 (1911); 25 pp.; (3) £ viscosity o hi
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Transformazione invertible della gomma e della gelatina dallo stato di
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Schaefar, C. & Frankenberg. The Influence of Temperature on the
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Schaefer, C. (1) tJber den Einfluss der Temperatur auf die Elasticitiit
der Metalle. Ann. Physik. 5, 220 (1901); 14 pp.; (2) tJber den Einfluss

INDEX

413

der Temperatur auf die Elastizitat der Elemente. Ann. Physik. 9,
665 (1902); 11 pp.; Ann. Physik. 9, 1124 (1902).

Schaeffer <fe Budenberg. Apparatus for the Determination of Viscosity.
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Schall, C. (1) Uber die Reibung von Losungen einiger Ester in unter-
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Schall, C. and van Rijn, W. Uber Reibung von Losungen in Glycerin.

Z. physik. Chem. 23, 329 (1897); 20 pp.

Schaufelberger, W. Bemerkungen zu der Arbeit des Herm Quincke:
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Schaum, K. Uber hylotrop-isomere Korperformen. Ann. 308, 18 (1899);

22 pp.

Scheitlin, W. Vergleichende Untersuchungen tiber die Blutviscosit&t bei
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Schenck, R. 209, (1) Untersuchungen uber die krystallinischen Fliis-
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Schetjer, O. Physikochemische Studien an binaren Gemischen mit einer
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Schidrowitz, P. <fe Goldsbrotjgh, A. H. (1) Die Viskositat von Gummi
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Schierloch. Uber dem Koefficienten der inneren Reibung von reinem
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Schiller, L. Determination of Absolute Viscosity by the Gtimbel Viscom¬
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Schiterg, J. The Importance of Viscosity Measurements for the Knowl¬
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Schlamp, A. Zur Dissociationtheorie der Losungen. Z. physik. chem.

14, 284 (1894); 14 pp. (Viscosity 1 p.).

Schlie. 179, Untersuchungen uber die Bewegung von Fliissigkeiten in
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Schlumberger, I. Sur la notice de M. le Dr. Sacc, au sujet de Tessai des

414

INDEX

gommes employees pour epaissir les couleurs. Bull. Soc. Ind. Mul-
house 26, 114, 2 pp. Cp. Sacc. Lunge, etc.

Schmidt, H. Capillary Rise in Filter Paper. Kolloid-Z. 26, 152 (1920);

8 pp.

Schmidt, M. and Jones, H. Conductivity and Viscosity in Mixed Solvents
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Schmidt, P. tlber die innere Reibung fester Korper. Wied. Ann. 2,
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Schmidt, T. 6, Bestimmung der Reibung von Flussigkeiten nach der
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Schmitt, K. The Viscosity of Certain Gases and Gaseous Mixtures at
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Schneebeli, M. La valeur absolue du coefficient de frottement de hair.

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Schneider. Diss. Rostock.

Schottner, F. 6, 179, (1) tlber die innere Reibung im Glycerin. Wien.
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Schroeder. Afhandlingen ar publicerad i en rysk tidskrift. Mining
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Schroeder, T. Experimentaluntersuchung liber den Einfluss der Tem-
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25 pp.

Schryver, S. B., Ramsden, W., Schidrowitz, et al. Discussion of
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Schott, K. tlber Zahigkeit und Festigkeit in der Oberflache von Flussig¬
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pp.

Schule, W. Air Resistance in the Light of Recent Experiments. Z.

Vereines duetsch. Ing. 54, 12, 54, 94 (1910); 18 pp.

Schulz. Ein schnell Viscosimeter. Chem. Ztg. 32, 891 (1908).
Schukowa-Florensowa, M. Die Viscositat des Blutes der Kinder bes-
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Schultze, H. 79, (1) Die innere Reibung von Argon und ihre Aenderung
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Schumann, O. 246, tlber die Reibungseonstante von Gasen und Dampfen
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Schuyten, M. Viscositatsbestimmungen von wasserig Antipyrinlosungen.
Chem. Ztg. 30, 18 (1906); 1 p.

INDEX

415

Schwartz, EL (1) Viscosity and its Importance for the Chemistry of Cellu¬
loid in Theory and Practice. Z. Chem. Ind. Kolloid. 12, 32 (1913);
10 pp.; J. Soc. Chem. Ind. 32, 191; Moniteur Scientifique, 49 (1915);
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Schwedhelm, H. (1) Calculating the Viscosity of Mixtures of Two Oils of
Different Viscosities from these Viscosities. Chem. Ztg. 44, 638 (1920);
(2) Die Zahigkeit von Olen und anderen Fliissigkeiten als Funktion
der Temperatur. Chem. Ztg. 46, 41 (1921); 1 p.

Schwedoff, T. Recherches expdrimentales sur la cohesion des liquides.
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Schweidler, E. von. 197, t)ber die innere Reibung und Elektrische
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Searle, A. B. Clays and Clay Products. Dept. Sci. Ind. Research, Brit.

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Searle, G. F. C. (1) Simple Viscometer for Very Viscous Liquids. Proc.
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Seaton, M., Probeck, E., and Sawyer, G. Viscosity of Varnishes. J.
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Seelis, K. Measurement of the Brownian Molecular Motion as a function
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Segel, M. 235, Uber eine Methode zur Bestimmung der inneren Iteibung
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Sellario, A. Viscosity of Insulating Liquids in the Electric Field. Nuovo
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Senter, G. Viscosity and Association in Binary Mixtures of Liquids.

Proc. Chem. Soc. 25, 292 (1910); 3 pp.

Shaefer, C. & Frankenberg, G. The Influence of Temperature upon
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Sharples, P. Relation Between the Melting Point and Viscosity of
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Shaw, W. Proc. Camb. Phil. Soc. 7, 21 (1892).

Sheppard, S. E. 329, Measurement of the Absolute Viscosity of Very
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Sheppard, S. E. and Sweet, S. S. The elastic properties of gelatin jellies.

J. Amer. Chem. Soc. 43, 539 (1921); 8 pp.

Shory, U. S. The Effect of the Method of Preparation on the Viscosity
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Shttkowski, N. E. (1) Hydrodynamische Theorie der Reibung gut gefett-
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416

INDEX

(2) Apparat zur Bestimmung der Koefficienten der Viskositat von
Flussigkeiten. Arbeiten d. Phys. Sekt. d. Moskauer Ges. von Freunden
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Shtjlenberger, F. W. (1) Viscometers. Paint, Oil and Chem. Rev.
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Do., 72, #9 (1921); 2 pp.

Sidgwick, N. Pickford, and Wilsdon, B. The Solubility of Aniline in
Aqueous Solutions of its Hydrochloride. J. Chem. Soc. 99, 1131
(1911); 2 pp.

Sidgwick, N. V. and Tizard, H. T. The Color and Ionization of Cupric
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Sidgwick, N. V. & Wilsdon, B. H. Conductivity and Viscosity of Aque¬
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Simeon, F. The Viscosity of Calcium Chloride Solutions. Phil. Mag.
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Simon, D. (1) Richerche sulla coagulazione delle albumine. I. Varia-
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4, 594 (1907); Do., 5,394 (1908); Do., 5, 402 (1908); Do., 5, 470 (1908);
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Simonis, M. 232, 233, 234, (1) Zur Zahigkeitsmessung von Tonbreien.
Sprech. 38, 597 (1905); 6 pp.; (2) Zettlitzer Erde und Alkali. Zur
Theorie des Giessverfahrens. Sprech. 38, 881 (1905); 3 pp.; (3)
Physikalische Definition der Tone und Magerungsmittel. Sprech.
38, 1625 (1905); 2 pp.; (4) Weitere Beitrage zum Verhalten von Tonen
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Singer, L. (1) The Lubrication of Machine Parts. Petroleum 7, 1307
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Skaupy, F. The Specific Heat of Liquid Mercury. The Heat Content of
Liquids, Especially Metals, at the Melting Point, and Its Relation to
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Slatowratsky, N. & Tammann, G. Erweichen Kristalle in der Nahe
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Slotte, K. 6, 128, 130, 142, 178, 179, (1) Tiber die innere Reibung der
Losungen einiger Chromate. Wied. Ann. 14, 13 (1881); (2) Tiber die
innere Reibung einiger Losungen und die Reibungsconstante des
Wassers bei verschiedenen Temperaturen. Wied. Ann. 20 , 257 (1883);
11 pp.; (3) Om den inre friktionen hos vatskor. Oefvers. af Finska
Vetensk. Soc. Forhandl. 32, 116 (1890); 33 pp.; (4) Tiber Reibungs-
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Finska Vetensk. Soc. Forhandl. 37, 11 (1895); Cp. Wied. Beibl. 16,
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(1908).

INDEX

417

Smith, A. and Holmes, W. and Hall, E. tJber den amorphen Schwefel.

Z. physik. Chem. 52, 606 (1905); 24 pp.

Smith, A. and Menzies, A. The Electrical Conductivity and Viscosity of
Concentrated Solutions of Orthophosphoric Acid. J. Am. Chem.
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von Smoluchowski. 96, 188, (1) Viscosity of Colloidal Solutions. Koll-
oid-Z. 18, 190 (1916); (2) tTber die innere Reibung in nicht wasserigen
Losungen. Wien. Sitzungsber (2A) 1136 (1893); 5 pp.; (3) Theory of
the Liquid State from the Standpoint of the Phenomena of Viscosity.
Kosmos 36 (Radziszevski Festband) 543 (1910); 7 pp.

Snyder, C. D. & Todd, M. H. The Viscosity of Body Fluids at Various
Temperatures within Physiological Limits. Am. J. Physiol. 28, 161
(1911); 6 pp.

Snyder, L. C. Theories Concerning the Plasticity of Clays. Mining Sci.
64, 106 (1911); 3 pp.

Sommerfeld, A. 264,268, (1) Zur hydrodynamischen Theorie der Schmier-
mittelreibung. Z. f. Math. u. Phys. 50, 97 (1904); 58 pp.; (2) Die
naturwissenschaftlichen Ergebnisse und die Ziele der modernen tech-
nischen Mechanik. Physik. Z. 4, 779 (1903); 3 pp.

Sondhatjs, C. Uber die Form von aus. runden Oeffnungen tretenden
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Sorkau, W. (1) Turbulence Viscosity of Water. Physik. Z. 14, 759
(1913); 7 pp.; Do., 14, 826 (1913); 5 pp.; (2) Influence of Temperature
Specific Gravity and Chemical Nature on the Turbulence Viscosity.
Physik. Z. 13, 805 (1913); 16 pp.; (3) Relation between Molecular
Weight and Turbulence Velocity Constant. Physik. Z. 14, 147 (1913);
6 pp.; Physik. Z. 16, 582 (1913); 5 pp.; (4) Experimental Investigation
of the Viscosity of Some Organic Liquids in the State of Turbulent
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South combe, J. E. Lubrication. Engineering 109, 184 (1920); 1 p. Cp.
Wells.

Southern Pacific R. R. Co. 258, A Long Rifled-Pipe Line for Pumping
Heavy Oils. Engineering Record 57, 676 (1908).

Spekanskii & Karaveyer. Viscosity of Certain Petroleum Distillates at
Low Temperatures. Nephtanoje Djelo, 14 (1912); Chem. Ztg. Rep.
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Spriggs, E. I. 286, Eine neue Methode zur Bestimmung der Pepsin-
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29 pp.

Spring, W. (1) Uber das Vorkommen gewisser ftir den Fliissigkeits
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Korper unter Druck, Diffusion der festen Material, innere Bewegungen
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Sproxton, F. Cellulose Esters. Dept. Sci. Ind. Research, Brit. Assoc.
Adv. Sci. Third Report on Colloid Chem. (1920); 10 pp.

418

INDEX

Sprung, A. 6, 178, 179, Experimentelle Untersuchungen iiber die Flus-
sigkeitsreibung bei Salzlosungen. Pogg. Ann. 159, 1 (1876); 35 pp*

Stables, W. and Wilson, A. 6, 254, Experiments on the Viscosity of a
Solution of Saponine. Phil. Mag. (5) 15, 406 (1883); 9 pp.

Staley, H. F. The Viscosity of Borate Glasses. Orig. Com. 8th Inter.
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Stange, M. Apparat zur 'Bestimmung der Zahflussigkeit. Chem. Ztg.
30, 643 (1906); 1 p.

St ankewitsch . Warschauer Universitats. Nachrichten (1887) (Russian).

Stanton, T. (1) Some Characteristics of the Flow of Water in Channels
of Varying Cross-Section. Engineering 74, 664 (1902); 2 pp.; (2)
The Law of Comparison for Surface Friction and Eddy-Making Resis¬
tance in Fluids. Nat. Phys. Lab. Coll. Res. 9, 1 (1913); 8 pp.; (3)
The Mechanical Viscosity of Fluids. Proc. Roy. Soc. London (A)
85, 366 (1911); 11 pp.; (4) Determination of the Absolute Viscosities
of Liquids at High Pressures. Engineering 108, 520 (1919); 2 pp.;
Cp. Do., 108, 758 (1919); 3 pp.

Stanton, T. & Parnell, J. Similarity of Motion in Relation to the Surface
Friction of Fluids. Phil. Trans. 214 (A), 199 (1914); 25 pp. Nat.
Phys. Lab. Coll. Researches 11, 295 (1914).

Steele, B., McIntosh, D., and Archibald. 183, Halogen hydrides as
Conducting Solvents. I. Vapor Pressures, Densities, Surface Energies,
and Viscosities of the Pure Solvents. Phil. Trans. London (A) 206,
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Stephan, C. Beitrage zu den Beziehungen zwischen Fluiditat und galvan-
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Stephan, J. 6, 14, (1) tlber die Bewegung flussiger Korper. Wien. Sitz-
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495 (1862); 26 pp.

Steudel, V. tlber Transpiration von Dampfen. III. Wied. Ann. 16,
369 (1882); 26 pp. Cp. Meyer.

Stevens, H. P. (1) Fractional Separation of Rubber by Solution in Benzene
and the Viscosity of the Fractions in Benzene solution. India Rubber
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on the Rate of Cure. Bull. Rubber Growers’ Assoc. 2, 214 (1920); 2 pp.

Stevens, J. Viscosity of Oils. J. Soc. Chem. Ind. 33,109 (1913); 2 pp.

Stewart, B. and Tait, P. (1) On the Heating of a Disk by Rapid Rota¬
tion in Vacuo . Proc. Roy. Soc. London 14, 339 (1865); 5 pp.; (2)
Do., Proc. Roy. Soc. London 16, 290 (1866); 10 pp.

Stewart, G. W. A Noteworthy Interrelation of Illuminating Power,
Density and Viscosity of Certain Kerosene Oils. Phys. Rev. 21, 513
(1912); 11 pp.

Stewart, J. The Plasticity of Clay. Orig. Com. 8th Interm. Congr.
appl. Chem. 16, 265 (1913); 6 pp.; J. Ind. Eng. Chem. 6, 421 (1913);

2 pp.

Stock. Bull, de l’Acad. des Sci. de Craeovie 1 (A), ,18 (1911).

INDEX

419

Stocks, H. B. Colloid Chemistry of Starch, Gums, Hemicelluloses, <?>
Albumin, Casein, Gluten and Gelatine. Dept. Sci. Ind. Research,
Brit. Assoc. Adv. Sci. First Report on Colloid Chem. (1917); 33 pp.
Stoel. 130, Measurements on the Influence of Temperature on the
Viscosity of Fluids between the Boiling-Point and the Critical State.
Diss. Leiden (1891); Physikalische Revue (Graetz) 1, 513 (1892);
Communications from the Laboratory of Physics, Leiden 2, (1891).
Stokes, G. 2, 6, 14, 188, 194, 264, (1) On Some Cases of Fluid Motion.
Phil. Trans. Camb. II 8, 105 (1849); (2) Supplement to a Memoir oh
Some Cases of Fluid Motion. Phil. Trans. Camb. II 8, 287 (1849);

(3) On the Theories of the Internal Friction of Fluids in Motion.
Phil. Trans. Camb. II 8, 409 (1849); (4) On the Effect of the Internal
Friction of Fluids on the Motion of Pendulums. Phil. Trans. Camb.

II 9, 8 (1851); 99 pp.; (5) Do. Phil. Mag. (4) 1, 337 (1851); (6) Note
on the Reduction of Mr. Crookes' Experiments on the Decrement of
the Arc of Vibration of a Mica Plate Oscillating within a Bulb Contain¬
ing a More or Less Rarefied Gas. Phil. Trans. London B 172, 435
(1881); Cp. Proc. Roy. Soc. London 31, 458 (1881); (7) Note on a Paper
by H. Tomlinson, just preceding. Phil. Trans. B 177, 786 (1886);

3 pp.; Cp. Mathematical and Physical Papers, Cambridge Univ. Press
(1883).

Stormer. 328, Viscometer. Drugs, Oils and Paints 27 (1911).

Stover, E. C. Die Fortpflanzung von Bakterien als Ursache der Plas-
tizitat der Tone. Deutsche Topfer-und Ziegler Ztg. No. 14 (1903).
Strasburger, C. F. Apparatus for Testing the Viscosity of Liquids.

U. S. Pat. 989, 822, April 18 (1909).

Strkintz, H. 237, (1) tlber die Dampfung der Torsionsschwingungen von
Drahten. Pogg. Ann. 153, 387 (1874); 25 pp.; (2) Wien. Sitzungsber.
(2A) 69, 337 (1874); (3) Wien. Sitzungsber. (2A) 80, 397 (1880).
Strevens, J. Viscosity of Oils. J. Soc. Chem. Ind. 33, 109 (1913); 2 pp.
Stribeck, R. Mitteilungen liber Forschungsarbeiten. Heft. 7 (1903);

Springer; Z. d. Ver. d. Ingen 46, 1341 (1902).

Strom, L. Separating Mixed Liquids of Different Viscosities. TJ. S. Pat.
968,206, Aug. 23 (1909).

Stull, R. T. Fluxing Power of Mica in Ceramic Bodies. Trans. Am. Cer.
Soc. 4, 255 (1902); 16 pp.

Sutherland, W. 195, 246, 247, 248, (1) The Viscosity of Gases and Mole¬
cular Force. Phil. Mag. (5) 507 (1893); 25 pp.; (2)Ionization in *
Solutions and Two New Types of Viscosity. Phil. Mag. (6) 14, 1
(1907); 35 pp.; (3) Phil. Mag. (6) 9, 781 (1905); Austr. Assoc. 10th
Meet., Dunedin (1904); p. 117 (1905).

Swedberg, T. Velocity of Diffusion and Size of the Particles in Disperse
Systems. Archiv. Kern. Min. Geol. 4, #12, 1 (1914); 6 pp.; J. Chem.
Soc. 102, II, 142 (1913).

Swedberg, T. & Swedberg, A. Diffusionsgeschwindigkeit und relative
Grosse geloster Molectile. Z. physik. Chem. 76, 145 (1913).

420

INDEX

Tait, P. On the Foundations of the Kinetic Theory of Gases. Trans.

Roy. Soc. Edinburgh 33, 259 (1887); 27 pp.

Tammann, G. 6, 188, (1) Velocity of Solidification of Liquids. Z.
physik. Chem. 23, 326 (1897); 3 pp.; (2) Uber die Viscositat unter-
kiihlter Flussigkeiten. Z. physik. Chem. 28, 17 (1898); 15 pp*; Cp.
Werigen, Lewkojeff, and Tammann; (3) Uber die Ausflussgeschwindig-
keit krystallisirter Stoffe. Ann. Physik. 7, 198 (1902); 27 pp.; (4)
Kristallisieren und Schmelzen, p. 158; (5) Sprechsaal (1904); p. 35; (6)
The Velocity of Crystallization. Z. Physik. Chem. 81, 171 (1913);
6 pp.

Tanzler, P. Koefficienten der inneren Reibung fur Gemische zwischen
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222 (1906); 13 pp.

Tappeiner. tlber die Wirkung der Mucilaginosa. Arch, internat. d.
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Taylor, F. W. On the Art of Cutting Metals. Am. Soc. Mech. Eng.,
N. Y., 248 pp.

Taylor, W. Note on the Standard of Relative Viscosity and on Negative
Viscosity. Proc. Roy. Soc. Edinburgh 25, 227 (1904); 3 pp.

Taylor, W. and Moore, T. On the Negative Viscosity of Aqueous
Solutions. Proc. Roy. Soc. Edinburgh 28, 461 (1907); 11 pp.

Taylor, W. and Ranken, C. (1) The Viscosity of Aqueous Solutions of
Chlorides, Bromides, and Iodides. Proc. Roy. Soc. Edinburgh 26,
231 (1904); 10 pp.; Cp. Ranken and Taylor; (2) Note on the Standard
of Relative Viscosity and on “Negative” Viscosity. Proc. Roy. Soc.
London 26, 227 (1904).

Temple, J. W. 297, 298, Thesis Lafayette College (1921).

Thiessen, M. Reibung vom Gasgemisehen. Verh. D. physik. Gesell.
8, 236 (1906); 2 pp. *

Thole, F. B. 112, (1) Viscosity and Association. I. The Association of
the Phenols. Proc. Chem. Soc. 26, 328 (1911); J. Chem. Soc. 97,
2596 (1911); 11 pp.; II. The Viscosity of Geometrical Isomerides.
Proc. Chem. Soc. 28, 51 (1912); J. Chem. Soc. 101, 552 (1912); 6 pp.;
III. Existence of Racemic Compounds in the Liquid State. J. Chem.
Soc. 103, 19 (1913); 8 pp.; Proc. Chem. Soc. 28, 286 (1913); IV. Viscosity
of Aromatic Amines. J. Chem. Soc. 103, 317 (1913); 6 pp.; Proc.
Chem. Soc. 29, 32 (1913); (2) Note of Anomalous Viscosity of Nitro¬
benzene. Proc. Chem. Soc. 25, 198 (1910); (3) The Viscosity of
Isodynamic and Moto-isomers. Z. Physik. Chem. 74, 683 (1910); 3
pp. Cp. Dunstan.

Thole, F. B., Mussel, A. G. & Dunstan, A. E. Viscosity Maxima and
Their Interpretation. J. Chem. Soc. 103, 1108 (1913); 1 p.;Proc.
Chem. Soc. 29, 174 (1913).

Thohlinson, H. Viscosity of Solids, Damping of Vibrations, Due to
Viscosity under Various Circumstances. Phil. Trans. London 177,
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Thomsen, E. The Viscosity of Gas Mixtures. Ann. Physik. 36, 815
(1911); 18 pp.

INDEX

421

Thomson, J. 6, 262, (1) On the Charge of Electricity Carried by Ions
produced by Rontgen Rays. Physic. Rev. 8 , 141 (1899); Phil. Mag.
(5) 46, 528 (1898); 17 pp.; (2) Do. Phil. Mag. (6) 5, 346 (1903); 10

pp.

Thorpe, E. Viscosity of Pure Liquids. Science Progress 12, 583 (1918); *
8 pp.

Thorpe, T. and Rodger, J. 2, 6, 63, 64, 70, 71, 72, 82, 90,107 et seq. 113,
122, 127, 130, 142, 160,166,169,175, (1) On the Relations between the
Viscosity of Liquids and their Chemical Nature. Phil. Trans. London
A 186, 397 (1894); 314 pp.; Cp. Proc. Roy. Soc. London 55, 148 (1894);
and Z. physik. Chem. 14, 361 (1894); 13 pp.; (2) Do., Part II. Phil.
Trans. London A 189, 71 (1897); 36 pp.; Cp. Proc. Roy. Soc. London
60, 152 (1896); Chem. News 76, 152 (1897); (3) On Some Recent
Results of Physico-Chemical Inquiry. Proc. Roy. Inst. Gt. Brit.
16, 641 (1898); 19 pp.; (4) The Viscosity of Mixtures of Miscible
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Thovert, J. 189, Relation entre la diffusion et la viscosity. Compt.
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Threlfall. Motion of Gases in Pipes. Proc. Inst. Mech. Engineers
(1904).

Thurston, R. H. (1) Friction and Lost Work. Wiley & Sons (1898); 6
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Tissot, R. Viscosity of Blood. Folia haematol. 4, #4 (1907).

Todhunter and Pearson. A History of the Elasticity and Strength of
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Tokar, E. Versuche uber den zeitlichen Verlauf der Viscositatsander-
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Tomlinson, H. 242, (1) The Coefficient of Viscosity of Air. Phil. Trans.
London 177, 767 (1886); 19 pp.; Cp..Proc. Roy. Soc. London 40, 40
(1886); (2) Do., Appendix. Proc. Roy. Soc. London 41, 315 (1886);

2 pp.; (3) The Influence of Stress and Strain on the Physical Properties
of Matter. Part I. Elasticity. The Internal Friction of Metals.
Proc. Roy. Soc. London 40, 240 (1886); 2 pp.; (4) Do., Part I. Elas¬
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(5) Do., Part I. The Effect of Change of Temperature on the Internal
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343 (1886); 2 pp.; (6) On Certain Sources of Error in Connection with
Experiments on Torsional Vibrations. Proc. Phys. Soc. 8 , 90 (1887); 5
PP-

Tower, B. 263, 278, Report on Friction Experiments. Proc. Inst. Mech.
Eng., 632 (1883); 27 pp.; Do., 29 (1884); 7pp.; Do., Ill (1891); Tower’s
Verfahren und Apparat zur Prufung von Schmiermitteln. Dingier’s
Polytech. J. 262, 12; 5 pp.; Do., 256, 129.

Traube, J. 58, 121, 123, (1) Ber. 19, 871 (1886); (2) The Viscostagono-

422

INDEX

meter. Methods for Determination of Surface Tension, Viscosity and
Adsorption. Bioehem. Z. 42, 500 (1912); 3 pp.; (3) On the Influence
of Viscosity and Surface Tension on Biological Phenomena. Internat.
Z. physik.-chem. Biol. 1, 260 (1914).

Trautz, M. and Henning, H. Die Winklersche Beziehung zwischen
innere Reibung und Gasabsorption. Z. physik. Chem. 57,251 (1907);
4 pp.

Treitschke, W. (1) Die innere Reibung des geschmolzenen Schwefels.
Z. physik. Chem. 58,433 (1907); 2 pp.; Cp. Beck; (2) tJber die Charakter-
isierung von Schmelzflussen mit Hilfe der Konstanten der inneren
Reibung. Diss. Leipzig (1905); 48 pp.

Tresca, H. 235, (1) M&noire sur Fdcoulement des corps solides. M6m.
Pr6s. k l’Acad. de Flnstitut de France 18, 773 (1868); 67 pp.; (2)
Compt. rend. 68 , 1197 (1869); 4 pp.; (3) M6moire sur F6coulement des
corps solides. M6m. pr6s. k l’Acad. de Hnstitut de France 20 , 75
(1872); 61 pp.; (4) Complement au M4moire sur F6coulement des corps
solides. M4m. pr£s. k FAcad. de Flnstitut de France 20 , 281 (1872);
8 pp.; (5) M4moire sur le poingonnage des mdtaux. M6m. prds. k
FAcad. de Flnstitut de France 20 , 617 (1872); 212 pp.; (6) M5moire
complementaire sur le poingonnage des m6taux. Mem. prds. k l’Acad.
de Flnstitut de France 20 , 829 (1872); (7) On Further Applications of
the Flow of Solids. Proc. Inst. Mech. Eng. 301 (1878); 45 pp.

Trommsdorf, F. Untersuchungen liber die innere Reibung des Blutes und
ihre Beziehung zur Albanese’sche Gummilosung. Arch. f. exp. Path,
u. Pharm. 45, 66 (1901).

Trotjton, F. 7, 226, 227, 228, 237, On the Coefficient of Viscous Traction
and its Relation to that of Viscosity. Proc. Roy. Soc. London (A)
77,426 (1906); 14 pp.

Trotjton, F. and Andrews, E. 7, 218, On the Viscosity of Pitch-like
Substances. Phil. Mag. (6) 7, 347 (1904); 9 pp.

Trtjmpp. Viseositat, Hamoglobin und Eiweissgehalt des kindlichen
Blutes. Munch. Med. Wochschr. #42 (1909).

Tasakalotos, D. 97, (1) Die innere Reibung in kritischen Zone.
Z. physik. Chem. 68 , 32 (1909); 6 pp.; (2) Sur la viscosity des melanges
binaires des composes organiques. Formation de combinaisons
mol6culaires k F6tat liquide. Bull. soc. chim. (4) 3, 234 (1908); (3)
Tiber die Viseositat binarer Gemische organischen Verbindungen.
Bull. soc. chim. (4) 3, 242 (1908); (4) Bull. soc. chim. (4) 6, 234, 397
(1903); 7 pp.

Tschwevsky. (1) Geschwindigkeit und Widerstand in der Strombahn der
Arteria carotis und cruralis ecc. Arch. f. d. Ges. Physiol. 97 , 210
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vaisseaux nourriciers du coeur. Rousski Vratch 441 (1904).

Turpin, S. S. and Warrington, A. W. Apparent Viscosity of Ice. Phil.
Mag. 18, 120 (1884).

Ubbelohde, L. 324, (1) Ehrenrettung des Kranzbrenners des Englerschen

INDEX

423

Schmierolvisko si meters. Chem. Ztg. 31, 28 (1907); 1 p.; (2) Tabelleri
zum Englerschen Yiskosimeter. Verlag. S. Hirzel, Leipzig (1907);
28 pp.; (3) Petroleum 4, Heft 15, 841; (4) EinigeNeuerungen am Engler¬
schen Schmierolviskosimeter und Tabellen fiir das Viskosimeter.
Chem. Zth. 31, 38 (1907); 2 pp.; (5) Theorie der Reibung. S. Hirzel,
Leipzig; (6) Theory of Friction of Lubricated Machinery. Petroleum
7, 938 (1912); Seifensieder Ztg. 39, 1009-10, 1045-6; Stahl und Eisen
32, 1685 (1912); 5 pp.; (7) The Theory of Lubrication. Petroleum
Rev. 27, 293, 325 (1913); 3 pp.; Cp. General Electric Rev. 18, 968
(1915).

Ubbelohde und Agthe. Diplomarbeit, Karlsruhe (1912); Cp. Engler
Erdol, p. 53, 54, 55, 56.

Ubbelohde, L. & Hofsass, M. A New Gas Meter, the “Capomesser;”
and a Viscometer for Gases. J. Gasb. 55, 557 (1912); 4 pp. Z.
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U chiyaura, K. TJber Viskositatsbestimmungen der Milch und der geb-
rauchlichsten Sauglingsnahrungen. Diss. Miinchen (1909); 38 pp.
Ulmer, A. Die Bestimmung des Volumes der Blutkorperchen auf viscosi-
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Ult£e, A. J. (1) Results of Viscosity Determinations of Rubber Solutions.
Arch. Rubbercult. 2, 331 (1918); 16 pp.; (2) Viscosity Determinations
and Uniformity. Do. 3, 24 (1919); 12 pp.

Umani, A. SulPattrito interno del mercurio. Cim. (4) 3, 151 (1896);

16 pp.

Upton, G. B. The Properties of Oils and their Relation to Lubrication.
Sibley J. of Engineering 30, 277 (1915); 7 pp.

Valenta, E. 6, Ein einfacher Apparat zur Bestimmung der Zahfliissigkeit
von Firnissen. Chem. Ztg. 30, 583 (1906); 1 p.

Valenti, A. L’influenza della viscosity sul comportamento delle soluzioni
saline verso il protoplasma vegetale e animale. Arch. di. farmacol.
sperim. e. scienze affini 3, 492 (1904).

Van der Bellen, E. tTber eine neue Methode der Bestimmung der Plasti-
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Vautier, T. (1) Sur la vitesse d'dcoulement des liquides. Compt. rend.
102, 165 (1886); 2 pp.; Cp. Phil. Mag. (5) 21, 285 (1886); 2 pp.; (2)
Do. Compt. rend. 103, 372 (1886); 4 pp.

Veazey, W. 183, The Conductivity and Viscosity of Solutions of Certain
Salts in Water, Methyl Alcohol, Ethyl Alcohol, Acetone, and Binary
Mixtures of these Solvents. Diss. Johns Hopkins (1907); 49 pp. Cp.
Jones.

Veinberg, B. P. (1) Study of Substances Having High Viscosity. J. Russ.
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Do., 44, 201 (1913); 29 pp.; IV. Resistance of a Viscous Medium to the
Motion of a Solid Body. Do. 44, 241 (1913); 11 pp.; V. Further

424

INDEX

Experiments on the Flow of a Viscous Liquid in a Canal. Do., (3)
44, 252 (1913); 5 pp.; (3) Phenomena in Liquids under Homogeneous
Friction. Do., 44, 514 (1913); 2 pp.; (4) Supplement to the Article
of S. I. Monstrov. Do., 44, 503 (1913); 1 p.; (5) Inner Friction of
Binary Systems. Do. 46, 701 (1913); 5 pp.; (6) Some Methods for
Studying the Viscosity of Solids. Proc. Physic. Soc. London 19,
472 (1904); 3 pp.; (7) Tiber die innere Reibung des Eises. Ann.
Physik. (4) 18, 81 (1905).

Veinberg, B. and Smirnov, I. Comparison of some Methods of Deter¬
mining the Viscosity of Pitch. J. Russ. Phys. Chem. Soc. (Phys. Pt.)
44, 3 (1912); 32 pp.

de St. Venant, Barr£. 6, (1) Rapport sur un M5moire de M. Kleintz
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25 pp.; (2) Sur Fhydrodynamique des cours d’eau. Do., 74, 570, 649,
693, 770; (3) Sur PintenshA des forces capables de ddformer, avec con¬
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Vernon, H. M. The conditions of action of “trypsin” on fibrin. J.
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Verschaffelt, J. E. The Viscosity of Liquefied Gases. VI. Observations
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dam 19, 1062 (1917); 12 pp.; VII. Torsional Oscillatory Motion of a
Body of Revolution in a Viscous Liquid. Do., 6 pp.; VIII. The Simi¬
larity in the Oscillatory Rotation of a Body of Revolution in a Viscous
Liquid. Do., 5 pp.; IX. Preliminary Determination of the Viscosity
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Vesely, V. The Viscosity of Glass. Sprechsaal 44, 441 (1911); 7 pp.

Vicat, L. G. A Practical and Scientific Treatise on Calcareous Mortars and
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Villari, von. Sull’efflusso del mercurio per tubi di vetro di piccolo diametro.
Mem dell’Acc. dello Instituto di Bologna (3) 6, (1876).

DE Villemontie, G. 192, Encyclopedic des Aide-Memoire, Section de
Plngenieur. Resistance eiectrique et Fluidite. Gauthier-Villars; 188

pp.

Vogel, H. 242, 247, The Viscosity of certain Gases and the Variation with
Temperature at Low temperatures. Ann. Physik. (4) 43, 1235 (1914);
37 pp.

Vogel, Coleman, and Fisher. TJntersuchungen der Mineral ole und
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Voigt, W. (1) Tiber innere Reibung fester Korper insbesondere der
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(1906); 2 pp.

INDEX

425

Vollmer, B. 194, Die elektrische Leitfahigkeit von einiger Salzen in
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Voris, 0. E. (Sweet milk as a cutting fluid). Machinery 22, 64 (1916).
Vriensniewski. J. Soc. Phys. Chem. St. Pet. 43, 1383 (191~).

Wagner, J. 3,128,179,184,186, (1) Tiber die Zahigkeit von Salzlosungen.
Wied. Ann. 18, 259 (1883); 33 pp.; (2) tJber die innere Reibung ver-
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Wagner, J. and Muhlenbein, J. Tiber die innere Reibung von Losungen.

Z. physik. Chem. 46, 867 (1903); 11 pp.

Walden, P. 179, 194, (1) Relation between Molecular Conductivity and
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lar Conductivity and Viscosity. Z. physik. Chem. 78, 257 (1912);
26 pp.; (3) Trans. Faraday Soc. 6, 75 (19—); (4) OrganischeLosungs-
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Walker, W. J. The Relationship between the Viscosity, Density and
Temperature of Salt Solutions. Phil. Mag. (6) 27, 288 (1914); 9 pp.
Warburg, E. 32, 237, (1) Tiber die Dampfung der Tone fester Korper
durch innere Widerstande. Berl. Monatsber. 538 (1869); 12 pp.;
Pogg. Ann. 139, 89 (1870); 15 pp.; (2) Tiber den Ausfluss des Queck-
silbers aus glasernen Capillarrohren. Pogg. Ann. 140, 367 (1870);
(3) tlber die Gleitung der Gase an Glaswanden. Pogg. Ann. 169,
399 (1876); 17 pp.; (4) Tiber das Gleichgewicht eines Systems ausge-
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Ann. 4, 232 (1878); 17 pp.; (5) Tiber die Torsion. Wied. Ann. 10,
13 (1880); 22 pp.; (6) Magnetische IJntersuchungen. I. Tiber einige
Wirkungen der Coercitivkraft. Wied. Ann. 13, 141 (1881); 24 pp.
Warburg, E. and Babo, L. von. 138, 244, (1) tjber eine Methode zur
Untersuchung der gleitenden Reibung fester Korper. Wied. Ann.
2, 406 (1877); 12 pp.; (2) tlber den Zusammenhang zwischen Viscositat
und Dichtigkeit bei fliissigen inbesonders bei gasformig fliissigen Kor-
pern. Wied. Ann. 17, 390 (1882); 37 pp.; Cp. Ber. Tiber Verhand-
lungen der naturforschenden Gesellschaft zu Freiburg 8, 1 (1862);
44 pp.

Warburg, E. and Sachs, J. 138, 140, Tiber den Einfluss der Dichtigkeit
auf die Viscositat tropfbarer Flussigkelten. Diss. Freiburg; Wied.
Ann. 22, 518 (1884); 5 pp.

Washburn, E. W. 196, 197, (1) The Laws of “Concentrated" Solutions:
II. The Estimation of the Degree of Ionization of Electrolytes in
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(1911); 18 pp.; Cp. Tech. Quarterly 21, 2023 (1908); (2) A Factory
Method for Measuring the Viscosity of Pot-Made Glass during the

426

INDEX

Process of Manufacture, together with Some Discuss on of the Value
of Viscosity Data to the Manufacturer. J. Am. Ceram. Soc. 3, 735
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Washbtjrn, E. W. & MacImes* D. A. The Laws of u Concentrated”

• Solutions. III. The Ionization and Hydration Relations of Elec¬
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(1911); 28 pp.

Washburn, E. W. & Williams, G. Y. (1) Precision Viscometer for
Measurement of Relative Viscosity and the Relative Viscosities of
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33 pp.; (2) The Viscosities and Conductivities of Aqueous Solutions
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Watson, F. 7, 266, Viscosity of Liquids as determined by measurement
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Way, J. T. On the Power of Soils to absorb Manure. Roy. Agric. Soc.
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Webb, J. The Viscous Dynamometer. Science (N. S.) 16, 338 (1902); 2

pp.

Weber, F. (1) Plasma Viscosity of Plant Cells. Z. allegemein Physiol.
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Z. 20, 169 (1917); 4 pp.

Weber, W. 237, (1) Vorlesung de fili bombycini vi elastica. Gotting.
Gelehrt. Anz., St. 8, 65 (1835); 12 pp.; (2) tJber die Elasticitat der
Seidenfaden. Pogg. Ann. 34, 247 (1835); 11 pp.; (3) Pogg. Ann. 34,

1 (1841); Cp. Comm. Soc. Gottingen 3, 45 (1841).

Weinberg. Cp. Veinberg.

Weinstein, M. B. The Internal Friction of Gases. I. The First Coeffi¬
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Second Coefficient of Friction, the Thermodynamic-Hydrodynamic
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(1916); 18 pp.

Weisbach. 18, Lehrbuch der Ingenieur und Machinenmechanik. Experi¬
mental hydraulik, 3d. Ed. 1, 736.

Wells, H. M. and Southcombe, J. E. Theory and Practice of Lubrication:
“Germ” Process. J. Soc. Chem. Ind. 39, 51 (1920); 9 pp.; Cp. South¬
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Welsh, W. N. Viscosity of the Blood. Heart 3, 112 (1912); 19 pp.
Wendriner, M. Ein einfaches Viscosimeter. Z. f. angew Ch. 545 (1894);

2 pp.

Wendt, P. Reply to Ubbelohdes’s Article “The Theory of the Friction
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Werigin, N., Lewkojeff, J., and Tammann, G. 236, Tiber die Ausfluss-
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8 pp. Cp. Tammann.

West, G. D. The Resistance to the Motion of a Thread of Mercury in a
Glass Tube. Proc. Roy. Soc. London (A) 86, 25 (1910); 11 pp.

INDEX 427

Wetzstein, G. 58, Vber Abweichungen von. Poiseuilleschen Gesetz.

Diss. Munchen (1899); Wied. Ann. 68, 441 (1899); 30 pp.
de Whalley, H. C. and Siegfried. A Gravimetric Method of Comparing
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Wheeler. Clay Deposits. Chap. V. Plasticity of Clay. Mo. Geol.
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Whetham, W. 31, 32, 213, (1) On the Alleged Slipping at the Boundary
of a Liquid in Motion. Proc. Boy. Soc. London 48, 225 (1890); (2)
On the Velocity of Ions. Phil. Trans. (A) 186, 507 (1896); 16 pp.
White, G. F. 97, 104, (1) Study of the Viscosity of Fish Oils. J. Ind.
Eng. Chem. 4, 106 (1912); 4 pp.; (2) Fluidity of Fish Oils as an Additive
Property. Do. 4, 267 (1912); 3 pp.; (3) Ein Neues Viscosimeter und
seine Anwendung auf Blut und Blutserum. Biochem. Z. 37, 482
(1911); 7 pp.; Cp. Bingham.

White, G. F. & Thomas, A. Studies on Fish Oils. III. Properties of
Fish and Vegetable Oil Mixtures. J. Ind. and Eng. Chem. 4, 878
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White, G. F. & Twining, R. H. (1) The Fluidity of Butter Fat and its
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of Undercooled Water as Measured in a New Viscosimeter. Am. Chem.
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Wiechert, E. 237, Ober elastische Nachwirkung. Diss. Konigsberg
(1889); 64 pp.

Wiedemann, E. 79, 246, (1) Arch. sci. phys. nat. 66, 273 (1876); (2) Ober die
Beziehung zwischen dem Reibungs und Leitungs-widerstand der
Losungen von Salzen in verschiedenen Losungsmitteln. Wied. Ann.
20, 537 (1883); 2 pp.

Wiedemann, G. & Verdet. 2, 6, 192, Me moire sur le mouvement des
liquides qui s'observe dans le circuit de la pile voltaique et au les rela¬
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Wijkander, 81, 92, Lunds Physiogr. Sallsk. Jubelskr. Lund (1878);
Wied. Bcibl. 3, 8 (1879).

Wilberforce, L. 17, On the Calculation of the Coefficient of Viscosity
of a Liquid from its Rate of Flow through a Capillary Tube. Phil.
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Wilkins. Elektrotechn. Zeitschr. 26, 135 (1904).

Willers, F. Viscosity Anomalies of Emulsions in the Conditions of
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Wilson, H. A. 7, 190, 191, On the Velocity of Solidification of Super¬
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Winkelmann. Handbuch der Physik. 578-582 (1891). Cp. Graetz & Jager.
Winkler, L. Gesetzmassigkeit bei der Absorption der Gase in Fltissig-
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10, 144 (1892); (2) Do., Z. physik. Chem. 56, 344 (1906); 11 pp.

428

INDEX

Wolff, H. (1) The Determination of the Viscosity of Varnishes. Farben.
Ztg. 17,2108; (2) Beitrag zur Kenntniss der Leitfahigkeiten gemischter
Losungen von Elektrolyten. Z. physik. Chem. 40, 222 (1902);
34 pp.

Wottdstra, H. 206, Uber die innere Reibung kolloidaler Silberlosungen
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Kolloide 8, 73 (1911); 8 pp.; (3) The Viscosity and Coagulation of
Caoutchouc Solutions. Z. Chem. Ind. Kolloide 6, 31 (1909); 2 pp.;
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Wright. 106. C. Kendall and Wright.

Wroplewski, S. von. (1) Uber die Abhangigkeit der Constante der
Verbreitung der Gase in einer Flussigkeit von der Zahigkeit der
letztern. Wied. Ann. 7, 11 (1879); 13 pp.; (2) Uber die Natur
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Wtjllner. Lehrbuch der Experimental physik. 4th Ed., 259 (1882).

Yen, Kia-lok. An Absolute Determination of the Coefficients of Viscosity
of Hydrogen, Nitrogen, and Oxygen. Phil. Mag. 38, 582 (1919); 16

pp.

Zahm, A. Atmospheric Friction on Even Surfaces. Phil. Mag. (6) 8,
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Zakrzewsktego, K. O oscylacyi krazka w plynie lepkin. Rozprawy
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Zand a, G. B. 286, (1) Viscosity of the Blood During the Absorption of
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punto di vista fisico-chimico. Giornale della R. Acc. di. Torino 10,
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Zaremba, S. Krakauer Anz. 380, 403 (1903); Rozpr. Akad. (A) 43, 14
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Zempl^n, G. (1) Bestimmung des Koefficienten der inneren Reibung der
Gase nach einer neuen experimentalen Methode. Ann. Physik. (4)
19, 783 (1906); 23 pp.; (2) Do., Ann. Physik. (4) 29, 869 (1909);
39 pp.; Cp. Math, natro. Ber. Ungarn 19, 74 (1904;; 7 pp. and Math.
Termt. Ert. Budapest 23, 561 (1905); (3) Investigations on the Vis¬
cosity of Gases. Ann. physik. 38, 71 (1912;; 54 pp.

Zeri, A. (1) La viscosith della bile umana. Arch, di farmac. speriment. c
scienze affini 4, 279 (1905); (2) Un Nuovo carattere differenziale tra
essudati e trasudati. II Politecnico. Sezione pratica 12 , 1373 (1905).

Zeuner. 18, Civilingenieur 1, 84.

Zimmer, O. (1) The Viscosity of Ethylene and Carbon Monoxide and
its Changes at Low Temperatures. Ber. deut. physik. Ges. 471
(1912).

Zoja, L. Physikalisch-chemische Untersuchung der Reaktionen zwischen
Eiereiweiss und Essigsaure. Koll. Z. 3, 249, 269 (1908); 20 pp.

INDEX

429

Zoller, H. F. (1) The Viscosity of Casein Solution. I. The Effect of Pjy.
Science 50, 49 (1919); (2) Casein Viscosity studies. J. Gen. Physiol.
3, 635 (1921); 16 pp.

Zschokke, B. (1) Untersuchungen fiber die Plastizitat der Thone. Bull.
Soc. d’Encouragement dTndustrie Nationale 103, 619 (1902); 40 pp.;
(2) Untersuchungen uber die Bildsamkeit der Thone. Baummaterial-
ienkunde 7, 377, 393 (1902); 7 pp.; (3) Untersuchungen liber die Plas-
tieitat der Thone. Do., #1, 2, 3, 4, 5, 6 (1903); 18 pp.

Ztjr Nedden, F. Induced Currents of Fluids. Proc. Am. Soc. Civ. Engi¬
neers 41, 1351 (1915); 54 pp.

SUBJECT INDEX

Decimals indicate the location of reference on the page. Two or more
references on the same page are indicated by a + sign.

A

Absorption 235, 259, 427.9, 428.3
Acetic anhydride, 410.7
acid, 402.2
Acetylene, 403.6
Acids, aliphatic, 115
Additivity of fluidities, 82, 83, 104,
412.8

Adhesion, 31, 221, 230, 257, 261,
268, 274

Adsorption, 378.9, 394.2

Adulteration, 370.4

Air, 361.5, 369.2, 373.5+, 374.2,

376.1, 382.5+, 383.1,

386.4, 396.1, 399.4, 402.6,

407.9, 409.1, 410.1, 414.3,

415.2, 421.6

Albumen, 404.6, 416.4, 419.1, 428.9
Alcohols, 116, 177, 349.7
Alloys, 349.4, 361.6, 424.9
Aluminium hydroxide, 372.1, 387.2
oleate in oil, 406.8
Amalgams, 415.4
Amides, 370.1

Amines, 366.5, 368.1, 371.3, 420.7
Ammonia, 371.3
Ammonium iodide, 186

nitrate, 180, 356.2, 374.6
thiocyanate, 374.6
Aniline, 416.2 +

Anisotropic liquids, 96, 209, 356.4,

359.6, 367.7, 390.4, 392.7,

401.6, 405.8, 407.7, 413.5
Annealing, 212

Antifriction metals, 278
Antimony chloride, 391.2
Antipyrine solutions, 414.9
Argon, 304.5, 389.8, 408.7, 409.3+,

413.7, 414.8, 420.2

Asphalt, 365.3, 399.5+, 410.5, 423.9
-base oils vs. paraffin-base
oils, 274

Association, 92, 112, 119, 161, 184,

378.1, 415.7, 420.6
Atomic constants, 108, 111, 126,

144, 186, 196
diameters, 253
weights and v. of gases, 250
Avogadro’s constant, 253

B

Barium sulphate, 349.8
Bath, 307
Beater-stock, 360.1
Belting, 283
Bent capillaries, 376.6
Benzene, 354.6, 366.3, 381.8
Benzyl benzoate, 354.6
Bile, 359.8, 428.8

Binders, sticking strength of, 347.2
Biology, 284

Blood, 284, 348.1, 349.3, 350.5,
352.6+, 353.1, 355.6+,

356.9, 359.3+, 360.2+ ,

361.8, 362.3+, 363.9,

364.1, 365.1+, 367.8,

368.4 + , 369.4+, 370.6+,
372.1+, 374.3, 375.2,

376.9, 379.1, 380.8,

381.6, 382.4, 383.2+,

385.1, 386.2+, 387.1,

389.9, 390.5, 392.2,

393.1+. 394.6+, 397.1,

400.5, 403.4, 404.7, 405.7,

406.3+ , 409.9, 411.6,

412.6, 413.4, 414.8, 421.5,

422.6+, 423.4, 426.3 + ,

427.3, 428.6

431

432

INDEX

“Body,” 269

Body fluids, 417.3

Boiling-point, 155

Brittleness, 403.7

Bromides, 114

Bromine, 386.8, 408.8

Brownian movement, 188, 190,

358.7, 415.5
Bunsen flame, 352.8
Butter, 281, 388.1, 427.4

C

Cadmium and zinc alloys, 424.9
iodide, 386.1
Caesium nitrate, 398.1
Calcium chloride, 416.4
Calculation of fluidity, 314
of plasticity, 323
Caoutchouc, cp. rubber.

Capillarity, 56, 70, 361.5, 370.8,

406.1, 414.1, cp. surface
tension.

Capillary tube method for plas¬
ticity, 222
Carbon black, 405.2

dioxide, 146, 383.2, 405.6, 408.1
monoxide, 428.9
tetrachloride and benzene, 167
Carbonyl sulphide, 359.4
Casein, 361.4, 419.1, 429.1
Castor oil, 363.7, 386.5, 408.4
Celluloid, 348.3, 415.2
Cellulose acetate, 351.1, 402.3

esters, 351.2, 360.3,365.2, 373.7,

403.7, 417.9, cp. nitro¬
cellulose.

Cements, 424.6
Centipoise, 61
Ceramics, 286
Chart for conversion, 401.7
Chemical composition, 106, 112,172,
249, 375.1, 387.6, 390.3,

407.4, 409.9, 422.7
Chloral solutions, 391.1
Chlorides, 381.3
Chlorine, 122, 144, 408.8

Chloroform and ether, 175, 364.4
Chromium salt solutions, 370.7,

374.9, 416.8

Clay, 221, 229, 281, 349.9, 355.3,

376.5, 388.4, 391.7, 397.7,

399.5, 403.6, 408.1, 410.6,

411.9, 415.44, 416.5,

418.9, 429.1

Close-packing, 228, 229
Cloud method, 399.4
Coagulation, 284, 371.9, 387.2,

396.8, 411.6, 428.2
Cohesion, 148, 212, 386.3, 400.1,

415.3

Cold working, 211
Collisions, 149, 200
Collisional viscosity, 147, 151
Colloidal solutions, 3, 198, 348.5,

351.8, 353.7, 360.6, 364.2,

369.1, 371.8, 372.7,

374.14-, 377.2, 378.8,

379.5, 380.7, 381.4,

392.9-f, 393.2-h, 403.9,
411.64”, 412.8, 413.8,

414.6, 417.2, 421.6, 428.1
Colloidoscope, 198
Colophonium, 52

Color of solutions, 416.3, cp,
chromium.

Comparable temperatures, 115,

410.4

Compressible fluids, 49
Conductivity, electrical, 191, 192,
193, 194, 349.1, 349.44,

353.7, 357.2, 360.6, 364.7,
367.2 4 , 368.1, 371.24,

374.6, 375.6, 376.34,

377.4, 379.7, 382.9, 383.4,

386.6, 389.7, 390.1,

392.74 , 394.34-, 396.34,

399.6, 402.3, 403.2, 405.9,

406.1, 412.1, 414.1, 415.4,
416.3, 417.1, 418.5, 423.8,

424.7, 425.14, 427.5

thermal, 252, 358.7, 360.8,

368.7, 380.3, 390.8, 406.7
Conjugate double bonds, 111

INDEX

433

Consistency, 235, 361.8
Constants of viscometer, 296, 313
Constitution, chemical, 121, 352.1,

354.9, 358.8, 359.1, 367.1,
366.4+, 372.8, 382.1,

388.8, 390.7, 405.1, 407.5,
413.4

Construction of viscometer, 315
Corresponding states, 403.4
Cream, 365.9
Criterion of Reynolds, 40
Critical-solution temperature, 94,
102, 364.8, 372.1

Critical state, 365.4, 419.1, 422.7
velocity of flow, 361.9
Crystalline liquids, 96, 208, cp. anis¬
otropic liquids.

Crystallization, 190, 371.8, 372.2,

375.9, 379.3, 420.2
Cutting fluids, 269, 272, 348.5,

354.3, 420.3, 425.1
Curcas oil, 348.9
Curved pipes, 368.9

D

Deflocculation, 208, 229, 231
Demonstration of Maxwell’s Law,
406.2, cp. lecture demon¬
strations.

Density determination, 309

tables, water, 309; mercury, 311
and v., 412.5, 425.5 +

Dextrine, 280, 381.1
Diastase, 347.3
Dielectrics, 350.4, 381.5
Diet, effect on v. of blood, 285
Diffusion, 188, 189, 214, 252, 360.8,

380.8, 387.2, 390.6, 402.8,

405.9, 419.9, 421.4
Diffusional v., 147, 150, 242

Disk method of v. measurement, 86
Displacement of particles, 400.3
Dissipation function, 401.1
Dissociation, 9, 161, 169, 184, 187,
195, 413.9
Double-bond, 118
28

Dough, 380.5

Dynamical theory, 396.6, 410.3
E

Eddies, 14, 39, 42
Effusion, 241

Elastic after-effect, 237, 355.9,

358.2 388.7, 389.5+,

390.9, 392.5, 398.5, 399.2,

401.5, 406.2, 408.5, 414.6,

425.6, 427.5

deformation, 4, 212, 217, 218,
350.2, 353.4, 357.1, 361.7,
365.5, 369.8, 375.4, 387.3,

400.9, 401.7, 412.9, 421.5,
424.9

limit, 211, 237

Electric field, v. in an, 34, 368.6,

404.4, 408.3, 413.3, 415.6
Electronic charge and Stoke’s Law,

411.8, 421.1
Electroosmosis, 371.9
Emulsions, 83, 89, 94, 100, 102, 210,

350.8, 354.8, 356.4, 367.9,
396.1, 405.8, 410.8

End correction, 21, et seq., 315,

353.2

Engine grease, 211
Enzyme reactions, 347.4
Ethane, 403.6

Ethers, 113, 364.4, 374.1, 381.8,
407.1

Ether-alcohol mixtures, 350.7
Ethyl acetate, 366.3

alcohol, 361.2, 366.3, 425.1
water mixtures, 341
Ethylene, 403.6, 428.9
Expansion, thermal, 363.5
External friction, 372.4, 376.2,

388.4, 389.2

• Eye fluids, 360.5, 393.7, 396.5
F

Falling sphere, cp. sphere.

Ferric hydroxide, 407.3

434

INDEX

Filling viscometer, 310, 312

Films as plastic substances, 255

Fineness of grain, 235

First regime, 39

Flashing, 39

Flotation, 361.7

Flour, 394.4 +

Flow through orifices, 233, 234, cp.
hydraulics,
in thin films, 380.2
of metals, 235, 236
theory, 365.2, 403.7, 411.3,

418.6, 419.3, 423.7, 427.7
Fluid defined, 215

Fluidity definition, 5, 364.9
table for reference, 318
in a magnetic field, cp.
magnetism.

in an electrostatic field, cp.
electric field.

Foam, 211, 229, 409.4
Formamide, 368.5, 397.9
Free volume, 142

Friction, cp. yield value, 238, 262,
280, 392.3, 415.9, 428.5
Fused salts, 193, 371.7, 374.5,

386.6, 394.1, 406.4

G

Gases, 241, 242, 351.4, -355.9,

358.4, 360.9, 361.1+,

362.6, 367.3+, 368.1+,

371.1,

374.9,

377.4,

384.9,

385.3,

388.6,

389.1,

392.2,

398.3,

403.5,

404.2,

410.8,

414.3,

419

.8, 420.6+,

422.1,

424

.5+,

426.6,

428.7

Gasoline, 380.9

Gelatine, 198, 212, 280, 289,

349.1,

352.4,

355.7,

375.8,

380.8,

384.6,

392.9,

393.6,

394.3,

400.2,

412.8,

414.5,

415.9,

419.1

Geophysics, 287

Glass, 286, 377.9,

384.7,

392.6,

418.1,

424.6,

425.9

Glue, 280, 355.7, 370.4
Gluten, 280, 419.1
Glycerol, 358.9, 413.3, 414.1 +

Gold value, 393.2
Graphite, 229
Greases, 281, 361.8, 395.3
Gums, 419/

H

Haemodynamics, 401.9
Halogens, 250
Hardness, 3, 235, 391.4
Heat of vaporization and v., 372.9
of fusion, 378.2
Hemoglobin, 356.8
Helium, 364.5, 403.5, 409.3, 413.7,

414.8, 420.2

Heptane as a standard for vapor
pressure comparisons, 157,
et seq.

Hexamethylene, 277
High temperature v., 349.1, 370.1
Homogenizing, 211, 281
Hydrate theory, 354.8
Hydraulic flow and plastic state, 231
Hydraulics, cp. also turbulence,
365.1, 369.9, 371.6, 372.3,

380.6, 394.2, 395.3+,

398.5, . 400.1, 407.7+,

408.9, 412.2, 417.4, 421.4

Hydrocarbons, 113, 351.3, 372.8
Hydrocellulose, 353.2
Hydrodynamics, 1, 212, 351.9,

353.6, 356.6, 358.1, 362.9,

368.8, 369.1, 373.9,

391.9+, 394.1, 395.9,

399.5, 401.3, 401.8, 406.7,

410.3, 415.9, 418.3+,

424.3, 426.6

Hydrogen, 353.5, 376.8, 389.8,

395.9, 403.5, 408.1, 409.3,

424.5, 428.4
bromide, 397.3
chloride, 397.3
iodide, 397.3
sulphide, 397.3

INDEX

435

Hydrogenation, 281
Hydrolysis, 359.4, 362.2
Hysteresis, elastic, 360.3, 368.9,

391.5

I

Ice, 239, 363.7, 371.6, 381.5, 395.4,

397.2, 400.7, 405.6, 422.9,

424.2

Ideal mixtures, 162

Immiscible liquids, 87, 211

Inflection curves, 178

Infusorial earth, 230

Interrupted flow, 28, CO

Iodides, 114

Ionic size, 357.1, 391.6

Ionic mobility, 358.3, 395.1, 381.7,

427.2

Ionization, 195, 350.9, 394.9
Iron and steel, 348.2, 375.4, 388.7,

407.6, 410.5

Iso-grouping, 108, 117,125, 144
Iso thermals of fluidity, 146

K

Kaolinite, 385.4

Kinetic energy, 2, 17, ct &eq., 59,

373.5, 384.9, 385.2,420.1

L

Law of Batsehinski, 142, 247

Poiseuille, 8, et seq., 365.4,

367.9, 374.2, 375.8, 377.2,
386.8, 400.9, 403.1, 406.5,

409.7, 427.1

Stokes, 188, 402.2, 411.8
lard oil as a cutting oil, 270
lava, 287

Lecture demonstrations, 397.8,
410.9

Lime, 281,388.4
Limiting volume, 142
Linear flow, 410.2

Liquid mixtures, 363.4+*, 364.4-f,

366.1, 367.2-b 368.6,

369.7, 370.8, 373.6, 387.1,

387.5, 391.3, 392.7, 393.5,

397.9, 398.2, 402.2, 412.3,

413.6, 415.7, 421.3, 422.7
Liquids, 359.4, 361.5, 374.8, 379.9,

386.2, 398.3, 400.44,

405.2, 406.9, 421.1, 426.3
Lithium chloride, 195, 374.6, 383.4

nitrate, 386.1

Logarithmic decrement, 236
homologucs, 45
viscosities, 104.

Lubricant, air as a, 388.2
Lubricants, 367.9, 384.1, 400.1,

412.8

Lubricating oils, 370.9, 386.8,

387.9, 388.5, 390.5, 391.8,

394.4, 395.6, 401.6, 401.9,

42.9, 406.8 •

•value, 269

Lubrication, 261, 264, et #eq., 347.8,

347.9, 348.2, 4, 5, 6, 9,

353.2, 363.6, 369.2, 377.1,

378.9, 879.1, 394.8, 396.2,

399.2, 405.3, 409.2, 410.2,

416.6, 417.4, 417.6, 419.6,

421.5, 421.9, 423.1, 426.7,

426.8

and adhesion, 268
M

Magnetism, 34, 350.1, 351.4, 360.4,
375.4, 389.3, 394.7, 401.6,

421.8

Manometer, 307
Marble, flow of, 347.5
Marine glue, 235
Mass of hydrogen atom, 253
Mayonnaise, 211
Mean free path, 243
Measurement of high v., 378.3
Medicine, 284

Melting point of tars, 415.8
Menthol, 235

Mercury, 352.1, 358.4, 351.9, 368.3,

436

INDEX

388.4, 389.2, 415.4, 416.7,

423.5, 424.6 425.5, 426.9
stabilizer, 294

vapor, 388.4, 402.4
Metal ammonia salts, 355.1
Metals, 348.1, 377.5, 383.2, 384.4,

385.4, 387.4, 394.3, 398.6,

404.9, 410.8, 416.9, 417.9,

419.6, 421.7, 424.9, 426.8
Metallurgy, 284

Methyl chloride, 130, 171, 371.3
Methylene group, 117, 123
Migration velocity, 185, 191
Milk, 284, 286, 351.8, 359.6, 359.7,

360.5, 372.4, 377.4, 389.2,
389.8, 390.2, 394.6, 395.2,

403.1, 404.2, 423.3
Mixtures, 84, 90, et seq. } 251, 349.7,

354.1, cp. liquid mixtures.
Mobility, 217, 218, 219, 220, 221,

226, 257, 280
of ions, 356.3
Moboil oil BB, 141
Molecular attraction and viscosity of
gases, 246

limiting volume, 142, 144
viscosity work, 107, 109, 110,
111

volume, 392.5
inner and outer, 145
Mortars, 368.2
Motor fuels, 400.1
Multiviscometer, 341.5

N

Naphthenic acids, 408.2
Negative v. and negative curvature,
160, 169, 178, etc., 183,
etc., 373 4, 386.1, 420.4,
etc.

Nickel, 372.7, 375.4
Nitric acid, 361.2
Nitrobenzene, 420.7
Nitrocellulose, 280, 291, 350.7,

373.7, 382.1, 393.2, 396.3,

397.6, 402.1, 405.9, 415.1,
cp. cellulose esters.

Nitrogen, 353.5, 354.7, 395.9, 400.6,
428.4 ,

Nomenclature, 7
Non-electrolytes, 400.7
Normal mixtures, 81

O

Ohm's Law, 83

Oil, films on water, 255

Oiliness, 269

Oils, 360.1, 360.4, 361.2, 362.1,

364.9, 366.6, 366.9, 368.4,

370.2, 372.9, 373.8, 380.5,

381.9, 382.7+, 383.9,

399.6, 401.8, 405.6, 408.5,
412.4, 412.6, 417.7, 418.8,

419.6, 423.5, 424.8
blown, 395.6
essential, 385.2

fish, 427.2, etc.
fixed, 406.3
flow in pipes, 407.3
from Oklahoma vs. Pa., 404.7
mixtures, 415.2
on metals, 409.6
Olive oil, 130, 359.3, 410.2
Opalescence, 94, 411.8
Orifices, 363.1

Ortho phosphoric acid, 417.1
Oscillatory motion, 397.7
Oxycellulose nitrate, 353.2
Oxygen, 124, 144, 353.5, 395.9,
428.4

P

Paint, 222, 282, 354.4, 400.2, 411.1
Pastes, 360.3, 381.1, 396.6
Pendulum method, 2, 6, 350.6,

374.2, 376.1, 398.5
Penetrance, 259, 351.9, 414.1
Pepsin action, 417.8
Periodic relationships, 185, 250
Pharmacy, 284

Phenol, 412.7, 420.6

INDEX

437

Phosphine, 397.3
Pigments, 282

Pitch, 216, 235, 406.4, 422.6, 423.9,

424.2

Plasma, 347.4, 426.4
Plastic flow, 4, 52, 228, 254
measurement, 321

Plasticity, 215, etc.; 349.3, 349.8,

350.1, 354.3, 355.3, etc.;

356.2, 390.3, 358.5, 358.6,

359.5, 362.7, 363.1, 367.7,

368.2, 376.5, 388.2, 388.6,

390.5, 391.3, 392.8, 399.3,

401.2, 403.7, 405.2, 406.4,

409.7, 411.2, 417.3, 418.9,

423.7, 427.1,-429.1
and bacteria, 419.5
calculation, 323

of clay, 387.8
definition, 216
and fusibility, 411.9
of ice, 239, cp. ice.
of salt rocks, 391.6
series of metals, 236
and solubility, 293
of steel and glass, 392.6
Plastics, 368.2, 374.2
Plastometer, 375.5, 406.8
Plate glass flow, 39
Pleural exudate, 390.3
Poise, 61

Polar colloids, 208, 212
Polarization and fluidity, 35
Polydispersed systems, 394.4
Positive curvature and chemical

combination, 172, 183
Potassium bromide solutions, 182
halide solutions, 381.2
iodide, 373.5, 374.6
' nitrate, 374.6
thiocynate, 374.6

Precipitation of colloids, 380.7,

384.5, 384.6

Pressure, 138, et seq., 243, 351.6,

354.2, 361.8, 369.7, 372.5,

379.7, 384.1, 385.5, 391.7,

410.9, 418.3

Pressure, corrections, 299
regulation, 294, 305
true average, 298

Proteins, 356.7, 361.3, 373.1, 393.6,

399.2, 400.3, 404.9, 410.7,

423.6, 426.4

Pseudoglobulin, 361.3
Pyridine, 379.3

Q

Quartz, viscosity of, 377.6
R

Racemic compounds, 112, 366.8
Radius of capillary, 321
Raffinose solutions, 426.2
Rare gases, 250, 378.8
Rate of crystallization, 190, cp.
solidification,
of hydration, 410.7
of reaction, 366.4, 376.3
Reciprocal properties, 83
Refractive index, 393.8
Regimes, 4, 142
Relaxation number, 128
Residual affinity, 112
Resistance, cp. conductivity.
Reynolds critical velocity, 40
Rigidity, 128, 218, 256, 384.4, 398.1
Ring grouping, 124
Road building, 282
Rocks, 352.8, 353.3
Roentgen rays as affecting viscosity,

410.1

Roughness of surfaces, 149
Rubber, 212, 280, 350.3, 352.6,

353.3, 371.5, 372.9, 388.3,
394.5, 410.4, 406.7, 411.5,
etc., 413.6, 418.7, 423.4,

428.2
Rupture, 229

S

Sagging beam method, plasticity,
227

Salt solutions, 347.2, 348.7, 349.4+,

438

INDEX

355.1, 359.1, 363.2, 368.7,

374.6, 376.4, 376.6, 378.4,

379.7, 381.2, 383.4, 383.7,

384.8, 385.6, 386.7, 396.3,

399.9, 402.3, 408.6, 412.2,

418.1, 420.5, 425.2 +,

427.5, 403.6

Saponine, 254, 401.5, 418.1
Saybolt Universal Viscometer, 324,
etc.

Scums, 256

Sealing wax as a viscous liquid,
216, 235
Seawater, 390.4

Second regime, see turbulent flow.
Seeding, 272
Seepage, 213, 223, 231
Separation of components of mixture
by flow, 257, 258, 259
Serum, 393.4, 396.9, 397.7, 411.6
Settling of suspensions, 188
Shales, viscosity of, 359.1
Shear, viscosity at low, 365.3
Sheet glass flow, 39
Shifting of minimum in fluidity,
cone, curves, 174

Silicate melts, 287, 364.3, 370.3,
375.7, 393.8

Silver nitrate, 181, 374.6
Size of molecules, 367.8, 384.9

of particles in colloid, 365.9,

380.6, 419.9, 428.1
Slags, 287, 370.3

Slipping, 14, 29, et seq., 148, 223,
225, 231, 244, 378.4, 380.4,

395.7, 425.5, 427.2
and superficial fluidity, 256

Slip, 367.3

Soap solutions, 254, 291, 357.1,
369.6, 374.4, 396.9, 397.1
Sodium chloride, 394.7
hydroxide, 357.3
nitrate, 374.6

salt solutions, of organic acids,

392.4

Softening temperature, 133
Soil moisture, 359.5

Solid, definition of, 215
friction, 262, 373.2
Solidification velocity, 190, 420.1,
427.8

Solids, 238, 239, 351.5, 353.9,

358.3, 363.8, 375.3, 375.4,

377.5, 378.6, 381.8, 407.2,

409.7, 414.2, 415.6, 416.8,

420.9, 422.3, 422.5, 424.1,

425.5

Solubility of glass, effect of on
viscosity, 377.3
and plasticity, 293
Solutions, 160, 280, 363.1, 400.5,

410.1

Sound and viscosity, 380.4
Specific volume differences, 164, 165
heat and viscosity, 368.7,

371.1

volumes of binary liquid mix¬
tures, 382.8, 388.8
Sphere, falling, 2, 6, 357.9, 362.6,

373.7, 397.4, 414.4
Stabilizer, 294

Standard substances, 354.6
Stannic chloride, 391.2
Starches, 373.4, 395.1, 406.8, 419.1
Steel, 351.8, 361.3, 392.6
Stereoisomerism, 420.6
Stokes’ method, 253, 329, 349.2
Strain, 235

Stress, influence of on properties,

364.5
Structure, 198
Sugar solutions, 407.1

Sulphur, 359.2, 369.6, 395.6, 411.8,

417.1, 422.2
dioxide, 371.3

Sulphuric acid, 361.2, 366.2, 388.9
Superficial fluidity, 254, 357.8,

401.5, 402.5,. 409.4, 414.7
Surface films as plastic solids, 255

tension, 35, 56, 96, 101, 211,
271, 356.8, 356.9, 359.3,

371.6, 376.7

Surtension and viscosity, 395.8
Suspensions, 102, 104, 203, 205,

INDEX

439

350.8, 367.6, 383.9, 385.2,
399.3, 400.8
of sulphur, 402.7
Sutherland’s equation, 247
Swelling of colloids, 404.2
Syrups, 371.7, 407.1, 407.6

T

Tables, fluidities and viscosities of
water, 339, 340
of ethyl alcohol water mix¬
tures, 341

of sucrose solutions, 341
logarithms, 345
radii limits for capillaries, 318
radius corresponding to weight
of mercury, 316
reciprocals, 342
values of K, 300
of M, 301
of N, 303

“Tackiness,” 411.5
Tallow as a plastic solid, 216
Tars, 415.8
Tautomerism, 111
Technical viscometry, 324
Temperature, 13, 92, 127, et seq.,
238, 245, 304, 350.1, 365.3,

376.9, 379.9, 409.8
Tensile strength and plasticity, 235
Tetraethylammoniumiodide, 194
Textiles, 282

Third or mixed regime, 35, 42
Thymol, 413.1
Time of relaxation, 128
measurement, 304
Tortion method, 226, 364.6
Traction method, 226
Tragacanth, 394.5
Transition points, 112, 293, 366.4
Transpiration, 2, 6, 241
Trypsin, 352.5, 424.4
Turbulence, 4, 35, 51, 97, 356.5,

357.9, 364.7, 371, 386.9,
388.1, 392.6, 399.2, 411.9,

412.9, 415.7, 417.5, 427.8
Turpentine, 53, 273

U

Ultimate electric charge, 252
Undercooled liquids, 420.1
Unsaturation, 366.8
Urea, 181
Urethane, 410.4
Urine, 356.2, 359.9

V

Vapor pressure, 155, 156, 276,

353.9, 406.9

Vapors, 246, 398.2, 407.2+, 409.1,

414.9, 418.7

Varnish, 358.8, 372.6, 400.2, 415.5,
423.6, 428.1

Velocity of crystallization, cp.

solidification.

Viscometer, 7

air bubble, 350.7
Barbey, 350.6, 405.1, 412.5
Clark, 368.2

constant pressure, 62, et seq.,

404.8, 416.2

Engler, 350.6, 367.5, 375.5,

380.9, 389.4, 397.5, 403.3,
405.1, 408.4, 423.1

Fischer, 370.9

Flowers, 371.4, 381.2

Giimbel, 413.8

Gurney, 377.2

Lunge, 394.5, 413.6

Mac Michael, 328, 416.1

Maxwell, 414.2

Ostwald, 403.8

Redwood, 397.5

Saybolt, 324, 380.9, 397.5, 375.5

Schulz, 414.7

Searle, 415.5

Stormer, 410.6, 411.1, 419.5
Washburn, 426.2
Viscose, 280

Viscosity definition, 5, 378.6
measurement, 6

Viscous liquids, 374.7, 391.7, 399.7,
402.6, 415.8, 418.2, 423.9
Volume, 141, 142, 184, 373.5 +

440

INDEX

W

Water, 347.6, 351.1, 364.8, 373.3,

375.8, 383.4, 383.5, 388.9,

391.4, 395.5, 399.8, 404.3,

408.3, 416.8, 427.4
Whipped cream, 211
Wide tubes, 397.8

Y

Yield value, 217, et seq., 237, 257
Z

Zero fluidity concentration, 54, 201,
203, 205, 220

Zinc-cadmium alloys, 424.9
Zinc sulphate, 394.3

FLUIDITY AND PLASTICITY

BY

EUGENE C. BINGHAM, Ph.D.

PROFESSOR OF CHEMISTRY AT LAFAYETTE COLLEGE, EASTON, PENNSYLVANIA.

First Edition

McGRAW-HILL BOOK COMPANY, Inc.
NEW YORK: 370 SEVENTH AVENUE

LONDON: 6 & 8 BOUVEEIE ST., E. C. 4

1922

To my sister

Anna

PREFACE

Our knowledge of the flow of electrical energy long ago de¬
veloped into the science of Electricity but our knowledge of the
flow of matter has even yet not developed into a coordinate
science. In this respect the outcome of the labors of the hydro-
dynamicians has been disappointing. The names of Newton,
Navier, Poisson, Graham, Maxwell, Stokes and Helmholtz with
a thousand others testify that this field has been well and com¬
petently tilled. Even from the first the flow of liquids has been
a subject of practical importance, yet the subject of Hydraulics
has never become more than an empirical subject of interest
merely to the engineer.

Unfortunately the theory is complicated in that the flow of
matter may be hydraulic (turbulent), viscous (linear), or plastic,
dependent upon the conditions. It was in 1842 that viscous
flow was first differentiated from hydraulic flow, and only now
are we coming to realize the important distinction between vis¬
cous and plastic deformation. Considering the confusion which
has existed in regard to the character of flow, it is not surprising
that there has been uncertainty in regard to precise methods of
measurement and that exact methods have been discovered, only
to be forgotten, and rediscovered independently later. As a
result, the amount of really trustworthy data in the literature
on the flow of matter under reproducible conditions is limited,
often to an embarrassing extent.

If we are to have a theory of flow in general, we must consider
matter in its three states. No such general theory has appeared,
although one is manifestly needed to give the breath of life to
the dead facts about flow. The author offers the theory given
in the following pages with the utmost trepidation. Although
he has given several years to the pleasant task of supporting its
most important conclusions, a lifetime would be far too short to
complete the work unaided. The author makes no apology for
any lack of finality. Parts of the theory which have already

X

PREFACE

found their way into print have awakened a vigorous discussion
which is still in progress. This is well, for our science thrives on
criticism and through the collaboration of many minds the final
theory of flow will be evolved.

Without going considerably beyond the limits which we have
placed upon ourselves, it is impossible to refer even briefly to all
of the important papers on the subject. References given in
the order that they come up in the discussion are not the best
suited for later reference. The novel plan has been tried of
placing nearly all of our references in a separate appendix which
is- also an author index and is, therefore, arranged alphabetically
under the authors’ names. In the text the name of the author
and the year of publication of the monograph is usually sufficient
for our purpose, but sometimes the page is also added. The titles
of the monographs are usually given in the hope that this bibli¬
ography may be of considerable service to investigators who are
looking up a particular line of work connected with this general
subject.

It is a pleasure to thank Dr. R. E. Wilson of the Massachusetts
Institute of Technology and Dr. Hamilton Bradshaw of the E. I.
bu Pont de Nemours & Company for reading over the manu¬
script and Dr. James Kendall for examining the proof. Profes¬
sor Brander Matthews of Columbia University, Professor James
Tupper and Professor James Hopkins of Lafayette College have
assisted in important details. The author gladly acknowledges
the valuable assistance of his colleagues and co-workers, Dr.
George F. White, Dr. J. Peachy Harrison, Dr. Henry S. Van
Klooster, Mr. Walter G. Kleinspehn, Mr. Henry Green, Mr.
William L. Hyden, Mr. Landon A. Sarver, Mr. Delbert F. Brown,
Mr. Wilfred F. Temple, Mr. Herbert D. Bruce, and others.

The author is especially indebted to the University of Rich¬
mond for the leisure which made possible a considerable portion
of this work.

Exjgene C. Bingham.

Easton, Pa.

Feb. 11, 1922.

' u _*

\'V

4 ^ Wk, x >■,

lio; LIBRARY

’'' " OOXT-ENfS'. r '- '

/O

y^j

Preface

Page

vii

Part I. Viscometry

Chapter

I. Preliminary. Methods of Measurement. 1

II. The Law of Poiseuille. 8

III. The Amplification of the Law of Poiseuille.17

IY. Is THE YlSCOSITY A DEFINITE PHYSICAL QUANTITY?.58

V. The Viscometer.62

Part II. Fluidity and Plasticity and Other Physical and Chemical Properties

I. Viscosity and Fluidity. 81

II. Fluidity and the Chemical Composition and Constitution

of Pure Liquids.106

III. Fluidity and Temperature, Volume, Pressure. Colli-

sional and Diffusional Viscosity.127

IV. Fluidity and Vapor Pressure.155

V. The Fluidity of Solutions.160

VI. Fluidity and Diffusion.188

VII. Colloidal Solutions.198

VIII. The Plasticity of Solids.215

IX. The Viscosity of Gases. ..241

X. Superficial Fluidity.254

XI. Lubrication. 261

XII. Further Applications of the Viscometric Method.279

Appendix A. Practical Viscometry.296

Appendix B. Practical Plastometry.320

Appendix C. Technical Viscometers.324

Appendix D. Measurements of Poiseuille.331

Viscosities and Fluidities of Water Fluidities of Ethyl Alco¬
hol and Sucrose Solutions.341

Reciprocals.342

Four-place Logarithms.345

Bibliography and Author Index.347

Subject Index.431

FLUIDITY AND PLASTICITY

PART I
VTSCOMETRY

CHAPTER I

PRELIMINARY. METHODS OF MEASUREMENT

Introductory. —What one may be pleased to call “dominant
ideas ” have so stimulated the work on viscosity, that it would
be entirely possible to treat the subject of viscosity by consider¬
ing in turn these dominant ideas.

Practically no measurements from which viscosities may be
calculated were made prior to 1842, yet very important work
was being done in Hydrodynamics, and the fundamental laws
of motion were established during this preliminary period. To
this group of investigations belong the classical researches of
Bernouilli (1726), Euler (1756), Prony (1804), Navier (1823),
and Poisson (1831). In the development of Hydrodynamics
much experimental work was done upon the flow of water in
pipes of large bore by Couplet (1732), Bossut (1775), Dubuat
(1786), Gerstner (1800), Girard (1813), Darcy (1858), but this
work could not lead to the elucidation of the theory of viscosity
as we shall see. Important work belonging to this preliminary
period was also done by Mariotte (1700), Galileo (1817), S’Grave-
sande (1719), Newton (1729), D’Alembert (1770), Boscovich
(1785), Coulomb (1801), Eytelwein, (1814).

It is to Poiseuille (1842) that we owe our knowledge of the
simple nature of flow in capillary spaces, which is in contrast
with the complex condition of flow in wide tubes, heretofore
used. He wished to understand the nature of the flow of the
blood in the capillaries, being interested in internal friction from
the physiological point of view. He made a great many meas-

1

2

FLUIDITY AND PLASTICITY

urements of the rates of flow of liquids through capillary tubes,
which are still perhaps unsurpassed. They lead directly to the
laws of viscous resistance and they will be described in detail
in a later chapter. The theoretical basis for these laws and a
definition of viscosity were supplied by the labors of Hagen
(1854), G. Wiedemann (1856), Hagenbach (1860), Helmholtz
(I860), Maxwell (1860). Since the velocity of flow through the
capillary may be considerable, a correction is generally necessary
for this kinetic energy, which is transformed into heat. Hagen¬
bach was the first to attempt to make this correction but
Neumann (1858) and Jacobson (1860) were the first to put the
correction into satisfactory form. Thus both the method of
measurement and the formula used in calculation of absolute vis¬
cosities were practically the same by 1860 that they are today.
Unfortunately, these important researches have not been suffi¬
ciently well-known, hence their results have been repeatedly
rediscovered, and there is an evident confusion in the minds of
many as to the conditions necessary for exact measurement.
The so-called “transpiration” or Poiseuille method was not the
only one which was worked out during this period of perfecting
the methods of measurement. The pendulum method was
developed by Moritz (1847), Stokes (1849), 0. E. Meyer (1860),
Helmholtz (1860) and Maxwell (1860). The well-known method
of the falling sphere was worked out by Stokes (1849).

During the period to which we have just referred, Graham
(1846-1862) had been doing his important work on gases, but
the development of the kinetic theory gave a great impetus to
the study of the viscosity of gases; and at the hands of Maxwell,
O. E. Meyer and others, viscosity in turn gave the most striking
confirmation to the kinetic theory. The work on the viscosity
of gases has continued on until the present, being done almost
exclusively by physicists.

To chemists, on the other hand, impressed by the relations
between physical properties and chemical composition, so forcibly
brought to their attention by the work of Kopp, the viscosity of
liquids has been an interesting subject of study. To this group
belong the researches of Graham (1861), Rellstab (1868), Guerout
(1875), Pribram and Handl (1878), Gartenmeister (1890),
Thorpe and Rodger (1893) and many others.

3

V ...U '

METHODS OF MEASUREMENT

The rise of modern physical chemistry resulted in an awaken¬
ing of interest in all of the properties of aqueous solutions.
Along with other properties, viscosity received attention from a
great number of physical chemists, among whom we may cite
Arrhenius (1887), Wm. Ostwald (1893), J. Wagner (1883-90),
Reyher (1888), Mutzel (1891). It must be admitted that our
knowledge of viscosity has not played an important part in the
development of modern physical chemistry. It is doubtless for
this reason that the subject of viscosity is left unconsidered in
most textbooks of physical chemistry. It is certainly not be¬
cause viscosity does not play an important role in solutions, but
rather that the variables in the problem have not been properly
estimated. That with the physical chemist viscosity has so
long remained in the background, makes it all the more promis¬
ing as a subject of study, particularly since it is becoming more
and more nearly certain that viscosity is intimately related to
many very diverse properties such as diffusion, migration of
ions, conductivity, volume, vapor-pressure, rate of solution and
of crystallization, as well as chemical composition and consti¬
tution, including association and hydration. It seems probable
that the ^ork in this field is going to expand rapidly, for it is
becoming imperative that the exact relation between viscosity
and conductivity, for example, should be clearly demonstrated.

With the recent advances in our knowledge of the nature of
colloids, there was certain to be an extended study of the vis¬
cosity of these substances, because no property of colloids is so
significant as the viscosity. This in turn has again stimulated
interest in viscosity on the part of the physiologist, so that the
viscosity of blood, milk, and other body fluids have been
repeatedly investigated under the most varied conditions during
the past few years.

The use of viscosity measurements for testing oils, paints,
and various substances of technical interest has given rise to a
series of investigations, that of Engler (1885) being among the
'earliest and most important in this group. These researches
have been devoted largely to devising of instruments and to a
comparison of the results obtained.

Quite unrelated to the above groups for the most part, are
the investigations which have undertaken to study the viscosity

4

FLUIDITY AXD PLASTICITY

of solids The study of elasticity has been the dominant- idea
in this group of researches.

Tory little work has been done upon the viscosity of matter
in the different states of aggregation taken as a whole. ^ If it
has been shown that our knowledge of viscosity consists of
somewhat unrelated groups, it is equally apparent that such a
separation Is artificial and that nothing could be more important
for our complete understanding of viscosity, than to bring these
groups together into an inter-related whole. We shall therefore
not make an attempt to follow the chronological method, where
It interferes with the consideration of the subject as a whole.
Nevertheless the groups of researches to which we have alluded
stand out rather clearly. The methods of measurement in use
will be first considered, after which we shall study the viscosities
of liquids, solutions, solids, and gases respectively.

Elastic Deformation, Plastic, Viscous, and Turbulent Flow. —
If a perfectly elastic solid be subjected to a shearing stress a
certain strain is developed which entirely disappears when the
stress is removed. The total work done is zero, the process is
reversible, and viscosity can play no part in the movement.
This is not a case of flow but of elastic deformation. If a body
which is imperfectly elastic as regards Its form be subjected to
shearing stress, it will be found that a part, at least, of the
deformation will remain long after the stress is removed. In this
ease work has been done in overcoming some kind of internal
friction. We may distinguish the kinds of flow under three
regimes* It is characteristic of viscous or linear flow that the
amount of deformation is directly proportional to the deforming
force, and the ratio of the latter to the former gives a measure
of viscosity. It has been questioned at times whether this ratio
is truly constant, but it appears that only one qualification is
necessary. In very viscous substances time may be necessary
for the flow to reach a steady state, aside from any period of
acceleration, because with substances like pitch the viscous
resstanee develops slowly, so that the above ratio gradually
increases when the load Is first put on, but even in this case the
ratio finally reaches a value which is independent of the amount
of the load. As, however, the deforming force is steadily in¬
creased, m point may be reached where the above ratio suddenly

METHODS OF MEASUREMENT

5

decreases. At this point the regime of turbulent or hydraulic
flow begins. This will be studied in detail at a later point in
the development of the subject. There are substances, on the
other hand, for which the value of the above ratio increases
indefinitely as soon as the deforming force falls below a certain
minimum. These substances are said to be plastic. In plastic
flow it is generally understood that a definite shearing force is
required before any deformation takes place. But whether this
is strictly true or not has not been established.

The Coefficient of Viscosity.—Consider two parallel planes A
and B } s being their distance apart. If a shearing force F per
unit area give the plane A a velocity v in reference to B , the
velocity of each stratum, between A and B , as was first pointed
out by Newton, will be proportional to its distance from B.
The rate of shear dv/ds is therefore constant throughout a
homogeneous fluid under the above conditions. The possibility
that it may not be constant near a boundary surface will be
considered later. Since the force F is required to maintain a
uniform velocity, this force must be opposed by another which
is equal in amount due to the internal friction. The ratio of
this force to the rate of shear is called the coefficient of viscosity
and is usually denoted by the symbol 77

The dimensions of viscosity are [MLr l T~ 1 ]. The definition of
viscosity due to Maxwell may be stated as follows: The vis¬
cosity of a substance is measured by the tangential force on a
unit area of either of two horizontal planes at a unit distance apart
required to move one plane with unit velocity in reference to
the other plane, the space between being filled with the viscous
substance. The coefficient of fluidity is the reciprocal of the
coefficient of viscosity, so that if the former is denoted by <t> we

have <t> = -• The coefficient of fluidity may be independently

defined as the velocity given to either of two horizontal planes
in respect to the other by a unit tangential force per unit area,
when the planes are a unit distance apart and the space between
them is filled with the viscous substance.

6

FLUIDITY AND PLASTICITY

Methods of Measurement. —Almost numberless instruments
have been devised for the measurement of viscosity, but the
greater part of these are suitable for giving relative values only _
There are, however, several quite distinct methods which are
susceptible of mathematical treatment so that absolute viscosities
may be obtained. The possible methods for measuring viscosity
may be classified under three heads as follows:

1 . The measurement of the resistance offered to a moving
body (usually a solid) in contact with the viscous fluid.

2 . The measurement of the rate of flow of a viscous fluid.

3. Methods in which neither the flow nor the resistance to
flow are measured.

1. The various methods for measuring viscosity while maintaining the
fluid in a nearly fixed position, together with the names of investigators
who have developed the method are as follows:

(a) A horizontal disk supported at its middle point by a wire and oscil¬
lating around the wire as an axis. Coulomb (1801), Moritz (1847), Stokes
(1850), Meyer (1865), Maxwell (1866), Grotrian (1876), Oberbeck (1880),
Th. Schmidt (1882), Stables and Wilson (1883), Fawsitt (1908).

(b) A sphere filled with liquid and oscillating around its vertical axis.
Helmholtz and Piotrowski (1868), Ladenburg (1908).

(c) A cylinder filled with liquid and oscillating around its vertical axis.
Miitzel (1891).

(d) Concentric cylinders. The outside one is rotated at constant velocity*
and the torque, exerted upon the inner coaxial cylinder which is immersed
in the viscous fluid, is measured. Stokes (1845), de St. Yenant (1847),
Boussinesq (1877), Couette (1888), Mallock (1888), Perry (1893).

(e) An oscillating solid sphere immersed in the viscous substance and
supported by bifilar suspension was used by Konig (1885).

GO A body moving freely under the action of gravity, e.g., falling sphere
of platinum, mercury, or water, a falling body of other shape than a sphere,
a rising bubble of air. Stokes (1845), Pisati (1877), Schottner (1879), de
Heen (1889), O. Jones (1894), Duff (1896), J. Thomson (1898), Tammann
(1898), Schaum (1899), Allen (1900), Ladenburg (1906), Yalenta (1906),
Arndt (1907).

2. The methods for measuring the rate of flow of a viscous fluid:

(a) Efflux through horizontal tubes of small diameter. Gerstner (1798),
Girard (1816), Poiseuille (1842), G. Wiedemann (1856), Rellstab (1868),
Sprung (1876), Rosencranz (1877), Grotrian (1877), Pribram and Handl
(1878), Slotte (1881), Stephan (1882), Foussereau (1885), Couette (1890),
Bruckner (1891), Thorpe and Rodger (1893), Hosking (1900), Bingham and
White (1912).

(b) Efflux through a vertical tube of small diameter. Stephan (1882),

METHODS OF MEASUREMENT 7

igler (1885), Arrhenius (1887), Ostwald (1893), Gartenmeister (1890),
sydweillex (1895), Friedlander (1901), McIntosh and Steele (1906)*
uikine (1910).

(c) Efflux through a bent capillary. Gruneisen (1905).

(d) Bending of beams and torsion of rods of viscous substance. Trouton
906), Trouton and Andrews (1904).

(e) Rate at which one substance penetrates another under the influence
capillary action, diffusion, or solution tension.

3. Other methods for measuring viscosity:

(a) Decay of oscillations of a liquid in U-shaped tubes. Lambert (1784).

(b) Decay of waves upon a free surface. Stokes (1851), Watson (1902).

(c) Decay of vibrations in a viscous substance. Guye and Mintz (1908).

(d) Rate of crystallization. Wilson (1900).

If ortienclature.—A great variety of names have been given to
struments devised for measuring viscosity, among which we
ay cite viscometer, viscosimeter, glischrometer, microrheom-
3 r, stalagnometer, and viscostagnometer. All but the first
0 are but little used and their introduction seems an unneces-
cy complication. Viscometer and viscosimeter are about
ually used in England and America, but such a standard work
Watt’s Dictionary uses only viscometer. Viscosimeter in its
;rman equivalent Viskosimeter is entirely satisfactory, but
English, viscosimeter is apt to be mispronounced viscos-
eter. Furthermore viscosimeter does not so easily relate
elf in one’s mind to viscometry which is the only word recog-
lied in the standard dictionaries to denote the measurement
viscosity. Professor Brander Matthews kindly informs me
at the formation of the word viscometer is quite as free from
jection as that of viscosimeter, and viscometer is in harmony
bh modern spelling reform. Hence viscometer should be
opted as the name for all instruments used for measuring vis-
sity. The different forms are distinguished by the names of
iir inventors.

W:

dc

CHAPTER II of

THE LAW OF POISEUILLE ^

Experimental Verification.—Prior to 1842 it had not been to

established as a fact that the movement of the blood through the lie

capillaries has its origin solely in the contractions of the heart. ■ . ca

There were theories current that the capillaries themselves at

caused the flow of blood or that the corpuscles were instrumental he

in producing it. Poiseuille reasoned that if the lengths and go

diameters of the capillaries are different in the various warm- ca

blooded animals and if the pressure and temperature of the blood aj:

vary in different parts of the body, light might be thrown upon fo

the problem by investigating the effects upon the rate of flow su

in capillary tubes of changes in (1) pressure, (2) length of capil- ol:

lary, (3) diameter of capillary, and (4) temperature. th

The results of Poiseuille’s experiments were of a more funda- be:

mental character than he anticipated for they proved that the bi;

conditions of capillary flow are much simpler than those in the su

wide tubes which had previously been employed, and by his m;

experiments the laws of viscous flow became established. Not co

only did Poiseuille perform experiments which resulted in the oil

law which bears his name, and therefore have affected all subse- to

quent work, but he measured the efflux times of water by the o u

absolute method taking elaborate precautions to insure accuracy, sp

and using capillaries of various lengths and diameters which are et<

equivalent to separate instruments—in all over forty in number.

Thus one is justified in studying his work in considerable detail, m<

not only for its historic interest, but on account of its bearing gi 1

upon questions which will arise later. In the Appendix his ur

measurements are reproduced in full. raj

In Fig. 1 is shown the most essential part of the apparatus of ' ne

Poiseuille. It consists of a horizontal glass capillary d joined op

to the bulb, whose volume between the marks c and e was accu- ub<

rately determined. The bulb is connected above with a tube tri:

which leads to (1) a 60-1 reservoir for keeping the pressure of the to

air within the apparatus constant, (2) a manometer, filled with sp:

THE LAW OF POISEUILLE

9

water or mercury, and (3) a pump which is used for giving the
desired pressure. The capillary opens into the distilled water
of the bath in which the bulb and capillary are immersed. After
the dimensions of the bulb and capillary have been found, it is
only necessary, in making a viscosity determination at any given
temperature, to observe the time necessary for a volume of
liquid equal to that contained in the bulb to flow through the
capillary under a determined pressure. Without going into detail
at this point, it need be merely stated
here that due means were taken for
getting the true dimensions of the
capillary and bulb, for filling the
apparatus with clean pure liquid, and
for estimating the mean effective pres¬
sure, which consists of the pressure
obtained from the manometer plus
the hydrostatic pressure from the
bottom of the falling meniscus in the
bulb to the level of the capillary, minus the hydrostatic pres¬
sure from the level of the capillary to the surface of the bath,
minus a correction for the capillary action in the bulb, and two
corrections for the pressure of the atmosphere, which may be
either positive or negative. One of these last corrections is due
to the air within the apparatus being more dense than that
outside, the other is due to the difference of pressure of the atmo¬
sphere upon the liquid surfaces in the upper arm of the manom¬
eter and in the bath, unless they happen to be at the same level.

Law of Pressures. —In obtaining this law all of the experi¬
ments were made at a temperature of 10°C. For a capillary of
given length and diameter, the time of transpiration was meas¬
ured for various pressures. For example, one capillary was 75.8
mm long, the major and minor axes of the end of the capillary
nearer the bulb were 0.1405 and 0.1430 mm and those of the
open end 0.1400 and 0.1420 mm respectively. The pressures
used are given in the first column of Table I and the times of
transpiration in column 2. One of these values is then employed
to calculate the others on the assumption that the times of tran¬
spiration are inversely proportional to the pressures } as given in
column 3.

Fig. 1.—Poiseuille’s viscome¬
ter.

10

FLUIDITY AND PLASTICITY

Table I.— Capillary A'

Pressure in
millimeters of
mercury at

10°C

Observed time
! for transpiration
of 13.34085 ce
of water

Calculated

time

Per cent
difference

97.764

10,361.0

147.832

6,851.0

6,851.91

0.01

193.632

5,233.0

5,231.22

0.03

3S7.675

2,612.5

2,612.84

0.01

738.715

1,372.5

1,371.20

0.09

774.676

1,308.0

1,307.55

0.04

In the above case it is certainly true that the rate of flow is
proportional to the pressure, but it is equally certain that this
relation no longer holds when the capillary becomes sufficiently
shortened. Thus when the length of the tube used above is
shortened to 15.75 mm, the values given in Table II are obtained.

Table II.— Capillary A. iv

Pressure in
millimeters of
mercury at

10°C

Observed time
for transpiration
of 13.34085 cc
of water

Calculated

time

Per cent
difference

24.661

8,646

49.591

4,355

4,299

— 1.29

98.233

2,194

2,170

-1.09

148.233

1,455

1,438 |

-1.17

194.257

1,116

1,097

-1.63

388.000

571

549

-3.85

775.160

298

275

-7.72

Not only is there a marked deviation from the assumed law
of pressures as soon as the capillary is sufficiently shortened, but
the percentage difference between the observed and calculated
values increases quite regularly as the pressure increases. But
in either case, whether the capillary is shortened or the pressure
increased, we note that the velocity is decreased. Whether the
irregularity here observed is due to the use of some of the avail¬
able work in imparting kinetic energy to the liquid, or it is due

THE LAW OF POISEUILLE 11

to eddy currents which appear under conditions of hydraulic
flow, we will reserve for later discussion. This question was not
considered by Poiseuille, yet with a great variety of tables show¬
ing an agreement like that in Table I above, Poiseuille was fully
justified in concluding that for tubes of very small diameters
and of sufficient length, the quantity of liquid which transpires
in a given time and at a given temperature is directly proportional
to the pressure , or V = Kp , where Ki s a constant, V the volume,
and p the pressure head, causing the flow through the tube.

Law of Lengths. —Poiseuille next studied the effect of the
length of the tube upon the rate of flow, but this problem pre¬
sented exceptional difficulty owing to the fact that tubes are
never of uniform cross-section. With the camera lucida he ex¬
amined and measured each section of the tubes, which had been
carefully selected from a large number, and finally corrections
were made for the small changes in diameter, assuming the law
of diameters to be given later. This seems justified since the
corrections were very small. In Table III the results are given
which Poiseuille obtained with capillary “B.” The lengths of
the capillary are given in column 1, the major and minor axes
of the free end in column 2, the time required for the transpiration

Table III.— Capillary B

Length of
tube in
millimeters

Major and
minor axes of
free end

Time of

transpiration of
6.4482 cc

Time

calculated

Per cent,
difference

100.050

f 0.11351
\ 0.1117 j

2,052.98

t

75.050

f 0.11401

1 0.1120]

1,526.20

1,539.0

0.85

49.375

\ 0.11421

1 0.1122]

998.74

1,004.0

0.53

23.575 .

f0.11451
\ 0.1123J

475.18

476.8

0.34

9.000

3.900

f0.11441
\ 0.1124 j
f 0.11451
\ 0.1125J

199.39

110.20

181.4

86.4

-9.05

-21.64

3.900

110.20

86.4

-21.64

12

FLUIDITY AND PLASTICITY

of the 6.4482 cc of water at 10°C contained in the bulb at a
constant pressure of 775 mm of mercury are given in column 3.'
Assuming that the time of flow is directly proportional to the
length of the tube, Poiseuille used the time of one experiment
to calculate the one immediately succeeding, and thus are ob¬
tained the values given in column 4. It is evident that the last
two lengths are too short, but the others fairly substantiate the
law. The agreement is still better when corrections are made
for the varying diameters of the tube. This correction is espe¬
cially important since, as will be shown, the efflux rate varies
as the fourth power of the diameter. From results like those
exhibited in Table III Poiseuille concluded that the quantity of
liquid passing through a tube of very small diameter at a given
temperature and pressure varies inversely as the length , and we have
that V = K n p/l where l represents the length. But the last
two observations show that this law has its limitations.

Law of Diameters. —To discover the relation between the
diameter of the capillary and the rate of flow, Poiseuille calculated
the quantity of water which would flow through 25 mm of the
different tubes at 10°C under a pressure of 775 mm of mercury
in 500 seconds, obtaining the values given in Table IV.

Table IV

Designation
of tube

Mean diameter
of tube in
centimeters

Volume efflux
in 500 sec. from
observations

i

Volume

calculated

Per cent,
difference

M

0.0013949

0.0014648

0.001465

+0.02

E

0.0029380

0.0288260

0.028808

-0.07

D

0.0043738

0.1415002

0.141630

+0.10

C

0.0085492

2.0673912

2.066930

-0.02

B

0.0113400

6.3982933

6.389240

-0.14

A \

0.0141600

15.5328451

15.547100

+0.10

F

0.0652170

6,995.8702463

The volumes calculated in the fourth column are obtained by
comparing each tube with the one following on the assumption
that the quantity traversing the tube is proportional to the fourth
power of the diameter , thus 0.002938 4 : 0.0013949 4 = 0.028826: x,
or x = 0.001465. The agreement is very satisfactory, hence the

THE LAW OF POISEVILLE

13

7)d^

formula becomes V — K y- • For water at 10°C he found the

value of K to be quite exactly 2,495,224, p being expressed in
millimeters of mercury at 10° and l and d in centimeters. He
experimented with alcohol and mixtures of alcohol and water
and for these we obtain different values of K . Poiseuille did
not use the terms viscosity or fluidity, nevertheless these values
of K are proportional to the fluidity.

The Effect of Temperature on the Rate of Flow.—Girard had
given a formula to represent the flow of water in a pipe as a
function of the temperature, but the constants had to be deter¬
mined for each pipe. Poiseuille gave a formula which was inde¬
pendent of the instrument used,

Q = 1,836,724,000(1 + 0.0336793 T + 0.0002209936!T 2 )^

where Q represents the weight of water traversing the capillary
in a unit of time. The adequacy of this formula to reproduce
the observed values is shown in Table V.

Table V. —Capillary A

l = 10.05 cm d — 0.0141125 cm p - 776 mm of mercury. Time of

flow 1,000 sec.

Temperature

Weight op efflux

OBSERVED

Weight of efflux

CALCULATED BY
FORMULA

0.6

5.74376

5.73955

5.0

6.60962

6.60381

10.0

7.64649

7.64435

15.0

8.74996

8.74705

20.0

9.91530

9.91191

25.0

11.14584

11.13892

30.1

12.45631

12.45423

35.1

13.80695

13.80710

40.1

15.21866

15.22184

45.0

16.67396

16.66860

Since the values calculated are weights and not volumes, the
values of Q are not proportional to the fluidity. This formula

remains empirical, but the expression V = K y- can be readily

derived from the fundamental laws of motion.

Theoretical Derivation of the Law. —Hagenbach (1860) appears
to have been the first to give a definition of viscosity. He made

14

FLUIDITY AND PLASTICITY

a very careful study of the earlier work on viscosity and gave a
theoretical derivation of the law of Poiseuille, which has had
very great effect upon the succeeding history of this subject.
Neumann gave the deduction of the Law of Poiseuille in his
lectures on Hydrodynamics in 1858, and thus prior to the publi¬
cation of Hagenbach’s paper in March, 1860. This deduction
was first published by Jacobson early in 1860 and the lectures
were published in full in 1883. In April, 1860 Helmholtz pub¬
lished the derivation of the law from the equations of motion.
J. Stephan (1862) and Mathieu (1863) gave independent deriva¬
tions of the law. Reference should also be made to the treat¬
ment of the flow in long narrow tubes by Stokes (1849).

Imagine a horizontal capillary whose bore is a true cylinder to
connect two reservoirs L (left) and R (right) there being a differ¬
ence of pressure between the two reservoirs, at the level of the
capillary, amounting to p grams per square centimeter. If the pres¬
sure in L is the greater the direction of flow through the capillary
will be from left to right. The total effective pressure p is used
up in doing various forms of work, several of which can be differ¬
entiated with a resultant gain in clearness of understanding of
the conditions of flow.

1 . Near the entrance to the capillary the particles of fluid
undergo a rapid acceleration; this absorption of kinetic energy
causes a fall in the pressure amounting to p*.

2 . Within the capillary, there may be a finite movement of
the fluid over the walls of the tube, due to slipping . Unless
the external friction is zero or infinity, work will be done and
there will be a fall of pressure p s .

3. Unless the external friction is zero, the layers of fluid
nearer the walls of the tube will move more slowly than the
layers nearer the axis of the tube, and an absorption of pressure
due to this internal friction will result. Let this be p v .

4. If the path of the particles through the capillary is not
perfectly linear, the additional distance travelled in the eddies ,
will give rise to a further drop in the pressure amounting to p e .
This turbulent flow is certain to occur when the velocity of flow
becomes sufficiently high.

5. But even before the velocity becomes turbulent it seems
possible that the stream lines at the extremities of the tube may

THE LAW OF POISEVILLE

15

be somewhat distorted, in which case there must be a drop in
pressure p 8 .

6 . Heat is produced as the fluid passes through the tube and
therefore the temperature may be different at different points of
the tube and since the temperature greatly affects the viscosity
of most substances, this may affect the amount of work done in
the passage through the tube. If the fluid is incompressible it
will have the same mean velocity through each cross-section of
the capillary, and the pressure must fall in a linear manner at least
so long as the flow is linear. If on the other hand the substance
is compressible, the velocity must increase as the fluid passes
through the tube, because of the expansion which results from
the decrease of pressure. With the expansion there is a lowering
of the temperature. Let the resultant effect of these changes in
the temperature upon the effective pressure be p T . It may be
either positive or negative.

At the exit of the capillary the fluid has no effective pressure
but it still possesses all of its kinetic energy which causes the
fluid to go for a considerable distance out into the reservoir R ,
dragging some of the fluid in R with it and producing eddies, so
that the kinetic energy is finally dissipated in overcoming viscous
resistance outside of the capillary, and not in adding to the effec¬
tive pressure, as Applebey (1910) has supposed.

The sum of these possible losses of effective pressure is then

V = Pk + Ps + Pv + Pe + Vs + Vt (2)

We shall consider first the case where p — p V} supposing that the
fluid is incompressible, as is nearly the case in liquids.

Let the radius of the capillary be R and the radius of a hollow
cylinder coaxial with the capillary be r. It is evident from the
symmetrical arrangement that at every point in such a cylinder,
the velocity must be identical. Let this velocity be v . The

dv

rate of deformation must be ^ and the tangential force due to

dv

the viscous resistance, acting from right to left, will be 77 ^ (c/.

Eq. ( 1 )). Over the whole surface of the cylinder whose length
is Z, this force must amount to

dv

16

FLUIDITY AND PLASTICITY

But the force duo to the frictional resistance on the outside of
the cylinder must be exactly balanced by a force due to the pres¬
sure and this is

— 7rr 2 pg

where p is the pressure in grams per square centimeter and g is
the acceleration due to gravity. The negative sign is used
because this force acts from left to right. We have then that

ML

2Zt?
pgr 2

but v = 0 when r = R, therefore the constant of integration K
can be evaluated

ygR 2

4Z?7

pg

dv — —

v =

rdr

+ K

K =

v =

4 bj

(R 2 - r 2 )

(3)

From Eq. (3) we may obtain the velocity in centimeters per
second at any point in the capillary. It follows that the liquid
flowing through the capillary in a given time has the volume of
a paraboloid of revolution. If the volume per second is U, then

f;/ _ rpg f;* *jyE

XI

(4)

which is the Law of Poiseuille. If V is the total volume of efflux
in the time t, the formula becomes

v _ ^gpRH
7 ~ 8l v

The mean velocity of the fluid, in cubic centimeters per second
passing through the tube, I, is

r JL _ VSE

rRH &lrj

Summary. —The simple law of Poiseuille was first discovered
experimentally, after which its theoretical deduction was quickly
made; There is, however, a considerable amount of data for
which the simple law is not sufficient. The law may be given
far greater usefulness by adding certain correction terms, which
are the subject of discussion in the following chapter.

( 6 )

CHAPTER III

THE AMPLIFICATION OF THE LAW OF POISETJILLE

The Kinetic Energy Correction—In deriving the law in the
preceding chapter, we limited ourselves to the simplest case,
where all of the energy is employed in overcoming viscous resis¬
tance within, the fluid, or p = p,. It is desirable however that
the law be given a wider application, and that the law be tested
under the most varied conditions. In the experiments which
Poiseuille used to verify his law, the kinetic energy correction was
negligible, bnt the time necessary for a single determination was
often excessiwe, consuming several hours. It is to be recalled at
this point that in some of his experiments, in which the rate of flow
was higher than in the others, the law was not verified. Poise¬
uille and others have been greatly troubled in their viscosity de¬
terminations by dust particles becoming lodged in the capillary.
If it were possible therefore to employ higher speeds, not only
would there he an economy in time but the dust particles would
be much more likely to be swept out from the tube. However
in using these higher velocities a correction for the loss in kinetic
energy must he applied.

Hagenbach (1860) is the first one to attempt to make this
correction, the results of whose work became generally known,
although it appears that Neumann prior to 1860 had made the
correction in nearly its present form. The work of Neumann
was reported by Jacobson in 1860 but his work has also remained
but little known to workers in this field. Gartenmeister (1890)
reported that Finkener had arrived at a correction which differed
from that of Hagenbach, but Finkener seems not to have pub¬
lished. any monograph on the subject stating why he considered
his correction superior. However Couette in the same year
(1890) published a very important paper in which he arrived
independently at the same correction as that given by Neumann
and Firrkener, and a year later Wilberforce (1891) independently
attacked the same subject and showed that there is a slip in the
2 17

18

FLUIDITY AND PLASTICITY

reasoning of Hagenbach. He showed that Hagenbach should
have reached a value which is identical with that given by the
others. The correction may be simply deduced as follows:

The kinetic energy of the fluid passing any cross-section of a
cylindrical tube per unit of time is

f R 2irrdr P v. - = ( R (R 2 - r*)Vdr = rpR'I*

Jo 2 64ZVJo

where p is the density of the fluid. Since the volume of fluid
passing any cross-section per unit of time is rR 2 I, the energy sup¬
plied in producing the flow is rR 2 Ipg, hence, the energy converted
into heat within the tube must be irR 2 I (pg — pP ). From Eqs. (2)
and (6) we have

72 P y2 rr>

W = Pi* = ( 7 )

Thus taking into account the loss in kinetic energy, the formula
of Poiseuille becomes

7i gpRH mp V
~~SVl 8 irlt

( 8 )

in which m is a constant which according to the above derivation
is equal to unity. The formula of Hagenbach differed only in
that the constant m is equal to or 0.7938.

It is of historical interest in this connection to note that Ber-
nouilFs assumption that all of the particles flowing through a pipe
have the same velocity, leads one to the conclusion that the
kinetic energy of the fluid passing any cross-section per unit of

tR 2 I 3

time is exactly one-half of that given above or —^— 3 an d the

value of m in that case would be only 0.50. This value was actu¬
ally suggested by Reynolds (1883) when the openings of the tubes
were rounded or trumpet-shaped, but m = 0.752 when the ends
are cylindrical. It may be added that Hagenbach compared his
value of 0.7938 with the observed values obtained by various
hydraulicians working with wide tubes, Hagen 0.76, Weisbach
0.815, Zeuner 0.80885, Morin 0.82, and Bossut 0.807, and he
found that his value was near the mean. But account should
have been taken of the fact that their results apply to the tur¬
bulent regime, but not necessarily to the regime of linear flow.

Boussinesq (1891) while admitting the correctness of the

19

AMPLIFICATION OF THE LAW OF POISEVILLE

method used by Couette—and as we have seen, also by Neumann,
Finkener, and Wilberforce—as a first approximation, gives a
more rigorous treatment of the subject on the basis of the kinetic
theory by which he finds m = 1.12.

Knibbs (1895) in a valuable discussion of the viscosity of water
by the efflux method has studied carefully the data of Poiseuille
and Jaeobson in the effort to find the value of m which would most
nearly accord with the experimental results. Throwing Eq. (8)
in the form

we observe that

. S v Vl
pt = —+

mpF 2

0 )

irjjiZ 4 1

since for a given tube and liquid only p and t

*Fxo. 2.—Finding the value of m for the kinetic energy correction.

vary, this is the equation of a straight line and may be written,

pt = a + ^ (9a)

where a and b are constants. Plotting the values of 1/t as abscis¬
sas and of pt as ordinates Knibbs obtained the curves shown in
JFig. 2, using the data for Poiseuille’s tubes A v , A VI1 , B v , and C v .
■When t becomes very great the corrective term vanishes and pt
— a. The values of a are given by the intercepts of the curves
with the axis of ordinates. The tangent of the angle which a
line makes with the axis of abscissas gives the value of b, from
which the value of m is obtained, since

20

FLUIDITY AND PLASTICITY

Using a combination of numerical and graphical methods td*
following values were obtained.

Table VI.— Values of m Deduced by Knibbs from Poiseuilde 7 ^
Experiments

Tube

Length in cent!-

Mean radius in

Values of wi-

meters

centimeters

A 111 .

l

2.55

0.00708

1.04

A™.

1.57

0.00708

1.02

A v .

0.95

0.00708

1.15

A VI ...

0.68

0.00708

1.08

A vn .

0.10

0.00708

1.12

B.

10.00

0.00567

1.23

B IV .

0.90

0.00567

1.14

B v .

0.39

0.00567

1.03

C v .

0.60

0.00427

1.87*

F 1 .

20.00

0.03267

1.08

•pn

9.97

0.03267

1.33

-pin

5.04

0.03267

1.16

-piY

2.60

0.03267

0.82*

F v ...

1.07 |

0.03267

0.82*

The mean is 1.14 or rejecting the values for C v , F^, and IF 1
1.13. Certain of the tubes, viz., A, A 1 , A 11 , B 1 , B 11 , B iri , C
C 1 , C“, C m , CP, D, D 1 , D“ D m , D 17 , E, E 1 , E“, and F giv
no satisfactory indication of the value of m. Knibbs deduce
the value of m from 34 series of experiments made by Jacobso
and obtained an average value of 1.14. This seems like a rema,r Ja
able justification of the deduction of Boussinesq. But it shonl
be added that the individual values vary from 0.82 to 1.44, yc
perhaps this variation in the values of m should not be ove)
emphasized since in some instances the amounts of the correction
are much smaller than the discrepancies among the observe
tions themselves. Knibbs thinks that the values do vary mox
than can possibly be accounted for by the experimental error an
that possibly the value of m is not a constant for all instruments
It is highly desirable that further experiments be undertaken t
determine whether m is a constant and equal to 1.12 or if it> j
not constant, the manner of its variation.

AMPLIFICATION OF THE LAW OF POISEVILLE

21

To the present writer it seems probable that the kinetic energy
correction is truly constant for all tubes which are perfect cylin¬
ders. Irregularities in the bore of the tubes will, however, have
very great influence in altering the amount of the correction,
since the correction, cf. Equation (7), depends upon the fourth
power of the radius of the tube. The shape of the ends of the
capillary has already been referred to in this connection, but it
seems preferable to consider the effect of the shape of the ends
of the tube as quite distinct from the kinetic energy correction.

There has been a tendency among many recent experimenters
to overlook the kinetic energy correction altogether, which is
quite unjustifiable. We have indicated that it is not practicable
to make the correction negligible. The only course open seems
therefore to be to select a capillary which has as nearly as possi¬
ble a uniform cylindrical (or elliptical) cross-section, to assume
that m for such a tube has the constant value of 1.12, but to
arrange the conditions of each experiment so that the kinetic
energy correction will not exceed 1 or 2 per cent of the viscosity
being measured. In this case an error of several per cent in the
value of the constant will not affect the result, unless an accuracy
is desired which is higher than has yet been attained. If such an
accuracy is desired the value of m should be found for each tube
by the method of Knibbs which has been discussed above, or by
the method employed by Bingham and White (1912), which will
be described below in dis¬
cussing the alteration in the
lines of flow at the ends of
the tube.

Correction for Phenomena
of the Flow Peculiar to the
Ends of the Tube.—If two

tubes of large diameter are
connected by a short capillary, the lines of flow will be as
represented in Fig. 3, the direction of flow being readily visible
in emulsions, suspensions, or when a strongly colored liquid is
allowed to flow out from a fine tube in the body of colorless
liquid near the entrance to the capillary, as was done by
Reynolds (1883). In the reservoir at the entrance A there is
apparently no disturbance until the opening of the capillary is

Fig. 3.-

-Diagram to illustrate
flow.

22

FLUIDITY AND PLASTICITY

almost reached, and there the acceleration is very rapid. Even
when the stream lines in the main part of the capillary are linear,
it seems theoretically necessary to assume that there is a choking
together of the stream lines near the entrance as indicated at c.
It has been suggested that this effect might be prevented by
using rounded or trumpet-shaped openings as indicated at d.

At the exit of the capillary, the stream continues on into
the reservoir B for a considerable distance with its diameter
apparently unchanged. However the fall in pressure of the
liquid passing through the large tube B is negligible, so that the
flow observed just beyond the exit takes place at the expense—
not of pressure—but of kinetic energy taken up at the entrance.
There is no distortion of the stream lines just within the exit end
of the capillary, and it is not clear that any correction at this end
is necessary, under the conditions which we have depicted. If
the capillary opens into the air, there will naturally be a capil¬
larity correction and the shape and material of the end of the
tube will be of importance— cf. Ronceray (1911).

That the stream should continue for some distance beyond
the exit with apparently constant diameter seems at first sight
quite surprising, as one might suppose that the stream would at
once drag along the adjacent fluid. The explanation is not far to
seek. In the first place one should remember that the velocities
even in the capillary are by no means uniform. Equation (3)
tells us that particles which at a given moment are in a plane
surface mno will after a certain time has elapsed be in a paraboloid
surface mpo . The transition from the stationary cylinder of
fluid in contact with the wall to the coaxial cylinders having
high speed is apparently abrupt. As the exit of the capillary is
passed, there is nothing to prevent the larger mass of liquid
from being drawn along except its own inertia. But the rate at
which the kinetic energy of the inner coaxial cylinders of fluid
passes out into the outer cylinders is proportional to the viscosity
of the medium and to the area of the cylinder. Thus in a fluid of
low viscosity a capillary stream will penetrate for some distance.
The stream disappears rather suddenly due probably to the
development of eddies.

Couette has attempted to evaluate the effects of the ends of
the tubes by supposing that they are equivalent to an addition to

the ac

correc

forro-iu

Accor

small c

the re

Sin<
diarrxc
differ c
There

To he
Poise \
the si:
times
the s
nearly
m =

Tab ms

AMPLIFICATION OF THE LAW OF POISEUILLE

23

the actual length of the capillary, which he represents by A.
corrected viscosity y c should therefore be calculated by
formula

_ irgpRH mpV

1?c “ SV(l + A) 8rr«7 + X)

The

the

( 10 )

According to Couette the corrected viscosity is always a little
smaller than that calculated by means of Eq. (8) and we obtain
the relation

y __ l + A

y c l

A = i iNZJh
Vc

Since A may be presumed to be the same for tubes of equal
diameter but of unequal lengths l and Z', one should obtain
different viscosities y and y ' by applying Eq. (8) to the same fluid.
There would thus be the relation

A - l (11)

yc r)e

To test out his theory, Couette used experimental results of
Poiseuille with tubes A 1V and A v which gave poor agreement with
the simple law, Eq. (5) c/. Table II, YII and VIII. The efflux
times are given in column 1, the viscosities y v calculated from
the simple Poiseuille formula (5), in column 2, the more
nearly correct viscosities y and y' } calculated from Eq. (8) taking
m = 1.00, in column 3.

Table VII.— Viscosity of Water Calculated from Poiseuillf/s
Experiments with Tube A iv

For dimensions cf. Appendix D, Table I, p. 331

Time

7jp Eq. (5)

rj Eq. (8), m = 1.00

8,646

0.01332

0.01328

4,355

0.01349

0.01339

2,194

0.01347

0.01332

1,455

0.01347

0.01324

1,116

0.01355

0.01325

571

0.01384

0.01325

298

0.01443

0.01330

24

FLUIDITY AND PLASTICITY

Table VIII.—Viscosity of Water Calculated from Poiseuille’s
Experiments with Tube A v
For dimensions cf. Appendix D, Table I, p. 331

Time

vp Eq. (5)

V Eq. (8), to = 1.00

3,829

0.01383

0.01363

1,924

0.01404

0.01363

994

0.01442

0.01363

682

0.01479

0.01364

537

0.01512

0.01366

291

0.01651

0.01382

165

0.01863

0.01388

The values of rj vary but little around the mean 0.01329, while
the values of Vp show a regular progression, thus demonstrating
the importance of the kinetic energy correction. The first three
values of ^ in Table VIII are constant and equal to 0.01363.
The last four values show a steady increase which may be due to
turbulent flow at such high velocities. From rj and V, which are
notably different in value, the corrected viscosity i\ c as well as the
value of A may be obtained by the use of Eq. (11). We get
7j c = 0.01303 and A = 0.041 cm. The mean diameter of these
tubes was 0.01417 cm hence, the fictitious elongation of the tube

is a little less than three times the diameter

Couette also obtained the corrected viscosity directly by
experiment, in a very ingenious manner. He employed two
capillaries simultaneously, which had the same diameter but
different lengths. The arrangement of his apparatus is shown in
Fig. 4, where T i and T 2 are the two capillaries connecting three
reservoirs M, N, and P. The pressure in each reservoir is
measured on the differential manometer FI. Since the volume of
efflux through both capillaries is the same and may be calculated
from the increase in weight of the liquid in the receiving flask D,
we obtain from Eqs. (7) and (9) the relation

V ^ TtyRHiVi - Pk) irjRH(p 2 - Vh)

8 Vrj c (h + A) 8Vrj c (h - A)
or

y = tqRH pi - p 2
8 Vrj c ' h - l 2 *

AMPLIFICATION' OF THE LAW OF POISEVILLE

By thus eliminating the correction, for the kinetic energy and the
e-nds of the tubes, Couette obtained, for the corrected viscosity
Qrio) of water at 10°, 0.01309 which is in excellent agreement with
value calculated above from Poiseuille’s experiments. If
on the other hand, the viscosity (77) is calculated by means of
El cl* (8) with m = 1.00 for one of Couette’s tubes, the apparent
vdscosity (77) is 0.01389. From the values of 77 and tj c the value
of A may be calculated as above. It is 0.32 cm and the dia,meter

Fig. 4. —Capillary-tube viscometer. Couette.

of the tube is 0.090 cm so that the fictitious length to be a.dded
is a little over three times the diameter of the tube.

In the experiments used by Couette to calculate • the value
of -A. the kinetic energy correction is very large, hence a consider¬
able error may have been introduced by taking m as equal to
1 .OO instead of the more probable 1.12. furthermore the range
of data used in establishing his conclusion is rather limited.
Hence, Knibbs has made an extended study of the same subject,
rf for A we substitute nR, Eq. (9) may be written

irgR* j mpV 2 \

TW V ir 2 gtfRV

t =

26

FLUIDITY AND PLASTICITY

but since from Eq. (9a) we have that

/ mpV 2 \ . _

and therefore

ic 2 gt 2 E 4

TrgR 4 a

~WT

. irgR A a

This is the equation of a straight line. If values of ^gyf & re

plotted as ordinates and those of R/l as abscissas, the intercept
on the axis of ordinates will give the corrected viscosity, i.e., the
value of the viscosity when l — & or R = 0; and the tangent of
the angle made by the line with the axis of abscissas when divided

by the viscosity will give the factor
l5 | RpH PTl n required. Figure 5, taken from

- /. Knibbs’ work, illustrates the method

w— -A. -as applied to the tubes used by Poi-

/_! _seuille B to B v and F to F IV . The

^ values of n are found to be —5.2 and

Z ;> +11-2 respectively. According to

—±—-Knibbs “these results challenge the

1 1 l 5 1 1 1 1 propriety of Couette’s statement that

Tig. 5.—Finding the value of A may be always regarded as positive
nfor the “end correction.” an d taken as nearly three times the

diameter of the tube.” In order to adequately test the question
Knibbs took the whole series of Poiseuille’s experiments at 10°
and reduced them rigorously on the basis of Eq. (8) taking into
account the peculiarities of the bore of the tubes used by Poiseuille
as indicated in his data. Whenever possible the value of pt ( cf .
Eq. (9)) was obtained by extrapolation since then the correction
term vanishes; in the other cases marked with a star, the value
of m was taken as 1.12. The results are arranged according to
increasing values of R/l, since if n has a positive value there
should be a progressive increase in the values of the viscosity.

Rejecting the last four values as uncertain, the general mean is
0.013107 which is almost identical with the mean for each group
of eight, whereas if n had a constant value there should be a
steady progression. On the other hand the values for the vis¬
cosity for the B series of tubes increase while those for the F series
decrease as we go down the Table. It appears therefore that no
general value can be assigned to n unless it be zero.

AMPLIFICATION OF THE LAW OF POISEUILLE

27

Table IX. —The Viscosity of Water at 10° Calculated by Knibbs from
Poiseuille’s Experiments, Using Eq. (8)

Tube

o

X

R 4 X 10 l °

1

D.

22

0.242840

0.013074*

M.

37

0.002367

0.013090*

C.

42

3.250400

0.013028*

D 1 .

44

0.233770

0.013020*

B.

56

10.235000

0.013202

C 1 .

57

3.265900

0.013071*

E.

64

0.047160

0.013242*

A.

70

24.941000

0.013145

Mean.

0.013109

B 1 .

75

10.276000

0.013134*

F.

85

11,207.000000

0.013147

C 11 .

86

3.298000

0.013151*

D 11 .

87

0.227870

0.013078*

A 1 .

93

25.059000

0.013109*

Bn.

115

10.303000

0.013070*

A 11 .

139

25.183000

0.013119*

F 1 .

163

11,187.000000

0.013065

Mean.

0.013109

E 1 .

174

0.048400

0.013588*

C 111 .

175

3.339400

0.013092*

D in .

219

0.224400

0.013045*

B m .

240

10.331000

0.013002*

A m .

277

25.231000

0.012946

F ir

326

11,233.000000

0.013249

c w .

421

3.339400

0.012498*

A^.

450

25.231000

0.013343

Mean.

0.013096

M 1 .

558

0.002367

0.013181*

B^.

630

10.357000

0.012742

F®

646

11,290.000000

0.013967

.

649

0.223310

0.012652*

E 1 .

706

0.048400

0.013222*

C v ..

709

3.339400

0.012015

A v .

742

25.231000

0.013515

A VI .

1,046

25.231000

0.013607

Mean.

0.013113

F IV

1,254

11,316.000000

0.014891

B v .

1,455

10.368000

0.012193

F v

3,034

11,316.000000

0.014851

A vn .

7,088

25.231000

0.016980

28 FLUIDITY AND PLASTICITY

Bingham and White (1912) have confirmed the conclusion of
Krnbbs by a study of interrupted flow. A capillary l = 9.38 cm
R = 0.01378 cm was used to determine the time of flow of a
given volume of water at 25° under a determined pressure. The
capillary was then broken squarely in two and the parts separated
by glass tubing, the whole being afterward covered with stout
rubber tubing. The time of flow was again determined under
the same conditions as before except that the corrections for
kinetic energy and for the effects of the ends of the tubes were
doubled by the interruption in the flow. The breaking of the
capillary was then repeated until the capillary was in six parts,
the corrections necessary being proportional to the number of
capillaries. For this case Eq. (10) becomes
irgR^pt mpVb

Vc &V(l+bA) 8 irf(I + 6A)

- c Tfu- c ’TTU <12)

where C and C ; are constants under the conditions of experiment,
and b is the number of capillaries, and A as before is the fictitious
length to be added to each capillary. Substituting in Eq. (12)
the values of the time of efflux and the pressure when the capillary
is unbroken ti and pi and when broken t 2 and p% respectively,
we obtain the relation

l + 6A _ Cp 2 t 2 — C'mb/U _ ~
l -f A Cpih — C'mi/ti ~

hence,

. K- 1 ,

Table X. —Experiments to Determine the “Fictitious Length ”
of a Capillary under Conditions of Interrupted Flow

Number of
capillaries 6

Pressure in
grams
per cm 2

1.12C'6

1.00 1+0.009
0.99 9-0.006
1.00 0 0.000
1.00 2+0.003

amplification of the law of poiseuille

20

Since the values of K are unity within the experimental error tho
addition to the length is zero. In no single instance does the
of A amount to even one-half the diameter of the tube. if
however the value of m had been taken as unity, A would have
appeared to have positive value.

Had A been found to have a definite value, It would
been necessary to consider the legitimacy of making the correc¬
tion by means of an addition to the length of the capillary instead
of by means of a correction in the pressure as suggested in Eq. (2) ,
but since no definite value can be assigned to this correction,
there is no need for raising the question.

The shape of the ends of the tube are of considerable impor¬
tance in determining the development of turbulent flow, under cer¬
tain conditions. Tubes with trumpet-shaped entrances appear to
promote linear flow (c/. Reynolds (1883) and Couette(1890) p. 48€>).

Slipping.—Coulomb (1801) made experiments with an oscillat¬
ing disk of white metal immersed in water, and he noted tha,t
coating the disk with tallow or sprinkling it over with sandstone
had no effect upon the vibrations. This seemed to prove that hire
fluid in contact with the disk moved with it, and that the property
being measured was characteristic of the fluid and not of the
nature of the surface. These observations were confirmed by
O. Meyer in 1861.

After the Law of Poiseuille had been experimentally and
theoretically established, it was still unsatisfactory that the
results of measurements of viscosity by the efflux method did
not agree with those by other methods. It was natural t>o
suppose that the discrepancy might be explained by the external
friction between the fluid and the solid boundary which had been
assumed by Uavier (1823), cf. also Margules (1881) and Hada-
mard (1903). Helmholtz in his derivation of the Law of Poi¬
seuille had taken into account the effect of slipping and obtained
the formula, which in our notation is

F = (13)

where X depends upon the nature of the fluid as well as upon that
of the bounding surface. In treatises on hydrodynamics this is
usually written

^ 4 s**]

A3A 3

V

30

FLUIDITY AND PLASTICITY

0 being the coefficient of sliding friction which is the reciprocal
of the coefficient of slipping.

From the experiments of Piotrowski upon the oscillations
of a hollow, polished metal sphere, suspended bifilarly and filled
with the viscous liquid, Helmholtz deduced a value for X of
0.23534 for water, but it is worth noting that he deduced a
value of the viscosity which was about 40 per cent greater than
that obtained by the efflux method. From some efflux experi¬
ments of Girard (1815) using copper tubes, Helmholtz deduced
the value X = 0.03984. More recently Brodman (1892) has
experimented with concentric metal spheres and coaxial cylin¬
ders, the space between being filled with the viscous substance.
He thought that he found evidence of slipping.

Slipping can be best understood in cases where a liquid does
not wet the surface, as is true of mercury moving over a glass
surface. If we consider a horizontal glass surface A, Fig. 6, as
being moved tangentially toward the right over a surface E ,

Fig. 6.

between which there is a thin layer of mercury C, then we can
imagine that the mercury is separated from the glass on either
side by thin films B and D of some other medium, usually air.
Points in a surface at right angles to the above indicated by
abed may at a later time occupy the relative positions aVc'd or if
the films B and D are more viscous than the mercury the section
may be better represented by a n b u c"d. But from Eq. (1)

dv oc <pds

so that in any case, the respective contributions to the flow by the
inner mercury layer or by the superficial films will depend upon
their relative fluidities and their relative thicknesses. Whether
the liquid wets the surface or not, anything which affects the

AMPLIFICATION OF TEE LAW OF POISEUILLE

31

fluidity of the surface film, whether it be surface tension, absolute
pressure, positive or negative polarization, static electricity, or
magnetism may therefore affect the amount of flow. And these
effects when detected experimentally would undoubtedly be
attributed to slipping or to the overcoming of external friction.
So while we might expect the effect of slipping to be more pro¬
nounced in cases where the liquid does not wet the surface, it is
quite possible that even when the liquid does wet the surface,
the fluidity of the liquid near the surface is not identical with
that within the body of the liquid.

On the other hand, it is important to remember that the
thickness of the layer of liquid affected by the forces of adhesion,
with which we are here chiefly concerned, is only molecular.
Even with mercury in a glass tube, the thickness of the layer of
air seems to be of molecular dimensions. One may get an idea of
the upper limit to this thickness by the following experiment.
A thread of mercury was placed in a narrow capillary so that
the air surface would be relatively large. Taking care that no
air-bubbles were present, the length of the thread was measured
with a dividing engine, in a determined part of the tube. The
tube was exhausted from both ends simultaneously and the
thread moved back and forth in order to sweep out the supposed
layer of air. When the mercury was finally brought back
to its former position no decrease in length could be detected.
In order to have slipping under ordinary conditions of measurement
it would appear that the surface film must he of very much more
than molecular thickness or else it must have practically infinite
fluidity . In view of the strong adhesion 1 between all liquids and
solids it seems improbable that the particular layer of liquid in
contact with the solid should show an amount of flow which is
comparable in amount with that of all of the other practically
infinite layers of liquid.

Nevertheless if the value deduced by Helmholtz for water
on a metal surface be correct, \ = 0.23534, the effect of slipping
ought to be readily observed. According to Whetham (1890), if
we take R = 0.051, Eq. (12) becomes

V = 117.67 X 10- 6

l Cf. Duclaux, 1872.

32

FLUIDITY AND PLASTICITY

whereas if there were no slip and therefore X = 0 we would have
V = £22 6.25 X 10- 6

brjl

Thus it would appear that the rate of flow through a polished
metal tube should be nearly 20 times more rapid than through a
tube in which there is no slip. Since Poiseuille’s experiments
prove that the viscosity is constant for tubes of very different
radius when calculated without regard to slipping, there can be no
slipping when water flows through glass tubes. This conclusion
is admitted by Helmholtz.

Jacobson (1860) criticised Helmholtz’s use of Girard’s experi¬
ments in that he failed to apply any correction to the pressure.
Jacobson himself experimented with copper tubes as well as glass
tubes but found no evidence of slipping.

Warburg (1870) investigated the flow of mercury in glass
tubes. He found that Poiseuille’s law of pressures and his law of
diameters were verified, which proved that slipping did not occur.
B£nard as reported by Brillouin (1907) page 152, has repeated
the work of Warburg using greater care, and he finds that \
cannot have a value greater than 0.00001.

Whetham (1890) caused water to flow through a glass tube
before and after being silvered, proper corrections being made for
changes in temperature and in the radius of the tube, due to
the silver layer. Different thicknesses of silver as well as
different pressures were used, but the difference in the times of
flow between the silvered and unsilvered tubes were all within
the limit of experimental error. Copper tubes were also used and
the results in all cases were in agreement with Poiseuille’s
observations. Cleaning the tubes with acids and alkalies, polish¬
ing with emery powder, coating with a film of oil and amalgamat¬
ing with mercury were all without effect in producing a deviation
which could be detected. Whetham repeated an experiment of
Piotrowski with an oscillating glass flask, plain and silvered.
Care was taken to make correction for temperature and to
prevent changes in the bifilar suspension which seems to have
been neglected by Piotrowski. Whetham found the ratio of the
friction of water on glass to the friction of water on silver to be
1 .0022, which may be taken as unity within the limits of experi¬
mental error. Couette (1888-1890) attacked the problem

AMPLIFICATION OF THE LAW OF POISEUILLE

33

independently but along much the same lines. He tried the
effect of a layer of grease and of silver on the inside of a tube.
He found invariably the same efflux time or even a little greater
which was due to the diminution in the radius of the tube. But
even this latter effect did not occur when the thickness of the
silver layer was a negligible fraction of the radius of the tube.
He then used tubes of white metal, copper, and paraffin using
rates of efflux close to the critical values, and obtained the
following results:

Table XI.— Couette’s Experiments on Slipping

Substance of tube

Temperature

7] Observed

r] Calculated from
Poiseuille

Copper.

15.5

0.01175

0.01130

Copper.

17.3

0.01073

0.01079

White metal.

18.2

0.01037

0.01055

White metal.

18.9

0.01064

0.01037

White metal.

18.3

0.01092

0.01052

Paraffin.

12.6

0.01241

0.01219

Paraffin.

12.9

0.01278

0.01209

Paraffin.

12.3

0.01276

0.01228

Couette goes further and gives reasons for the conclusion that
slipping does not occur even after the flow becomes turbulent.

More recently Ladenburg (1908) has carefully repeated the
experiments of Piotrowski under as nearly as possible the same

Table XII. —Ladenburg’s Experiments with an Oscillating Glass
Flask, Showing Absence of Slipping, at 19.0°

Flask

Logarithm dec¬
rement

Period of
vibration

Remark

A.

0.019570 ±2

11,973+2

TJnsilvered

A.

0.019642 + 3

12,049+2

Silvered

A.

0.019620 + 3

11,990 + 1

TJnsilvered

B.

0.025026 + 25

11,716+4

TJnsilvered

B.

0.025011 + 15

11,688+2

Silvered

C.

0.025162 + 2

11,870+2

Silvered

3

34

FLUIDITY AND PLASTICITY

experimental conditions. He used plain and silvered oscillating
glass vessels and a hollow metal sphere. Table XII prove **
conclusively that slipping was absent in the former case.

Using the hollow metal sphere filled with water, Ladenburg
obtained values of the viscosity which agree with the values found
by other methods, and shown in Table XIII.

Table XIII.— A Comparison op the Viscosity op Water as Obtained
by Different Methods (Ladenburg)

Method

tj at 17.5

v at 19.2°

Observer

Efflux glass.

0.01076

0.01031

Poiseuille (1846)

Efflux glass.

0.01065

0.01027

Sprung (1876)

Efflux glass.

0.01075

0.01030

Slotte (1883)

Efflux glass.

0.01067

0.01025

Thorpe and Rodger (1894!

Oscillating solid sphere.

0.01099

0.01054

W. Konig (1887)

Oscillating hollow cylinder..

0.01082

0.01037

Mutzel (1891)

Oscillating hollow sphere.. .

0.01065

0.01032

Ladenburg (1908)

Ladenburg indicates how Helmholtz erroneously obtained
his large coefficient of slipping by overlooking a point in th«
theory, and recalculating Piotrowski’s data he finds that instead
of the viscosity being 40 per cent greater than the general^
accepted value, this difference becomes only 3 per cent and thi
slipping becomes negligible.

It was stated above that the verification of the Law c>!
Diameters of Poiseuille is a proof that slipping does not occua
between glass and water. Knibbs (1895) has collected ai
extensive table of observations of the viscosity of water at 10*
for tubes of various materials having radii varying from 0.0140 t*
0.6350 cm or nearly a thousand-fold, but there is no evidence oj
progressive deviation as the radius increases.

In experimenting on the possible effect of an electrical oa
magnetic field upon viscosity, W. Konig (1885) obtained *i
negative result. Duff (1896) seemed to detect an increase in the
viscosity of castor oil of 0.5 per cent using the falling droj
method and a potential gradient of 27,000 volts per centimeter
but for the most part the results were negative. Quincke (1897^

AMPLIFICATION OF THE LAW OF POISEUILLE

35

i definite effect on the viscosity in an electrical field, which
elberger (1898) attempted to explain on the basis of
;sis. However, Pacher and Finazzi (1900) obtained results
were contrary to those of Dnff and Quincke finding that
ing liquids under the action of an electrical field do not
o any sensible change in viscosity. Ercolini (1903) made
aents along the same fine and concluded that the effect
ss than his experimental error. He used petroleum,
e, turpentine, olive oil, and vaseline. Carpini (1903)
•ed the viscosity of magnetic liquids in a magnetic field but
no certain effect. Koch (1911) tried the effect of oxygen
rogen polarization at the boundary using a platinum tube
. oscillating copper disk. No change in the viscosity was
ed and Koch regards this as strong evidence against
g. Honeeray (1911) has studied the effect of surface

l.

se results seem to make it quite certain that, whether
uid wets the solid or not, there is no measurable difference
n the velocity of the solid and of the liquid immediately in
t with it, at least so long as the flow is linear.

'Transition from Linear to Turbulent Flow.—It is well
that the formulas which have been discussed do not
to the ordinary flow of liquids in pipes. Under ordinary
.ons we know that the flow is undulatory, instead of being
is is assumed in the simple laws of motion. It is important
?e know under what conditions these sinuous motions
• so that they may be properly taken into account or
d against. An extended study of the flow of water in
laving a diameter varying from 0.14 to 50 cm was made by
(1858). He found the hydraulic resistance proportional
here n had a value nearly equal to 2 (1.92). He saw moie
than any of his predecessors that hydraulic flow is very
at in character from the viscous flow studied by Poiseuille,
be viscous resistance is proportional to the first power of
*an velocity (I). Darcy paid little attention to the tern—
re at which his experiments were carried out, probably as
Ids remarks, because “the resistance after eddies have
established is nearly, if not quite, independent of the
ty.” Since Darcy's work was approved by the Academy

Efflux Cubic Inches

36 FLUIDITY AND PLASTICITY

in 1845, he is probably the first to distinguish clearly between
the two regimes.

Hagen (1854) investigated the effect of changes m temperature
upon the rate of efflux in tubes of moderate diameters. Figure 7

Fig. 7. —Transition from linear to turbulent flow. The effect of temperature.

exhibits the results of his experiments. The abscissas are
degrees, Reaumur, the ordinates the volumes in cubic inches
(“Rheinland Zollen ”) transpiring in a unit of time. The pressure
to which each curve corresponds is given at the right of the figure,
being expressed in inches of water. Hagen used three tubes of
varying width as follows:

N arrow
]Vlean - -

\\Tide- ' *

1

Inspec
the sane
excep>t
there i
pressxa:

AMPLIFICATION OF THE LAW OF POISEUILLE

37

Radius, inches

Length, inches

Narrow,
Mean...
Wide...

0.053844

0.077394

11.391400

18.092

41.650

39.858

Fig. 8.—Apparatus of Reynolds for studying the critical regime.

Inspection of the figure shows that with the lowest pressure and
the smaller tubes the efflux is a linear function of the temperature
except at the highest temperatures. With the wide tube, however,
there is a maximum of efflux at about 37° even at the smallest
pressure. As the pressure is increased the maximum appears at a

38

FLUIDITY AND PLASTICITY

I

lower and lower temperature and the maximum appears even in
the smallest tube used. There is a minimum of efflux after
passing the maximum but then the efflux becomes again a linear
function of the temperature. Brillouin (1907) page 208, has
confirmed the experimental results of Hagen.

A clear picture of the phenomena connected with the passage
from one regime to the other has been given by Reynolds (1883).
One form of apparatus used by him is depicted in Fig. 8. It

Fig. 9.—Linear flow.

consists of a glass tube BC, with a trumpet-shaped mouthpiece
AB of wood, which was carefully shaped so that the surfaces
would be continuous from the wood to the glass. Connected
with the other end is a metal tube CD with a valve at E having
an opening of nearly 1 sq. in. The cock was controlled by a long
lever so that the observer could stand at the level of the bath,
which surrounded the tube BC. The wash-bottle W contained
a colored liquid which was led to the inside of the trumpet¬
shaped opening. The gage G was used for determining the level

Fig. 10.—The beginning of turbulent flow.

of water in the tank. When the valve E was gradually opened
and the color was at the same time allowed to flow out slowly, the
color was drawn out into a narrow band which was beautifully
steady haying the appearance shown in Fig. 9. Any consider¬
able disturbance of the water in the tank would make itself
evident by a wavering of the color band in the tube; sometimes it
would be driven against the glass tube and would spread out,
but without any indication of eddies.

As the velocity increased however, suddenly at a point 30 or
more times the diameter of the tube from the entrance, the color

AMPLIFICATION OF THE LAW OF POISEUILLE

39

band appeared to expand and to fill the remainder of the tube
with a colored cloud. When looked at by means of an electric
spark in a darkened room, the colored cloud resolved itself into
distinct eddies having the appearance shown in Fig. 10. By
lowering the velocity ever so slightly, the undulatory movement
would disappear, only to reappear as soon as the velocity was
increased. If the water in the tank was not steady the eddies
appeared at a lower velocity and an obstruction in the tube
caused the eddies to be produced at the obstruction at a consider¬
ably lower velocity than before. “Another phenomenon which
was very marked in the smaller tubes was the intermittent char¬
acter of the disturbance. The disturbance would suddenly come on
through a certain length of the tube, pass away, and then come
again, giving the appearance of flashes, and these flashes would
often commence successively at one point in the pipe.” The ap¬
pearance when the flashes succeeded each other rapidly is shown

Fig. 11.—Flashing.

in Fig. 11. “This condition of flashing was quite as marked
when the water in the tank was very steady, as when somewhat
disturbed. Under no circumstances would the disturbance occur
nearer the funnel than about 30 diameters in any of the pipes,
and the flashes generally, but not always commenced at about
this point. In the smaller tubes generally, and with the larger
tube in-the case of ice-cold water at 4°, the first evidence of
instability was an occasional flash beginning at the usual place
and passing out as a disturbed patch 2 or 3 in. long. As the
velocity further increased these flashes became more frequent
until the disturbance became general.”

Reynolds further noted that the free surface of a liquid indi¬
cates the nature of the motion beneath. In linear flow, the sur¬
face is like that of plate glass, in which objects are reflected without
distortion, while in sinuous flow, the surface is like that of sheet
glass . A colored liquid flowing out into a vessel of water has the
appearance of a stationary glass rod in the first rdgime, but as the

40

FLUIDITY AND PLASTICITY

velocity is increased the surface takes on a sheet glass appearance
due to the sinuous motions, and finally the stream breaks into
eddies and is lost to view (cf. Collected Papers 2, 158).

Reynolds reasoned from the equations of motion that the
birth of eddies should depend upon a definite value of

pRIcp

where R is a single linear parameter, as the radius of the tube,
and I is a single velocity parameter, as the mean velocity of flow
along the tube. Reynolds found the value of the constant to be
approximately 1,000, hence, the maximum mean velocity in
centimeters per second for which we may expect linear flow, may
be taken to be

1,000

pRcp

(14)

In Table XIY we have calculated the value of the product
pRI<p from some of Reynolds’ observations near the critical
velocity.

Table XIV.— Calculation of the Critical Velocity Constant

Flow

Temper¬

ature

p

R centi¬
meters

<p

I centi¬
meters per
second

pRI<p

Steady.

9

1.00

0.3075

74.4

44.26

1,012

Unsteady...

8

1.00

0.3075

72.4

48.65

1,113

Steady.

5

1.00

0.3075

66.2

51.06

1,039

Unsteady...

5

1.00

0.3075

66.2

54.33

1,106

Steady.

8

1.00

0.6350

72.4

22.60

1,039

Unsteady...

8

1.00

0.6350

72.4

22.60

1,039

Reynolds tried plotting the mean velocity against the fall in
pressure per unit length of the tube as shown in Fig. 12. It is
to be observed that the resistance increases as a linear function
of the velocity according to Poiseuille’s law up to a certain definite
point, and that from that point on the resistance varies as some
higher power of the velocity. This power as we shall see is
constant and is, according to Reynolds equal to 1.723 over a
wide range of pressures. Formulated this relation becomes

P = KI* (15)

AMPLIFICATION OF THE LAW OF POISEUILLE

41

^here P is the pressure gradient and n is the constant. In the
first regime n = 1 and we have

P = K1

I-t may be remarked here that hydraulicians have usually

Log Velocities

Fia. 13.—A bother method for bringing out the characteristics of the viscous,

hydraulic, and critical regimes.

employed the expressions

P = KP
or

P = AI H- BP

Putting Eq. (15) in the form
log P = n log I + log K

we observe th.at the relations may be presented more forcefully by
plotting the logarithms of the pressure gradients as abscissas

42

FLUIDITY AND PLASTICITY

and the logarithms of the mean velocities as ordinates. For
tubes 4 and 5, Reynolds obtained the curves given in Fig. 13.
Linear flow exists along the line ABC, or A f B'C f , the hydraulic
regime exists along the line BDEF or B'D'E'F'. It is evident now
that n is constant for each part of the curves for the two tubes
and that it is the same for both, Le. the curves can be exactly
superimposed by merely a rectangular shift. The line ABC is
inclined at an angle of 45° so that n = 1, and the line BDEF is
inclined at an angle 3G°-8' so that n = 1.723. Except for
the unstable region BCD, the formula P = KI n will represent the
viscosity in both regimes, it being necessary to merely change
the value of n in passing from one regime to the other.

In passing from B to C it is evident that the linear flow becomes
increasingly unstable, and thus is explained why the eddies
appear suddenly and full-fledged, when the disturbance is
sufficiently great. The more undisturbed the liquid is, the
farther is it possible to go from B.

The points along the curve CD (or C'D') correspond to the
mixed regime where the flashing occurs, the turbulent movement
alternating with the linear. Light has been thrown upon the
cause of the flashing by Couette (1890) and Brillouin (1907),
using a horizontal tube opening directly into the air. At high
pressures the surface of the jet had a sheet-glass appearance
indicating that the flow was hydraulic but the amplitude of the
jet was constant. As the pressure was lowered the velocity fell
to a point where linear flow began. But as the resistance was
much less in linear flow, the velocity increased as shown by the
increased amplitude of the jet, and the condition of hydraulic
flow was reestablished. The rapidity of these fluctuations
gradually increased as the pressure was further reduced, passed
through a maximum and gradually declined as the linear flow
came to predominate. Finally the jet became regular with a
plate glass surface.

It has been suggested that there are really three regimes, one
for velocities with which only linear flow can exist, a second for
velocities with which only turbulent flow can exist, and a third
where linear and turbulent flow alternate. The linear r4gime is
sharply marked off from the hydraulic regime by the point B
where the lines ABC amj BDE intersect. While the period of

Log velocities

44

FLUIDITY AND PLASTICITY

Number

Diameter

Temper-

Surface

ature

0.0014

0.0270

0.0650

0.615

1.270

1.400

2.700

4.100

2.600

8.260

19.600

28.500

8.190

13.700

18.800

50.000

24.320

24.470

4.968

10

10

10

5

5

12

21

21

15

15

Reynolds

Glass j

Glass \ Poiseuille
Glass J
Lead No. 4
Lead No. 5
Lead
Lead
Lead

Varnished
Varnished
Varnished
Varnished
Cast iron, new
Cast iron, new
Cast iron, new
Cast iron, new
Cast iron, incrusted
Cast iron, cleaned
Glass

Darcy

the oscillations in the mixed regime is entirely characteristic, it
seems hardly probable that we can always sharply differentiate the
mixed from the hydraulic regime. Indeed Couette (p. 486)

Fig. 15.—Coaxial cylinder viscometer of Couette.

found that with a tapering tube the oscillations do not appear at
all. One may draw the conclusion from Reynolds’ observations

AMPLIFICATION OF TEE LAW OF POISEUILLE

45

that the formula P = KP may be used in the critical regime.

Reynolds has compared the data of Darcy for large tubes,
that of Poiseuille for small tubes, with his own, plotting the
logarithmic homologues as in Fig. 13. The result is shown in

Fig. 14. Each line represents the logarithmic homologue for
some particular tube, described in the figure. It is at once
apparent that, for the most part, experiments have been made
well below or else well above the critical values. In the small
tubes of Poiseuille the velocities were below the critical values.

46

FLUIDITY AND PLASTICITY

The smallest tube with which he experimented, A, gives a
curve, only part of which is shown in the figure. It should be

Velocity

-Fig* 17. The transition from viscous to hydraulic flow with coaxial cylinders.

added that Reynolds corrected Poiseuille’s data for the loss in
kinetic energy.

For pipes ranging in diameter from 0.0014 to 500 cm and for
pressure gradients ranging from 1 to 700,000, there is not a
difference of more than 10 per cent in the experimental and

AMPLIFICATION OF THE LAW OF POISEUILLE

47

calculated velocities and, with very few exceptions, the agree¬
ment is within 2 or 3 per cent, and it does not appear that there
is any systematic deviation.

Couette (1890) has strongly confirmed the work of Reynolds
by his measurements with coaxial cylinders. The external
appearance of the apparatus used is shown in Fig. 15 where V
is the outer cylinder of brass which can be rotated at a constant
velocity by means of an electric motor around its axis of figure T .
The inner cylinder is supported by a wire attached at n.
A section through a part of the apparatus in Fig. 16, shows the
inner cylinder $ while g and g f are guard rings to eliminate the
effect of the ends of the cylinder. The torque may be measured
by the forces exerted on the pulley r which are necessary to
hold the cylinder in its zero position. Plotting viscosities as
ordinates and the mean velocities as abscissas, he obtained
Fig. 17. Curve I represents the results for the coaxial cylinders,
curve II represents the same results on five times as large a
scale in order to show better the point where the regime changes.
Curves III and IV are for two different capillary tubes. It is
clear from the figure that the viscosity is quite constant up to
the point where the rdgime changes. The apparent viscosity
then increases very rapidly, and finally becomes a linear func¬
tion of the velocity. The dotted parts of the curves where the
viscosity increases most rapidly, represents the region of the
mixed rdgime, and the measurements were very difficult to ob¬
tain with precision.

He proved that pRI<p = a constant by a series of experiments.

(1) The mean velocity at the lower limit of the oscillations is
independent of the length of the tube. He used a glass tube
R = 0.1778 and obtained the efflux per minute V', thus:

Table XV. —Law of Lengths

Length, centimeters

V' mean

86.5

388

71.5

367

57.9

365

41.8

376

25.7

394

48

FLUIDITY AND PLASTICITY

(2) The mean velocity at the lower limit of the oscillations is
inversely proportional to the radius of the tube.

Table XVI.— Law of Eadii

R

Temperature

V'

V'

R

0.04998

12.7

103.6

2,073

0.09036

13.6

214.9

2,378

0.13070

13.6

344.0

2,632

0.17780

13.6

377.0

2,121

0.21080

13.6

542.0

2,570

0.27620

13.6

701.0

2,538

0.29690 (Copper)

15.0

648.0

2,182

0.45000

15.0

1,205.0

2,678

(3) The mean velocity at the lower limit of the oscillations is
inversely proportional to the fluidity. For both mercury and
water an elevation of the temperature caused a lowering of the
mean velocity at the lower limit of the oscillations. The in¬
crease in the temperature causes an increase in the fluidity in
both cases.

(4) Experiments with air and water confirm the law that the
mean velocity at the lower limit of the oscillations is inversely
proportional to the density of the medium. The number of
turns of the outer cylinder per minute is taken as proportional
to the mean velocity, a being a constant.

Table XVTI.— Law of Densities

Substance

V

p

al

CKppI

Water.

0.91096

1.0900

56

Ah ..

0.90018

0.0012

800

5,300

There would be still some doubt whether the critical velocity
is inversely proportional to the fluidity, but this doubt is re¬
moved by the work of Coker and Clement (1903) to test this
very point. They used a single tube Z = 6 ft. R = 0.38 in.
measuring the flow of water over a range of temperatures from

AMPLIFICATION OF THE LAW OF POI SEVILLE

49

4° to nearly 50°. Plotting the logarithmic homologues they
obtained a family of curves exactly similar to those in Fig. 14,
so that it is unnecessary to reproduce them. The points of
intersections between the curves for linear and for turbulent
flow lie on a perfectly straight line as is true in Fig. 14. This
proves that the critical velocity is directly proportional to the
viscosity. Indeed plotting the critical velocities read from their
curves against the temperatures, one obtains a curve which is
almost identical with that obtained by calculation from the
viscosities according to the assumed law.

Compressible Fluids.—As a compressible fluid flows through
a capillary under pressure, expansion takes place as the pressure
is relieved. The expansion may give rise to several effects which
must be taken into consideration. ( 1 ) The velocity increases as
the fluid passes along the tube. ( 2 ) There must be a component
of the flow which is toward the axis of the tube. (3) The expan¬
sion may cause a change of temperature. This may affect the
flow in two ways (a) by changing the volume and consequently
the velocity and (J) by changing the viscosity of the medium and
consequently the resistance to the flow. (4) As the density
changes, the viscosity may also change, unless the viscosity is
independent of the density. (5) We must also consider whether
the kinetic energy correction is changed when the velocity
increases as the fluid passes along the tube.

For incompressible fluids, we have seen that the viscosity
measurement may be made without reference to the absolute
pressure. But with compressible fluids this is not the case,
because the rate of expansion depends upon the absolute pres¬
sures, in the two reservoirs at the level of the capillary, Pi and P 2 .
We will first suppose that Boyle’s law holds, the flow taking place
isothermally. For this case, as we shall see, page 243, the vis¬
cosity is independent of the density. Let U, P, and p represent
the mean velocity, absolute pressure, and density at any cross-
section of the tube. Since at any instant the quantity Q of the
fluid passing every cross-section is constant, we have from Eq. (4)

n rT 7 igR* dp _ rgR* p dP 2
y “ pU ~ 8t; p dl ~ 8v 2 p dl

( 16 )

But

161J p

is constant aad therefore

dP 2
dl ~

Pi 2 -P z 2

l

50

PLVIDITY Atfb PLASTICITY

and we obtain

7i git 4 Pi

irgRH f-0 i z> t /p 2_p.21 fi7'i

v =mWi ( 1 Ps) ~ 16lP 2 V 3 (Pl Fs) W

where Vi and V 2 are the volumes corresponding to pressure
Pi and Pi-

Since p = Pi — P 2 we observe that the ratio between the

values of the viscosity calculated by Eqs. (17) and (5) is ——

where P may have any value between Pi and P 2 depending upon
the value of F which is employed. If F be taken as
2Pi v rr 2P 2
Pi+p 2 1 Pi+p 2 2

(17) becomes identical with Eq. (5) and becomes unnecessary.

The derivation of the law for gases was made by 0. E. Meyer
(1866) and by Boussinesq (1868). With Fisher (1903) we may
regard the above case where PV is constant as extreme, and that
more generally we may take PV n as constant. Equation (16)
becomes on integration

,.Tg» Pi * ~Pj n

817, ' J 1

When n = co this becomes identical with Eq. (4), for incom¬
pressible fluids. When n = 1 the flow is isothermal and we
obtain Eq. (16a). Ordinarily the value of n will lie between these
two extremes, thus in adiabatic expansion n = C p /C v = 1.0 to
1.7, the ratio of the specific heats. Hence, it seems probable that
the Law of Poiseuille as given in Eq. (5) may be used, irrespective
of whether the fluid is compressible or not, but in every case the
volume of flow must be taken as

(‘ +;) p ‘" v ‘

P*+Pl* 1 p 2

. . -Pi

In the extreme ease where n = 1, if p is not greater than P 2 /IO

AMPLIFICATION OF THE LAW OF POT SEVILLE

51

V will not differ from

7i + 7 2
2

by much over 0.2 per cent.

This means that working at atmospheric pressure, with a hydro¬
static pressure of oyer 100 cm of water, one may take the volume
7i + V 2

of flow as -g— without any very appreciable error. It is

therefore extremely improbable that an appreciable error is
incurred through our lack of knowledge in regard to the exact
value of n in a given case. The effect of the temperature upon
the viscosity will be discussed later, page 246, as a temperature
correction.

The kinetic energy of the fluid increases as it passes along the
tube, but we are interested only in the total amount of the
kinetic energy as the fluid leaves the tube. This is 7rp 2 .K 2 iVb
The total energy supplied in producing the flow is tR 2 I 2 (Pi —P<i)g
and the difference between the two is the energy converted into
heat 7r/J 2 I 2 [(Pi — P*)g — g^ 2 ]- The loss of head in dynes per
cm 2 in imparting kinetic energy to the fluid is therefore

' p k g = P 2 / 2 2

P 2 7 2 *
t rW

With this correction, but neglecting the slipping, we obtain
__ tt gRH Pi 2 — P 2 2 mp 2 F 2 , .

V 8 V 2 l 2 Pi Sir If l j

Substituting V for V 2 and remembering that p 2 7 2 is constant,
Eq. (19) becomes identical with the complete formula for the
viscosity as given in Eq. (17).

Although it is admitted that the flow of compressible fluids
is not quite linear, no correction for this has yet been attempted.
However it is certain that the correction is negligible if p is small
in comparison with P 2 . The correction for slipping in gases
plays an important part in the literature. The correction is the
same as for incompressible fluids.

Turbulent Flow in Gases.—The distinction between viscous
and turbulent flow in gases has been investigated by several
workers, among whom we may mention particularly Grindley
and Gibson (1908) and Ruckes (1908). Ruckes discovered that
the criterion for gases was greatly raised if the capillary was blown
out into a trumpet shape.

52

FLUIDITY AND PLASTICITY

Plastic Flow or the Fourth Regime— When a mixture of liquids,
such as petroleum, is allowed to flow through a tube of large
diameter Silled with finely porous material like Fuller’s earth,
Gilpin 1 and others have shown that there is a tendency for the
more volatile, i.e. the more fluid substances, to pass through
the maze of capillaries first, leaving the more viscous substances
behind. Naturally this effect is greatest when the pressure is
very small. It is easy to see that under such conditions of flow
the fluidity as calculated might appear quite abnormal. Just
as the fluidity appears abnormal when the velocity exceeds a
certain value and we pass into the second regime, so it appears
that the fluidity may appear abnormal when the velocity drops
below a certain critical value, and we pass into what may be
called the u Fourth Fegime.”

With homogeneous liquids or gases of high fluidity it is diffi¬
cult to work at excessively low velocities, particularly on
account of the interference of dust particles. Very little work
has been done upon such substances having low fluidity, so that
for aught we know now the lower critical velocity may be ob¬
servable only in mixtures.

Glaser (1907) measured the viscosity of colophonium-turpen-
tine mixtures by the transpiration method with the object of
testing the law of Poiseuille for very viscous and plastic sub¬
stances. With one tube R = 0.49 cm, l = 10.5 cm he found

Table XVIII.— The Viscosity of an 85 Pee cent Colophonttjm— 15

Per cent Turpentine Mixture at 11.3° and Under a Constant
Pressure op 2,040 cm Water in Tubes of
Various Diameters (Glaser)

R

l

t

y

V X 10 7

<e X 10-»

1.525

25.1

600

2.28

4.20

2.38

1.019

15.9

1,800

2.30

4.21

2.37

0.746

16.0

900

0.329

4.25

2.35

0.576

15.1 i

18,000

1.972

4.22

2.36

0.364

15.8

46,800

0.755

5.22

1.80

0.257

15.2

43,200

0.149

6.59

1.51

0.15S

15.1

173,500

0.023

19.90

0.50

0.117

15.4

3 weeks

0.000

oo

0.00

1 ^ Chem. J., 40, 495 (1908); 44, 251 (1910); 50, 59 (1913).

AMPLIFICATION OF TEE LAW OF POI SEVILLE

53

the product of the pressure multiplied by the time of efflux to
be constant. The velocities ranged from 0.00011 to 0.00175
cm per second. From these experiments Glaser concluded that
“ The velocity of efflux in this mixture is within very wide limits
without influence upon the magnitude of the viscosity.” But
in experimenting with tubes of varying diameter, he obtained
remarkable results a part of which are given in Table XVIII.

We observe with Glaser that the fluidity rapidly falls off
as soon as the diameter falls below a certain limit. But this
limit depends upon the fluidity of the mixture, as was proved

Figk 18.—Eighty, eighty-five and ninety per cent mixtures of colophonium
and turpentine give fluidities, multiplied by 10 -6 , 10~ 8 and 10“ 10 respectively,
which, vary with the radius of the tube. Such mixtures are apparently -plastic
and do not obey the laws of viscous flow.

by working in the same way with 80 and 90 per cent colopho-
nium mixtures, the true fluidities of which are approximately
2 X 10“ 6 and 2 X 10~ 10 respectively. Since with such rapidly
increasing values, the viscosities are inconvenient to plot, we
have changed his viscosities to fluidities. All of his values are
plotted in Fig. 18, using apparent fluidities as ordinates and
radii as abscissas. . We note that the points lie, for the most

54

FLUIDITY AND PLASTICITY

part, on a smooth curve indicating that the phenomenon under
consideration is not one of mere clogging, as by accidental dust
particles in ordinary measurements. The effect is pronounced
in a tube of about 0.8 cm radius with a 90 per cent colopho-
nium mixture, but the effect is not noticeable in a tube of 0.1
cm radius with an 80 per cent mixture. It is therefore no
wonder if this effect is not noticeable in ordinary liquids which
are millions of times yet more fluid.

The fact which seems to have been overlooked by Glaser
and is of prime importance in explaining the phenomenon, is
that the shearing stress and the mean velocity of efflux is very
much less in the smaller tubes. Obtaining the critical value of
the radius for each mixture by the graphical method, wc have
calculated the mean velocity by means of Eq. (6). It is of the
same order of magnitude in all three cases being around 0.000,01
cm per second. It seems probable that had these experiments
been repeated at a very greatly different® pressure, it would have
been discovered that the viscosity is dependent upon the shear¬
ing force rather than upon the radius of the tube, and the con¬
clusion that the viscosity is independent of the velocity would
have been amended. It is highly desirable that experiments
be made to establish this point.

It is important to observe that each mixture used by Glaser
gave a zero fluidity when the radius of the tube fell below a
certain well-defined limit. Bingham and Durham (1911)
have studied various suspensions of clay, graphite et cetera in
different liquids over a range of temperatures, using a single
capillary and a nearly constant pressure. As shown in Fig. 19,
the fluidity-concentration curves are all linear and at all concen¬
trations and temperatures they point to a well-defined mixture
with zero fluidity, at no great concentration. This mixture
apparently sharply demarcates viscous from plastic flow, for
be it noted that the mixture having “zero fluidity” was not a
hard solid Biass, but rather of the nature of a thin mud. In
the mixture of “zero fluidity” it appears that with the given
instrument all of the pressure is required for some other purpose
than to produce viscous flow. The amount of pressure used up
in this way is zero for the suspending medium alone but increases
in a linear manner with the concentration of solid. If this view is

A *n>LJFTCATlOH OF rrrv t a™ ~

F TRB LAW OF POISEUILLF er

worroct, a tii-1 of thr, LE 55

flow and the rest in ^Sf****-*

22>0 c._ ‘ xt hgheT pressures

Percenfnge Volume of forth '° ^ ' 3 U ° Jo

iq. 19.—The fluidity of suspension of infusorial earth in water.

wouldT/e/) e ugr/? S su?e/ o y ir OUl l be Changed because there

resi S t ance in the mixture tha?? 6 ° vercome the plastic
mixture that formerly had “zero fluidity.”

56

FLUIDITY AND PLASTICITY

Further work is therefore demanded in order that we may clearly
define and separate the coefficients of plasticity and fluidity
which are here measured together.

Surface Tension and Capillarity. —Several investigators have
attempted to measure viscosity by means of a capillary opening
directly into the air. Poiseuille (1846) found that whether drops
were allowed to form on the end of the capillary or the end of the
capillary was kept in contact with the wall of the receiving
vessel, he was unable to obtain consistent results. The effect
of surface tension varies with the rate of flow, with the tempera¬
ture, and it also depends upon the shape and position of the end
of the capillary, so that as a whole the effects are quite indeter¬
minate. That the effects are large and variable, may be inferred
from the measurements of Ronceray (1911) with a capillary,
l = 10.5 cm, R = 0.0275, immersed under water or opening
into the air, given in Table XIX.

Table XIX .— Effect of Surface Tension on the Flow of Water

(Ronceray)

P centimeters,
water

Time of flow of
10 ml in air at 17°

Time of flow im¬
mersed

Difference

10

1,132.0

1,089.44

42.6

20

559.5

550.4

9.1

30

373.0

368.5

4.5

40

280.6

277.4

3.2

50

224.9

222.7

2.2

60

187.9 |

186.1

1.8

70

161.8

159.5

2.3

Poiseuille recorded similar results. The irregularity is com¬
pletely removed by having the end of the capillary immersed.
Nevertheless in an apparatus like that used by Poiseuille there
may still be a correction for capillary attraction within the bulb
which is considerable (c/. p. 66).

Summary. —From the foregoing, it appears that under proper
conditions, the only correction that it is necessary to make
to the simple Law of Poiseuille is that for the kinetic energy

of the fluid as it leaves the capillary Other sources

AMPLIFICATION OF THE LAW OF POISEUILLE

57

of error such as surface tension, slipping at the boundary,
necking in of the lines of flow at the entrance of the capillary,
eddy currents inside of the capillary, resistance to flow outside
of the capillary, peculiar shapes to the ends of the capillary
affecting the magnitude of the kinetic energy correction have
all been considered in detail. They may all be eliminated
by using long, narrow capillaries with a low velocity of flow.

The fluidity of compressible fluids may be obtained by the
same Law of Poiseuille but the volume of flow is approximately
the mean of the volume at the entrance and at the exit of the

capillary

F1+F2

Plastic solids in their flow do not obey the Law of Poiseuille
and their study is deferred until Chapter VIII. Many attempts
have been made to measure the viscosity of soft solids. The
fluidity of such a substance is not a constant quantity but
falls off rapidly although regularly as the radius of the capillary
falls below a certain point. This is not stoppage of the capillary
of the ordinary sort due to extraneous particles, but rather a new
type of flow. The terms fluidity and viscosity will therefore be
avoided when referring to plastic substances in order to avoid
confusion and a sharp criterion given by which a soft solid may be
distinguished from a true fluid, just as Reynolds’ criterion
enables one to distinguish between viscous and turbulent flow.

CHAPTER IV

IS THE VISCOSITY A DEFINITE PHYSICAL QUANTITY?

So long as the theory was so imperfectly worked out that
the values for the viscosity of a well-defined substance like
water were different when obtained with different forms of
instruments, it was inevitable that the whole theory and practice
of viscosity measurement should have been called into question.
Among numerous researches, we may cite in this connection
those of Traube (1886), Wetzstein (1899) for liquids,- Fisher
(1909) for gases, and Reiger (1906) for solids. Since the limita¬
tions and corrections discussed in the preceding chapter have
evolved very gradually, many of these researches are now of
historical interest only, and their discussion here would be as
tedious as it is unnecessary. Enough material has already been
given to prove that viscosity is an entirely definite property for
liquids. Table IX proved that tubes of quite diverse dimen¬
sions give entirely harmonious results. This has been confirmed
repeatedly, especially by Jacobson, 1860, working with tubes of
considerably larger bore. Not only are the results obtained
with the transpiration method in agreement, among themselves,
they also agree with the results from various other methods, as
shown in Table XIII.

Knibbs (1896) has made a critical study of the existing data
for water, recalculating and using the corrections suggested
in the last chapter. The result was not satisfactory. Many of
the measurements were found to be uncertain and as a result of his
study Knibbs doubted whether it was yet possible to determine
the viscosity of a substance like water with an error of much less
than 1 per cent from 0 to 50°, or 5 per cent from 50 to 100°.
During the last 20 years investigations have been carried out,
which give thoroughly satisfactory and concordant results, as is
shown by Table II in Appendix D. The improvement is due
to a happier disposition of apparatus for controlling the different
correction factors.

58

IS THE VISCOSITY A DEFINITE PHYSICAL QUANTITY? 59

Among gases air may be regarded as the standard substance
as water is among liquids. And if we compare the numerous
values for the viscosity of air obtained prior to 20 years ago, the
result is discouraging and has been often commented upon.
These values are given for 0° in Table XX.

Table XX.—Viscosity of Air at 0°C.

Method

<p

<p

Observer

Transpiration.

5,942

Graham (1846)

Oscillating disks.

5,325

Maxwell (1866)

Oscillating disks.

5,814

Meyer and Springmuhl (1873)

Oscillating disks.

5,590

Puluj (1874)

Oscillating disks.

5,556

Puluj (1874)

Transpiration.

5,854

Obermaycr (1875)

Oscillating disks.

5,489

Puluj (1876)

Transpiration.

5,951

Obermayer (1876)

Transpiration.

5,650

E. Wiedemann (1876)

Transpiration.

5,988

Obermayer (1876)

Transpiration.

5,952

Obermayer (1876)

Transpiration.

5,848

O. Meyer (1877)

Transpiration.

5,882

O. Meyer (1877)

Transpiration.

5,747

O. Meyer (1877)

Oscillating disk.

5,714

Puluj (1878)

Transpiration.

5,650

Hoffman (1884)

Oscillating disk.

5,955

Schumann (1884)

Oscillating disk.

5,838

Schneebeli (1885)

Oscillating cylinder....

5,831

Tomlinson (1886)

Transpiration.

5,770

Breitenbach (1899)

Oscillating cylinder....

5,659

F. Reynolds (1904)

Transpiration.

5,761

Tanzler (1906)

The transpiration method appears to give higher values for the
fluidity than are obtained by the other methods but the results
are not very consistent among themselves. However the follow¬
ing table of recent values for the viscosity of air at 15° is very
satisfactory.

We have the authority of Fisher (1909) page 150, for the state¬
ment that “No experimenter has made the attempt to apply
a correction to his measured pressures to allow for the kinetic
energy of the emerging gas.” It appears probable that the exist-

60

FLUIDITY AND PLASTICITY

Table XXL— Viscosity op Air at 15°

Method

<p

Observer

Transpiration.

5,507

Breitenbach (1899)

Transpiration..

5,502

Schultze (1901)

Transpiration.

5,528

Markowski (1904)

Transpiration.

5,502

Schmitt (1909)

Transpiration.

5,531

Knudsen (1909)

ing data might be improved by a critical study for the purpose
of making the needed corrections. It is a curious fact that the
kinetic energy correction has been so little understood and appre¬
ciated. Even in the case of liquids it is very commonly neglected
although it may amount to several per cent of the viscosity to
he measured. It is sometimes stated that no kinetic energy
correction is necessary when liquids how through a capillary from
one reservoir to another and not into the air. The Ostwald
viscometer (cf. p. 75), is used more than any other but it appears
that no kinetic energy correction is ever applied. It is true that
the instrument is used for relative measurements only, but this
fact does not cause this correction to be without effect in the cal¬
culation, for the reason that the kinetic energy correction is not
proportional to the viscosity.

There may be those who would maintain that the correction
for kinetic energy, as given above, is not the correct one to apply
to gases, c/. Fisher (1909). But that a correction is unnecessary
cannot he maintained even in the case of gases, in view of Hoff¬
mann^ work on the interrupted flow of gases. One of Ms capil¬
laries was cut into 28 pieces without loss. In flowing through the
interrupted capillary, the kinetic energy correction would be
increased twenty-eight fold, as has been already indicated on page
28. As a matter of fact the time of flow was considerably
greater in the interrupted capillary, proving the importance of the
correction. It would be particularly interesting to see whether
an experimental verification of the correction would be obtained
by an intensive study of the data of Hoffmann (1884).

The term “ specific viscosity ” has been very largely used and
may here receive brief comment. Water at 0° has been taken
as a standard with a specific viscosity of 100, but water at 25°

IS THE VISCOSITY A DEFINITE PHYSICAL QUANTITYf 61

has also been taken as the standard and equal to unity. Still
other standards have been employed. The principal advantages
in this form of expression are the saving of labor in calculation,
the avoidance of inconveniently small fractions, and the use of a
common liquid as standard. This advantage however is more
than offset by the disadvantages. The proper corrections are
never applied to specific viscosities and consequently the values
are not really comparable among themselves. Certainly they are
inconvenient to use for reference and for comparison with vis¬
cosities calculated in other ways. The time necessary for the
preparation of the substances is nearly always great enough to
justify the inconsiderable expenditure of time necessary for the
proper reduction of the data to absolute units.

Of course much depends upon the disposition of the apparatus
used in the measurement. Some forms of apparatus will not
permit accurate estimations of the viscosity to be made. But
given an apparatus which is well-suited for precise measurements,
the time required for making a measurement is no greater, and
may be much less, than in the less accurate forms of apparatus.

The Centipoise. —In expressing viscosities, it is possible to
secure simultaneously the advantage of expression in absolute
units with the advantages of viscosities relative to some common
substance as standard. It is proposed to name the absolute unit
of viscosity after Poiseuille the “poise,” and consequently the
submultiple of this unit which is one-hundredth as large the
“centipoise” (cp). It so happens that one centipoise is almost ex¬
actly the viscosity of water at 20°C, hence absolute viscosities
expressed in centiposes are also specific viscosities referred to water
20°C as standard. To be sure the viscosity of water is not ex¬
actly one centipoise at 20° C but it is 1.005 which is unity within
the limits of possible experimental error in ordinary measurement
(cf. Appendix D, Table II).

All fluidities are expressed in absolute units, water at 20°
having of course a fluidity of 100 units.

CHAPTER V

THE VISCOMETER

A very full discussion, has been given of the theory of the trans¬
piration method. Much matter of great historical interest in
regard to the various other methods has been passed over. This
has been done in order to present the matter which will be of
greatest use to the worker. At present the transpiration method
is by all odds the most important, this superiority being based
upon the following advantages: (1) It is susceptible of simple
mathematical treatment. (2) It is rapid. (3) Only a small
amount of fluid is required. (4) It can be used under the widest
variety of conditions as regards temperature, pressure et cetera .
(5) The preliminary measurements and adjustments are not
difficult to make. (6) Finally, it has been tested out most
thoroughly and found to be capable of the highest degree of
precision.

For certain purposes other methods must apparently continue
to be used. Thus the pendulum method seems best suited for
investigating superficial viscosity and the viscosity of solids like
steel. The fall method is of great use also for certain purposes.

It is quite impracticable to discuss here the almost innumerable
forms of instruments which have been suggested for use, so we
propose to consider briefly some of the instruments which have
shown the greatest advance toward meeting the conditions of
an ideal disposition of apparatus.

Naturally all transpiration instruments are based upon that
of Poiseuille; but for general purposes, his instrument was defi¬
cient, since the capillary terminated directly into the bath and
hence the apparatus had to be refilled after each measurement.
This difficulty was overcome in two forms of apparatus designed
by Pribram and Handl (1880), which consisted of a capillary placed
between two tubes of larger bore. In the better form shown in
Fig. 20, the two tubes are vertical, the capillary being bent.
The advantage of this arrangement is immediately apparent

62

THE VISCOMETER

63

because as soon as a measurement has been made in one direction,
the apparatus is ready for an observation in the opposite direc¬
tion. With this apparatus Pribram and Handl made very
numerous observations on organic liquids over a range of tempera¬
ture. They used a constant pressure head.

The apparatus of Bruckner (1891) marked another step in

advance. He used a horizontal capillary K,
Fig. 21, connected to the two limbs of the
apparatus by means of short pieces of rubber
tubing. Two reservoirs R and R f served for
the deposition of any dust particles that might
have found their way into the liquid. The
volumes of flow were accurately measured by
the volumes of the bulbs V and V', the tubes
leading from these bulbs being constricted

Fig. 20.—Viscometer of Fig. 21.—Viscometer of Bruckner.

Pribram and Handl.

in order to give a sharp reading. Either limb could be turned
to pressure at H or H\ or to air at Hi or H\.

In the study of organic liquids rubber connections become
objectionable, hence Thorpe and Rodger (1894) in their monu¬
mental work on the relation between the viscosity of liquids and
their chemical nature, employed an instrument, Fig. 22, similar
to that of Bruckner except that the capillary was placed inside
of a wider tube which was itself subsequently sealed to the two
limbs of the viscometer. The middle of this tube was heated at

64

FLUIDITY AND PLASTICITY

its middle point until it attached itself to the capillary all of the
way around, the greatest care being taken not to decrease the
diameter of the capillary or change it in any way.

Another marked improvement was the introduction of the
traps T 1 and T 2 , Fig. 22, for the purpose of easily adjusting the
total volume of liquid within the instrument, which they denoted
as the ‘ ‘ working volume ’ 7 to distinguish it from
] [ the volume of efflux V. By keeping the work¬

ing volume constant, the correction for the
hydrostatic pressure within the instrument is
greatly simplified.

Unfortunately Thorpe and Rodger's instru¬
ment has not come into general use. This is
Jl Jl probably due to the following disadvantages:

7 1 I I y 2 The sealing of the wide tube to the middle of

][ the capillary is difficult to accomplish; and

0 Q 3 according to Knibbs (1895) and Blanchard

m Jl IT 7 (1913) one cannot be sure that the bore of the

capillary has not been altered in spite of the
v utmost precaution. It is practicable to get

the dimensions of the capillary and the other

V y constants of the instrument only after the

JT 7 sealing has been completed. To get them then

1 f hr 2 is a ma ^ er some difficulty if not uncertainty.

The instrument is difficult to clean and dry
k K on account of the narrow spaces between the

( \—,,-A capillary and the wider tube at either side of

Y R p the constriction R. At the same time the

Pig. 22.—' Viacom- instrument is rather fragile. These difficulties

etet of Thorpe and may all be overcome by using ground glass
' Rodser ‘ joints between the capillary and the two

limbs. Over the ground-glass joints rubber tubing may be
stretched and tied and thus any danger of a leak guarded
against. The absence of a leak can be proven very easily at any
time by simply testing the working volume. By having good
ground-glass joints there can be no change in volume due to the
change in the expansion of the rubber under the changing head
and there can be no solvent action except that on the glass itself.
On the other hand, by planing the ends of the capillary off at

THE VISCOMETER

65

right angles to the axis of the tube, the dimensions of the capillary
may be most accurately ascertained. There is the further
advantage that other capillaries may he used without changing
the other constants of the apparatus. It is perhaps unneces¬
sary to add that this instrument must be modified to make it
suitable for the measurement of gases.

The Most Suitable Dimensions for a Viscometer. —As we
shall see in Part II, it is more convenient to compare fluidities
than viscosities. Combining Eqs. (8) and (12) we obtain for
the general formula for the fluidity of all fluids

__ _ SmVlt _ fork)

91 r 2 gpi 2 (R 4 + 4Xi£ 3 ) -mpV*’ { J

where R is the radius and l the length of the capillary in cen¬
timeters, and V is the volume in cubic centimeters, all being
reduced to the same temperature. No correction is necessary for
changes in these dimensions later since the changes just neutralize
each other as can be proved by introducing the coefficient
of expansion into the above formula. The pressure p, expressed
in grams per square centimeter, is the difference between the
absolute pressures at the level of the capillary at the entrance, Pi
and exit, P 2 . The time of flow in seconds is t, for the fluid whose
density is p at the temperature of observation T. r = 3.1416,
g is the acceleration due to gravitation, m = 1.12, X is the
coefficient of slipping, which is negligible for all liquids but is
of importance in rarefied gases, cjf. page 244. In the measure¬
ment of the fluidity of gases the volume of efflux must be cal¬
culated according to the formula of page 50.

V =

2Pi

Pi+P :

Fx =

2 P 2

Pi-bPs

F 2 .

where Vi is the volume of flow as measured before flow under
pressure Pa and Y 2 is the volume after expansion to the pressure

Pi.

By the proper choice of the dimensions of the apparatus
an accuracy of one-tenth of 1 per cent may probably be attained.
With a stop-watch reading to 0.2 sec. the time of flow may be
made as small as 200 sec. The volume of flow should be small
for the following reasons. (1) The kinetic energy correction
mpV 2 in Eq. (20) should be kept from becoming inconveniently

66

FLUIDITY AND PLASTICITY

large. (2) Tlie time of flow should be small for the sake of
economy and also that the temperature may be more readily kept
constant during the time of flow. (3) Small masses of fluid come
to the temperature of the bath more quickly, and (4) there is
an economy in material, which is sometimes very important.
The minimum of flow is determined by onr ability to read the
volume with the desired accuracy. This in turn is determined by
the diameter of the constricted portions of the instrument above
and below the measured volume V. If, however, the constricted
parts are of very small bore, the capillary action becomes dis¬
turbing. Yery viscous liquids will not drain out properly and
they may even form a meniscus across the capillary which will
prevent the transmission of the pressure and will render the
results quite valueless. It may he remarked that the troubles
due to had drainage may be minimized by having the drainage
surfaces everywhere as nearly vertical as possible. In other
words, the change from constricted portion to the tube of large
diameter should be made gradually. If the constricted part of
the instrument has an inside diameter of 0.25 cm we believe
that the capillary correction will not cause trouble. The
volume per centimeter of the constricted tube is then 0.05 ml
and if we assume that the meniscus can be read to 0.01 cm as it
passes a mark on the tube, it is only necessary to have a volume
of 0.5 ml to obtain the desired accuracy. To provide a margin
of safety in the construction and use of the apparatus we select
about 3 ml as the minimum.

To detect any error due to faulty drainage, it is only necessary
to test the flow of the most viscous liquid to be measured using
very different rates of transpiration by varying the pressure.
Lack of perfect drainage will be made evident, by the substance
appearing to be more viscous at the lower rate of flow. Natur¬
ally the more viscous liquids must be allowed to flow slowly
enough so that the drainage will appear to be perfect. If in the
instrument depicted in Fig. 23 the flow were to begin with the
upper meniscus at the point marked 3, it would be necessary for
all of the liquid of the measured volume V to have drained out at
the expiration of the time t. This is not necessary, however, if
the flow begins at some point considerably higher up, as for exam¬
ple in the neighborhood of the trap-opening F } for then a certain

68

FLUIDITY AND PLASTICITY

amount of liquid may flow into V from E after the record of the
time has begun, and this will tend to offset the effect of any liquid
left in V at the end of the time of flow. To make these amounts
as nearly equal as possible, the lower part of E should be exactly
similar in shape to the lower part of V.

The pressure should be variable at will so that the time of
flow may be kept reasonably constant. For gases, high pressures
are as unnecessary as they are undesirable. Tor incompressible
fluids, there need be no upper limit set to the pressure. A pres¬
sure of 50 g per square centimeter can easily be read to 0.1 per
cent on a water manometer, and the various pressure correc¬
tions—to be discussed—may be ascertained well within this
limit, hence this may be taken as a lower limit.

The measurement of the radius of the capillary offers the great¬
est difficulty in viscosity measurement by this method. Since
the flow is proportional to the fourth power of the radius, any
error in this measurement is multiplied four times. Careful
weighing of the quantity of mercury required to fill the tube is
perhaps the best means for obtaining the mean radius, R =
VWM l but for a capillary such as that used by Thorpe and
Rodger, Z = 4.94- cm R = 0.00824- cm, the weight of the mer¬
cury is only about 0.013 g so that the desired accuracy is diffi¬
cult to obtain with the ordinary balance. If the radius is
increased, the time of flow may be kept constant by increasing
the length so that the ratio l/R 4 is constant. Fortunately both
of these changes tend to increase the volume of the capillary.
At the same time the increase in length diminishes the effect
of any possible alteration in the stream lines near the ends; and
the increase in the radius diminishes the possible effect of slip¬
ping and probably also the effect of dust particles.

The formula (20) applies only to a capillary which has the
form of a true cylinder, but usually the capillary is elliptical and
it may at the same time be conical. To determine the coni city,
the tube must be calibrated with a mercury thread. To deter¬
mine the ratio of the axes, the micrometer microscope should
be used. In using the micrometer microscope it is somewhat
difficult to see the exact circumference to be measured, owing to
various causes. Poiseuille found it best to grind off and polish
the end of the tube and then attach a cover-slip to this end by

THE VISCOMETER

69

means of Canada balsam which is warmed slightly until it fills
the end of the capillary.

If the capillary is elliptical, R 4 in Eq. (20) must, according
to Rucker (cj*. Thorpe and Rodger (1893)), be given the value
2B Z C Z

where B and C are the major and minor axes of the ellip¬
tical cross-section. If the capillary is the frustrum of a circular
cone, Knibbs has shown that R A must be replaced by

SR^RS _

Ri 2 -f- R\R 2 4“ R 2 2

where R and R 2 are the radii of the two ends. If the capillary
is at the same time elliptical, E 4 becomes

3fl 3 3 fl4 3 (1 - e 2 )*

R 3 2 *4" RzRi 4" R* 2 1 4 s 2

where i? 3 and i2 4 represent the arithmetical means of the major

_ £Y

and minor radii at their respective ends, and e = g ._p ^ where

B and C represent the mean semi-axes. Knibbs has also con¬
sidered the corrections necessary for other peculiarities in the
bore of the tube which need not be considered here.

There is no special advantage in using a variety of viscometers
for liquids of not very different fluidity. Tor liquids below the
boiling-point the fluidity never exceeds about 500. Assuming
this value as the maximum the lengths necessary for a capillary
of a given radius have been calculated by means of Eq. (5) and
plotted curve A in Eig. 24. It is not always possible to obtain
a capillary of an exactly specified radius, but with one having an
approximately satisfactory radius, the necessary length can be
read off from’ the curve. For gases the maximum fluidity must
be taken as 10,000. If only very viscous liquids are to be meas¬
ured the maximum may be taken as less than 500, curve B or C.
(cf. also Appendix A, Table IX.)

Construction and Calibration of Apparatus. —A point of great
importance in the construction of the viscometer is to have the
volume V (1) as nearly equal to that of V' as possible, (2)
similar to it in shape, and (3) at the same height from the hori¬
zontal capillary. This construction greatly facilitates the esti¬
mation of the correction for hydrostatic pressure, within the

70

FLUIDITY AND PLASTICITY

instrument. Finally the small bulbs B, E, B f , and E* should
have nearly the same volume. By having the surfaces nowhere
depart greatly from the vertical, the drainage is improved. It
is impracticable however to use long, cylindrical bulbs, since
then the true average pressure, due to the hydrostatic head within
the instrument, becomes awkward to determine. (Cf. Appendix
A, page 298.) The best form for the bulbs V and V' is therefore
obtained by making them so that each resembles as much as

Fig. 24.—Chart for use of instrument maker in selecting capillary for vis¬
cometer, knowing the approximate radius of the capillary and the maximum
fluidity to be measured, the length to be used may be read off. V =3ml t =200
sec., p =50 g per cm 2 .

possible a pair of hollow cones, placed base to base as shown in
Fig. 23.

The marks at l and V are so placed that the volume from l to
F r is exactly equal to that from V to F. If the two limbs of the
apparatus are similar there will be no correction for capillarity.
Poiseuille has given a method for estimating this correction when
that is necessary. The volumes V and V' may be easily deter¬
mined by the weight of volumes of mercury.

The appearance of the complete apparatus used by Thorpe
and Rodger is shown in Fig. 25. The viscometer is shown in the
bath B which has transparent sides. Water in the vessel R
exerts pressure upon the air in the large reservoir L. The gas

72

FLUIDITY AND PLASTICITY

is dried by passing over sulfuric acid in a smaller bottle \ ^
whence tubes lead to the three-way stop cocks Z and Z' ai
thence to the two limbs of the viscometer. The pressure
measured on the water manometer D. The bath is stirred
means of a motor connected with the mechanism shown al /
Since the fluidity of a substance like water changes from 1 in
per cent with a change of 1° in the temperature, it is neceswtf v
that the temperature be controlled to a few hundredths of r *

degree. Since they were workis "

A over a wide range of temperaluy

Thorpe and Rodger controlled ft- r
temperature by hand.

A word may be added here :
regard to stop-watches. The c <tu¬
rnon form of stop-watch in whi* -
: the whole mechanism starts or st<q ^

j simultaneously with the time recur'

# may not give consistent results, ev*

u though it appears to neither v

SP nor lose during a long period of tiny
|| This is the fault of the mechanic
„ . . , The watches whose movements ei**-

Fig. 26.—Apparatus of

Thorpe and Rodger for obtain- tinue, whether the time is
mg dust-free liquid. recorded or not, seem to be fry *

from this defect.

The Measurement.— In preparing substances for measurem*i« **•
as well as in cleaning and drying the instrument, many invent *
gators have strongly emphasized the importance of avoiding
presence of dust particles. Both Poiseuille and Thorpe itiyl
Rodger took elaborate precaution in this regard. Figure ##'*■
shows the apparatus used by the latter for distilling pure liquid**

It has the advantage of allowing a good determination of fly
boiling-point to be made while the liquid is being fractional r*f
To avoid contamination by dust and moisture in filling the \ r*
eometer, Thorpe and Rodger used a special apparatus, Fig. 27
The liquid was placed in the bottle H and forced over into f ft ^
right limb of the viscometer M by means of the pressure of
mercury head A . The viscometer was held in a frame afc !
supported on the vertical rod by means of the setscrew X

TEE VISCOMETER

73

The left limb of the viscometer was evacuated by means of
the mercury head Q in order to draw the liquid through the
capillary.

Having run in a little more than the required amount of
liquid, the viscometer and frame were placed in the bath B of
Fig- 25 and the limbs of the viscometer were connected to the

pressure outlets on either side.
With the temperature main¬
tained constant at the lowest
point at which measurements
were desired, the cock Z f (or Z)
was turned to air and the cock Z
(or Z r ) to pressure. As the liquid
rose in the left limb, it finally
overran into the trap T/ Fig. 22.
At the instant that the meniscus
in the right limb reached the
point h 2 , the cock Z was turned
to air. Thus the working volume
was adjusted.

A measurement of the fluidity
is made by turning the cock Z'
to pressure and immediately read¬
ing the pressure on the manom¬
eter as well as the temperature of
the manometer, while the liquid
is flowing out of the bulb F. As
the meniscus passes the point
m’ the time recorded is begun.
Keeping the temperature constant
the time is taken as the meniscus

passes the point m 2 . The pres- Fig. 27.—Filling device of Thorpe
. , and Rodger.

sure is then read as before, and

before the meniscus reaches the point h f the left limb is again
turned to air. The apparatus is then ready for a duplicate

observation in the opposite direction.

The Calculation.—The corrections to the time and temperature
are not peculiar to viscosity measurements and need no special
comment. In obtaining the pressure, several corrections must

74

FLUIDITY AND PLASTICITY

be made. (1) The pressure on the manometer must be calculated
to grams per square centimeter from the known height of the
liquid and its specific gravity at the temperature observed. A
correction to the observed height of the liquid is avoided by
having the long limb of the manometer doubly bent at its middle
point so that the upper half is vertical and in the same straight
line with the lower limb of the manometer. The levels on both
limbs may then be read on the same scale, which may con¬
veniently consist of a steel tape mounted on a strip of plate-glass
mirror placed vertically. Similarly a correction for capillary
action may be avoided if the bore of the manometer is large
enough so that it may be assumed to be uniform. (2) The pres¬
sure must be corrected for the weight of the air displaced by the
liquid in the manometer. (3) Unless the surface of the liquid in
the lower limb of the manometer is at the same height as the
average level of the liquid in the viscometer, a correction must be
made for the greater density of this enclosed air, than of the
outside air which is not under pressure. (4) Finally a correction
must be made for the average resultant hydrostatic head of the
liquid within the viscometer. If the two volumes V and V'
in Tig. 23 are exactly equal in volume, similar in shape, and
at the same elevation above the capillary, when the viscometer is
in position, in the hath, it is evident that the gain in head during
the first half of the flow will be exactly neutralized by the loss
in head during the last half of the flow. Since this cannot be
exactly realized, a correction may be made as follows: Duplicate
observations in reverse directions are made upon a liquid of
known density and viscosity at a constant temperature and
pressure. Let U be the time of flow from left to right and U the
corresponding time from right to left. Let be the pressure as
corrected, except for the average resultant head of liquid in the
viscometer. Suppose this latter correction to amount to x cm
of the liquid as the liquid flows from left to right. In this case
the total pressure becomes equal to p 0 + and when the
liquid flows from right to left, it becomes equal to p 0 — px.
Since Eq. (8) when used for a given viscometer may be written
in the form

y = Cpt — C'p/t

( 22 )

THE VISCOMETER

75

where C and C' are constants, which can be calculated, we obtain

V -f C'pf h

Po + =

Vo - pv =

Ck

v -f C'pf u
Ct 2

whence,

= JL[1 - ll

x 2CpUi <J " r 2CLii 2 <2 2 J

In subsequent calculations it is necessary to know the specific
gravity of the liquid whose viscosity is desired, in order to make,
the necessary pressure correction and in order to make the kinetic
energy correction, but it is to be noted that if the
instrument has been constructed with that end in
view, these corrections will both be small, and there¬
fore the specific gravity need be only approximately
known, which is a great advantage.

Relative Viscosity Measurement. —On account
of the labor involved in obtaining the dimensions
of the viscometer, many investigators have followed
the example of Pribram and Handl in disregarding
these dimensions, and calibrating the instrument
with some standard liquid. The most important
instrument of this class, is that of Ostwald, Fig. 28.

It consists essentially of a U-tube with a capillary
in the middle of one limb above which is placed a
bulb. A given volume of liquid is placed in the
instrument and the time measured that is required
for the meniscus to pass two marks one above and
one below the bulb under the influence of the hydrostatic
pressure of the liquid only.

If 770 is the viscosity of the standard liquid and 77 that of liquid
to be measured, we have from Eq. (22)

1 = c & ~~ CW*

Va Cpot 0 — C'po/tc’

and if tj is very nearly equal to rjo or if t and t 0 are very large, this
may be written

n _ Pi }

Vo

Fig. 28 .—
The Ostwald
viscometer.

( 23 )

76

FLUIDITY AND PLASTICITY

Pig. 29 . —Viscom¬
eter suitable for the
relative measure¬
ment of not too
viscous liquids.

The pressure in this instrument must be
proportional to the densities so that
2i _ pt
Vo PoU

which is the formula suggested by Ostwald.
The formula is true for dilute solutions when
water is taken, as the standard, for rj is then
nearly equal to %.

It is inconvenient to make the time of flow
very large both on account of the lack of
economy and because of the increased danger
of clogging. Unfortunately this formula has
been used where neither of the necessary con¬
ditions was complied with and the results are
therefore of uncertain value. It is much
better to make the correction for the kinetic
energy, in such cases, than to attempt to make
the correction negligible.

It is a disadvantage of the Ostwald instru¬
ment that the pressure is not variable at will,
because if the time of flow is sufficient in one
liquid, in another more viscous liquid the time
of flow may be intolerably long, practically
necessitating the use of a variety of instru¬
ments. Furthermore the total pressure is so
small that a small error in the working volume
may introduce considerable error into the
result and the density of the liquid must be
known with considerable accuracy.

A. form of instrument which has the mani¬
fest advantages of the Ostwald instrument
and overcomes the above objections is shown
in Fig. 29 . The volume K is made as nearly
as possible equal in volume, similar in shape,
and at the same height as C. The working
volume is contained between A and H and
the volume of flow between B and D, the
measurement being made as the meniscus
passes either from B to D or from D to B

THE VISCOMETER

77

depending upon the direction of the flow. The corrections are
made as for absolute measurements and the viscosity calculated
from formula (22). In obtaining the pressure correction due to
the average resultant hydrostatic pressure in the viscometer C'
can be estimated accurately enough by means of rough measure¬
ments. The value of C can be
obtained accurately enough for the ^^ |u

calculation of this correction by ,

* i /% i |~0~* -Tf

assuming p 0 = p. After obtaining i

the value of the hydrostatic head x ®l p U

. . i „ -S Top View tit

m this way, the true value of C may |

be calculated from an observation I

upon the time of flow of any liquid ^li®) jp 1
whose viscosity is accurately known. 1

In the use of any relative instru- section w-m. |
ment, it is important that two stand- p 1

ards be employed so as to obtain a

check upon the method. For this B

purpose a single liquid may be used ll ||

at widely different temperatures or It n B

two or more liquids may be used of sedionm-iz ^ ^

widely different viscosities. While |nl D a

this test is very simple and its I 1

importance is obvious, it does not I 1

appear to have been frequently 1 LU

employed. |||® 5ectl0n 111

Viscosity Measurements of secfionv-Yi
Liquids above the Boiling-point.— Fig. 30.—Piastometer. For use

If the viscosity of liquids is to be ^^^ 8 J i80O,lB 01 with plasti °
measured above the ordinary boiling

temperature, one must work at pressures above the atmospheric
pressure. The three-way cocks in Fig. 22 must lead to a low-
pressure reservoir, this pressure being measured by a second
manometer. The rubber connections must of course be replaced
by others capable of withstanding the desired pressure.

Viscosity Measurement of Very Viscous Substances.—Sub¬
stances like pitch which are excessively viscous can yet be
measured by the efflux method by the use of very great pressure
(c/. Barus ( 1893 )). On account of the lack of proper drainage,

Fig. 30. —Piastometer. For use
■with very viscous or with plastic
substances.

78

FLUIDITY AND PLASTICITY

the apparatus described above is unsuited. But in this case the
volume may very properly be obtained from the weight of the
efflux into air, because the effect of surface tension would he

negligible at these high pressures. A viscometer designed for*
very viscous substances is shown in Big. 30 . The use of this
form of apparatus is described in detail in connection with,
plastic flow (qf. Appendix B, p. 320 ).

The Viscosity Measurement of Gases.—A very satisfactory
apparatus for the measurement of the viscosity of gases by the

THE VISCOMETER

79

efflux method has been worked out through the labors of Graham
(1846-1861), 0. E. Meyer (1866-1873), Puluj (1876), E. Wiede¬
mann (1876), Breitenbach (1899), and Schultze (1901). We may
describe briefly the form used by Schultze as illustrating the
modifications which are necessary in the apparatus used for
liquids. In Pig. 31 the glass capillary, 1 = 52.54 cm, R =
0.007572 cm, is contained in the upper chamber of the bath /,
which is maintained at constant temperature by water, water
vapor, or aniline vapor. A. condenser is shown at 6 and SS is a
shield to protect the rest of the apparatus from the radiation.
On either side of the bath the apparatus is exactly similar, so that
only the right side is shown in the figure. The gas is contained
in the bulbs P and Q (and P r and Q' on the left side) surrounded
by a separate bath. The lower bulbs are each connected with
two stop cocks B and C (or B* and C'); from B (or B') a rubber
tube leads to the mercury reservoir G (or G'), and from C (or C')
there is a glass tube drawn out into a capillary. Adj acent to both
the capillary and the bulbs, considerable lengths of glass tubing
are put in connection and immersed in the respective baths in
order that the gas in the capillary or bulbs may be at the desired
temperature at the time of measurement. In each tube leading
from the bulbs to the capillary there is a stop cock A (and A')
and a connection with a manometer K (and A 7 ). By means of
stop cocks at E and E' the two manometers may he connected
together or gas admitted to the apparatus from outside. Since
the presence of water vapor is objectionable and gases are
more or less soluble in water, the manometer contains both mer¬
cury and water, and is calibrated before use.

In makin'g a measurement, enough gas is admitted into
the evacuated apparatus so that at atmospheric pressure, the
surface of the mercury is in the lower part of the bulb Q and in
the middle part of the bulb Q\ The stop cock A is then closed
and the mercury reservoirs G and Q* raised, but the former
enough higher than the latter so that a pressure head is estab¬
lished which is a few millimeters greater than is desired in the
measurement. The mercury fills the two bulbs Q and Q'. When
the temperature is constant the stop cock A is opened. The
pressure is immediately adjusted and thereafter maintained
constant by means of the screws F and F' which serve to slowly

80

FLUIDITY AND PLASTICITY

raise or lower the mercury reservoirs. When the mercury passes
into the bulb P, contact is formed with a platinum point and an
electrical signal given. At this moment the chronometer is
started. After the elapse of sufficient time, the stop cock B is
closed and thus the current is broken between the two platinum
electrodes at either side of this stop cock, and a signal is given.
The mercury is now allowed to run out through the stop cock C
until the signal is given when the mercury loses connection with
the platinum point in the bulb P. From the weight of this
mercury, the volume of flow is calculated.

PART II

FLUIDITY AND OTHER PHYSICAL AND
CHEMICAL PROPERTIES

CHAPTER I

VISCOSITY AND FLUIDITY

It has been tacitly assumed by the great majority of workers
that when two liquids are mixed, the viscosity of the mixture
is normally a linear function of the composition. This appeared
as early as 1876 in the work of Wijkander. In a great many
mixtures, including practically all of those in which water is a
component, the viscosity is certainly very far from being a linear
function of the composition, there being often a maximum in the
viscosity curves. However water mixtures should not be con¬
sidered as “normal,” but since it is difficult to decide what shall
be considered normal mixtures, the question whether the
viscosities are additive or not is admittedly difficult of solution.
Dunstan (1905) classifies as normal those mixtures whose vis¬
cosity-weight concentration curves do not show a maximum or
a minimum. This classification is not satisfactory not only
because it lacks a theoretical justification but also because many
of the so-defined normal mixtures give curves which depart
considerably from the linear, so that the suspicion is aroused that
the occurrence of a maximum or minimum may depend upon
accidental circumstances such as the nearness to equality of the
viscosity of the components. The accidental character of such
a classification is very striking in mixtures which fall into the
normal class at one temperature but at a slightly different tem¬
perature must be classified as abnormal.

Such light as can be gained from a study of the viscosities
of mixtures, seems to lead to the conclusion that viscosities are
not additive, as has been assumed. Thus Dunstan (1904)
remarks, “The law of mixtures is never accurately obeyed and
6 81

82

FLUIDITY AND PLASTICITY

divergences from it seem to be more clearly marked out in the
case of viscosity than with other properties, such as refractive
index,” Thorpe and Rodger (1897) say, “The observations
described in this paper afford additional evidence of the fact indi¬
cated by Wijkander and supported by Linebarger, that the vis¬
cosity of a mixture of miscible and chemically indifferent liquids
is rarely, if ever, under all conditions, a linear function of the
composition. It seldom happens that the liquid in a mixture
preserves the particular viscosity it posesses in the unmixei
condition. To judge from the instances heretofore studied, the
viscosity of the mixture is, as a rule, uniformly lower than the
mixture law would indicate, but no simple relation can yet be
traced between the viscosity of a mixture and that of its constit¬
uents.” Thorpe and Rodger were so struck by the absence of
linearity in the viscosity curves, that they thought that an ex¬
planation was needed for the fact that the viscosity curves of
some mixtures measured by linebarger (1896) are indeed linear.
“The observed viscosities in general are less than those calculated
by the mixture rule, except, possibly, in the case of mixtures of
benzene and chloroform and mixtures of carbon disulfide with
benzene, toluene, ether, and acetic ether, where, possibly, the
temperature of observation (25°) was too near the boiling-point
of the carbon disulfide to make any specific influence, which that
liquid might exert at lower temperatures, perceptible /' 1

lees (1900) showed what are the necessary assumptions in
regard to the nature of flow in mixtures, so that the viscosities
should be additive, but by making a careful study of existing
data, he found little justification for these assumptions. Simi¬
larly lees tried the assumptions that fluidities or logarithmic
viscosities are the characteristic additive property, but he was
unable to obtain a satisfactory verification of either from the
experimental results.

The question before us seems to be: “Is viscosity or fluidity
or some function of one of them the characteristic additive prop¬
erty?” The answer to this question is imperative before we
can intelligently discuss the relation of viscosity to other proper¬
ties. This statement requires no proof in view of the statements
which we have quoted to show that in some cases the viscosity
concentration curve is linear according to assumption, but in the

VISCOSITY AND FLUIDITY

83

great majority of cases it is sagged and there is no known law to
account for the peculiarity. Surely any discussion of chemical
combination or of dissociation on the basis of deviation from the
"normal ” curve under such conditions would be of very uncertain
value.

There are numerous reciprocal relations besides viscosity and
fluidity, such as electrical resistance and conductance, or specific
heat and heat capacity, or specific gravity and specific volume.
It has been repeatedly pointed out 1 that if one of these is additive,
its reciprocal cannot be. It is singular enough that among all
of these reciprocal relations, viscosity is the only one for which
the decision has not been reached as to whether viscosity is
additive or not, or if it is, under what conditions. In electricity
for example we have absolutely no doubt but that resistances
are additive under certain conditions, viz., when the conductors
are in series, and likewise that conductances are additive under
other equally definite conditions, viz., when the conductors
are in parallel. It seems probable that the present unsatisfactory
condition as regards viscosity has arisen due to the extraordinary
sensitiveness of this property to molecular changes in fluids,
either combination or dissociation. We shall attempt to reach
a solution of the problem from a consideration of the nature of
viscous flow and then test this solution by means of the experi¬
mental f acts. After we have reached a conclusion in regard to the
true additive property under given conditions, it may well turn
out that the present unsatisfactory condition will prove to be a
blessing in disguise, for it may then be shown that viscosity is of
the greatest importance in physiochemical investigations.

The fundamental law of viscous flow

dv _ F
dr 7}

is the analogue of the well-known electrical law of Ohm. In
fact Elie in 1882 suggested a modification of the Wheatstone's
bridge method for the measurement of viscous resistance.

Case I. Viscosities Additive—Emulsions.—We will first con¬
sider the very simple case of a series of vertical lamellae of viscous
material arranged alternately, as in Tig. 32, and subj ected to a

FLUIDITY AND PLASTICITY

t- ’

84|

horizontal shearing stress. For convenience suppose that all
of the lamellae j$f the one substance A have the same thickness
&i and that the laminae of the substance B have- the uniform
thickness $ 2 , etc. Let the viscosities of the substances be 771 ,
ri 2 . . . and the shearing stress# per unit area pi, . - .
respectively; then if R is the distance between the horizontal
planes, the velocity of the moving surface is

—

" “ ~H ~ V ~ In’

where H is the viscosity of the mixture, and P is the average
shearing stress over the entire distance S .
D } But

PS = pi$i + P 2 S 2 + . . . ,
hence

u R (P 1 S 1 + P 2 S 2 + . . A

v \ S ) m

Fig. 32.—Diagram to illus- .

trate additive viscosities. But since Si/b is the fraction by volume
of the substance A present in the mixture,
which we may designate a, and similarly s%/S = b, etc.,

H = arji -f by 2 + . '. . (24)

This case is of particular interest in connection with emulsions
and many other poorly mixed substances. The formula tells
us that the viscosity of the mixture is the sum of the partial
viscosities of the components, provided that the drops of the
emulsion completely fill the capillary space through which the
flow is taking place.

Case n. Fluidities Additive—Fluid Mixtures.—If the lamellae
are arranged parallel to the direction of shear, as shown in Fig.
33, we have a constant shearing stress, so that

yv 1 _ w
n r 2

(24a)

where v 1} v 2 , . . . are the partial velocities as indicated in the
figure.

There are two different ways of defining the viscosity of a
mixture, and it becomes necessary for us to adopt one of these
before we proceed further.

1 . If we measure viscosity with a viscometer of the Coulomb

VISCOSITY AND FLUIDITY

85

or disk type, we actually measure the velocity v, BS in the figure,
and we very naturally assume that

2 . It is more usual, however, to calculate the viscosity from
the volume of flow, as in the Poiseuille type of instrument,
let v f , £ S' in the figure, be the effective velocity which the
surface £8 would have, were the series of lamellae replaced by

a homogeneous fluid having the same volume of flow. The
effective velocity is related to the quantity of fluid U passing per
second in a stream of unit width, as follows :

Let the viscosity as calculated from the flow,
geneous fluid, be H then

W 2H'U

1 “ R R 2

as for a homo-
(24b)

It is to be noted that had the less viscous substance been in
contact with the surface A E, the effective velocity of flow would
have been represented by the distance BS". We shall take the
former of these for our definition of the viscosity of a mixture,

86

FLUIDITY AND PLASTICITY

since, as we shall now show, by using it the viscosity is indepen¬
dent of the number or arrangement of the lamellae.

Since v = Vi + v 2 + . . .
we obtain from Eqs. (24a) and (24b) that

PjB4> = P(ri<pi +r 2 <p2 + . . .)

or since 0 = % b = etc.

lb JlC

the fluidity of the mixture is

= Q<Pi "f” . . . (25)

The fluidities are, according to this definition, strictly additive
and entirely independent of the number and arrangement of
the layers. Since, however, the viscosities are usually calculated
by means of the Poiseuille formula based on the volume of flow,
it is important to determine for a given arrangement of lamellae
what correction must be made to the effective viscosity, as calcu¬
lated from the volume of flow, to make it accord with the true
viscosity, as defined above and as obtained by the disk or other
similar method for the measurement of viscosity.

Reverting again to the figure, we find that

I t ^2' 2

+ Vir 2 + -g-
+ viTx + v 2 ri H—g~

+ v x r 2 + v 2 r 2 + Vi r 2 + ~ 2 ~

If there were n pairs of alternate lamellae of the two substances
A and B

U = An 2 v iri + n(n + l)vir 2 + m\r 2 + n(n — l)^] (26).

Since n = ~ — - ■, on substituting into Eq. (26) the values of

Vi and v 2 , we get

'[ ' a< Pl + + ~(<Pl ”“^2)J

and if 4> ; = ~, we obtain from Eq. (24b)

Jtl

$ l = atpi + btpi + ~(<Pi — V2)

( 27 )

VISCOSITY AND FLUIDITY

87

and when n = °°, the fluidity becomes simply

cup i -j" bcp 2

and in this case

& = (28)

In a homogeneous mixture it appears, therefore, that the two
definitions lead to the same fluidity, and experimental results
lead us to believe that this is the case usually presented in liquid
mixtures, since the disk method and the capillary tube method
give the same fluidity so far as we
have certain knowledge. If, however,
the number of lamellae is small, which
may well be the case in very imperfect
mixtures, or when the flow takes place
through very narrow passages, the
effective fluidity as calculated from
the volume of flow may be either
greater or less than the sum of the
partial fluidities of the components,
depending upon the order of the
arrangement of the lamellae in refer¬
ence to the stationary, surface. The
amount to be added or subtracted
from the effective fluidity in order to
obtain the true fluidity is represented
by the term, corresponding to the
areas ACD , etc. or AFD, etc., Fig. 33.

A combination of the cases I and II would lead to a checker¬
board arrangement, but it may be shown now that such an
arrangement tends to reduce itself to the case II where fluidities
are additive.

If the arrangement considered in Fig. 32 is subjected to
continued shearing stress, the lamellae will tend to become
indefinitely elongated as indicated in Fig. 34; and unless the
surface tension intervenes, as may be' the case in immiscible
liquids, the lamellae will approach more and more nearly the
horizontal position. Thus, so far as we can determine without
going into the complicated problem of the molecular motions,
it seems certain that the fluidities will become more and more

Fig. 34. —Diagram to illus¬
trate how, in incompletely
mixed but miscible fluids, flow
necessarily brings about com¬
plete mixing, so that even when
the viscosities were originally
additive the fluidities finally
become additive. In immis¬
cible fluids, the layers A and
B resist indefinite extension
and emulsions are the result.

88

FLUIDITY AND PLASTICITY

nearly additive as the flow progresses and the mixture becomes
more and more nearly complete. This result takes place further¬
more irrespective of the original arrangement of the parts of the
mixture.

Some one may object that a perfectly homogeneous mixture
—in itself a contradiction of terms—is not made up of layers
such as we have considered in these greatly simplified cases.
There can be no doubt whatever of the existence of layers during
the process of mixing. No one has watched the drifting of
tobacco smoke in his study without noting how it is drawn out
into gossamer-like layers. 1 Since the fluidity is least when
fluidities are additive, there would have to be a sudden drop in
fluidity as the mixture became perfect, if the fluidities were no
longer additive. This is not supported by any experimental
evidence.

We have already noted that when there is no chemical action
between the components of a mixture, the viscosity-concentration
curves are usually but not always sagged. Dunstan (1913) has
put it: “It can therefore safely be predicted that wherever the
two components show little tendency for chemical union a
sagged curve, or one departing hut slightly from linearity, will
be found.” If the fluidities of such mixtures are additive, these
facts ought to be accounted for by the theory, peculiar as they
may seem to he. We shall first prove that according to the
theory that fluidities are additive, we should expect the viscosity-
concentration curves to be sagged.

Equations (25) and (24) represent the two assumptions
that fluidities are additive and that viscosities are additive
respectively but for convenience we shall assume that only two
components are present in the mixture. From Eq. (23) we get
that

1 _ _a_ b_

<fi’ <Pl <P2

b<fii -j- o,(p2

When a = 0 or 1, and b = 1 or 0 respectively, must be equal to
<f>'. For all intermediate values of a, and l we desire to learn

VISCOSITY AND FLUIDITY

89

whether <p must be invariably greater than, equal to, or less
then V'* Multiplying Eq. (25) by unity, we obtain

__ {ci<Pi + b(p 2 ) ( b<pi + d(p 2 )
b<pi + ®<p2

— ( a2 + b 2 )<Pi<P2 + ab((pi 2 +• <p2 2 ) y <pi<p2 _ ,
bcpi -f- dept ^ b(pi -f- d<p2 ^

Since b = 1 — a,

2 a(a — l)(pi<p2 + d{l — a) ((pi 2 + <p2 2 )^0

Discarding the known roots, a — 0 and a = 1 , we get
(p\ 2 — 2<P\(p2 + (p 2 2 ^ 0,

which is a perfect square and therefore must be positive. Hence,
when cpi is equal to <p 2 , <p is equal to <p f , but for all other values
(p must be greater than Our conclusions may be stated as
follows:

1 . The viscosity of a thorough mixture of chemically indifferent
fluids must always be less than would be expected on the assump¬
tion that viscosities are additive, but this inequality will approach
zero as the difference between the viscosities of the components
approaches zero.

2 . So, on the other hand, the viscosity of an emulsion must be
greater than that of a perfect mixture of the same composition,
because in emulsions the viscosities tend to become additive.

Equation (25) may be expressed in the form

(p — <pi + (^2 — (pi)b , (29)

where <pi and (p 2 are constant and (p and b are variable. The'
corresponding viscosity equation is

77 <pi + (<P2 ~ <pi)b ^ ^

It is important to note that Eq. (29) is the equation of a
straight line, but that Eq. (30) is the equation of a hyperbola.
If we replace <p\ +(<p 2 — ^ 1)62 by (<p% — <pi)m } where

which is the equation of an equilateral hyperbola, whose X-axis is

90

FLUIDITY AND PLASTICITY

at a distance <pi/(<p 2 — <pi) to the left of the origin to which.
Eq. (27) is referred.

3. We conclude therefore that the curve obtained by plotting
viscosities against volume concentrations is not normally linear
but a part of an equilateral hyperbola.

From Eq. (30)-we find the curvature for any mixture to be

if __ 2(<P2 — <P\) 2 [<Pi + ( < P2 — <?Ui)i ] 3 _

* “ {[*>1 + (?* “ ^i)a] 4 + (*> a - ^i) 2 } g C j

By differentiating this curvature in respect to the concentration
a and equating to zero, we find the concentration where the
curvature is a maximum to be

<p 2 — <pl

Substituting this value in Eq. (31), the amount of the curvature
where the curvature is a maximum, is found to be

4. The curvature of the viscosity-volume concentration curves
is greatest when the difference between the fluidities, i.e., <pz — <pi }
is large, and becomes zero when <p 2 — <p i = 0 .

5. The curvature must continually decrease as the concentra¬
tion increases unless the square root of <p 2 — (pi is greater than <pi,
in which case the point of greatest curvature will be found at
some positive concentration (cf. Eq. (32)).

6 . Mathematically considered, the curvature is dependent only
upon the difference in the fluidities of the components, i.e.,
<P 2 — <pi and not upon <pi, but since we can only realize positive
values of a\, it follows that for a given value of <p 2 — <pi the
curvature at any concentration will be greatest when <pi is very
small.

Liquid Mixtures

The first conclusion is confirmed by the repeated observation
that viscosity-concentration curves of homogeneous mixtures are
normally sagged. The fourth conclusion offers an explanation
of the fact that they are sometimes very nearly linear. In
particular, this conclusion is confirmed by Thorpe and Rodger,
who, in commenting on the data of Linebarger, make the interest-

VISCOSITY AND FLUIDITY

91

mg observation “As a rule, the greater the difference between the
viscosities of the pure liquids, the greater is the difference between
the calculated and the observed values of the mixtures. 77 The
second conclusion is confirmed by the sudden drop in the fluidity
of a mixture as it is cooled below its critical-solution tempera¬
ture. This has often been noted and commented upon, and will
be discussed more fully at a later point. In undercooled liquids

Weight percentage of second component of mixture.

Fig. 35.—1. Fluidity curve of nitrobenzene and etliyi acetate at 25°; 2.
Fluidity curve of ethyl alcohol and acetone at 25°; 3. Fluidity curve of benzene
and ethyl acetate at 25°; 4. Fluidity curve of benzene and ethyl ether at 25°;
5. Fluidity curve of carbon bisulphide and ethyl ether at 25°; 2rj. Viscosity curve
(dotted) of ethyl alcohol and acetone at 25°. Were viscosities additive, this
curve would be linear (dashes).

and other very viscous substances it has been often noted that the
viscosity curves have a very high degree of curvature, at least
during a part of their course. This is in harmony with the fifth
and sixth conclusions.

Were the fluidity-volume concentration curves invariably
linear, it would constitute an experimental verification of our
fourth conclusion. Unfortunately for this purpose, the fluidity
concentration curves are rarely, if ever, perfectly linear, for the
reason that has been indicated; viz,- that there is perhaps nearly
always some molecular change on mixing, even though very
feeble, and to this change fluidity is very sensitive. These

92

FLUIDITY AND PLASTICITY

changes are of no interest to us at this point, but the fact is very
important for us that the fluidity-volume concentration curves
are much more nearly linear than the viscosity concentration
curves. To become convinced of this the reader should plot
the fluidity concentration curves of several of the mixtures given
by Wijkander (1878), Linebarger (1896), Dunstan et cet. (1904)
and others and compare them with the viscosity curves given
by those authors. A few of these curves are given in Fig. 35. 1
Kendall (1913) has gone over the whole range of available data
and finds that the percentage deviation between the observed
and calculated values is 11.1 for the viscosity curves and 3.4 for
the fluidity curves.

Fluidity and Temperature

The conclusion that fluidities are additive has far-reaching
consequences, so that there arise tests for the conclusion which
were at first quite unsuspected. For example, it is evident that
the reasoning which has been found to hold for mixtures of fluids
must also hold for mixtures of the same fluid at different tempera¬
tures; for a fluid at any temperature may be thought of as a
mixture of appropriate amounts of portions of the fluid main¬
tained at the extreme temperatures. Hence, we are led to the
hypothesis that the fluidity-temperature curves of pure fluids
should normally be linear, and the viscosity-temperature curves
hyperbolic. This relation cannot hold through a change of state
because a new cause of viscosity then enters in. Furthermore the
fluidity of liquids is closely related to their volumes, as we shall
see later, and the volumes of liquids do not generally increase in
a linear manner with the temperature. Then, too, association and
dissociation may play a disturbing factor, so that as in mixtures,
a perfect verification can scarcely be expected. In Fig. 36 there
are given the fluidity-temperature and viscosity-temperature
curves of mercury and of water from 0 to 100°C. Both of the

1 We have here but a rough test of the truth of the hypothesis that
fluidites are additive in homogeneous mixtures, because the fluidites of the
components are too close together, all of the components are certainly not
inert, and volume concentrations should have been studied. More rigorous
tests of the hypothesis will be made after the law of Batschinski has been
considered.

VISCOSITY AND FLUIDITY

93

fluidity curves are much more nearly linear than are the viscosit
curves, the true linear curve being represented in each case bv ^
series of dashes. Mercury is an ideal substance in this conn e{ f*
tion for it is far removed from the critical temperature, it is no t
highly associated, and its volume increases in a linear manner
with the temperature. 1 The fluidity curve is almost perfect!
linear, what curvature there is being in a direction opposite to
that of every other known substance, so that it can hardly be

Temperature Centigrade.

Fig. 36. —Fluidity (continuous) and viscosity (dotted) temperature curves for

mercury and water.

regarded as certain that this deviation is not due to experimental
error. An extensive study of the fluidity-temperature curves of
pirre liquids leads to the conclusion that even when the expansion,
is not linear and there is association, the curves approach linearity,
as is seen to be the case with water in the figure. The extent to
which this is true can be best judged by an algebraic analysis
of the data to be given later. However it may be stated here
that the fcrst approximation of Meyer and Rosencranz (1877)

v ~ 1+aT

1 Landolt and Bomstein, Tabellen, 3d. ed., p. 41.

(34)

94

FLUIDITY AND PLASTICITY

when put in the form

v = A + BT . (35)

is but an algebraic expression of the law that the fluidity of a
liquid is a linear function of the temperature. The law is only
approximately true, but even with the alcohols where the curva¬
ture is greatest, there is an approach to linearity at high tempera¬
tures. Like the Law of Boyle, we may assume that this law holds
in ideal cases, and that the theory underlying it is valid.

Emulsions

The study of the viscosity of mixtures near their critical-
solution temperatures affords another very sharp and distinct
means for testing the theory which has been outlined. It has
been pointed out that the fluidities should be additive in the per¬
fect mixture but the viscosities additive in the emulsion.
According to the second conclusion page, 89, there should be a
sudden drop in the fluidity at or near the critical solution tempera¬
ture. We do not propose to discuss in detail here the viscosity
of colloids but it is appropriate here to seek an answer to the ques¬
tion “Has such a drop in fluidity ever been observed?"

Ostwald and Stebutt (1897) observed an abnormally large vis¬
cosity mixture of isobutyric acid and water in the neighborhood
of the critical-solution temperature. This was attributed by
them—not to the reason given above—but to the fact that at the
critical-solution temperature, the surface energy becomes zero.

Friedlander (1901) investigated the phenomena which are
peculiar to the critical-solution temperature in an intensive
manner. He found a very marked increase in the viscosity as
the solution was cooled to temperatures where the opalescence
became evident and the critical-solution temperature was
approached. He observed the opalescence with particular care.
His investigation was extended to include phenol and water,
and the ternary mixture of benzene, acetic acid, and water.
Similar relations were found in all proving that the phenomena
are quite general. He concluded that the temperature coefficient
of viscosity is greatest where the opalescence and the tendency
to foam are greatest. He says, 1 “Der Trlibungsgrad und Tcm-
peraturkoefficient der inneren Reibung zeigen eine starke Zu-

1 Friedlandier, 439.

VISCOSITY AND FLUIDITY

95

nahme im kritischen Gebiete und stehen mit einander in einem
innigen Zusammenhange.” Friedlander also observed that the
expansion coefficient and the coefficient of electrical conductivity
as -well as the refractive index remained normal. He believed
that it was necessary to go farther than had Ostwald and Stebutt
in order to reach an explanation, and that a definite radius of
curvature of the separating surfaces must correspond to each
temperature, otherwise the degree of opalescence could not be
definitely determined. He therefore attributed the increase in
viscosity to the formation of drops, but he was puzzled by the
fact that when a solution of colophonium in alcohol is poured into
a large quantity of water, a highly opalescent liquid is obtained
which has, nevertheless, practically the same viscosity as pure
water. This theory of Friedlander is apparently an outgrowth
of the theory of “ halbbegrenzte Tropfen” of Lehmann. Fried¬
lander also discussed the electrical theory of Hardy that an
increase of work would be required to move the particles of a liquid
among charged particles, so that if the “drops” were charged
an increase in viscosity might result. But by experiment Fried¬
lander found that an electrical field was without noticeable
effect upon an opalescent liquid.

Friedlander’s values are expressed in relative units. Searpa
(1903) and (1904) has measured the viscosity of solutions of
phenol and water, expressing his results in absolute units. For a
given temperature, he plotted the viscosities against the varying
concentrations, and obtained a point of inflection in the curves
at the critical-solution temperature. He tried to explain the
irregularities on the assumption that hydrates are formed. He
was apparently unfamiliar with the work of Friedlander.

Rothmund (1908) started from Friedlander’s work to make a
study of the opalescence at the critical temperature. He meas¬
ured the times of flow of butyric and isobutyric acid solutions in
water, noting particularly the effect upon the opalescence of
adding various substances, both electrolytes and non-electrolytes.
He objected to the hypothesis of Friedlander in that, according
to the well-known formula of Lord Kelvin, small drops are less
stable than large ones, so that the former must tend to disappear.
Furthermore he remarked upon the entirely analogous opales¬
cence which is observed in a single pure substance at its ordinary

96

FLUIDITY AND PLASTICITY

critical temperature. Rothmund therefore called to his aid
Dorman’s hypothesis that when drops are very small their surface
tension is very different from that of the liquid in bulk and is a
function of the radius of curvature. Since at the critical tem¬
perature the surface tension is normally zero, it was thought that
the small drops might thus exist in a state of stable equilibrium
in the neighborhood of the critical-solution temperature. As the
temperature is raised the opalescence would become less and less,
due to the solution of the drops. Rothmund found that the
addition of naphthalene to his solutions greatly increased the
opalescence, while the addition of grape sugar decreased it very
greatly, although the effect of these additions upon the viscosity
was negligible. He reasoned that the refractive index of butyric
acid is greater than that of water and sugar and electrolytes
raise the refractive index of water, hence they make the presence
of small drops less evident. Naphthalene does not dissolve in
water but does dissolve in butyric acid, raising its refractive
index, and therefore it makes the opalescence more apparent.

Von Smoluchowski (1908) regards Rothmund’s hypothesis as
superfluous, believing that the kinetic theory is sufficient to
explain the opalescence. According to him, differences in
molecular motion, local differences in density, and therefore
differences in surface tension cause the critical temperature to be
not entirely definite. Due to this indefiniteness in the critical
temperature, rough surfaces are formed, which must have a
thickness of less than a wave length of light, since, greater thick¬
nesses would not reflect the light. The inequalities in the density
would reach their maximum at the critical temperature.

Bose and his co-workers (1907—1909) have also verified
these earlier observations that abnormally large viscosities are
obtained at the critical-solution temperature. Bose regards this
as due to the rolling of drops of liquid along the capillary. They
did a considerable amount of work to prove that “ cry stallin’’ or
“ amstropie ” liquids are similar to the emulsions here discussed.
Bose proved that these liquids have abnormal viscosities near the
clarifying point and they also possess marked opalescence.
Vorlander and Gahren had found that a crystallin liquid may
result from the mixing of two liquids neither of which is itself

cry stallin’’ in the pure condition. The mixture therefore

HtPBWf

VISCOSITY AND FLUIDITY 97

resembles an emulsion. Bose regards all “ crystallin” liquids as
emulsions of very long life, i.e., they settle out with extreme
slowness, and he proposes an extension of the kinetic theory to
account for them. According to van der Waals, the molecules
are to be regarded as spheres; however, the molecules of sub¬
stances known to form crystallin liquids do not approximate
to a spherical form but consist of two benzene rings united in
such a way as to make a rather elongated molecule. Hence, Bose
thinks that they may be better represented by ellipsoids of
revolution. As the temperature is lowered, these molecules
naturally arrange themselves with their long axes in parallel
planes. As the molecules unite to form the so-called “ swarms,”
the viscosity is increased. This orderly arrangement also
causes the liquids to show double refraction.

It was shown that quite often the viscosity increases rapidly
as the temperature is raised at the clarifying point, but there is
also then an increase in the density.

It occurred to Bose, Willers, and Rauert (1909) that the
orderly “swarm” arrangement might be destroyed by measuring
the viscosity under conditions for turbulent flow. It was shown
by them in fact, that the abnormalities at the critical-solution
temperature do decrease as the transpiration velocity is increased.
But these results are not very conclusive since the measurement
of viscosity under conditions for turbulent flow has been but
little investigated. Pure liquids were studied by them under
conditions for turbulent flow and it was found that there is not a
complete parallelism between the viscosities as measured by the
two methods. In fact, there are several cases where one sub¬
stance has a higher viscosity than another substance under
conditions for linear flow, but a lower apparent viscosity under
conditions for turbulent flow. No explanation seems to have
been given for this.

Tsakalotos (1910) has studied mixtures which show a lower
critical-solution temperature, triethylamine and water, and
nicotine and water, as well as amylen and aniline, and isobutyric
acid and water. He used only one or two temperatures so that
the peculiarity with which we are here concerned did not appear.

Bingham and White (1911) investigated phenol and water
mixtures with the following results. (1) The fluidity decreases

7

98

FLUIDITY AND PLASTICITY

unusually rapidly as the solutions are cooled toward the critical-
solution temperature. (2) But this abnormality appears before
the critical-solution is reached and continues on and through the
critical-solution temperature. (3) In the region where the
abnormality appears, it is very difficult to obtain concordant
values for the apparent fluidity. It may be added that this is to
be expected since according to the theory, the apparent fluidity
depends upon the size of the drops. (4) By reflected light the solu¬
tions in this region appear opalescent: by direct light the liquid
shows unequal refraction, the images of objects being distorted.

Drapier (1911) studied two mixtures in which water is not
a component, viz . hexane and nitrobenzene, and cyclohexane and
aniline. The fluidity-temperature curves and the fluidity-weight
concentration curves of the latter are shown in Fig. 37. Drapier
states that the relations are similar when volume-concentrations
are employed. According to his experiments, the contention
that fluidities are normally additive in homogeneous mixtures is
fully sustained.

“II semble done que dans un intervalle assez etendu de varia¬
tion de temperature on puisse consid4rer la fluidity comme une
fonction lin^aire de la temperature, sauf pour les corps tr&s
associes comme beau,.

“Pour les melanges, loin de la temperature critique la variation
de la fluidite est encore lineaire. Mais plus on approche de la
r6gion critique, moins les formules lineaires sont exactes. Elies
ne peuvent meme plus pretendre k un semblant d ? exactitude,
ainsi que le montrent bien les lignes de fluidite des melanges k
concentrations voisines de la concentration critique: elles sont
tout k fait courbes et concaves vers Faxe des temperatures.
D’ailleurs, d6j pour des concentrations eioignees de la concentra¬
tion critique, au voisinage de la temperature de demixtion le coeffi¬
cient de fluidite varie tr£s fort. Mais le changement est plus
graduel pres de la concentration critique.

“Si l J on examine Failure des isothermes de fluidite, on voit
que pour les melanges de corps normaux la loi d’additivite*.

ip = Clipi b<f2

est assez bien satisfaite a des temperatures superieures k la
temperature critique de dissolution. J’ai porte en abscisses les

VISCOSITY AND FLUIDITY

99

concentrations on poids, mais on prenant les concentrations on
volume la loi d’additivite n’est pas mieux verifiee. Ce n’est que

dans le voisinage dc la temperature critique qu’il se prdsente des
6 carts singulars, resultant de la courbure des lignes d’egale
concentration et se traduisant par une double inflexion des
iso thermos, ...”

100

FLUIDITY AND PLASTICITY

Commenting on the theory of v. Smoluchowski by way of
explanation he remarks, “II est probable que de pareilles h6t6ro-
g6n£it6s produirairent une augmentation de la viscosity et
pourraient done expliquer la courbure, toujours de meme sens,
des courbes d’6gale concentration et par consequent les hearts k
loi d^additivite.”

These researches make it perfectly clear that there is a decrease
in the fluidity near the critical-solution temperature as predicted
and that in some way this decrease is connected with the dis¬
appearance of homogeneity in the mixture. Most of the in¬
vestigators have concerned themselves with the explanation of
disappearance of homogeneity before the critical-solution tem¬
perature is reached, rather than of the increase in viscosity.

Fig. 38.—Diagram illustrating the flow of emulsions.

But we are here only interested in the fact that heterogeneity
does occur simultaneously with the abnormal increase in viscosity,
and not in the cause 1 of the heterogeneity itself.

Scarpa and Bose however offered explanations of the abnormal
increase in the viscosity. In regard to Scarpa’s assumption
that the decrease in fluidity is due to the formation of hydrates,
it is very possible that hydrates are formed between phenol and
water, with which he worked; but he has not given any facts to
prove that the hydration suddenly increases as the critical-
solution temperature is approached even in this favorable
case. In the cases studied by Drapier (cjf. Fig. 34), such a
chemical action seems to be out of the question, because if solva¬
tion occurred the fluidity-concentration curves would be sagged
even above the critical-solution temperature.

In order to understand the explanation of Bose, we refer to
Fig. 38 which may be taken to represent the hypothetical

1 For an attempted explanation cf. Am. Chem. J., 33 , 1273 (1911).

VISCOSITY AND FLUIDITY

101

appearance of the drops of an emulsion as they pass through a
capillary tube. Due to the friction against the walls, the rear
end of each drop is flattened and the front end is unusually convex.
It is to be especially noted that when the drops are small in
diameter as compared with the diameter of the tube and yet
large enough to occupy the whole cross-section of the tube,
the motion of the liquid is by no means entirely linear, being
transverse as well as horizontal as indicated by the arrows. The
effect of this transverse motion is to increase the apparent
viscosity of the liquid. If, however, the drops are very large in
comparison to the diameter of the tube, the importance of this
transverse motion may become vanishingly small. Thus if the
drops of an emulsion are large enough to fill the cross-section of a
tube, the viscosity, as measured by the rate of efflux, will be at
least as great as the sum of the component viscosities, but it
may be greater due to the transverse motions. We grant that
below the critical-solution temperature a part of the increase in
viscosity may be due to these transverse motions, but Bose
would seem to account for all of the abnormal increase in the
viscosity in this way. This however is not warranted, for the
reason that at the center of the capillary the liquid has normally
a high velocity while at the boundary the velocity is zero, so that
there is a considerable tendency for any drops to become dis¬
rupted and drawn out into long threads. It is impossible to
believe that above the critical-solution temperature the surface
tension of the “drops” is sufficient to prevent disruption, for
we are accustomed to think that the surface tension at the critical
temperature is zero, and the abnormality in the fluidity is a
maximum at this temperature. We conclude therefore that
neither the explanation of Scarpa nor of Bose is sufficient, but
that the explanation based upon the nature of viscous flow in
a heterogeneous mixture is both necessary and sufficient.

The theory requires that if the fluidities of the two components
of the mixture are identical, it makes no difference whether
fluidities or viscosities be considered additive; hence there should
be no irregularity in the fluidity curves of such a pair of sub¬
stances even in the vicinity of the critical-solution temperature.
No case has been examined, so far as we know, in which
the components have approximately the same fluidity and

102

FLUIDITY AND PLASTICITY

the mixture has a critical-solution temperature. The nearest
approximation is in the case of isobutyric acid, <p 2 o° = 76.0, and
water, <p 2 o° = 99.8, examined by Friedlandcr. As can be seen
from Fig. 39 taken from the work of Drapier, the irregularity is
very slight. The calculated deviation is

a<pi + b<p 2 - TTT — (36)

arn + br ]2

which for a 50 per cent mixture corresponds to 27.6. The irregu¬
larity is greatest in the case of hexane, <p 2 o° = 314.0, and nitro¬
benzene, <p 2 o° = 50.1, where the fluidities are also the most
unequal. The calculated deviation is in this case 95.6 for a
50 per cent mixture. The deviations actually read from the
curves for isobutyric acid and water and hexane and nitro¬
benzene are of the order of 11 and 25 respectively. That these
numbers are so much smaller than the calculated values may be
easily accounted for on the supposition that the drops are not all
sufficiently large to fill the cross-section of the capillary, and
hence the viscosities are not strictly additive.

Suspensions

According to the view that viscosities are always additive,
the viscosity of all suspensions should be infinite. On the con¬
trary, as already stated, Friedlander found that colophonium
suspended in water had practically no influence on the viscosity
of water. Similarly Bose measured the viscosity of suspensions
of finely-divided quartz, whose viscosity may be taken as infinite,
in bromoform and water, and he found that the viscosity of the
medium was but little changed. Had they measured the visco¬
sities at increasing concentrations of the solid, they would have
undoubtedly found that the viscosity was altered and in a
perfectly definite manner. As already indicated, page 55 and
Fig. 19, the fluidity curves of such suspensions are normally
linear. Since, however, suspensions and emulsions are closely
allied, it is important to inquire why viscosities arc not additive
in suspensions as well as in the emulsions already considered.

In suspensions, we have practically the same conditions as in
emulsions in which the drops are so small that they do not nearly
fill a cross-section of the capillary tube. In this case the viscosi¬
ties are not strictly additive.

VIKCOfUTY AhTD FLTUDITY

1 c

104

FLUIDITY AND PLASTICITY

If for simplicity we imagine the solid particles of a suspension
to be all united into sheets parallel to the direction of flow, as the
shaded areas in Fig. 33 then it is evident that the flow will be the
sum of the flows of the unshaded areas, i.e. } the fluidities will be
strictly additive or

(p = a<pi 4“ b(p 2 = b(p 2
since <pi is practically zero.

But if these solid sheets were broken up into fragments, one of
which is shown as the cross-section of a cube at F in Fig. 40,
the deformation of the liquid would tend to change the form of
the cube into that of a parallelopiped as shown at G, but as the
solid is rigid, this cannot take place; so that the shearing force
can only rotate the cube around its center as shown at H. But

the failure of the solid to change its
shape with the flow of the liquid
will necessitate transverse motions
in the liquid by way of readjust¬
ment, hence the viscosity of a
suspension will always be greater
than it would be were the fluidities
strictly additive. If, as we believe can be proved to be the
case, the amount of transverse motion in a suspension is pro¬
portional to the number of suspended particles of a given size,
and for each particle the amount of transverse motion bears a
constant ratio to the amount of shear, it will follow that the
fluidity curves of suspensions must be linear, as has already
been shown to be generally true (c/. p. 55 and Fig. 19).

Enough evidence has been given to indicate that the theory of
the subject and the most diverse sorts of experimental data are
in accord in supporting the fundamental hypothesis that fluidities
are normally additive in homogeneous mixtures and fine sus¬
pensions, hut not in heterogeneous mixtures. Much additional
evidence could he given, but not without taking up subjects out
of their natural order. This evidence will appear as we proceed.
For further confirmation of these views cj. Drucker and Kassel
(1911), White (1912).

James Kendall (1913), working in the Nobel Institute of Phys¬
ical Chemistry under the direction of Professor Arrhenius,
concluded that “the logarithmic viscosity (or fluidity) of a solu-

JD D' V D D D,

A A/

Fig. 40.—Diagram illustrating the

flow of suspensions.

VISCOSITY AND FLUIDITY

105

tion is the characteristic additive property, and not these quan¬
tities themselves.” This conclusion was based upon data which
for the reasons already given was not well suited for reaching a
final decision of the matter. As the result of more recent study
of the matter with Monroe (1919) and Wright (1920) Kendall has
come to the conclusion that no formula tested by him will repro¬
duce the observed data. The present author is in hearty accord
with this conclusion of Kendall. It cannot be emphasized too
strongly, to the novitiate particularly, that no single formula will
reproduce faithfully any considerable portion of the observed
data on the fluidity of mixtures. Moreover it is useless to look
for such a formula in the present state of our knowledge. A much
better plan is to assume the additivity of fluidities, which also
has the virtue of being the simplest hypothesis that we can make,
and then try to account for the deviations from the exact law
on the basis of well-established physical and chemical evidence.
If the fundamental hypothesis is incorrect, incongruities will soon
develop to put us on the right track. If correct, we should pro¬
ceed as rapidly as possible to exploit the new knowledge which
fluidity measurements place in our hands.

CHAPTER II

FLUIDITY AND THE CHEMICAL COMPOSITION
AND CONSTITUTION OF PURE LIQUIDS

Attention was first strongly drawn to the desirability of study¬
ing the viscosity of homogeneous liquids in relation to their other
properties by Graham in 1861. He himself measured the
viscosity of several organic liquids at the uniform temperature of
20° and noted that the times of flow increase with the boiling-
point, from which he inferred that there is a connection between
viscosity and chemical composition similar to that which exists
between the boiling-point and the chemical composition. By
comparing the times of flow of “equivalent amounts,” obtained
by multiplying the times of flow of equal volume by the molecu¬
lar weights and dividing by the density (rjM/p), Rellstab (1868)
sought to gain a more intimate knowledge of this relation. He
measured the viscosity over a range of temperatures from 10 to
50° and then compared the substances at temperatures at which
their vapor-pressures are equal, as well as at a given temperature.
No simple quantitative relationship was found between his times
of flow and the molecular weight or vapor-pressure, but he stated
several qualitative relationships. Thus he noted that the time
of flow always decreases as the temperature rises, that an incre¬
ment of CH 2 in a homologous series is in general accompanied
by an increase in the time of flow, but that metameric substances
may have very different efflux-times. Without attempting a
complete summary of his observations, the above suffice to show
that he regarded temperature, chemical composition and con¬
stitution as all important in determining the rate of flow.

Pribram and Handl (1878-1881) studied a large number of
pure liquids over a range of temperatures from 10 to 60° express¬
ing their results in “specific viscosities” taking water at 0° as 100.
Their researches marked a great step in advance but only to the
extent of confirming and extending the qualitative observations

106

FLUIDITY AND THE CHEMICAL COMPOSITION 107

of Rellstab and also of Gucrout, and not in establishing quantita¬
tive relationships.

Struck by the fact that metameric substances sometimes have
such widely different viscosities, e.g., isobutyl alcohol 0.03906
( and ethyl ether 0.002345 at 20°, Bruhl (1880) noted that the one
with the higher viscosity generally had the higher boiling-point
and index of refraction. But to this observation Gartenmeister
(1890), testing a large number of substances at 20° or over a
range of temperatures, found numerous exceptions.

It was at this point that Thorpe and Rodger (1894) decided
to make an intensive study of the whole subject of the relation
between the viscosity of liquids and their chemical nature. Their
first care was to work out a method which would give them a far
greater precision of measurement than had been obtained by
many of their predecessors. They then carefully purified
some 87 substances and measured their viscosities from 0°C to
the boiling-point of each substance. An exhaustive search was
then made for a basis of comparison which would bring out the
quantitative connection between the viscosity and the chemical
nature of the liquids. In this search they compared the viscosity
coefficients (rj), the “molecular viscosities” the

“molecular viscosity work” (rjM/p), and in order to make the
comparison under comparable conditions they made the com¬
parisons at the boiling temperatures, at “corresponding”
temperatures, at temperatures where the slopes of the viscosity-
temperature curves are equal, and at slopes varying under speci¬
fied conditions. They furthermore compared the constants in
the empirical equations which they found to best reproduce the
observed viscosities as a function of the temperature, and they
also compared the temperatures corresponding to a given slope
in the viscosity-temperature curves. Their choice of tempera¬
tures of equal slope as a basis of comparison deserves a word in
explanation. They found that on comparing the viscosity
curves of substances which gave the best physico-chemical rela¬
tionships at the boiling-point that the general shape of these
curves was the same, or in other words the slopes of the substances
at their boiling-points were practically identical. On the other
hand, alcohols and other substances, which gave little evidence
of physico-chemical relationships, had invariably a different

108

FLUIDITY AND PLASTICITY

slope. It therefore occurred to them to compare their substances
at temperatures of equal slope and they seemed to find theoret¬
ical justification in this proceeding, since at a given slope the
temperature is exercising the same effect upon the viscosity of
different substances, i.e., drj/dt is constant.

They were able to establish the most nearly quantitative
relationship in the comparison between molecular viscosity work
and chemical composition and constitution using a constant
slope, arbitrarily selected as 0.000,032,3. We shall now examine
the nature of this relationship.

By comparing the values for the homologues given in Table
XXII they observed that the addition of a methylene group
to a compound increases the observed value of the molecular
viscosity work by (80 ± 5) X 10~ 3 c.g.s. units. They assume
that CH 2 = 80, the factor 10~ 3 being understood. Similarly an
iso-grouping is found to lower the value observed for the normal
compound by 8 ± 3 provided that the highly associated butyric
acids are left out of the calculation.

The value of H 2 was found by subtracting the value of nCH 2 ,
as calculated from the above constants, from the observed
values of the paraffins whose general formula is C n H 2w + 2 as
shown in Table XXIII. The mean value of H 2 is —68 and since
CH 2 = 80, C = 148.

Comparing normal propyl with allyl compounds, it was
found that the occurrence of a double linkage and the loss of two
hydrogen atoms lower the molecular viscosity work by 27 ± 1;
hence the value of a double linkage was assumed to be —95.

Using the values thus obtained, they determined the value
of oxygen in ketones to be —19, excluding acetic aldehyde and
dimethyl ketone from the calculation because they are the first
members of their respective series, and are probably associated.
In the aliphatic acids the two oxygen atoms have a value of
81 ± 4, but since one of these is a carbonyl oxygen, the value of
hydroxyl oxygen must be 100. On the other hand, oxygen when
united as in ether was found to be 43. It seemed to them possible
that oxygen might have yet other values such as the carbonyl
oxygen in aldehydes as distinguished from ketones. Comment¬
ing on the different values which it seemed necessary to give to
the same atom in differently constituted compounds, Thorpe

FLUIDITY AND THE CHEMICAL COMPOSITION 109

Table XXII. —Molecular Viscosity Work (yM/p) in Ergs X 10“ 3 at
a Slope of 0.000,032,3

Substance

Observed

Difference

Calculated

Difference,
per cent

329

86.0

332

- 0.9

415

80.0

412

0.7

Heptane.

495

574

79.0

492

572

0.6

0.3

320

84.0

324

- 1.2

404

78.0

404

0.0

482

484

- 0.4

284

72.0

278

2.1

356

358

- 0.5

255

86.0

264

- 3.5

341

84.0

344

- 0.9

425

424

0.2

417

88.0

416

0.2

505

496

1.8

399

397

0.5

282

71.0

277

1.8

353

357

— 1.1

346

87.0

349

433

427

1.4

327

330

- 0.9

450

76.0

456

- 1.3

526

88.0

536

- 1.9

614

608

1.0

418

409

2.0

Pro'nvl rhlnnd©...

294

295

- 0.3

TflAvwomjl pll mfirlft . .. 1

290

74.0

287

1.0

lSUpiCJp^i ....

IsoDutv* chlond© • ..

364

367

- 0.8

AUyl chloride.

• 268

268

0.0

Methylene chloride.

P+llvl^TIA A'Plfip

241

326

85.0

244

324

- 1.2

0.6

JUjIjI] y ItfilC* LHJUl iuc* »

Ivlctlijd sillfido.

240

76.5

236

1.7

Pflivl ftiilfid© ..

393

396

- 0.8

X J vu j i ljuuiv*'» ..

Methyl ethvl ketone.

302

81.0

301

0.3

Methyl propyl ketone.

383

381

0 . 5

Dipfhvl Kfitono.*

376

381

- 1.3

Formic istcid *.-.

160

77.0

n

159

0.6

Ar^A’f’IA ar*id .. ...

237

oo * u

*7 a n

239

— 0.8

Propionic acid.

323

319

1.2

- 0.5

Butyric acid.

397

399

Isobutyric acid..

398

391

1.8

Acetic anhydride.

394

74.0

393

0.3

- 2.0

Propionic nnhydnd a , .. r .

542

553

Ethyl ether...

295

295

0.0

Benzene...

314

81.0

315

- 0.3

0.0

0.0

Toluene .

395

80.0

395

Ethyl benzene .

475

475

Ortho-xylene T .

483

475

1.7

Meta-xylene....

474

47 5

— 0.2

Para-xylene.

467

475

— 1.7

110

FLUIDITY AND PLASTICITY

Table XXIII.— The Value oe Uydiiouen

Substance

Normal paraffins

Iso-paraffins.N

n

Ou2lI?i-f 2

ftCHa

II 2

5

329

400

-71

6

415

480

-G5

7

495

560

-65

8

574

640

-66

5

320

392

-72

6

405

472

-67

7

482

j 552

-70

and Rodger remark (p. 643), “If such differences are confirmed
by more numerous observations, viscosity will rank ’as one of the
most useful properties in determining the constitution of oxygen
compounds.” They then add, “It is, of course, to be remem¬
bered here that the value of hydroxyl oxygen as it is derived from
the acids is no doubt affected by molecular complexity.”

Using the constants obtained as above, and grouped together
in Table XXIV for reference, Thorpe and Rodger calculated the
values of the molecular viscosity work for the substances given
in Table XXII, and reproduced in column 4. The average dif¬
ference between the observed and calculated values is less than
1 per cent, but it is to be remarked that water and the alcohols
do not enter into comparison at this particular slope. At a differ¬
ent slope they were able to bring these substances into the
comparison, and they found a very great divergence between
the observed and calculated values amounting to 44 per cent in
the case of dimethyl ethyl carbinol and 47 per cent in that of
water. Again the difference was partly attributable to constitu¬
tive influences, since it was noted that the divergence is least in
the primary and greatest in the tertiary alcohols. But at the
same time they note that these compounds are most certainly
associated and the theoretical values of the molecular weight were
used in place of the actual values. They conclude their study
of molecular viscosity work at equal slope with the following
noteworthy statement: “The results here obtained are of
precisely the same nature as those discussed under molecular
viscosity. More detail has been given to show that the sub¬
stances which give deviations from the calculated values fall

FLUIDITY AND THE CHEMICAL COMPOSITION 111

into two classes. In the first the deviations arc to be attributed
to chemical constitution, as similar disturbing effects may be
detected in the magnitudes of other physical properties which do
not seem to be affected by molecular complexity. In the second
arc those substances like the acids, water, and the alcohols, for
which the disturbing factor is, no doubt, molecular complexity,
the effect produced in this way, in the case of the alcohols, being
dependent upon their chemical nature.” Thorpe and Rodger
have done great service in stating the problem before us so
clearly. At a subsequent point in our discussion, we will show
how by a different method of comparison it is possible to largely
avoid the first cause of discrepancy given above, and how then
with only one unknown quantity remaining, it is possible to get a
proximate solution of the problem.

Tajble XXIV.— Molecular Viscosity Work Constants at Slope

0.000,032,3

Hydrogen. — 34

Carbon. 148

Hydroxyl-oxygen, C-O-H. 100

Ether-oxygen, C-O-C. 43

Carbonyl-oxygen, C = O. — 19

Sulfur, C-S-C. 144

Chlorine (in monochlorides). 89

Chlorine (in dichlorides). 82

Bromine (in mo nobromides). 151

Bromine (in dibromidcs). 148

Iodine. 218

Iso-grouping. — <8

Double linkage. — 95

Ring-grouping. —369

The effect of chemical constitution upon viscosity has been
employed to good effect in the solution of several much-mooted
chemical problems by Dunstan and Thole and their co-workers.
Thus Thole (1910) observed a steady increase in the viscosity of
freshly distilled ethyl acetoacetate owing to the gradual enoliza-
tion of the kctonic form. Hilditch and Dunstan (1911) have
observed that the presence of Thiele's “ conjugated double bonds”
in compounds produces a great increase in the viscosity. Thole
(1912) has shown that the viscosity method can be used to dis¬
tinguish between geometrical isomeridcs like maleic and fumaric

112

FLUIDITY AND PLASTICITY

acids. But while they attribute these effects to the constitution
of the molecules, it should be noted that the immediate cause of
the increase in viscosity may in each case be association , which
is the same as saying that it may be due to chemical composition
as distinguished from chemical constitution. Certainly com¬
pounds containing the hydroxyl radical are often associated and
these same compounds are noted for their high viscosity, so that
in the case of ethyl acetoacetate the way seems open to explain
the greater viscosity of the enol form on the basis of an associated
molecule, quite as well as on the basis of symmetry or other
constitutive influence. At first sight it seems as though consti¬
tutive influences must solely and immediately determine the
viscosity values in each of the above examples, hut Thole (1912)
seems to realize that this is not the actual case with maleic and
fumaric acids, the latter of which gives the higher viscosity in
methyl alcohol solution. He says,“ The viscosities of the isomers
depend not only on the relative positions of the unsaturated
groups but also on the degree of residual affinity ” which causes
molecular association. Thus the “adjacent” maleic acid may
have the lower viscosity due to slighter association. This view
is borne out by the fact noted by Thole that “ barium fumarate
crystallizes with three molecules of water while barium maleate,
in which the residual affinities of the carboxyl groups are more
nearly mutually satisfied, combines with only one molecule of
water.”

To what extent different constitutive influences affect the
association of compounds is an exceedingly important subject
but it is not relevant to our discussion of viscosity. Our problem
is to study the immediate effects of constitutive influences and
the chemical composition of the molecule upon the viscosity and
to estimate their relative importance.

Regardless of how much uncertainty there may be in regard
to the importance of constitutive influences on viscosity, there
can be no doubt about the importance of chemical composition.
All evidence shows that this factor is of great importance. Dun-
stan and Langton (1912) have made use of this for the determina¬
tion of transition points, and Thole (1913) in the detection of the
presence of racemic compounds in the liquid state, and many other
instances might be cited.

FLUIDITY .AND THE CHEMICAL COMPOSITION 113

Comparison of Fluidities.—We have already given reasons for
believing that if liquids were completely unassociated and
expanded in a linear manner with the temperature, the fluidity-
temperature curves would be straight lines. To compare a
family of curves which are straight lines is a simpler task than
the comparison of a family of hyperbolas, hence it seems a justi-

Fig. 41.—The fluidities of vari¬
ous hydrocarbons at different
temperatures and extrapolated
to their boiling temperatures.
4. Pentane; 5. Isopentane; 6.
Hexane; 7. Iso hexane; 8. Hep¬
tane; 9. Isoheptane; ID. Octane;
11. Trimethylethylene; 12. Iso-
prene; 13. Diallyl; 56. Benzene;
57. Toluene; 58. Ethylbenzene;
59'. (o)-Xylene 60. (m)-Xylene;
61. (p)-Xylene.

Fig. 42.—The fluidities of various
ethers and acid anhydrides at differ¬
ent temperatures and extrapolated
to their boiling temperatures. 53.
Acetic anhydride; 54. Propionic
anhydride; 55. Diethyl ether; 83.
Methyl propyl ether; 84. Ethyl
propyl ether; 85. Dipropyi ether;
86. Methylisobutyl ether; 8 7.
Ethylisobutyl ether.

fiable expectation that we may be able to find simpler relations
by a suitable comparison of fluidities. Before deciding on a
basis of comparison let us inspect the fluidity-temperature
curves as obtained from the observations of Thorpe and Rodger
as given in Figs. 41 to 46. Confining our attention first of all
to the aliphatic hydrocarbons in Fig. 41 we see that near their
boiling-points, indicated by small circles in the figure, the fluidity
8

114

FLUIDITY AND PLASTICITY

curves are nearly straight and parallel lines. However as we get
away from the boiling-temperature, there is a curvature present
so that it is probable that the fluidity curve would reach the tem¬
perature axis asymptotically as the temperature were lowered.
Broadly speaking, the curves of a given homologous series near
their boiling-points consist of a scries of parallel straight lines,
which are therefore completely defined mathematically by their
slopes and intercepts. We find the same thing in other series,

Fig. 43.—The fluidities of vari¬
ous bromides at different tem¬
peratures. 20. Ethyl bromide;
21. Propyl bromide; 22. Iso¬
propyl bromide; 23. Isobutyl
bromide; 24. Allyl bromide; 25.
Ethylene bromide; 26. Propy¬
lene bromide; 27. Isobutylene
bromide; 28. Acetylene bro¬
mide.

Fig. 44.—The fluidities of various
iodides at different temperatures.
14. Methyl iodide; 15. Ethyl iodide;
10. Propyl iodide; 17. Isopropyl
iodide; 18. Isobutyl iodide; 19.
Allyl iodide.

as the ethers and acid anhydrides given in Fig. 42, but it is clear
that the slope is different in the two classes. The slope then is
dependent upon the class to which a compound belongs and the
intercepts are dependent upon the chemical composition.
According to this broad aspect of the case it should make no dif¬
ference whether we compare fluidities at a given temperature or
temperatures corresponding to a given fluidity. But there are
several reasons for choosing the latter basis of comparison rather
than the formero

FLUIDITY AND THE CHEMICAL COMPOSITION

115

1. The slopes of the fluidity-temperature curves for a given
homologous series are more nearly the same when the fluidities
are equal.

2. When the fluidities are the same, the vapor-pressures are
nearly equal, and experience has shown that substances are
comparable at temperatures which correspond to equal vapor-
pressure.

3. The fluidity curves of associated substances like the alcohols,
Fig. 46, depart widely from linearity at low fluidities, although
they approach linearity at high fluidities, as do the curves of
other compounds.

4. A yet more cogent reason grows out of the fact that exact

Fig. 45.—The fluidities of various organic acids at different temperatures.
48. Formic acid; 49. Acetic acid; 50. Propionic acid; 51. Butyric acid; 52.
Isobutyric acid.

parallelism in the curves of a given class is not to be expected
since all fluidity-temperature curves must undoubtedly meet at
the absolute zero of temperature. Hence while it may require
a constant increment of temperature to produce a given fluidity
as each methylene group is added to the molecule, it is absolutely
certain that a constant decrement of the fluidity at a given tem¬
perature cannot be expected as each methylene group is added.
Thus a methylene group added to pentane, Fig. 37, lowers the
fluidity at 0° by a certain amount, but the effect of adding a

116

FLUIDITY AND PLASTICITY

methylene group to heptane is less and the effect of adding
other methylene groups must be still less, otherwise it would
require no very high molecular weight to give a negative fluidity,
which is inconceivable.

The fluidity of 200 is chosen as a basis of comparison in order
that as large a number of substances as possible may be included.
The absolute temperatures and slopes of several unassociated

Fig. 46.—The fluidities of various alcohols at different temperatures. 62.
Methyl alcohol; 63. Ethyl alcohol; 64. Propyl alcohol; 65. Isopropyl alcohol;
66. Butyl alcohol; 67. Isobutyl alcohol; 68. Trimethyl carbinol; 69. Active
amyl alcohol; 70. Inactive amyl alcohol; 71. Dimethylethylcarbinol; 72.
Allyl alcohol.

compounds corresponding to the fluidity of 200 are given in
Table XXV. The third column of this table shows that the
value of a methylene grouping varies around a mean value of
22.7, the mean deviation from this value being 3. The effect
of an iso-grouping is to decrease the temperature required by
about 7.6°, as shown in Table XXVI.

FLUIDITY AND THE CHEMICAL COMPOSITION 117 ‘

Table XXV.— Absolute Temperatures and Slopes op Non-associated
Substances Corresponding to a Fluidity op 200 c.g.s. Units

Substance

Absolute
tempera¬
ture (<j> =
200) ob¬
served

Differ¬

ence

ch 2

Slope at
(<£ =
200)

Absolute
tempera¬
ture (<j>

« 200)
calculated

Per

cent,

differ¬

ence

Hexane.

(255.1) J \

(21.0)

(2.88)

254.6

0.2

Heptane.

276.1 /

277.3

0.4

Octane.

299.1 }

23.0

2.44

300.0

0.3

Isohexane.

(249.0) 1

(20.2)

(2.79)

247.0

0.8

Isoheptane.

269.2 /

2.68

269.7

0.2

Methyl iodide.

290.2 }

19.0

1.92

287.4

1.0

Ethyl iodide.

309.2 \

23.5

1.80

310.1

0.3

Propyl iodide.

332.7 f

1.82

332.8

0.0

Isopropyl iodide.

324.5 \

21.0

1.92

325.2

0.2

Isobutyl iodide.

345.5 J

1.86

347.9

0.7

Allyl iodide.

330.5 .

1.82

328.8

0.5

Ethyl bromide.

268.7 1

27.9 .

2.22

273.5

1.8

Propyl bromide.

296.6 f

2.08

296.2

0.1

Isopropyl bromide.

289.4 \

25.6

2.22

273.5

1.8

Isobutyl bromide.

315.0 f

2.08

311.3

1.1

Ethyl propyl ether.

(255.0) \

(24.0)

(2.70)

256.1

0.5

Dipropyl ether-'.

279.0 j

2.62

278.8

0.1

Methylisosbutyl ether...

(251.1) 1

(19.0)

(2.75)

248.5

1.0

Ethylisobutyl ether.

270.1 /

2.68

271.2

0.4

1 Values in parentheses are extrapolated.

Table XXVI.— The Value op the Iso-grouping

Substance

Temperature ob¬
served, normal
grouping

Temperature ob¬
served, iso¬
grouping

Difference

Hexane.

255.1

249.0

6.1

Heptane..

276.1

269.2

6.9 .

Propyl iodide.

332.7

324.5

8.2

Propyl bromide.. .

296.6

289.4

7.2

Propyl chloride....

261.5

255.2

6.3

Butyric acid.

381.6

371.6

10.0

Methyl butyrate. .

304.2

295.8

8.4

The value for the hydrogen atom is calculated as follows:

118

FLUIDITY AND PLASTICITY

Table XXVII.— The Value of the Hydrogen Atom

Substance

Temperature

observed

nCH 2

calculated

Difference

Hexane.

255.1

136.2

118.9

Heptane.

276.1

158.9

117.2

Octane.

299.1

181.6

117.5

Isohexane.

249.0

128.6

120.4

Isoheptane.

269.2

151.3

117.9

The value for H 2 is 118.4 ± 1.0. The hydrogen atom has
therefore a value of 59.2 and the carbon atom of —95.7.

The value of the “ double bond ” in allyl compounds is obtained
from Table XXVIII.

Table XXVIII.— The Value of the Double Bond

Substance

Temperature ob¬
served, normal
propyl

Temperature ob¬
served, allyl

Difference

Iodides.

332.7

330.7

2.2

Bromides.

296.6

292.2

4.4

Chlorides.

261.5

256.0

5.5

To raise the fluidity of an allyl compound to 200 it is only
necessary to raise it to a temperature which is some 4° lower than
is necessary for the corresponding normal compound, containing
two more hydrogen atoms. Thus the “double bond” has a
value of 114.4, the absence of the hydrogen atoms being nearly
compensated for by the “condition of unsaturation.”

Assuming that the ethers are unassociated, we may obtain the
value of the oxygen atom.

Table XXIX.— The Value of the Oxygen Atom

Substance

Temperature

observed

C*H 2 „ + 2

Oxygen

Ethylpropyl ether.

254.9

231.9

23.0

Dipropyl ether...

279.0

254.6

24.4

Methylisobutyl ether.

251.4

224.3

27.1

Ethylisobutyl ether.

270.3

247.0

23.3

FLUIDITY AND THE CHEMICAL COMPOSITION 119

This gives an average value for oxygen of 24.2 with an average
divergence of 1.3 from this mean. From these values, the
absolute temperatures corresponding to a fluidity of 200 may be
calculated. Some of these calculated values are given in the
fifth column of Table XXV. A comparison between these
calculated and the ‘observed values for 35 substances shows an
average percentage difference of less than 0.8 per cent.

Association.—In attempts to establish a relation between
viscosity and chemical composition it has been customary to
disregard entirely the fact that certain classes of substances are
known to be highly associated, and hence the molecular values as
calculated from the atomic constants cannot be expected to agree
with the observed values. A more logical method of procedure
would be to use known values of the association in arriving at
the calculated molecular temperatures. The difficulty of this
method is that the values of the association as given by different
methods do not agree very closely and even the methods of
getting these values have been subjected to criticism. It seems
best therefore to reverse the method and use our atomic constants
to calculate the association, which can then be compared with the
values of the association obtained from the surface tension,
volume, et cetera.

In the calculation of the atomic constants as given above,
it was assumed that the compounds chosen were non-associated.
This is not entirely warranted, but they must be associated to
approximately the same extent since the agreement between the
calculated and observed values is generally satisfactory, and it is
the general belief that some of these compounds are indeed
unassociated. It is highly probable that association or constitu¬
tion is responsible, in part at least, for the uncertainty in the
so-called “constants/ 7 but this uncertainty can be removed by
further amplification of our data.

Since the atomic constants are additive, it follows directly
that the association will be given by the ratio of the observed to
the calculated values of the temperature corresponding to the
given fluidity. Thus for water (H 2 0)x at the fluidity of 200
the absolute temperature is 328.9, while the value calculated
from the gas formula H 2 0 is 2 X 59.2 + 24.2 = 142.6. The
association factor (x) at the temperature of observation (328.9°

120

FLUIDITY AND PLASTICITY

absolute) is therefore 328.9/142.6 = 2.31. In Table XXX are
given the observed and calculated absolute temperatures corre¬
sponding to the fluidity of 200 and the association calculated
therefrom for some typical associated compounds. The slopes
of these curves are also given in the fourth column.

Table XXX.— Absolute Temperatures and Slopes of Some Asso¬
ciated Compounds Corresponding to a Fluidity of 200 C.G.S. Units

Substance

Absolute
temperature
for (<f> — 200)
observed

Absolute
temperature
for (<f> — 200)
calculated

Slope for
(4> = 200)

Association

Water.

328.9

142.6

3.04

2.31

Formic acid.

(380.2)

185.5

(2.18)

2.05

Acetic acid.

363.8

208.2 1

2.06

1.77

Propionic acid.

362.0

230.9

1.92

1.57

Butyric acid.

381.6

253.6

1.92

1.57

Isobutyric acid.

371.6

246.0 |

2.00

1.51

Methyl alcohol.

305.2 i

165.3

2.78

1.84

Ethyl alcohol.

343.4

188.0

3.24

1.83

Propyl alcohol.

365.6

210.7

3.76

1.74

Butyl alcohol.

377.0

233.4

3.44

1.62

Ethyl formate.

273.8

230.7

2.40

1.19

Ethyl acetate.

284.0

253.4

2.50

1.12

Ethyl propionate.

298.1

275.1

2.44

1.08

The test of our complete process of reasoning comes now
when we compare the association obtained in this way with the
values which have been obtained by other methods. The results
of this comparison are shown by Table XXXI.

So far as one is able to judge, the f-esult seems to be all that
could be desired. There are almost invariably values given by
other methods which are both higher and lower than our values
and such a degree of association is certainly not inconsistent with
our knowledge of the chemical conduct of these substances.
The fluidity method of obtaining the association factor seems to
be freer from assumptions, to which questions maybe raised, than
other methods which have been proposed, and it is to be hoped
that it may prove useful in calculating this very important fac¬
tor* If eventually we are able to obtain thoroughly consistent

FLUIDITY AND THE CHEMICAL COMPOSITION 121

Table XXXI. —A Comparison of the Values of Association as Deter¬
mined by Different Investigators

Substance

Water...

Dimethyl ketone.

Diethyl ketone.

Methyl propyl ketone. ..
Formic acid.

Acetic acid.

Propionic acid.

Butyric acid.

Isobutyric acid.....

Benzene.

Toluene...

Methyl alcohol.

Ethyl alcohol.

Propyl alcohol.

Isopropyl alcohol.

Butyl alcohol.

Isobutyl alcohol.

Active amyl alcohol.

Allyl alcohol.

Methyl formate.

Ethyl formate.

Methyl acetate.

Ethyl acetate.

Propyl acetate.

Ethyl propionate.

Methyl butyrate.

R. & S.,i
16-46°

R. & S.,
corrected
by Traube

Traube, 2

15°

Longi-

nescu 8

1 B. & H., 4 tem¬
perature of ( <j>

= 200)

f 3.55

1.79

3.06

4.67

2.31

\ 1.64
1.26

1.18

1.53

1.60

1.23

1.25

1.16

1.11

1.10

1.43

1.25

1.14

3.61

2.41

1.80

1.80

2.05

f 3.62

2.32

1.56

1.75

1.77

(2.13

1.77

1.45

1.46

1.55

1.57

1.58

1.35

• 1.39

1.36

1.51

1.45

1.28

1.31

1.51

1.01

1.05

1.18

....

>1.17 <1.31

0.94

1.01

1.08

....

>1.08 <1.517

f3.43

2.53

1.79

3.17

1.84

12.32 1
(2.74

1.80

1.67

2.11

1.83

\ 1.65
2.25

1.70

1.66

1.67

1.74

2.86

2.00

1.53

1.75

1.94

| 1.47

1.62

1.95

1.53

1.54

1.66

1.97

1.54

1.53

1.54

1.88

1.50

1.55

1.80

1.69

1.06

1.07

(1.60)

1.12

1.25

1.07

1.08

1.39o°

1.19

1.00

1.04

1.48o°

1.09

1.17

0.99

1.04

1.25

| 1.00

1.12

0.92

1.00

1.31

1 1.00

1.11

0.92

1.00

1.27

0.94

1.08

0.92

1.00

1.30o°

1.00

1.10

1 Ramsay and Shields, Zeitschr.f. phyaik. Chem., 12, 464 (1893); 15, 115 (1894).
8 Traube, Ber. d. deutach. chem. Gesell., 30, 273 (1897).

* J. chim. Phys., 1, 289 (1903).

* Bingham and Harrison, loc . cit .

results from the different methods, it is interesting to observe
that it should be possible to calculate the volume, surface
tension, et cetera , even of associated liquids from their atomic
constants and their fluidities.

Fluidity and Chemical Constitution.—Dunstan and Thole
(Viscosity of Liquids , page 31) have very properly called attention
to the fact that the differences between the calculated and
observed values of the fluidity in Table XXV “are due not only

122

FLUIDITY AND PLASTICITY

to association but to want of sufficient data for calculating accu¬
rately the atomic ‘constants’ and also to constitutional effects,
such as the mutual influence of groupings in the molecule, sym¬
metry and so forth.” As was intimated earlier in this chapter,
to chemical constitution has generally been attributed a very
large effect on viscosity, but it often turns out on investigation
that this supposed constitutive influence occurs in substances
that are known to be associated and this association was not taken
into account, and in other cases the supposed constitutive influ¬
ence is almost certainly purely a hypothesis framed to explain
an unnoticed defect in the method of comparison. We shall now
give some facts to support these bare statements and we shall
then investigate the important question as to whether this dwind¬
ling constitutive effect, as distinct from the effect of association,
can safely be disregarded altogether.

In assigning values to the halogen atoms, Thorpe and Rodger
(p. 669 et seq .) found it necessary to give a different value to
chlorine in monochlorides, dichlorides, trichlorides and tetra¬
chlorides, but even then the results are not satisfactory since in
ethylene and ethylidene chlorides the value which must be
assigned the chlorine atom is certainly different. How the effect
of the chlorine atom varies at the fluidity of 200 is shown in the
fourth column of Table XXXII.

Table XXXII. —The Value of the Chlorine Atom

Substance

Absolute tem¬
perature (<j> =
200), observed

Hydro¬

carbon

residue,

calculated

Chlorine

!

Associa¬

tion

Propyl chloride.

261.5

127.3

134.2

1.105

Isopropyl chloride.

255.2

119.7

135.5

1.11

Isobutyl chloride.

285.2

142.4

142.8

1.13

Allyl chloride.

256.0

123.3

132.7

1.10

Ethylene chloride.

336.5

45.4

145.5

1.27

Ethylidene chloride....

291.2

45.4

122.9

1.10

Methylene chloride....

279.1

22.7

128.7

1.15

Chloroform.

' 305.3

- 36.5

113.9

1.04

Carbon tetrachloride. .

347.0

- 95.7

110.7

1.01

Carbon dichloride.j

356.3 !

- 77.0

108.3

i

0.99

FLUIDITY AND THE CHEMICAL COMPOSITION 123

There is then a somewhat regular decrease in the apparent
value of chlorine as the number of atoms in the molecule are
increased. How much of this is due to constitutive influence
directly and how much can be explained on the ground of asso¬
ciation? Ramsay and Shields and Traube agree that carbon
tetrachloride is very little associated if at all, Ramsay and
Shields giving the value 1.01 and Traube 1.00i 5 °. If then we take
the average of the closely agreeing values of the two compounds
containing four chlorine atoms we obtain as the value of the
chlorine constant 109.5 and with this we can calculate the asso¬
ciation of the other compounds. The values thus obtained are
given in the fifth column of Table XXXII. Ethylene chloride
is seen according to this method of calculation to be highly
associated, but Traube has given a still higher value for the asso¬
ciation at 15° of 1.46. Data for the other chlorides is lacking,
but calculating the association of propyl chloride by the method
of Traube, the author obtains the value of 1.11 which agrees
excellently with our value of 1.105. The mono-halides seem to
be usually associated according to Traube for he gives for methyl
iodide 1.30, for ethyl iodide 1.19 and for ethyl bromide 1.28. It
is greatly to be regretted that our available data is so meager,
but for the present we can only conclude that the effect of' con¬
stitution upon the value of the chlorine atom is too small to be
detected.

In reference to the lack of constancy in the value of a methyl¬
ene group in Table XXV, it seemed desirable to take the average
of as large number of values as possible, but with the limited
data on hand this made it necessary to include a number of
compounds which are certainly associated. This does not mean
that the value of the methylene group is therefore certainly in
error because associated compounds can give this as well as others,
provided the homologues are equally associated; and even if they
are unequally associated, the average value for the methylene
grouping may not be greatly in error although the individual
differences may be large. Finally the fact that the calculated
values in Table XXV differ from the observed values by less than
1 per cent seems to put a maximum limit upon certain kinds of
constitutive influences.

Hitherto it has been deemed necessary to give oxygen a differ-

124

FLUIDITY AND PLASTICITY

ent value depending upon whether the oxygen was in a carbonyl
group, hydroxyl, ether, et cetera . We will now attempt to show
that this was necessary so long as viscosities formed the basis of
comparison, but it was not an evidence of constitutive influence,
and in comparing fluidities only one value for oxygen is obtained
irrespective of the manner in which it is combined, and yet we
have seen that satisfactory association factors are obtained.
Let AB and A'B 1 in Fig. 47 represent two fluidity curves, parallel
to each other and therefore presumably representing members
of the same class of substances, and let a third fluidity curve
CD be at an angle to the other two to represent a substance in
another class. Since we have elected to compare absolute

temperatures at a fluidity of 200, this
amounts to comparing the intercepts
of the curves on the line AD, whose
equation is <p = 200. The corre¬
sponding viscosity curves obtained
by taking the reciprocal values of
the above fluidities and multiplying
by 10,000 are represented by ab ,
aV, and cd; and ad is of course the
reciprocal of the line AD . But the
point a and the point where the
curve a'b' crosses ad are points of
equal slope on the viscosity curves, hence within a given class
it makes no great difference whether we compare temperatures
corresponding to a given fluidity or temperatures corresponding
to a given slope on the viscosity curves. The latter is exactly
the method of comparison which Thorpe and Rodger found
very advantageous. But between different classes they found
difficulties which they attributed to constitutive influences. But
their difficulty is now easily explained, for d is the true reciprocal
of the point D which we believe should be used in the com¬
parison; on the other hand, they selected the point e , which has
the same slope as the point a, and for this choice we think that
there is not adequate reason.

It has been customary to assume that different kinds of
groupings should have special values assigned to them, and
particularly important among these was the “ring grouping.”

Pig. 47.—Diagram illustrating
the relationship between, viscos¬
ity- and fluidity-temperature
curves.

FLUIDITY AND THE CHEMICAL COMPOSITION 125

But it is not clear that this is unavoidable, for in the case of
benzene we note that the compound differs from hexane by eight
hydrogen atoms, and since we found in the allyl compounds that
the absence of a pair of hydrogen atoms is compensated for to the
extent of 114.4 we obtain for the calculated value, 59.2 X 6 -
95.7 X 6 + 114.4 X 4 = 238.6. The observed value is 311.9;
hence the association is 1.30, which is somewhat larger than the
value obtained by Traube of 1.18. It has usually been believed
that the more compact and symmetrical the molecule was, the
lower would be the temperature required to give it a certain
fluidity. In disregarding constitutive influences entirely for the
time being, as we have done here, we suppose that benzene would
require the same temperature as a straight chain hydrocarbon
containing four “double bonds.” If Traube’s value of the
association or some other value less than 1.30 is correct, we will
be compelled to assign a positive value to the ring grouping in
order to increase the calculated value. In other words, the
evidence at hand indicates that the effect of the ring grouping
is not to make the compound less viscous but more so. This is
so contrary to earlier belief and to the probabilities of the case
that it seems preferable to await further data before assigning
any value to the ring grouping.

Having been unable to detect the effect of constitutive
influences upon fluidity with the data at hand in the halogen,
oxygen, or ring compounds, we have left remaining one positive
evidence in the value which we have found it necessary to give
to the “iso-grouping.” This effect is not large but it is fairly
uniform and quite outside of the observational error. We cannot
believe that normal hexane and heptane are sufficiently associated
to account for the higher temperature above their isomers
required to give them a fluidity of 200. If then an iso-grouping
affects fluidity it is probable that there are other constitutive
influences, but the solution of this problem evidently requires
more data, particularly among the higher homologues. In this
connection the reader should have regard for the relation of
fluidity to volume, to be discussed later.

Before closing the chapter on fluidity and chemical composition
and constitution, we may add that constants calculated for a
fluidity of 300 give an association which is invariably a little

1

126 FLUIDITY AND PLASTICITY

lower than at a fluidity of 200. This is as one would expect.
From these values of the association the temperature coefficient
of the association factor can be obtained. (Bingham (1910),
page 306.)

From the constants at different fluidities one may conceivably
obtain a series of points ori the fluidity-temperature curve
heretofore unknown perhaps, and using these points the whole
curve may evidently be drawn.

The constants for fluidities 200 and 300 are for convenience
grouped together in Table XXXIII.

Table XXXIII.— Temperature Constants at Fluidity 200 and
Fluidity 300

Atom or grouping

v = 200

o

o

CO

II

Carbon.

-95.7

-110.2

Hydrogen.

59.2

67.8

Oxveen.

24.2

27.1

Iso.

-7.6

-8.2

Double bond.

114.4

131.3

Sulfur.

76.5

Chlorine.:.

109.5

CHAPTER III

FLUIDITY AND TEMPERATURE, VOLUME AND PRES¬
SURE ; COLLISIONAL AND DIFFUSIONAL VISCOSITY

~We have established in the preceding chapter that the vis¬
cosity of a substance is closely dependent upon the magnitude
of the molecules making up the substance. In this and succeed¬
ing chapters we will investigate the relation between viscosity
and various physical properties.

Temperature.— Prior to 1800, water was considered to be
perfectly fluid, but by causing equal volumes of water at corre¬
sponding pressures to flow through tubes of given dimensions
Gerstner in that year proved that the fluidity (Flussigkeit)
of water varies considerably with the temperature.

We have already seen that Poiseuille expressed this change in
tlie form of the parabolic equation

A = a + bT c + cT c 2 . (36)

A.fter viscosity had been defined, 0. E. Meyer (1861) introduced
tire viscosity coefficient into the formula which then became

V l+aT°+pT c * (37)

where is the viscosity at 0°C and T c is the temperature
Centigrade. In spite of the fact that the two equations are not
interchangeable, the latter formula is usually associated with the
name of Poiseuille. We will refer to it as the Meyer-Poiseuille
formula. It holds for water from 0 to 45° with a maximum
deviation of 1 per cent. For temperatures above 45° Meyer and
Rosencranz (1877) proposed the formula

Vo

1 +aT c

(38)

"V~arious investigators have employed the Meyer-Poiseuille
formula and confirmed the fact of its limited applicability. We

may mention Grotrian (1877), Noack (1886), Thorpe and
Rodger (1893), Knibbs (1895).

127

128

FLUIDITY AND PLASTICITY

In 1881 Slotte gave a formula to cover the entire range of
viscosities from 0 to 100°

__ a ~ ^Tc

* - ^TjT (39)

1

which accords with the values of Sprung to 0.7 per cent.

Most of the formulas which have been proposed have been
applied primarily to water. But Koch (1881) and Wagner (1883)
found that a formula of a different type is necessary for mercury
and Koch proposed the formula

v = a + pTc+yT^ + dTc* (41)

which holds from —20 to 340°C with a maximum deviation of
less than 2 per cent. But Slotte has applied the much simpler
formula of Meyer and Rosencranz with good results. On the
other hand, Batschinski (1900) has given a formula for mercury

„ = ± + b + cT (42)

where T is the temperature absolute. As a first approximation

vT = a, (43)

which can be deduced from Jaeger's theory of fluid friction.

Graetz (1883-5) is one of the few who have attempted to derive
a formula from theoretical considerations. We may therefore
give his argument in some detail.

According to Maxwell (1868) the viscosity of a body is the
product of two factors, the modulus of rigidity E and the time
of relaxation r. The time of relaxation was defined as the time
necessary for the strain after deformation in a body to sink to
1 JE of its original value. The reciprocal of the time of relaxation
is called the relaxation number, n, or

1

n = -
r

This is the number of times per second that the strain will sink
to 1 JE per second if the strain is renewed. For absolutely rigid
solids the value of r is infinite and for ductile solid bodies which

show elastic after-effect the relaxation may continue for hours or
days. But if, through raising the temperature, the substance is

FLUIDITY AND TEMPERATURE 129

changed to a liquid or gas, the time of relaxation becomes smaller
and smaller, and for air Maxwell has given the value

r = 1/5,099,100,000 sec.

With rising temperature, the value of r increases, and according to
Graetz, one may write

n = + ri 2 & 2 + W3# 3 + . . .

= 7&1#(1 -(- CKl# -f* ■ • •)

where # is the temperature reckoned from the temperature at
which the viscosity is infinite, i.e., the temperature of solidifica¬
tion.

In gases the modulus of rigidity is known to be equal to the
gas pressure at the critical temperature and is of the order of
magnitude of a hundred atmospheres. In solids, where the
modulus of rigidity is known, it has a value from 100,000 to
1,000,000 atmospheres. Since E decreases in passing from the
solid through the liquid into the gaseous condition, its value
approaches the critical pressure P at the critical temperature # 0 ,
and we have according to Graetz

E = P + 6 i(*o - tf) + b 2 (&o - ^) 2 + 63 (#o - tf) 3 +. . .
where 61 , 62 , 63 , . . . are constants.

From Maxwell, we get

r, = Er = -
n

or

= ^ + -*) + ■■■

^ -f- “f" CL2 $2 “f" . . .)

= (1 + Bid- + fad 2 + • • •)

where a, fa, /3 2 , . . . are constants. Since the formula with a
large number of constants is of little practical use, Graetz neg¬
lected the constants fr, j8 2 , . . . which are of small magnitude
and thus obtained

77 = a -J->

or if the temperatures are changed to the absolute scale,

( 44 )

130

FLUIDITY AND PLASTICITY

where T cr is the critical temperature and T s is the temperature of
solidification.

Applying this formula to the data of Rellstab, and Pribram
and Handl, Graetz found that it was satisfactory in some fifty
eases, but it is inapplicable to the fatty alcohols.

This formula is a particular form of the one already given by
Slotte. Slotte (1883-90) reached the conclusion that none of
the preceding formulas gives satisfactory results with substances
whose viscosity changes rapidly with the temperature, as is
generally true of very viscous substances. He proposed the
formula

’ = <FW- <«

where T is the temperature centigrade and a, b , and n are arbi¬
trary constants. Slotte found that this formula gave better
results than any other and Thorpe and Rodger adopted it in
their great work as the most satisfactory formula at their disposal,
but this formula like the others breaks down when applied to the
alcohols. In the case of several of the alcohols it was necessary
to apply the formula three times with different constants over
different parts of the curves in order to reproduce the observed
values with anything like the desired accuracy. The values of
n vary from 1.4 to 4.3.

Several exponential formulas have been proposed. Reynolds
(1886) and Stoel (1891) suggested the formula

7] = ae “ yTe (46)

which Reynolds found to apply to olive oil and Stoel and De Haas
(1894) found to apply to methyl chloride between the boiling-
point and the critical temperature. In this formula e is the
natural logarithmic base and a and y are constants. Heydweiller
in 1895 investigated benzene and ethyl ether over a similar
range of temperature, and found that the formula holds between
the reduced temperatures of 0.62 and 0.87 for the purposes of
interpolation; but that similar systematic deviations occur for all
substances, which is due to the fact that the temperature coef¬
ficient of the viscosity is not constant but passes through a point of
inflection. Below the boiling-point the viscosity-temperature
curve is convex toward the temperature axis while near the

FLUIDITY AND TEMPERATURE

131

critical temperature it is concave, and the deviations from the
formula are considerable. Heydweiller made the interesting
observation that within the series of compounds with which he
worked the temperatures of equal viscosity are in the ratio of
their critical temperatures. But to this rule water and the
alcohols are exceptional.

De Keen in his Theorie des Liquides (1888) found the following
formula satisfactory

V = 7] 0 (1 + ae~ hTe ) c (47)

The constant c varies little from liquid to liquid from 2.65 to 2.85.

Jaeger (1893) has worked out a kinetic theory of liquids on
the ground that the transfer of momentum takes place by the
molecules passing back and forth from one layer of molecules to
another. He gives the expression

v =

2 r 2 pv
~3X“

(48)

where r is the radius of a molecule, v its mean velocity, X its
mean free path, and p is the density of the liquid.

Similarly Kamerling Onnes has derived a formula from the
theory of corresponding condition of van der Waals,
v V **

vm .' CoBStont <48)

where V is the molecular volume, M the molecular weight,
T C r the critical temperature, and rjV ^ is the molecular surface
friction. The formula does not apply at low temperatures and
perhaps only perfectly as the critical temperature is reached.

Perry (1893) states that sperm oil cannot be represented by
any single formula since a discontinuity occurs in the viscosity
and density curves at 40°. It should be added that he took care
that the velocity of flow did not exceed the critical value for
viscous flow. He employed two sets of constants in the formula
v = a{T - &)-* (50)

Examining a considerable number of the formulas which
had already proved of value and given above, Duff (1896)
obtained the following formula by integrating the curve of
subtangents derived from them:

( 51 )

132

FLUIDITY AND PLASTICITY

This formula reproduces observed values of all classes of com¬
pounds with remarkable fidelity, but it contains four constants.

Finally Batschinski (1901) has indicated that the viscosity
varies inversely as the cube of the absolute temperature

where E is dependent upon the nature of the liquid and is called
“the viscosity parameter.” Using the data of Thorpe and
Rodger, Pribram and Handl, Gartenmeister and others, Bats¬
chinski has tested his expression fully. While it gives very good

Temperature absolute

Fig. 48.—Diagram illustrating ideal fluidity-temperature curves.

agreement in many cases, there are numerous exceptions, particu¬
larly water, the alcohols, acids and anhydrides (cf. Eq. (41)
for mercury).

Fluidity and Temperature.—From the evidence given in Chap¬
ter I to prove that fluidities are normally additive in homogeneous
mixtures, it would appear probable that fluidity-temperature
curves in their simplest form are straight lines, and since every
liquid is theoretically capable of existing in an undercooled
condition, these curves meet at absolute zero. The equation for
the fluidity-temperature curves of all substances should be

cp— aT

which is the same as Eq. (43). A family of curves would present
the appearance shown diagrammatically in Fig. 48.

FLUIDITY AND TEMPERATURE

133

Unfortunately, fluidity-temperature curves are not generally
linear throughout. Even the aliphatic hydrocarbons, which are
supposedly unassociated, give curves which depart considerably
from linearity. For this peculiarity two possible explanations
suggest themselves. The effect of expansion is not linear except
in the case of mercury and there may be changes in the molecular
weight, either association or dissociation. Later each of these
causes will be considered in detail. Suffice it for the present
to note that the alcohols, Fig. 46, give fluidity-temperature
curves which are strongly curved at low fluidities, but the curves
tend to become linear at high fluidities as is true of the other

classes of compounds. In a _

given homologous series the evi¬
dence at hand shows a tendency
for the fluidity-temperature
curves near the ordinary boil- ^
ing-points to become parallel §
to each other with equal dis- g
tances between them. But
they must all meet at absolute
zero of temperature, which

requires that in the higher Temperature absolute

members of the series at least Fig. 49.—Diagram illustrating the

there should be a region of fluidity-temperature curves of a “homol-
° . ogous senes” of liquids.

rapid curvature. I his is the

region of “softening” which has been observed in many viscous
substances. According to the views here presented the phe¬
nomenon of softening may not occur in any substance con¬
sisting of monatomic molecules but will certainly become
manifest as we increase the complexity of the molecule. These
ideas are represented diagrammatically in Fig. 49. If they
represent accurately the actual behavior of substances, the
fluidity-temperature curves have the following properties: (1)
They all approach linearity as the fluidity increases, i.e., each
tends to become asymptotic to a line which forms an acute
angle with the temperature-axis. (2) As the fluidity becomes
very small, each curve tends to become asymptotic to the tem¬
perature-axis itself. Thus the general form of the fluidity-
temperature curve must be a hyperbola, since this is the

134

FLUIDITY AND PLASTICITY

curve which contains these properties. The linear curve for
substances like mercury is of course but a special and extreme
case. It is very important that we find out whether other mon¬
atomic liquids, such as, for example, liquid sodium, argon, et ceL
have a linear volume temperature and fluidity-temperature curve.
. Given the properties of a general fluidity-temperature curve
it is easy to obtain the curve which contains these properties.
The simplest is

T = A<p + C - - (53)

<P

Tor the simplest case, which we have in mercury, the constants
£ and C are each zero and our equation becomes that of Bats chin-
ski (41); for other substances at high temperatures the term B/<p
becomes negligible and the equation becomes that of the asymp¬
tote

T = Acp ’+ C

This linear equation can be transformed into the formula of
Meyer and Hosencranz. Thus while our formula (51) agrees
with' some of the simpler formulas which have been proposed,
it is at variance with those which have had the widest usefulness.
Tor example, the Meyer-Poiseuille formula may be written

<p = A + BT C + C2V (53a)

which is the equation of a parabola. Were it valid as a general
equation, the fluidity would have a minimum value at 0°C,
and at high temperatures the fluidity curve would tend to be¬
come parallel to the fluidity-axis. It needs no argument to show
that this equation cannot be general.

By increasing the number of terms, the formula of Meyer-
Poiseuille becomes the second formula of Slotte,

<P = A(B + T c ) n

The effect of the additional terms is to increase the curvature
but it does not remedy the defects noted. The equation of
Stoel and others may be regarded as a special case of Slotted
equation where the number of terms is infinite.

The formula of Graetz is defective in that it assumes that the
fluidity becomes zero at a definite positive temperature and that
at the critical temperature the fluidity is infinite. Neither of
these assumptions can longer be held.

FLUIDITY AND TEMPERATURE

135

Table XXXIV.—The Constants in the Equation T ~ A<p - f- < y ___

ip

Substance ,

A

|

B !

C

^Vlean per
cent differ¬
ence

1

Water.

0.27727

1,263.3

278.80

O. 17

2

Bromine.

0.79098

1,376.3

227.16

0.03

3

Nitrogen peroxide.

0.32274

3,837.6

231.92

0.00

4

Pentane.

0.16544

14,937.0

256.61

0.03

5

Isopentane. . .

0.18327

10,454.0

234.17

0.03

6

Hexane.

0.21892

9,137.8

254.03

0.04

7

Isohexane.

0.20708

9,562.8

252.85

0.05

8

Heptane.

0.23203

8,681.2

272.70

0.09

9

Isoheptane .

0.22445

869.84

267.35

0. 12

10

Octane.

0.27055

6,332.6

277.25

0. 16

11

Trimethyl ethylene .

0.21942

6,770.3

203 . 56

0.06

12

Isoprene .

0.21505

7,126. 5

208.74

0.09

13

Diallyl.

! 0.19818

9,774.4

247.75

0.06

14

Methyl iodide.

0.44916

3,107. 1

215.85

0.01

15

Ethyl iodide.

; 0.43828

4,348.2

243.29

0.04

16

Propyl iodide . 1

0.46571

3,474.4

255.85

0 . 11

17

Isopropyl iodide.

0.41279

4,108.8

262.07

0. 14

18

Isobutyl iodide.!

0.46881

2,727.2

264.30

0.23

19

Allyl iodide. (

0.47443

2,994.0

249.80

0.11

20

Ethyl bromide.j

0.35853

4,073.2

217.39

0.02

21

Propyl bromide.

0.36084

4,950. 6

248.95

0.04

22

Isopropyl bromide.

0.33069

5,009.9

248.57

0.03

23

Isobutyl bromide .

0.34098

4,754.4

270.75

0.20

24

Allyl bromide .

0.36448

4,467. 7

241.77

0.06

25

Ethylene bromide .

0.68897

1,421. 6

277.92

0.23

26

Propylene bromide .

0.64567

1,463.0

i 278.53

0.34

27

Isobutylene bromide .

0.65358

1,060.2

288.40

0.04

28

Acetylene bromide .

0.63374

2,275.3

249.46

0.12

29

Propyl chloride ..

0.25540

7,465.2

246.74

0.05

30

Isopropyl chloride .

0.24993

5,881.9

234.22

0.01

31

Isobutyl chloride .

0.28656

4,973.0

253.00

0.02

32

Allyl chloride.

0.26292

6,377.2

234.10

0.03

33

Methylene chloride.

0.39806

2,666.5

212.97

0.04

34

Ethylene chloride.

0.44121

2,219.3

258.83

0.08

35

Ethylidcne chloride.

0.33277

4,651.6

247.99

0.04

36

Chloroform.

0.40697

4,400.0

245.73

0.07

37

Carbon tetrachloride.

0.47337

1,807.5

262.15

0.08

38

Carbon dichloride.

0.55768

3,081.6

259.13

0.20

39

Carbon disulfide.

0.26901

16,751.0

282.23

0.17

40

Methylsulfide.

0.25514L

8,387.6

230.57

0.01

41

Ethylsulfi.de.

0.28517

6,447.9

258.00

0.11

42

Thiophene.

0.38204

2,967.2

254.95

0.09

43

Dimethyllceto ne.

0.23871

8,905.0

247.64

0.03

44

Methylethylketone.

0.27275

5,572.6

252.32

0.08

45

Diethylketone.

0.28145

6,179.3

262.41

0.16

46

Methylpropyl ketone.

0.29251

5,544.3

263.21

0.15

47

Acetaldehyde.

0.15205

18,364.0

265.13

0.006

48

Formic acid.

0.52465

963.4

281.57

0.19

136

FLUIDITY AND PLASTICITY

Table XXX.IV.—(Continued)

Substance

A

B

C

Mean per
cent differ¬
ence

49

Acetic acid.

0.42437

2,716.8

291.81

0.22

50

Propionic acid.

0.43533

2,908.9

287.53

0.41

51

Butyric acid.

0.45359

1,951.9

296.43

0.69

52

Isobutyric acid.

0.45841

2,143.6

288.41

0.43

53

Acetic acid anhydride.

0.40620

2,747.8

273.61

0.31

54

Propionic acid anhydride.

0.39705

2,444.6

287.45

0.61

55

Diethy ether.

0.16574

14,674.0

256.72

0.07

56

Benzene.

0.32052

2,633.1

260.82

0.05

57

Toluene.

0.32688

4,193.5

262.66

0.12

58

Ethyl benzene.

0.34180

4,540.8

273.54

59

Ortho xylene.

0.37738

3,009.3

271.96

0.28

60

Meta xylene.

0.34134

4,542.9

266.82

0.19

61

Para xylene.

0.32087

5,127.4

277.17

0.20

62

Methyl alcohol.

0.24316

4,498.9

279.01

0.14

63

Ethyl alcohol.

0.28395

2,398.6

298.39

0.12

64

Propyl alcohol.

0.31496

1,211.7

308.41

0.35

65

Isopropyl alcohol.

0.29810

738.16

300.17

0.36

66

Butyl alcohol.

0.33610

877.08

311.94

0.72

67

Isobutyl alcohol.

0.33648

512.18

309.72

0.81

68

Trimethyl carbinol.

0.29657

260.86

305.73

0.33

69

Amyl alcohol, active.

0.35020

376.48

311.50

1.17

70

Amyl alcohol, inactive.

0.36060

513.18

312.40

1.08

71

Dimethyl ethyl carbinol.

0.29578

259.87

307.20

0.87

72

Allyl alcohol.

0.28815

1,935.7

299.53

0.23

73

Methyl formate.

0.34444

1,292.0

198.31

0.01

74

Ethyl formate.

0.29418

4,858.1

239.32

0.02

75

Propyl formate.

0.29797

4,800.6

260.44

0.06

76

Methyl acetate.

0.26047

6,475.7

249.48

0.03

77

Ethyl acetate.

0.27056

5,361.2

257.20

0.06

78

Propyl acetate.

0.29534

4,262.6

267.50

0.15

79

Methyl propionate.

0.27300

5,954.3

261.08

0.15

80

Ethyl propionate.

0.29125

4,846.4

264.51

0.12

81

Methyl butyrate.

0.30210

4,315.3

265.89

0.15

82

Methyl isobutyrate.

0.28615

5,073.2

264.44

0.15

83

Methyl propyl ether.

0.21797

7,206.3

224.27

0.03

84

Ethyl propyl ether.

0.20872

8,933.6

255.80

0.05

85

Dipropyl ether.

0.23579

6,858.5

266.34

0.14

86

Methyl isobutyl ether.

0.21201

8,748.8

250.76

0.03

87

Ethyl isobuty ether.

0.22545

7,188.2

260.96

0.09 ‘

We will now see how far the fluidity Eq. (53) can be used to
reproduce the experimentally observed values. In Table
XXXIV we give the constants for the 87 substances investigated
by Thorpe and Rodger and in the last column of the table we
give the average percentage difference between the observed
and calculated values.

The mean percentage difference between the calculated and

FLUIDITY AND TEMPERATURE

13 7

observed values is 0.17 for the 87 substances and based on some
1,000 duplicate observations. If we omit the alcohols, this
difference falls to 0.09 for 70 substances. This is much better
Agreement than Thorpe and Rodger obtained with Slotted
Equation, since the percentage difference is nearly twice the
above, viz., 0.15 per cent for 64 substances. But the real test
is with substances which give fluidity curves departing widely
from the linear type and here Slotted equation breaks down
completely.

For this type of substances, the fluidity Eq. (53) with three
constants does not reproduce the observed values to the limit of
experimental error, but a great improvement can be made by
introducing another constant and writing the equation

T - Ar + C -Un> cm>

For example, the mean divergence between the observed and
calculated values for the eight substances, which gave the largest
percentage difference, was 0.77 per cent with the simpler for-
mula; the Eq. (54) with four constants reduces this to only
0.07 per cent which is nearly within the limits of the experi¬
mental error. In the case of water, which gave a mean difference
of only 0.17 per cent with the simpler formula, the difference is
reduced to 0.01 per cent and similarly in the case of octane it is
reduced from 0.16 to 0.02 per cent. For reference, the constants
for these substances are given in Table XXXY.

T^jble XXXV. — Thk Constants in the Equation T — A<p -f- C -r~r^

<P 4 - L>

j

Mean

Substance

A

B

C

D

per cent
difference

W ater.

0.23275

8,676.8

309.17

120

0.01

Octane.

0.14507

100,745.0

438.10

400

0.02

TEixutyric acid.

0.10154

70,630.0

504.44

250

0.02

Isot>utyric acid.

0.23862

43,665.0

433.17

200

0.06

JPropionic acid anhydride.

0.23019

52,294.0

425.82

250

0.07

I8xrtyl alcohol..

0.23095

4,802.0

349.71

40

0.04

Isobutyl alcohol.

0.23700

2,993.7

340.06

30

0.09

.Active amyl alcohol.

0.24050

2,942.8

346.82

30

0.08

Inactive amyl alcohol.

0.24191

3,908.7

354.17

35

0.09

X>i methyl ethyl carbinol.

0.22988

2,124.0

328.84

30

0.09

138

FLUIDITY AND PLASTICITY

Fluidity and Pressure.—To find the effect of pressure on
viscosity, Coulomb in 1800 measured the rate of oscillation
of a disk in a liquid both under atmospheric pressure and when
the space above the liquid had been evacuated. It was found
that the viscosity is independent of small changes of pressure.
This conclusion was confirmed by Poiseuille in 1846, using his
transpiration method.

However, quite the opposite conclusions must be drawn
from the experiments of Warburg and Babo (1882), but they
employed liquid carbon dioxide at 25.1°C, which is quite near the
critical temperature, and they used pressures from 70 to 105
atmospheres. It is worth noting that under these conditions
the compressibility of carbon dioxide is 0.00314 which is about
18 times as great as that of ether at the same temperature. They
found an increase in the viscosity which amounted to over 25
per cent, and it therefore seemed possible that the effect was
caused by the change in density and that a similar effect would be
observed in other liquids if high enough pressures were employed.

Warburg and Sachs (1884) continued the previous investigation
and indeed found that liquid carbon dioxide, ether, and benzene
all suffer an increase in viscosity on increasing the pressure, but
they also noted that water is exceptional in that an increase in
pressure actually lowers the viscosity. They sought to connect
the viscosity and pressure by means of the following linear
formula,

7] = 170 (1 + Ap). (55)

The values of the constant A are given in Table XXXVI.

Table XXXVI.— Constants in Equation (53)

Substance

Carbon

dioxide

Ether

Benzene

Water

Temperature of experiment.

.... 25.1

20

20

20

Critical temperature.

.... 30.9

190

280.6

365

A X 10".

.... 7,470

730

930

-*170

The striking fact that water is peculiar in this as in so many
other respects was discovered independently by Rontgen (1884).
It was made the subject of a special study by Cohen in 1892,

FLUIDITY AND TEMPERATURE

139

working at 1, 5, and 23° at pressures ranging from 1 to 600
atmospheres and using pure water and four solutions of sodium
chloride of different concentrations. The nature of his results
is shown in Figs. 50 to 54. In Fig. 50 the percentage change
in the time of flow, [(ti — t p )/ti] 100, is plotted as ordinates

Fig. 53.

Fig. 51. Fig. 52. Fig. 54.

The effect of pressure on the viscosity of aqueous solutions. (After Cohen.)

against the pressures as abscissas. It is observed that the vis¬
cosity continues to decrease for all pressures up to 900 atmos¬
pheres, but the decrease becomes yery slight at 23°. In Fig. 51
the ordinates are the same as before but the temperatures are
plotted as abscissas. It is evident that the'curves are approach¬
ing each other and the zero axis, and thus they indicate the
possibility that at some higher temperature the curves will cross

140

FLUIDITY AND PLASTICITY

the zero axis and the viscosity will then increase with the pressure
as in other liquids.

In Fig. 52 the percentage change in the time of flow is plotted
against the pressure as in Fig. 50. A saturated solution (25.7
per cent) is seen to behave unlike pure water but like other
liquids in that the pressure causes an increase in the time of
flow. The curves for other concentrations lie between those
for pure water (0 per cent) and those for the saturated solution.
The continuous curves represent measurements at 14.5° and the
dotted curves represent measurements at 2°. From a compari¬
son of these it is evident that the temperature coefficient of the
percentage change in the time of flow decreases rapidly as the
concentration of the solution is increased. This is shown more
clearly in Fig. 53 where the temperatures are plotted as abscissas
against the percentage change in the time of flow for a pressure of
600 atmospheres, i.e., the percentage change in the time of flow
is nearly constant in the most concentrated solution. Further¬
more in an 8 per cent solution at a pressure of 600 atmospheres,
the effect of pressure on the time of flow is zero at 11°. Below
that temperature, pressure decreases the time of flow; above that
temperature, it increases it.

The relation of the percentage change in the time of flow at
600 atmospheres pressure as ordinates to the percentage con¬
centration as abscissas is indicated in Fig. 54. At 22.5° the per¬
centage change of the time of flow is a linear function of the
concentration, but at 2° this is no longer true, the effect of the first
additions of the salt to water being much greater than subsequent
additions. The curves do not cross, hence the effect of pressure
in the concentrated salt solutions is greatest at the high tempera¬
tures even up to the point of saturation. Cohen found the oppo¬
site to be true of turpentine, viz ., the effect of pressure is greatest
at low temperatures. According to Warburg and Sachs, ether
behaves like turpentine and benzene like sodium chloride
solutions.

Hauser (1901) found that the effect of pressure upon the vis¬
cosity of water continually decreases as the temperature is
raised until it becomes zero at 32° up to 400 atmospheres. Above
this temperature the viscosity increases with the pressure as in
other liquids, and the effect becomes more pronounced as the

FLUIDITY AND TEMPERATURE

141

temperature is raised, amounting to 4 per cent for a pressure of
400 atmospheres at 100°.

Faust (1913), using pressures as high as 3,000 kg per square
centimeter, found that the viscosity of ether, alcohol, and carbon
disulfide were each increased by about fourfold. This result
has important bearing upon the theory and practice of lubrica¬
tion. And very recently J. H. Hyde (1920) has reported to
the Lubricants and Lubrication Committee of the Department of
Scientific and Industrial Research the results of an investigation
of the viscosity of a variety of lubricating oils, using pressures
up to 7 tons per square inch. He made the important deduction
that the mineral oils increase in viscosity far faster with the
pressure than do the fixed oils. Thus the viscosity of Mobil® BB
increases over twenty-six-fold, whereas with the same increase
in pressure the fixed oils increase in viscosity about fourfold.

Fluidity and Volume.—We have now before us the two follow¬
ing generalizations: (1) An increase in pressure is usually
associated with a decrease in fluidity, and (2) an increase in
temperature is usually associated with an increase in fluidity.
To be sure, there are prominent exceptions to both generaliza¬
tions, as, for example, water, in its behavior under pressure and
sulfur, as affected by temperature. But water and sulfur are
highly associated in the liquid state so that an explanation of
these exceptions is possible on the basis of changing molecular
weights.

Lowering of pressure or raising of the temperature of a liquid
have one thing in common in addition to their similar effect
upon the fluidity—they both produce an increase in the volume,
to which there are very few exceptions. It is worth while there¬
fore to investigate the question of how much of the change in
fluidity can be attributed primarily to a change in volume. If
one has in mind the fact that in gases, where the volume changes
are large, the fluidity is nearly independent of the volume, one
would naturally expect the changes in the volume of liquids to
be responsible for only a small part of the fluctuations in fluidity
which actually exist. But the viscosities of gases and liquids
arise from entirely different causes, hence reasoning by analogy
is useless.

The parallelism between fluidity and volume may be followed

142

FLUIDITY AND PLASTICITY

in another direction, for generally speaking whenever ’
are mixed and a contraction takes place, there seemH * *'
decrease in fluidity. Alcohol and water, acetic acid *
and chloroform and ether are a few examples. On tin
when liquids mix with an expansion in volume, the mi*
greater fluidity than we would expect from the linear ft 15
ume concentration curve. Methyl iodide and carbon * 1 j
furnish an example of this sort (Bingham et al., 1913). ^

facts have suggested to various workers (Brillouin {\* H
2, p. 127; Dunstan and Thole (1909), p. 204; Bingh^ f *'
p. 270) that fluidity and volume are intimately relate* * -
in fact than fluidity is to either temperature or presm** 4 "
In spite of this intimate relationship, it has been lift ^ *
gated. Slotte (1894) stated that the logarithms of tin 1
are proportional to the logarithms of the specific veil 11 *
from this observation he deduced his second Eq. (43).
most important discovery was made by Batschinski in I 04
found that the relation between the molecular volume $
fluidity may be expressed in the following formula:

<j> = C(V - G)

where 0 and C are constants. The constant Q may be*
the limiting value which the molecular volume of any *|
have as its fluidity approaches zero, and it is thereby
the “molecular limiting volume.” Consequently the
V — 0 may be called the “molecular free volume” and ft#
relation may be very simply expressed as follows: Th*'
varies directly as the free molecular volume . Sixty-six * # i
substances investigated by Thorpe and Rodger exhibit f I
tionship and the* values of the fluidity, as calculated f r
volume, seldom deviate from the observed values by m#*
1 per cent. The 21 substances for which the agreem^t$1
good include the alcohols, water, some of the acids, the art*!
drides, and some of the halides. These substances arc
regarded as associated and it may well be that the i**«;
weight is not constant as the temperature is raised. Th**
of the very remarkable agreement obtained is shown %$%
XXXVII containing data for benzene obtained by Thr»r
Rodger between 0° and the boiling-point and by Hey*l

FLUIDITY AND TEMPERATURE

143 .

from the boiling-point up to 185.7°. This agreement is shown
graphically for a number of substances in Fig. 55.

Table XXXVII.— Calculation of the Fluidity of Benzene from its
Volume by Means of the Formula <p = (V — 81.76)/0.045,35

Temperature

<p Observed

Specific

volume

ip Calculated

Difference

0.0

110.8

1.1124

311

0

10.0

131.5

1.1242

i 132

0

20.0

154.1

1.1377

155

1

30.0

178.0

1.1514

179

1

40.0

203.1

1.1661

204

1

50.0

228.8

1.1812

230

1

60.0

256.1

1.1966

256

0

70.0

284.9

1.2124

283

-2

80.0

305.8

1.2278

311

5

78.4

314.0

1.2253

310

—4

100.5

383.7

1.2624

385

1

131.8

504.8

1.3255

510

5

161.4

646.8

1.3957

649

2

185.7

797.4

1.4661

794

-3

Batschinski tested his formula with the recently obtained
data of Phillips (1912) on the viscosity of carbon dioxide under
varying pressures. He thus proved that at least while the
substance remains liquid the fluidity varies directly as the free
volume.

Table XXXVIII.— Calculation of the Fluidity of Carbon Dioxide

from its Volume by Means of the Formula <p —

V - 33.2
0.0169

Pressure,

atmospheres

Observed

Specific

volume

Calculated

Per cent,
difference

110.5

1,299

1.259

1,300

0

96.0

1,443

1.316

1,450

1

82.0

1,689

1.397

1,640

—3

76.0

1,890

1.471

1,850

-2

72.0

2,183

1.575

2,080

-5

144

FLUIDITY AND PLASTICITY

If we could always compare the fluidities of liquids at a defini
multiple of the molecular limiting volume, it is evident that t.
effect of the volume, and therefore of temperature and pressu
in so far as they affect the volume, would be eliminated. Such
procedure would enormously simplify fluidity relationship
As a matter of fact Batschinski has shown that the molecuL
limiting volume possesses an additive character, the values

Fig. 55.—Relative volume-fluidity curves after Batschinski using the data c
Thorpe and Rodger.

4, Bromine; 5, nitrogen peroxide; 9, isohexane; 11, isoheptane; 13, trimethy.
ethylene; 14, isoprene; 15, diallyl; 16, propylchloride; 17, isopropylchloridc
18, isobutylchloride; 19, ally 1 chloride; 20, methylenechloride; 21, ethylenechlc
ride; 22, ethylidenechloride; 23, chloroform; 25, perchlorethylene; 68, ethyl
benzene; 69, orthoxylene; 70, metaxylene; 71, paraxylene.

the atomic constants being H = 4.3, 0 = 8.6, C = 8.8, Cl =
19.2, Br = 24.8, I = 32.0, S = 19.0, a double bond = 3.3
and an iso-grouping = 0.7. For 53 of the substances studied b]
Thorpe and Rodger the differences between the observed anc
calculated values of the molecular limiting volume do not exceec
2 per cent. The limiting specific volume is approximately 0.3(T<

. of the critical volume which is close to the parameter b of var
der Waals’ equation.

We are now in a position to get a very clear understanding
of the law that the fluidity is proportional to the free volume.

FLUIDITY AND TEMPERATURE

145

When the molecules of a liquid are closely packed, the volume
reaches its minimum value and the fluidity is zero. With
tetrahedral close-packing of the molecules, shear would require
rupture of the molecules themselves. If there are pore spaces
between or within the molecules, they do not give rise to fluidity,
so that the molecules somewhat resemble close-fitting solid
figures. As the fluid expands, due to molecular agitation, the
volume of the molecules themselves, i.e., the inner molecular
volume may remain the same, but the ordinary, i.e., the outer
molecular volume increases. The law states that the fluidity
originates solely in the free space which is the difference between
the outer molecular volume, or the volume occupied by the
molecules, and the inner molecular volume or the spac e filled by
the molecules in the sense indicated above. Given two sub¬
stances with the same outer molecular volume, it is evident that
the one with the larger molecular kernel will have the smaller
fluidity. It is therefore natural to expect that the limiting
molecular volumes should be additive as Batschinski has found
to be the case. This opens the way to a study of the relation
between fluidity and chemical composition and constitution
which, is most fascinating. It is very simple to measure the
outer molecular volume, and if this with the fluidity will give a
certain and easy method for determining the inner molecular
volume, it is a result much to be desired. It is apparent that
density and fluidity determinations should go hand in hand.

If the above reasoning held true for gases as well as liquids,
the fluidity isothermals of carbon dioxide should closely resemble
the familiar volume isothermals. By substituting for the volume
its value in terms of the fluidity given by Batschinski’s law, we
would obtain a modified van der Waals’ equation. As a matter
of fact van der Waals’ equation may be written

T - P -„ - +- a L

R R Rv' Rv 2

but since viscosities are ordinarily measured at constant, i.e.,
atmospheric pressure, this may be written

T = A<p + C —

B

E

<p -f- D (^> -f- D ) 2
where A, B, C y D and E are constants. This is identical with
Eq. (54) which is entirely satisfactory for liquids, except that it

144

FLUIDITY AND PLASTICITY

If we could always compare the fluidities of liquids at a definite
multiple of the molecular limiting volume, it is evident that the
effect of the volume, and therefore of temperature and pressure
in so far as they affect the volume, would be eliminated. Such a
procedure would enormously simplify fluidity relationships.
As a matter of fact Batschinski has shown that the molecular
limiting volume possesses an additive character, the values of

Fig. 55.—Relative volume-fluidity curves after Batschinski using the data of
Thorpe and Rodger.

4, Bromine; 5, nitrogen peroxide; 9, isohexane; 11, isoheptane; 13, trimethyl-
ethylene; 14, isoprene; 15, diallyl; 16, propylchloride; 17, isopropylchloride;
18, isobutyl chloride; 19, allyl chloride; 20, methylenechloride; 21, ethylenechlo-
ride; 22, ethylidenechloride; 23, chloroform; 25, perchlorethylene; 68, ethyl¬
benzene; 69, orthoxylene; 70, metaxylene; 71, paraxylene.

the atomic constants being H = 4.3, 0 = 8.6, C = 8.8, Cl =
19.2, Br = 24.8, I = 32.0, S = 19.0, a double bond = 3.3,
and an iso-grouping = 0.7. For 53 of the substances studied by-
Thorpe and Rodger the differences between the observed and
calculated values of the molecular limiting volume do not exceed
2 per cent. The limiting specific volume is approximately 0.307
of the critical volume which is close to the parameter b of van
der Waals’ equation.

We are now in a position to get a very clear understanding
of the law that the fluidity is proportional to the free volume.

FLUIDITY AND TEMPERATURE

145

When the molecules of a liquid are closely packed, the volume
reaches its minimum value and the' fluidity is zero. With
tetrahedral close-packing of the molecules, shear would require
rupture of the molecules themselves. If there are pore spaces
between or within the molecules, they do not give rise to fluidity,
so that the molecules somewhat resemble close-fitting solid
figures. As the fluid expands, due to molecular agitation, the
volume of the molecules themselves, i.e., the inner molecular
volume may remain the same, but the ordinary, i.e. } the outer
molecular volume increases. The law states that the fluidity
originates solely in the free space which is the difference between
the outer molecular volume, or the volume occupied by the
molecules, and the inner molecular volume or the space filled by
the molecules in the sense indicated above. Given two sub¬
stances with the same outer molecular volume, it is evident that
the one with the larger molecular kernel will have the smaller
fluidity. It is therefore natural to expect that the limiting
molecular volumes should be additive as Batschinski has found
to be the case. This opens the way to a study of the relation
between fluidity and chemical composition and constitution
which is most fascinating. It is very simple to measure the
outer molecular volume, and if this with the fluidity will give a
certain and easy method for determining the inner molecular
volume, it is a result much to be desired. It is apparent that
density and fluidity determinations should go hand in hand.

If the above reasoning held true for gases as well as liquids,
the fluidity isothermals of carbon dioxide should closely resemble
the familiar volume isothermals. By substituting for the volume
its value in terms of the fluidity given by Batschinski^ law, we
would obtain a modified van der Waals’ equation. As a matter
of fact van der Waals’ equation may be written
m P vb a . ab

T ~ 11 R Rv + Rv 2

but since viscosities are ordinarily measured at constant, i.e.,
atmospheric pressure, this may be written

T = A<p + C

B

+ •

E

<p + D 1 (<p + D) 2
where A , B, C, D and E are constants. This is identical with
Eq. (54) which is entirely satisfactory for liquids, except that it

10

146

FLUIDITY AND PLASTICITY

contains an additional term, which becomes negligible when the
fluidity is large.

That the fluidity isothermals of carbon dioxide do not in any
way resemble the volume isothermals as we pass into the gaseous

Fig. 56.—The fluidity isothermala of carbon dioxide based on the measurements

of Phillips.

condition is sufficiently obvious from inspection of Fig. 56,
plotted from Phillips’ data. The continuous curves are drawn
between observed points. The broken lines are added for
diagrammatic purposes. The left half of the figure, correspond¬
ing to low fluidity and temperature, presents a strong similarity

FLUIDITY AND TEMPERATURE

147

to the familiar pressure-volume diagram. At the highest
pressures, the fluidity is not greatly affected by a change in
pressure, e.g., at 32° and a pressure of 120 atmospheres, a lowering
of the pressure by 4 atmospheres causes an increase in fluidity
of less than 4 per cent. At a lower pressure the fluidity becomes
extremely susceptible to changes in pressure, a lowering of the
pressure by 4 atmospheres at 76 atmospheres causing an increase
in the fluidity of a full 100 per cent at 32°. The gaseous and liquid
phases are both present inside of the curve kbmcl. But the right side
of the figure is entirely different from the familiar pressure-
volume diagram. Instead of the fluidity being highly susceptible
to changes in pressure, as is the volume, it is but slightly affected,
e.g., at 32° and 50 atmospheres pressure, a lowering of the pressure
by 4 atmospheres causes only a 10 per cent increase in the fluidity.

Let us follow in detail the isothermal of carbon dioxide at 20°
which is well below the critical temperature. At high pressures,
the fluidity increases nearly linearly from a to b; there is then
a sudden increase in the fluidity from 1,500 to 5,300 absolute
units, as the substance passes from the liquid to the gaseous
condition. We should expect the fluidity to continue to increase
as the pressure is further lowered, giving the curve cd f , but the
curve actually obtained is cd. We have seen that the fluidity of
liquids increases with the temperature, while, on the other hand,
the fluidity of gases decreases with the temperature, hence, the
pressure-fluidity curves for different temperatures must intersect
each other. The figure proves that not only is this true, but, •
when the temperatures are sufficiently high, the curves all tend to
pass through the particular point n, so that at this point the
fluidity is independent of the temperature; for the lower tempera¬
tures, the curves seem to intersect each other on the curve ncl.

Collisional and Diffusional Viscosity.—That the pressure-
fluidity curves do not follow an equation of the van der Waals
type as the fluidity becomes large may be due to the appearance
of a new type of viscous resistance. We must therefore now
investigate more particularly into the nature of viscous resis¬
tance. One's first impulse in looking for a cause* of viscosity is to
assume a cohesion between the particles which is exerted during
motion and acts in opposition to motion, 1 but with the develop-

1 “Kinetic Theory of Gases,” Meyer, p. 171.

148

FLUIDITY AND PLASTICITY

ment of the kinetic theory of gases, it became evident that
viscous resistance arises from the diffusion of the particles
of high translational velocity into layers whose translational
velocity is lower, and vice versa . According to this explanation,
the loss of translational velocity must increase with the tempera¬
ture, which accords with the fact that the fluidity of a gas
decreases as the temperature is raised.

But in liquids the fluidity increases with the temperature
and it is generally agreed that there is a second cause of viscous
resistance, which, without any very good reason in its favor,
has been repeatedly attributed to the attraction between the
molecules. According to Batschinski 1 “ If we think of two parallel
layers of liquid as of two rows of men, the men moving in place
of molecules, we must assume that these men take hold of their
nearest neighbors and hold them for a time.” This explanation is
however inadequate, for a particle A, coming within the range
of attraction of a particle B in an adjacent layer supposed to be
possessed of slightly less translational velocity, will be accel¬
erated and only after passing B will the retardation take place.
Apparently the two actions exactly neutralize each other, or if
they do not there must result a destruction of energy in violation
of the first law of thermodynamics. No reasonable hypothesis
has been proposed to extricate us from this dilemma, on the
basis of cohesion, hence, we are forced to look for some other
cause. Whatever -the explanation, it must show how transla¬
tional or ordered motion is being continuously transformed
into heat or disordered motion.

To get a clearer idea of the nature of the two causes of viscous
resistance, we imagine two parallel planes A and B , the former
moving to the right parallel to itself in respect to the second
plane, which for convenience only may be assumed to be at rest.
We will first assume that between 1 the planes there is a highly
rarefied gas. If the walls are smooth and unyielding and the
particles of gas perfectly elastic spheres, we will not have a
model of viscous flow; for as the particles collide with the sur¬
faces, the angle of rebound will be equal to the angle of incidence,
there will be no translational velocity transmitted to or from the
walls and the so-called “slipping”, would be perfect. In order

1 ( 1913 ) p. 643 .

FLUIDITY AND TEMPERATURE

149

to obtain a model of viscous flow it is therefore necessary to
assume that the surfaces are not perfectly smooth. In view of the
known discontinuity of matter, one could hardly assume a
smooth surface, and the least degree of roughness which one
could well assume would be one made up of equal spheres
whose centers lie in the same plane and as closely packed
together as possible. That there is a greater degree of roughness
in all ordinary surfaces is probable, but it suffices for our present
purposes to show in what follows that this simple assumption
in regard to the nature of the surfaces gives a workable model of
viscous flow.

It becomes necessary to show that momentum is being con-

Fig. 57.—A diagram illustrating how translational motion becomes changed into
vibrational motion by striking a rough surface.

tinually taken from the surface A and changed into heat. That
the model meets the requirements depends upon the truth of the
following theorem: When a series of elastic particles strike a
rough surface , the resultant component of velocity along the surface
will be diminished. Let M y N , and P in Fig. 57 represent the
section through the centers of three of the greatly magnified
spheres supposed to make up the surface. It is evident that if a
small particle were to strike such a surface at an angle 9 , its
possible paths in striking the sphere N would all lie between A and
G. Considering the directions of the particle before and after
collision, assuming that the angle of rebound at any point of
the surface is equal to the angle of incidence, we find that for
possible paths between B and D the average resultant velocity on

150

FLUIDITY AND PLASTICITY

rebound is exactly opposite in direction, although diminished in
amount. For paths between A and B a particle would collide
with M on rebounding from N but the component of the velocity
in the direction NP is diminished. Also for paths between D and
E, as well as between F and G, the component of the velocity
in the direction NP will be diminished. Only between E and F
is the component in the direction of the flow greater after collision
than before. But the distance EF becomes zero when 9 = 90°
and it has its maximum value when 9 = 0°, i.e., when the trans¬
lational motion is zero. Since all of the paths between A and G
are equally likely, it is clear that for this seotion at least the
average translational velocity is diminished by collision, irrespec¬
tive of the size of the angle or of the velocity of the particle, and
the same would be true even if the particle were of considerable
size. The same must be true a fortiori for sections other than the
one passing through the centers of the spheres, for then there
must, after collision, be a component velocity at right angles
to the plane of the paper and therefore to the direction of flow.
The section would be similar to the one given except that the
circles would not touch, the spaces between them corresponding
to the pores of the surface in which the translational velocity
would quite certainly be changed to disordered motion.

It follows from the above that a fluid in contact with a rough
surface tends to have a translational velocity identical with that
of the surface 1 . Reverting to our model, the theorem explains
how molecules striking the surface A receive its translational
velocity and how this translational velocity becomes trans¬
formed into disordered motion at the surface B. If the motion
of the surface A were suddenly stopped, all of the flow would
cease in a time which, for gases made up of particles whose
velocity is expressed in kilometers per minute, must be quite
inappreciable. * It is to be particularly noted that collisions
between molecules of a gas are unnecessary for this type of viscous
resistance. This type of resistance is caused solely by the diffu¬
sion of the molecules and it therefore may be appropriately
referred to as diffusional viscosity.

For the opposite extreme we may take for consideration a
very viscous liquid. The molecular free path is so greatly reduced

1 Cf. Jeans (1904) and Dushman (1921).

FLUIDITY AND TEMPERATURE

151

that diffusion between adjacent layers is comparatively slight,
whereas the volume of the molecules themselves is a consider¬
able portion of the total volume of the liquid. Given a layer of
molecules C whose translational velocity is higher than that of
another layer D , there must of necessity occur collisions between
the two layers due to the flow, and quite irrespective of any
diffusion, provided only that the diameter of the molecules is
greater than the distance between the layers. On collision the
translational velocity is partly communicated to the slower
moving molecules of the layer D, so that the molecules of the
layer D have a mean resultant velocity in the direction of the flow,
the remainder of the translational motion being converted' into
disordered motion or heat. When 'the system has reached a
steady state, any layer D imparts to the layer E below it the same
amount of translational momentum that it has received from the
layer C above it, except for the amount of energy which is being
continuously changed into heat, and it is this disappearance of
translational momentum which gives rise to the new type of
viscous resistance known as collisional viscosity. Since each
layer is able to transmit but a portion of the translational momen¬
tum which it receives to the adjacent more slowly moving layers,
there results a steady diminution in the velocity of flow from the
most rapidly moving layer A to the layer which is at rest B .

From this model of viscous flow in liquids it is possible to deduce
the effects of changes in concentration, pressure, temperature, and
size of the molecules. The number of collisions of the particles
of one layer with those of another layer, due to translational
velocity, will be directly proportional to the rate of shear between
the layers. This is the fundamental law of viscous flow. It will
also be directly proportional to the number of molecules in each
layer, i.e. to the concentration. It is a confirmation of this
prediction, that we find that the fluidity is decreased in just the
proportion that the concentration is increased either by lowering
the temperature or by raising the pressure. It is significant
that the temperature by itself is without effect on collisional vis¬
cosity. The reason for this is evidently that the mere vibration
of the molecules without diffusion through successive layers does
not affect the rate at which the molecules of one layer overtake
the molecules of an adjacent layer moving more slowly. It is

152

FLUIDITY AND PLASTICITY

clear that collisional viscosity will increase not only with the
concentration but also with the size of the molecules. If the
particles were mere points, there would be no collisions and
therefore no collisional resistance to flow. On the other hand, if
the molecules completely filled the space they occupy, collisions
would be most rapid and the collisional resistance a maximum.

The discovery of Batschinski, that in unassociated liquids
the fluidity is directly proportional to the free volume, seems to
indicate that collisional viscosity is almost entirely responsible
for the viscosity of ordinary liquids and it must be highly impor¬
tant in compressed gases. It is also clear why associated liquids
are exceptional. For the breaking down of association, as by
heating, would doubtless decrease the size of the molecules without
a corresponding decrease in the space which they occupy.

The Mixed Regime.—It has been indicated that in rarefied
gases viscous resistance is certainly diffusional and in very viscous
liquids it is collisional. In fluids at ordinary temperatures and
pressures the viscous resistance is evidently the sum of the
diffusional and the collisional resistances. The total viscous
resistance is in every case given by the equation

v = Vd + Vc (57)

where rj d is the diffusional viscosity and rj c is the collisional
viscosity.

According to Maxwell, as discussed in Chapter XIV, the
viscosity of a gas varies as the absolute temperature, so

Vd — BT

where B is a constant. Later experimenters have found that this
formula does not accord with the experimental facts, and they
have therefore given to the temperature T an exponent n with
values varying from the theoretically deduced 0.5 to 1.0. The
discrepancy, however, may be due to the fact that collisional vis¬
cosity has been overlooked. For diffusional viscosity we here
assume as a first approximation that n = 1.

We have seen that Batschinski’s formula represents collisional
viscosity only, which we may now write in the form

A

FLUIDITY AND TEMPERATURE

153

Table XXXIX. —The Fluidity of Carbon Dioxide as Calculated by
Means of the Formula cp = (v — w)[A 4- BT (v — «)] Where w -
0.841, A = 0.000,257,8, and B = 4,998 Compared with the
Values Observed by Phillips at Various Temperatures
and Pressures

Temperature,

absolute

Pressure in
atmospheres

r

<p observed

<p calculated

Per cent,
difference

293

83.0

1.198

1,215

1,152

- 5

72.0

1.232

1,297

1,241

- 4

59.0

1.302

1,435

1,418

- 1

56.0

5.263

5,376

4,890

- 9

50.0

6.897

5,650

5,299

- 6

40.0

10.000

6,024

5,734

- 5

20.0

27.780

6,410

6,421

0

1.0

546.400

6,757

6,907

+ 2

303

110.5

1.258

1,299

1,300

0

104.0

1.280

1,364

1,355

- 1

96.0

1.316

1,443

1,441

0

90.0

1.346

1,555

1,512

- 3

82.0

1.397

1,689

1,519

-11

80.0

1.416

1,770

1,668

- 6

76.0

1.471

1,890

1,784

- 6

74.0

1.506

2,020

1,855

- 8

73.0

1.531

2,092

1,906

- 9

72.0

1.575

2,183

1,992

- 9

70.0

3.484

4,367

4,021

- 8

60.0

5. 650

5,348

4,887

- 9

40.0

10.870

5,952

5,654

- 5

20.0

28.250

6,289

6,086

- 3

1.0

565.000

6,536

6,594

+ 1

305

120.0

1.266

1,269

1,318

+ 4

112.0

1.287

1,350

1,369

+ 1

104.0 '

1.316

1,439

1,439

0

93.0

1.372

1,595

1,568

- 2

87.0

1.429

1,706

1,693

” 1

84.0

1.466

1,786

1,771

” 1 •

80.0

1.527

1,894

1,894

O'

76.0

1.675

2,232

2,168

- 3

75.0

1.802

2,463

2,379 1

- 3

74.0

2. 778

3,937

3,506

- 8

70. 0

3. 922

4,673

4,240

- 9

60. 0

5.882

5,348

4,918

- 8

40.0

11.Ill

5,714

5,641

- 1

20.0

28.410

6,173

6,201

0

1.0

568.100

6,452

6,553

+ 2

308

114.5

1.324

1,443

1,455

+ 1

109.0

1.349

1,515

1,512

0

96. 0

1.437

1,706

1,706

0

88.0

1.531

1,957

1,896

- 3

85. 0

1.597

2,193

2,021

- 8

80.0

2.024

2,770

2,690

- 3

75.0

3.460

4,219

3,966

- 6

70. 0

4.405

4,673

4,425

- 5

60. 0

6.135

5,618

4,943

-12

40.0

11.765

5,747

5,642

- 2

20.0

28.740

6,135

6,135

0

1.0

574.700

6,410

6,487

4- 1

313

112.0

1.431

1,751

1,686

- 4

108. 0

1.466

1,852

1,759

— 5

100.0

1. 572

2,070

1,965

— 5

94. 0

1.718

2,415

2,221

- 8

85.0

2.597

3,717

3,302

-13

80.0

3.436

4,587

3,920

-14

70.0

4.902

5,000

4,553

- 9

60.0

6. 536

5,348

4,964

- 7

40.0

12.050

5,682

5,585

- 2

23. 8

24.510

5,917

5,987

4. i

1.0

578.000

6,369

6,385

0

154

FLUIDITY AND PLASTICITY

where A and a> are constants and v is the specific volur
per gram. We have then

V

BT H—

V — CO

or

= v—u (

* A+BT(v -«) ’ v

It is truly remarkable that so simple an equation t

can be employed with success to reproduce so complex
that for the fluidity isothermals of carbon dioxide passing
the critical state. To what extent it does do this is
Table XXXIX. Since the calculated values are nearly ■

small, it is evident that a better concordance could ha^*^ fo<
secured by a happier choice of constants, but consider* 3:1 £5
difficulties in these measurements, the percentage of
between the calculated and the observed values is not la/rg><^.

Having established a fairly exact relationship between-
and volume, and indirectly with temperature and p* e ss\
the problem of associated substances again presses itself i:
the foreground as it tends to do so often. A means xxxizs-fc
found for bringing these substances into conformity witfo
others, but the solution is not yet forthcoming.

Dr. Kendall inquires in regard to the foregoing:— Is

formula of Batschinski of such great importance as your eX*ko:n<
treatment of it would lead the readers to believe? Is it; :
merely an interpolation formula? Would it not be *woll
mention something about the alternative formula of Arrfcion.i'u.
(1918). The expotential formula of Arrhenius (1918) does
lead us to a definite mental picture, and like many anot
frankly empirical formula was omitted in this brief treatm
of the subject. The relation of Batschinski fills a need wl
was felt in many minds, cf. p. 142. It leads us at onco t
definite mental picture which is neccessary in buildirxg xx]
consistent theory, so that we are now able to explain the relat
of fluidity to volume, temperature and pressure et cei. i:
manner which is so natural, so unexpectedly simple and so foe
tifully in accord with observed facts that it is hard to see "W
more evidence is needed to carry conviction.

CHAPTER IY

FLUIDITY AND VAPOR PRESSURE

All physical and chemical properties will perhaps in time
be shown to be related so that the knowledge of a certain set of
facts in regard to a substance, such for example as its chemical
structure, will enable one to deduce it multitudinous properties.
Thus having established a direct causal dependence of the fluidity
upon the volume, it is also important to study other properties
which depend upon the fluidity, or which together with the
fluidity depend upon a common cause. Migration velocity and
electrical conductivity of solutions are examples of properties
which are directly dependent upon the fluidity. There arc
properties which are not dependent upon the fluidity directly but
which with the fluidity are dependent upon the same property
and therefore are indirectly related. The boiling-point, the
critical temperature and the vapor-pressure are properties of this
latter type, which we will now consider.

Fluidity and Boiling-point.—On examination of the fluidity-
temperature curves of the aliphatic hydrocarbons, Fig. 41, and
ethers, Fig. 42, we note that the fluidities of these substances at
their boiling-temperatures—shown by small circles—are nearly
identical. It is perhaps of no Special significance that the flu¬
idities are identical, but it is important that the line connecting
the fluidities at the boiling-points is linear. This linear character
of the fluidity-boiling-point curve is exemplified by the aliphatic
chlorides, bromides and iodides as well as by the ethers and
hydrocarbons. The acids and alcohols are again exceptional.

The meaning of this relation may be most easily grasped
by reference to Fig. 42. If we assume that the curves of the
members of a given class have the same slope and the same
degree of curvature at the boiling-point, it is evident that the
addition of a methylene group to a molecule causes a rise in the
boiling-point T measured by AC 1 or CE, but at the same time

1 Of. Fig. 42.

156

FLUIDITY AND PLASTICITY

the addition of a methylene group causes a decrease in the fluidity
A which is AB or CD. It appears that AC/AB = CE/CD , $0
the ratio between the effect produced on the boiling-point to th<*
effect produced on the fluidity T/A is constant, for this particular
homologous series. This relation does not apply to the alcoholH
and acids. A reason is that as the temperature is raised, th< ?
association is lowered, T becomes smaller and at the same time A
becomes greater, so that their ratio may vary.

Fluidity and Vapor Pressure.—If there is a relation between
the fluidity and the boiling-point, it is evident that a more

Fig. 58.—Fluidity-vapor pressure curves of a series of ethers.

general relation than the above can be obtained by comparison
at other vapor pressures than at the ordinary boiling temperature.
Thus if all of the aliphatic ethers have the same fluidity at the
ordinary boiling-point, and the same were true at other vapor
pressures, it follows that, when the vapor pressures corresponding
to a given temperature are plotted against the fluidities corre¬
sponding to that temperature, and this process is repeated for a
series of temperatures, the curves of all of the substances of the
class should fall together; in other words, the flui dity vapor-pressure
curve of one ether ought to be the curve of all the other members
of the class. Conversely, if either the vapor pressure or the
fluidity of an ether is known for a given temperature, the other
quantity, supposedly unknown, can be determined by means of

FLUIDITY AND VAPOR PRESSURE 157

the fluidity vapor-pressure curve of the class. Not only do all
the members of this class fall together in a single parabolic curve,
shovm in Fig. 58, but substances of other classes give a curve of
similar form, as will now be demonstrated.

If we take a single substance, such as heptane, as our standard

Table XL. Fluidities and Vapor Pressures Corresponding to the
Standard Fluidity Vapor-pressuee Curve

Fluidity

reduced

Vapor
pressure
in mm

Fluidity-

reduced

Vapor
pressure
in mm

Fluidity

reduced

Vapor
pressure
in mm

100

0.5

310

111.1

420

396.7

110

0.7

315

119.0

425

415.7

120

1.0

320

127.8

430

434.9

130

1.7

325

136.5

435

455.5

140

2.2

330

146.1

440

476.1

150

3.0

335

155.4

445

497.3

160

4.3

340

165.2

450

520.0

170

6.1

345

175.5

455

542.2

180

8.2

350

186.2

460

566.8

190

11 .0

355

197.6

465

589.5

200

14.2

360

210.4

470

614.7

210

17.7

365

223.2

475

635.9

220

22.3

370

236.6

480

660.1

230

27.7

375

251.0

485

683.7

240

34.1

380

266.0

490

711.7

250

41.2

385

281.1

495

736.3

260

40.3

390

296.2

500

760.0

270

58.8

395

311.4

505

783.8

280

69.1

400

328.3

510

810.0

290

81.8

405

344.7

515

838.8

300

95.9

410

360.7

520

869.5

305

103.2

415

378.4

525

902.0

substance with a fluidity of approximately 500 at the boiling-
point, we can compare other substances with this one. Since
other substances do not have the same fluidity at the boiling-
point, we multiply the fluidity at the boiling-point by a factor
so that the product or the reduced fluidity will be 500. The
fluidities for other temperatures are also reduced by similarly

158

FLUIDITY AND PLASTICITY

multiplying by the same factor. Since the fluidity vapor-pressure
curves pass through the origin and we by this process bring them

Fig. 59 .—The vapor-pressure curves of some associated substances compared

with heptane.

together at the boiling-point, a comparison of the reduced values
for other temperatures with the values for heptane will show
how nearly similar are the curves for different substances, even

FLUIDITY AND VAPOR PRESSURE

159

when, they belong to very different classes. For 17 substances
compared by this method, the average deviation between the
observed and calculated values is approximately 3 per cent,
so long as the vapor pressure is above 10 cm.

A few examples will serve to make the use of the method clear.
Thus the fluidity of ethyl acetate at its boiling-point (77.2°C)
is 395.6 and the reduction factor is therefore 500/395.6 = 1.264.
The fluidity of the substance as determined by Thorpe and
Rodger for 30° is 249.9 and the reduced fluidity is therefore
249.9 X 1.264 = 315.8, and the vapor pressure read from the
Table XL for the standard curve corresponding to this
fluidity is 120.4 mm, while the vapor pressure observed by Young
is 118.7 mm. As a further example, ethyl propyl ether has a flu¬
idity of 479.9 at its boiling-point (63.4°). The factor is 500/479.9
= 1.042. The fluidity at 20° as determined by Thorpe and
Rodger is 314.9 and this reduced is 314.9 X 1.042 = 328.1, and
according to the Table XL this corresponds to 142.5 mm which
is practically identical with the experimental value of 142.6.

Associated substances do not show the relation between
fluidity and vapor pressure shown elsewhere. The greatest
deviation is shown by isobutyl alcohol and formic acid and ethyl
alcohol. In Fig. 59, there is plotted the vapor-pressure-tempera-
ture curve of heptane (not reduced) with several “associated”
substances which show large deviation from the standard. We
note that isobutyl alcohol, formic acid, and ethyl alcohol show
the most rapid increase in the vapor pressure. This is added
evidence of the breaking down of association. (C/. pp. 276
and 277.)

Before concluding our consideration of vapor pressure it may
be remarked that since fluidity is related to volume and at the
same time to vapor pressure, there is necessarily a relation be¬
tween volume and vapor pressure. The volume is doubtless
affected by the internal forces between the molecules which we
ordinarily call cohesion and vapor pressure naturally depends
upon the same. So there may quite possibly be a connection
between fluidity and cohesion, even though it is not the connec¬
tion which is often supposed. 1

Cf. p. 147 et seq.

CHAPTER V

THE FLUIDITY OF SOLUTIONS

The fluidity curves of solutions are most logically considered
under four types: I. In the simplest case the fluidity of the
mixture can be calculated from the fluidities of the components.
There is no volume change on mixing, and it is assumed that the
components neither dissociate nor interact with each other on
mixing. The method of calculation of the fluidity of the ideal
mixture has been the subject of much discussion and it will be
discussed presently. Examples of this simplest type are carbon
tetrachloride and benzene, and diethyl ether and benzene.

Thorpe and Rodger (1897) found that there was a very slight
contraction on mixing carbon tetrachloride and benzene, thus
confirming the earlier observation of F. D. Brown. In the case
of methyl iodide and carbon disulfide there was a very slight
expansion which decreased as the temperature was raised.
Ramsay and Aston found that the surface tension of mixtures of
carbon tetrachloride and benzene followed the mixture rule.
Zawidsti furthermore observed that the vapor pressures of these
same mixtures showed but a slight deviation from the mixture
rule, due, according to Dolazalek, to association of the carbon
tetrachloride. This is the sort of parallelism which needs much
further investigation because it affords the most nearly indis¬
putable evidence to aid the investigator in the selection of ideal
mixtures. In much of our physico-chemical reasoning, it would
beyond any question be a great advantage if we could assume
certain mixtures as ideal in the sense defined above.

The fluidity-volume concentration curves of this class are
nearly but not quite linear, as will be explained.

II. There are instances where there is a well-defined expansion
on mixing, accompanied with heat absorption, and in such mix¬
tures we generally find the fluidity greater than calculated. The
fluidity-volume concentration is convex upward, i.e ., the curva¬
ture d 2 (p/dc 2 is negative. The increase in fluidity may be

THE FLUIDITY OF SOLUTIONS

161

attributed to breaking down of association or to dissociation
which, also give rise to the increase in volume. Benzene and
ethyl acetate may be cited as an example of this type.

III. When two liquids are mixed, perhaps more often than
not, there is a decrease in the volume, particularly in aqueous
solutions. With this decrease in volume there goes a positive
heat effect and a decrease in the fluidity, so the fluidity-volume
concentration curves are convex downward, i.e. } dSip/dc 2 is
positive. Since the fluidity changes some 2,000 times as rapidly
as the free volume, as shown by Batschinski’s law, the effects
of the solvation which is presumed to be present in this case
manifest themselves far more prominently in the fluidity data
than in the data on volume. Examples of this type are very
common, making up the greater portion of aqueous mixtures,
such as ethyl alcohol and water, acetic acid and water and many
mixtures of organic liquids such as chloroform and ether. In
some cases there is incontestible proof that a chemical compound
is formed on mixing, as when aniline is mixed with acetic acid.
In other cases, the formation of hydrates or solvates is very
probable. Whether there is a sharp line to be drawn between the
forces of cohesion of a purely physical nature and the forces
which bring about actual chemical combination must be decided
by future experiment.

IV. When associated solvents break down, to later unite
with each other, we have a combination of the second and third
types. The resulting curve may then show positive curvature
over part of its course and negative curvature over a part,
there being a point of inflection. Examples of this type are
found in both aqueous and non-aqueous solutions, ammonium
nitrate solutions being a good example of the former.

Instances of this type are instructive, since they put us on our
guard against assuming that because a given mixture displays
strong positive curvature, dissociation is not a factor. A pair
of liquids may fall into type II and yet have a tendency to unite
together chemically, provided merely that the effect of dissocia¬
tion predominates in all mixtures.

Again the opposing effects may be nearly equal at all concentra¬
tions, as is true of ethyl alcohol and acetone, and formic acid and

water. These mixtures evidently fall under type IV and not
11

162

FLUIDITY AND PLASTICITY

typo I, but in oases of doubt resort may be had to a study of
the other physical properties.

I. The Ideal Mixture

If there is no contraction on mixing two liquids

Fig. 60. —Specific volume-weight concentration curve of mixtures of benzene
and ether. (After D. F. Brown.)

V = mv i + nv 2

1

(59)

apt + bp2

where v is the specific volume and p the density of the mixture,
containing m weight fraction or a volume fraction of the com¬
ponent A whose specific volume is v± = — and n weight frac-

P i

tion or b volume fraction of the component B whose specific

volume is v 2 = -- So a = --From the above equa-

P2 mvi + nv 2

THE FLUIDITY OF SOLUTIONS

163

tion it follows that if we plot volumes against weight concentra¬
tion, we will obtain a linear curve such as curve I in Fig. 60;
but if we plot specific volumes against volume concentrations,
we will obtain not the linear curve III, Fig. 61, but curve IV.

We have seen that the fluidity of a liquid is directly propor¬
tional to its free volume, but the fluidities are additive (Eq. (25))

Fig-. 61.—Specific volume-volume concentration curve of mixtures of benzene
and ether. (After D. F. Brown.)

only when we use volume percentages; hence it follows that if a
pair of liquids on mixing gave a linear specific volume-volume
concentration curve (curve III) they would also give a linear
fluidity-volume concentration curve, curve VI, Fig. 62. Since,
however, the ideal mixture gives a volume-volume concentration
curve which shows positive curvature, the fluidity-volume con¬
centration curve of the ideal mixture will also show positive
curvature, curve VII, Fig. 62.

164

FLUIDITY AND PLASTICITY

Since this sag in the fluidity curve is due to the mathematical
necessities of the case and not to chemical combination or
dissociation, it is evidently possible to calculate the fluidity of
the mixture from the fluidities and volumes of the components.

Benzene Volume Concenfra+ion of Ether E * her

Fig. 62 .—Fluidity-volume concentration curve of mixtures of benzene and ether.
(After D. F. Brown.)

We have seen that the observed specific volume of the mixture
is

mv i + nv 2

whereas the specific volume should be

av\ + bv 2

in order to give a linear fluidity-volume concentration curve
(Eq. (25)), so the specific volume is too small by an amount
represented by the specific volume difference } Av.

THE FLUIDITY OF SOLUTIONS

165

Av = avi + bv 2 ~ mvi — nv 2
— (a — m)v x — (n b)v 2
= (a — m) (v x — v 2 ) (60)

since a — m = n — b.

If the fluidity is directly proportional to the free volume (Eq. 56),
it seems reasonable to assume that if the volume is decreased
for any reason by an amount Av , the fluidity will be decreased
by an amount which is some function of this fAv. Since in the
ideal mixture the fluidity is only slightly less than that given
by the linear formula (Eq. 25), we may assume as a first approxi¬
mation that the decrease in fluidity is directly proportional to the
specific volume difference. We then obtain as our formula for
the true fluidity 4>

4> = k(v — w) — KAv
= ^ — KAv

= a<pi + b(p 2 — K(a — m) (vi — v 2 ). (61)

It may be possible later to evaluate the above function, but
it is only necessary to know the fluidity of one or more mixtures
in order to determine a value for the constant K, from which the
fluidities of all other mixtures of the two components may be
calculated. The physical significance of K will be explained later.

We may take for an example carbon tetrachloride and benzene,
the mixtures of which were studied carefully by Thorpe and Rodger,
and at a single temperature by Linebarger. (C/. Table XLII.)
In the first line at each temperature are given the fluidities
observed. In the second line are given the fluidities (3>) as
calculated with the use of Eq. (61), using 40 as the value for K .
The fluidities (<?) as given by the simple additive formula,
Eq. (25), are given in the third line. These last are invariably
higher than the observed values, but when corrected for the
volume the agreement is very close, the average deviation
being less than half of 1 per cent.

It is worth noting in passing to what extent Batschinski’s Law
applies to the ideal mixture, with which we assume to be dealing
in this case. Using the limiting specific volumes of carbon
tetrachloride and benzene as 0.5782 and 1.0476 respectively,
we have calculated the values of w by the admixture rule to be
0.683, 0.784, and 0.896, using weight percentages. This accords

166

FLUIDITY AND PLASTICITY

perfectly with the values 0.683, 0.785, and 0.897 obtained from
the observed fluidity data for each mixture. The values of k
in the Batschinski formula are 2,019, 1,937, 1,845 as calculated
from the pure solvents, as compared with 2,034, 1,993, and 1,876
as obtained from the data for the mixtures. The values of
the fluidities calculated by means of the former set of
constants are not so close to the observed values as are the values
calculated by the corrected fluidity formula. But they are at
least as close to the observed values of the mixtures as the cal¬
culated fluidities of pure carbon tetrachloride are to the observed.

It is impracticable here to consider in detail all of the examples

Table XLI.—Specific Volumes in Milliliters per Gram op Mixtures
of Carbon Tetrachloride and Benzene, from Thorpe
and Rodger

Temperature

Per cent benzene by weight

0 22.37 43.79 67.71 100

Per cent benzene by volume

1.1109 Observed
. Calculated

1.1242 Observed
. Calculated

1.1377 Observed
. Calculated

1.1514 Observed
. Calculated

0.6435 0.7602

. 0.7604

1.1661 Observed
. Calculated

0.6518 0.7700

. 0.7702

1.1812 Observed
. Calculated

0.6604 0.7801

. 0.7803

1.1966 Observed
. Calculated

0.6694 0.7907

. 0.7909

1.2124 Observed
. Calculated

THE FLUIDITY OF SOLUTIONS

167

Table XLIL—The Fluidities of Mixtures of Carbon Tetrachloride
and Benzene from Thorpe and Rodger 1 and from Linebarger 2

Per cent benzene by weight (100 h)

Temperature

0

22.37

43. 79

67.71

100

Per cent benzene by volume (1006)

0

34.30

58. 54

79. 17

100

0

74. 1

83.6

91.9

100.6

110.8

Fluidity observed

84.3

92.7

100.9

<t> calculated, Eq. (61)

86.7

95.6

103.2

<p calculated, Eq. (25)
Fluidity calculated Eq. (54)

72.6

82.2

90.4

99.1

108.9

10

! 88.2

100. 0

110.1

120.2

131. 5

Observed

100. 6

110.6

120.2

<f> calculated, Eq. (61)

103.0

113.5

122.5

<p calculated, Eq. (25)

88.4

99.9

110.0

119.9

131.8

Calculated Eq. (56)

20

103.2

117. 6

128.9

141.4

154.1

Observed

118.3

130.0

141.2

<t> calculated, Eq. (61)

120.7

133.0

143.5

(p calculated, Eq. (25)

104.4

117.9

130.0

141.7

155. 1

Calculated Eq. (56)

30

118.9

136.2

149.0

163.4

178. 0

Observed

136.8

150.5

163.4

<t> calculated, Eq. (61)

139.2

153.5

165.7

<p calculated Eq. (25)

120.6

136. 7

150.3

163.5

178. 0

Calculated Eq. (56)

40

135.5

156. 0

171.5

186.6

203.1

Observed

156.2

172.0

186.6

<t> calculated, Eq. (61)

158. 7

175.1

189.0

<p calculated, Eq. (25)

137.5

155.7

171.8

186.3

203.9

Calculated Eq. (56)

50

153.0

176.7

194.9

211.4

228.8

Observed

176.5

194.2

210.6

4> calculated, Eq. (61)

179.0

197.4

213.0

<p calculated, Eq. (25)

154.9

175.5

193.5

210.3

229.9 !

Calculated Eq. (56)

00

171.5

198.8

219.3

237.0

256. 1

Observed

198.0

217.8

236. 1

. 1

4> calculated, Eq. (61)

200.5

221.0

238.5

<p calculated, Eq. (25)
Calculated Eq. (56)

173.0

195.9

215.8

234.3

256.4

70

191.0

243.3

263.9

284.9

Observed

242.8

262.9

<f> calculated, Eq. (61)

<p calculated, Eq. (25)

246.0

265.4

192.0

239.0

260. 1

283.6

Calculated Eq. (56)

Value of k .

Value of eo.

2,105*

0. 5782*

2,019
0.6831

1,937
0.7837

1,845
0.8960

1,721*
1.0476.

Temperature

Per cent benzene by weight

0

13.73

40.78

58.60

100

25

113.2

111.0

123.8

121.8

141.6 1
137.2

151.5
147. 1

166.9

166.7

Linebarger observed

Thorpe and Rodger observed

'J. Chem. Soc. (London), 71, 364 (1897).

2 Am. J. Sci. (4). U, 331 (1896).

3 Batschinski, Zeitachr. f. phyaik. Chem., 84, 643 (1913). Calculated on the basis of
Young’s specific gravities.

168

FLUIDITY AND PLASTICITY

which have been studied lately. Kendall and Wright and many
others have done valuable service; they have chosen inert liquids
whose individual fluidities are widely separated, hence these
mixtures are suited to give a crucial test of the mixture formulas.

Delbert F. Brown has studied this data to determine (1) whether
the volume difference Av is greatest when the specific gravities
of the components are most widely different, (2) whether the
fluidity difference A <p is greatest in the same mixtures in which the
volume difference is greatest, (3) whether those pairs of liquids
showing the greatest volume difference also show the greatest
fluidity divergence, and (4) whether the fluidities of the mixtures
can be calculated from the fluidities and volumes of the compo¬
nents. Table XLIII gives a summary of part of his results.
The table is so arranged that the differences between the specific
volumes of the components, column 2, are in the order of increas-

Table XLIII. —Fluidity and Volume Relations in Certain Ideal
Mixtures (Using Data of Kendall and Wright)

Components

Specific vol¬
ume differ¬
ence of com¬
ponents

. —

Specific vol¬
ume differ¬
ence A v

Fluidity dif¬
ference A <p

Average de¬
viation in
per cent

Ethyl benzoate and benzyl
benzoate.

0.0594

0.0010

5.04

1.0

Phenetol and diphenyl ether....

0.1056

0.0028

5.58

0.56

Ethyl acetate and ethyl benzoate

0.1616

0.0062

25.9

1.2

Ethyl acetate and benzyl benzoate

0.2183

0.0118

55.05

9.2

Diethyl ether and phenetol.

0.3610

0.0268

60.3

1.2

Diethyl ether and diphenyl ether

0.4666

0.0458

110.5

3.8

ing magnitude. The third column shows that the sag in the
volume-volume concentration curve follows exactly this order
of increase, and column 4 shows that the fluidity divergence Acp
follows the same order of increase. Moreover, the maximum
divergence in both the volumes and the fluidities occurs in the
same mixture in every case, except that of diethyl ether and
diphenyl ether, although it is not possible to bring this out in
the table. The last column shows the average deviation of the
values of the fluidity, as calculated by Brown from the data of
Kendall and Wright by the formula (61). In only two cases is

THE FLUIDITY OF SOLUTIONS

169

this deviation much over 1 per cent. Brown found that the
deviation is usually larger in mixtures which contain an ester as
one or both of the components. This, however, is not shown by
this table very well, but if the conclusion is correct, the deviation
would be explained by the chemical character of the components.
This brings us to the consideration of the non-ideal types of
mixtures.

The reader will perhaps ask whether the fluidities of ideal
mixtures would be additive if plotted against weight concentra¬
tions. The curves for carbon tetrachloride and benzene have
been published, 1 using both volume and weight concentrations.
Using volume concentrations the curves are slightly sagged as
already pointed out, but using weight concentrations they show
marked negative curvature particularly at the higher tempera¬
tures. The very slight contraction of carbon tetrachloride and
benzene on mixing in no way accounts for this negative
curvature.

II. Negative Curvature and Dissociation by Dilution

We will now consider a pair of substances which expand on
mixing, using the data of Thorpe and Rodger for methyl iodide
and carbon disulphide. The curvature of the fluidity-volume con¬
centration curves is negative and greatest at the lowest tempera¬
tures. This is in accordance with the view that the components
are less associated at the higher temperatures and therefore can
show less dissociation on mixing.

The expansion on mixing amounts to as much as 0.2 per cent
of the volume, as may be seen by comparing the observed specific
volumes with those calculated by the admixture rule, Table
XLV. The fluidities are given in Table XLIV and it is seen that
Batschinski’s Law applies to each mixture, but the values of the
limiting specific volumes co cannot be calculated by the admixture
rule as in the normal mixture. The actual limiting volume is
some 2 per cent less than the calculated value, presumably due
to the dissociation. The values of k 7 which measure the slope

.^ °f the fluidity-specific volume curves are very much less

1 Zeitschr. f . physik. Chem 83 , 657 (1913).

170

FLUIDITY AND PLASTICITY

than the calculated values. This is also in marked contrast to
the case of carbon tetrachloride and benzene.

Table XLIV. —The Fluidities of Mixtures of Methyl Iodide and

Carbon Disulphide (from Thorpe and Rodger)

Temper¬

ature

Per cent carbon disulphide by weight

0

21.60

38.81

48.11

68.81

82.39

100

Per cent

carbon

disulphide by volume

0

32.22

53.39

62.61

79.94

89.42

100

0

168.3

193.1

207.5

213.2

222.7

228.3

232.8

Observed

168.6

193.0

207.5

212.5

223.1

228.7

233.1

Calculated, Batschinski

192.3

207.6

213.3

223.0

228.4

Calculated, Gibson

10

186.6

211.4

225.7

225.9

242.1

248.1

253.0

Observed

186.2

211.3

225.6

230.5

241.6

248.0

252.1

Calculated, Batschinski

211.1

225.7

230.0

242.0

248.0

Calculated, Gibson

20

205.3

230.9

243.9

248.1

261.8

268.1

272.5

Observed

205.3

230.4

243.8

248.5

260.6

268.0

271.9

Calculated, Batschinski

229.2

244.0

248.2

262.0

268.0

Calculated, Gibson

30

224.8

250.6

263.2

267.4

280.1

288.2

293.5

Observed

224.7

250.2

262.8

267.0

280.0

288.2

292.2

Calculated, Batschinski

250.8

263.2

268.0

281.0

288.0

Calculated, Gibson

40

244.7

271.0

282.5

285.7

299.4

309.6

314.0

Observed

244.8

270.2

282.3

286.2

300.1

309.8

313.4

Calculated, Batschinski

270.8

282.6

286.0

300.0

309.5

Calculated, Gibson

Value of u>

from mix¬

tures. . ..

0.38081

0.4405

0.4855

0.5096

0.5722

0.6161

0.6642

from sol¬

vents ....

0.4420

0.4908

0.5171

0.5758

0.6143

Value of k

from mix¬

tures

3,527i

3,040

2,633

2,465

2,341

2,322

2,123

from sol¬

vents. ...

3,224

2,982

2,852

2,561

2,370

1 Using the densities of Thorpe and Rodger we have recalculated these constants, instead
of taking the values of Batschinski.

When there was a volume change on mixing, Gibson assumed
that the specific volumes v% and v 2 were not the same as for the

TIIE FLUIDITY OF SOLUTIONS

171

components. He assumed that the free volume per unit of
limiting volume was the same for each kind of molecule, so that
from the equations

and

Vj _ 2

COi C0 2

v = MV I + nv 2

the values of in and y 2 could be calculated. He then calculated
the fluidities of the mixture by means of the simple additive
fluidity formula (25). That the values calculated by Gibson
agree well with the observed is shown in the third line of fluidities
for each temperature given in Table XLIV,

Table XLV. —The Specific Volumes in Milliliters per Gram of
Mixtures of Methyl Iodide and Carbon Disulppide (from
Thorpe and Rodger)

Temper-

Per cent carbon disulphide by volume

ature

0

33.22

53.39

62.61

79.94

89.42

100

0

0.4285

0.5040

0.5031

0.5643

0.5624

0.5959

0.5945

0.6675

0.6659

0.7146

0.7128

0.7740

Observed

Calculated

10

0.4336

| 0.5100

0.5712

0.6031

0.6754

0.7229

0.7830

Observed

20

0.4390

! 0.5163
0.5152

0.5781

0.5759

0.6104

0.6087

0.6835

0.6817

0.7315

0.7297

0.7923

Observed

Calculated

30

0.4445

0.5228

0.5853

0.6179

0.6918

0.7412

0.8018

Observed

40

0.4502

0.5295

0.5282

0.5927

0.5904

0.6257

0.6242

0.7004

0.6987

0.7495

0.7475

0.8118

Observed

Calculated

We have come now to the case where there is chemical com¬
bination on mixing. There is generally a decrease in volume
and the specific volume-weight concentration curve, curve II,
Fig. 60, is sagged as well as curve V, Fig. 61, representing the
specific volume-volume concentration curve. Since new sub¬
stances are formed, no method given thus far can be depended
upon for calculating the fluidity-volume concentration curve.

172 FLUIDITY AND PLASTICITY

III Positive Ctjrvation and Chemical Combination

Before considering the meaning of positive curvature in detail,
it is necessary to emphasize the fact that a Minimum in the fluid¬
ity-volume concentration curve is not necessary to indicate that
chemical combination is taking place and when a minimum does
occur, its location, according to numerous investigators, notably
Findlay (1909) and Denison (1913), does not correspond to the
exact composition of the compound formed. This is proved, if
proof be needed, by the fact that the minimum usually changes
with the temperature and may disappear altogether. The ques¬
tion then is, assuming that a chemical combination is formed by
the mixing of the two components of a binary mixture, how can the
data be used to show what this compound is? To answer this,
we will present three cases of increasing complexity, in all of
which there is the same amount of chemical combination, it being
assumed that in the feeble combination with which we are dealing
the two components A and B are always in equilibrium with a
small amount of the compound C so that

A + B<=±C

Case I .—The fluidity-temperature curves of two closely related
substances are represented by the curves A and B in Fig. 63a.
If there were no combination between the components on mixing,
the curve for the 50 per cent mixture would lie half-way between
the curves A and B (dotted). Let it be assumed that this mix¬
ture does show the maximum amount of combination and that
the curve is thereby lowered to 0.5B. Using the data plotted in
Tig. 63a it becomes possible to plot the fluidity-volume concen¬
tration curves for the various temperatures ti, t 2 , h, etc., as shown
in Fig. 635. In this case there is a well-defined minimum in the
fluidity-volume concentration curve in the 50 per cent mixture
and the deviation of the curves from the normal (dotted) curves
is constant in amount.

Case II. Let us now assume that we are dealing with two
substances whose fluidities are widely different, although they
still run parallel to each other. With the same amount of combi¬
nation as before, the curve 0.5 B falls between the curves A and B f
Fig. 64a. As a result the fluidity-volume concentration curves.
Fig. 646, no longer exhibit a minimum although, by assumption,

THE FLUIDITY OF SOLUTION'S 173

the hydration is the same as before both in relative composition
and amount. However, it is clear that the deviation of the fluidity

Temp.

a

Fig.. 63.— Diagram to illustrate the fact that when two substances A and B
of similar fluidity are mixed, the formation of a solvate produces a minimum in
the fluidity-concentration curves.

Fig. 64.—Diagram illustrating how when two substances A and B are mixed
whose fluidities are very different, the formation of a solvate produces no mini¬
mum in the fluidity-concentration curve.

volume concentration curves from the linear curves, which would
be expected were there no combination, and as indicated in the
figures by the distance MN, is the same as in the preceding case.

174

FLUIDITY AND PLASTICITY

Case III .—In the usual ease in practice, the fluidity-tempera¬
ture curves are not parallel, so that the fluidities may be identical
at one temperature but very different at another- We then
obtain a series of curves as shown in Fig. 65a and 656. At low
temperatures there is a good minimum in the fluidity-volume
concentration curves, but it gradually shifts to the right as the
temperature is raised, until at the highest temperatures it dis¬
appears altogether. It is manifestly erroneous to assume that
the composition of the hydrate changes on this account. On the
other hand, the deviation from the expected linear curves as

a

Fig. 65.—Diagram illustrating how the minimum in the fluidity-concentra¬
tion curve may shift with the temperature. The maximum deviation from the
linear curve is the significant quantity. This quantity does not vary with the
temperature and it indicates the composition of the solvate.

measured vertically is everywhere the same as in the simpler
cases. In practice, the hydration is generally less at the higher
temperatures so that the deviation should grow less as the tem¬
perature is raised, but the cases already given are sufficient to
show that the deviation of the observed fluidity-volume concentra¬
tion curve from the linear curve, which would be expected were
there no combination between the components of the solution,
can alone furnish trustworthy information.

Were the components of the mixture non-associated, it seems
possible to calculate not only the composition of the solvate
formed but also the percentage of it existing in the solution.
But substances which form feeble combinations on mixing are
usually themselves associated, and it is quite likely that this

VITE FLUIDITY OF SOLUTION’S

175

association is altered in the mixture, so that the result is consid¬
erably complicated thereby. We have, however, a fairly simple
case in mixtures of ether and chloroform studied by Thorpe and
Bodgcr. Chloroform, like carbon tetrachloride, is probably
slightly associated but ether may be regarded as unassociated.

So far as can be learned from their measurements the maximum
contraction on mixing occurs in a mixture containing less than
40 per cent of ether and perhaps less than 39 per cent; the maxi-
m uin deviation of the fluidity-volume concentration curve from the
linear curve occurs in the 58 volume per cent mixture ± 3 per cent.
This corresponds to 39.8 per cent by weight. A mixture corre¬
sponding to the formula C 4 II 10 Q.CHCI 3 contains 38.30 per cent
ether by weight. Guthrie has noted that heat is evolved on
mixing and that it is a maximum when the components are in
molecular proportions. The vapor-pressure, refractive index
and the freezing-point curves all give evidence of the formation
of a compound C 4 H 10 O.GHOI 3 .

In the mixture containing 56.26 volume per cent of ether, or
one molecule of ether to one of chloroform, we will now calculate
the percentage combined. From the atomic constants already
given, p. 126, it appears that the compounds C 4 H 10 O.CHCI 3
should have a fluidity of 200 at the absolute temperature of 538.6°.
But actually a mixture of this composition has a fluidity of 200
at 282.9° absolute (9.9°G). Pure ether and pure chloroform
have fluidities of 200 at 216.5° and 305.3° absolute respectively,
so that if the mixture were wholly uncombined, the absolute
temperature necessary for a fluidity of 200 would be 216.5 X
0.5626 •+ 305.6 X 0.4374 = 255.4°. Letting x represent the
fraction of the volume of the mixture which is combined, we ob¬
tain the equation

538.6a; + 255.4(1 - a?) = 282.9

and x = 0.0971. Since at this temperature (9.9°C) less than 10
percent of the volume of the mixture is actually in combination,
it seems reasonable to assume that a dynamic equilibrium exists
between the combined and the uncombined portions. If the
Maas Law holds, we have

(C 4 H 10 OI [CHClsl = jr
[C4H10O . GHCb]

176

FLUIDITY AND PLASTICITY

where the concentrations are molecular and not volume concen¬
trations.

In the above equimolecular mixture, if we let y represent the
number of milliliters of ether which are combined in every 100
ml of mixture, the volume of the chloroform combined will be
0.7366 X 119.36y 7777

74.08 X 1.526 J

where the specific gravities of ether p e and chloroform p c are
taken as 0.7366 and 1.526 respectively and their molecular
weights, m e and m c , 74.08 and 119.36. Since the sum of the two
volumes y + 0.7777?/ is 9.71, the volume of the ether combined
is 5.46 ml and the volume of the chloroform is 4.25 ml.

Substituting the molecular concentrations in the above formula

K _. [(56.26 — 5.46)p fi /m e ][(43.74 — 4.25)p c /m c ] ^ ggg
9.71p/(m e +m c )

where p is the density of the compound calculated by averages
to be 1.082.

With this value of K, it is possible to calculate the absolute
temperature corresponding to a fluidity of 200 for any mixture
on the assumption that only one compound is formed and that
the Law of Mass Action is obeyed. Thus for any mixture if a
is the volume percentage of ether and z is the fraction of the
ether which is combined,

®Pe _ \ l~ (I &)pc _ QrPe 1

■ - L -L 4 696

m e

For the 28.21 volume per cent ether mixture, z = 0.157. The
volume of ether in 100 ml which is combined is az = 4.43 ml,
and the volume of chloroform combined is 0.7777 X 4.43 =
3.44 ml. Hence the calculated absolute temperature correspond¬
ing to a fluidity of 200 is

0.2378 X 216.5 + 0.6835 X 305.3 + 0.0787 X 538.6 = 302.5°
which is in fair agreement with the value read from the curve
of 297.4°.

Ethyl Alcohol and Water Mixtures

We will now take up a case in which the components of the
mixture are highly associated. Ethyl alcohol and water are a

the fluidity of solutions

177

particularly good example as there is a very strongly pronounced
minimum in the fluidity curves. The greatest deviation, from
the linear is in a mixture corresponding to the formula C 2 H 6 0.-
4H 2 0, containing 44.79 per cent alcohol by volume. To obtain
a fluidity of 200, ethyl alcohol requires an absolute temperature
of 343.6° and water a temperature of 328.9°, so if there were no
chemical change on mixing we should expect a temperature
corresponding to the fluidity of 200 in the 44.79 volume per cent

C x H&.ir*0 CJl.O.mfi

Pig. GO.—Mixing solutions of ethyl alcohol in. water (corresponding to the
o,ornposition. OaHuO.SHaO) with solutions of acetic acid in water (corresponding
to the composition O2H4O2.H2O) brings about an increase in fluidity.

mixture of 0.4479 X 343.6 X 0.5521 X 328.9 = 335.5°. On the
other hand, from the constants already given, the temperature
required to give the pure hydrate CzHjOdHjO a fluidity of 200
would be 14 X 59.2 + 5 X 24.2 — 2 X 95.7 = 758.4°. But the
observed absolute temperature at which the 44.79 volume per
cent mixture has a fluidity of 200 is 362.3°. Hence, if we let
x represent the fraction by volume of the mixture combined as
C 2 H a 0.4H 2 0, the rest remaining unchanged, we have
335.5 (1 - *) + 758.4x = 362.3

and re = 0.0634.

12 '

1

178 FLUIDITY AND PLASTICITY

That ethyl alcohol and water are less than 7 per cent combined
is surprising in view of the higher amount of combination in
chloroform and ether, but the temperature of comparison is
very much higher, in the case of water and alcohol being 89°C.
But there is another important disturbing factor which must be
considered, in that water and alcohol are both highly associated,
2.31 and 1.83 respectively, so that when the two are mixed there
is almost certainly dissociation.

That dissociation does occur can be proved as follows: We
have seen that when ethyl alcohol and water are mixed there is
a lowering of the fluidity. There is also a pronounced lowering
of the fluidity when acetic acid and water are mixed. There is
furthermore a lowering of the fluidity when acetic acid is mixed
with ethyl alcohol. Yet when acetic acid solution (C 2 H 4 O 2 .H 2 O)
is mixed with ethyl alcohol solution (C 2 H 6 0 . 3 H 2 0 ), having
practically the same fluidity of 43 absolute units at 25°C, there
is a very noticeable increase in the fluidity as seen in Fig. 66 from
the paper by Bingham, White, Thomas and Cadwell (1913).

IY. Inflection Curves

The discussion of simultaneous dissociation and chemical
combination brings us naturally to the consideration of the
fourth type of fluidity-volume concentration curves. There
are several pairs of non-aqueous mixtures which fall into this
class, such as ethyl alcohol and benzene discovered by Dunstan
(1904); but by far more important are certain aqueous solutions of
electrolytes, notably the salts of potassium, rubidium, caesium
and ammonium. That potassium nitrate added to water lowers
the time of flow was discovered by Poiseuille (1847) although
priority is usually attributed to Hubener (1873). The list of those
substances which lower the viscosity of water has been added to
by Sprung (1875), Slotte (1883) and many others and is given in
Table XLVI. The phenomenon has been often referred to as
“negative viscosity,” but since viscosity is a result of friction,
which is never negative in fact, the use of the term is not happy.
The term “negative curvature,” d 2 <p/db 2 < 0 , where b is the
volume concentration, is not open to similar objection when dis¬
cussing the fluidity-volume concentration curves of these solutions.

TEE FLUIDITY OF SOLUTIONS

179

Apparently all of those aqueous solutions which exhibit negative
curvature fall into the class of mixtures showing inflection curves.

Table XL VI. —Substances Which Appear to Exhibit the So-called
“Negative "Viscosity” in Aqueous Solution

Substance

Observer

Bromic acid.

Hydroforomic acid.

Hydrocyanic acid.

Hydriodic acid..

Hydro sulfuric acid.

Nitric acid. . -..

Ammonium acetate.

Ammonium bromide.

Ammonium chloride.

Ammonium chromate.

Ammonium iodide.

Ammonium nitrate.

Ammonium thiocyanate.

Caesium chloride.

Caesium nitrate.

Ferrous iodide.

Mercuric chloride.

Mercuric cyanide..

Potassium bromide.

Potassium chlorate.

Potassium chloride.

Potassium cyanide.

Potassium ferricyanide.

Potassium iodide—.

Potassium nitrate..

Potassium thiocyanate.

Rubidium bromide.

Rubidium chloride.

Silver nitrate.

Sodium iodide.

Tefcramethylammonium iodide

Thallium nitrate.

Urea.

Poiseuille (1847)

Poiseuille (1847)

Poiseuille (1847)

Poiseuille (1847
Poiseuille (1847
Poiseuille 1847)

Poiseuille (1847)

Sprung (1876)

Poiseuille (1847)

Schlie (1869)

Hubener 1873), Wagner (1890)

Sprung (1876), Gorke (1905), Walden (1906)
Sprung (1876)

Schottner (1878)

Bruckner (1891)

Poiseuille (1847)

Poiseuille (1847)

Slotte (1883), Ranken and Taylor (1906)
Poiseuille (1847)

Poiseuille (1847)

Poiseuille (1847)

Poiseuille (1847)

Hubener (1873), Kanitz 1897)

Poiseuille (1847)

Poiseuille (1847)

Sprung (1876), Gorke (1905)

Davis, Hughes and Jones (1913)

Wagner (1890)

Poiseuille (1847)

Poiseuille (1847)

Schlie (1869)

Schottner (1878)

Miitzel (1891)

Mlost workers have confined their attention to dilute solutions
and they have studied viscosity relations almost exclusively,
so that the positive curvature in concentrated solutions has

180 FLUIDITY AND PLASTICITY

Per Cent*

Fig. 67. Fluidity-volume concentration curves for aqueous solutions of ammo¬
nium nitrate, showing both positive and negative curvature.

THE FLUIDITY OF SOLUTIONS

181

remained undiscovered. That positive curvature is general even
when it is quite unsuspected in dilute solutions is proved by
all of the data available, as may be seen by inspection of Figs.
67, 68, 69 and 70. The negative curvature is in each case most

■ Fig. 68.

Fig. 69.

pronounced at the lowest temperatures and in solutions contain¬
ing not over 20 per cent of the salt by volume. The negative
curvature disappears in concentrated solutions and at high
temperatures. If the negative curvature is strongly marked,
as with ammonium nitrate or potassium chloride, the positive
curvature is unimportant, but when the negative curvature

182

FLUIDITY AND PLASTICITY

is weak, as with potassium iodide, silver nitrate and urea, the
positive curvature quickly shows itself.

The first important attempt to explain the lowering of the
viscosity of water was made by Arrhenius (1887), who thought
that it might be due to electrolytic dissociation. Wagner and

Fig. 70.—Fluidities of potassium halide solutions in water at various tempera¬
tures. The curves show negative curvature which is most marked for the
chloride, and at low temperatures and at low volume concentrations of the salt.
At high concentrations or at high temperatures all of these solutions may show
positive curvature, but the nitrate and iodide most readily. {After Gorhe.)

Muhlenbein (1903), however, showed that the dissociation
hypothesis was by itself insufficient as an explanation since salts
like NaN0 3 and K 2 SO 4 are highly ionized and yet do not show
negative curvature as does KN0 3 . Now that it appears that
urea and mercuric chloride solutions both show negative curva¬
ture, it would seem probable that electrolytic dissociation is
not necessary for the phenomenon. Since these substances in
solution are practically unionized.

THE FLUIDITY OF SOLUTIONS

183

Jones and Veazey (1907) observed that potassium, rubidium
and caesium are the elements with the largest atomic volume and
they therefore reasoned that their salts would also be relatively
fluid. From what has preceded we are prepared to find relations
between fluidity and volume, but as a matter of fact the fluidity
of the pure salts in the molten condition is very low. For
example, Foussereau (1885) found the fluidity of ammonium
nitrate to be 0.505 at 185°C and 0.4037 at 162°C, so that at ordi¬
nary temperatures the fluidity of the salt in the undercooled
condition would certainly be very low, probably negligible as
compared with water. Furthermore, there are salts which show
negative curvature but in which the metal has a small atomic
volume such as silver nitrate, mercuric chloride, and thallium
nitrate. In view of the periodic relationship of the elements,
the same coincidence noted by Jones and Veazey would occur
with many other properties. Finally there are several salts of
potassium and ammonium which have not been found to show
negative curvature hence the explanation proposed by Jones and
Veazey is not satisfactory.

Explanation of the Inflected Curve

As to the reason for positive curvature, it seems probable
from what precedes that it is due to combination between
the solvent and the solute. That many of the salts of potassium,
rubidium, caesium and ammonium exhibit so slight positive
curvature is due to their smaller tendency to form hydrates
than is usually the case in aqueous solution. In contrast with
the salts of potassium, no sodium salts show “ negative viscosity.”
Perhaps the most striking difference between the salts of sodium
and potassium, generally so similar, is the greater affinity for
water on the part of sodium salts. None of the salts which show
negative curvature crystallize from water with water of crys¬
tallization, and the few salts of potassium and ammonium which
do not show negative curvature do exhibit a tendency to form
hydrates. Examples are potassium carbonate, ferrocyanide and
sulfate, and ammonium sulfate. It is true that hydrobromic
acid solutions are probably hydrated, but according to the
measurements of Steele, McIntosh, and Archibald (1906) anhy¬
drous liquid hydrogen bromide has a high fluidity. The small-

184

FLUIDITY AND PLASTICITY

ness of the positive curvature is then due to the small amount of
hydration which is well-nigh universal in aqueous solution.

The negative curvature, on the other hand, must be due to
dissociation either (1) of the salt or (2) of the associated water.
Since the negative curvature occurs in dilute solution, the
electrolytic dissociation is immediately suggested. If the fluidity
of the anhydrous salt in the form of an undercooled liquid is
negligibly small, it is hard to conceive of how the dissociation of
the salt into two, or at the most a few, ions would increase the
fluidity so remarkably, for it must be remembered that there
must be a substance present whose fluidity is higher than that of
water. Then, as already pointed out, there are substances
which give negative curvature which are very slightly dissociated
into ions, such as urea.

We are then compelled to seek further in our explanation
and admit that water itself is dissociated by the presence of the
salt or its ions. There is nothing inherently improbable in this
since water is highly associated (2.3 at 56°C). The association
is less at high temperatures and in concentrated solutions so
that under these conditions negative curvature would be less
apparent as we have already seen to be the case. It is often
assumed that electrolytic dissociation is brought about by union
of simple water molecules with the ions of the salt, but if the ions
have low fluidity, the fluidity of the solution will evidently not be
raised by uniting with even simple water molecules, hence
hydration will not explain the phenomenon. In other words, the
formation of larger molecules does not tend to raise the fluidity.

Wagner (1890) has measured the volume of water required to
make a liter of normal solution of the chlorides of various salts.
In the cases of silver and thallium the nitrates were used instead.
Salts like calcium chloride, which unite strongly with water to
form hydrates, produce a contraction on going into solution, so
that a comparatively large volume of water is required. But
rubidium and caesium chlorides expand on going into solution so
that the volume of water required is correspondingly small. The
difference between the volume of water required and 1 1. is the
volume of the salt together with the expansion. Calculating
the volume of the salt from its specific gravity the expansion is
obtained. The resulting numbers, plotted in curves IV and V in

Migration*Velocity Fluidify Atomic Volume 6ait Volume-*- Expansion Expansion

THE FLUIDITY OF SOLUTIONS

Fig. 71.—Some “periodic” relationships.

186

FLUIDITY AND PLASTICITY

Fig. 71, show that in general the salts which occupy the largest
volume in solution correspond to those having the highest fluidity
curve II, but silver seems to be strongly exceptional. Here
again we have evidence that fluidity is proportional to the free
volume. The cause of the volume change is also the cause of
the negative curvature.

Ammonium iodide according to Getman (1908) and Ranken
and Taylor (1906) shows negative curvature but it goes into
solution with contraction, according to Schiff and Monsacchi.
There is thus a lack of parallelism between the two properties of
whioh one further example may be cited. In ammonium nitrate
solutions, the expansion is least in a 7-weight per cent
solution and yet the fluidity is a maximum in this solution at
some temperature between 25 and 40°C. Since we are dealing
with inflected curves signifying simultaneous dissociation and
chemical combination, these anomalies are to be expected. The
limiting volume is continually changing and the specific volume is
for that reason no measure of the free volume. There is need
for further work in this very important field.

Attempts have been made by Wagner and others to assign
to each element a specific viscosity effect in solution. The fluidi¬
ties of nitrates, chlorides, and sulfates of certain metals in normal
solution at 25°C are given in Table XLVII as modified from
Wagner. The table shows that the fluidity of the nitrates is

Table XLVIL —A Comparison op the Fluidities of Various Metals
and Acid Radicals in Normal Solution at 25°C (after Wagner)

NO;

Cl 3

S0 4

NQ 3

Cl

NO,

S0 4

K.

K/H..

H.

K/Na.

Na.

K/Zn.

Zn.

K/Mg

Mtr

114.7
1.053

108.9

1.095

104.7
1.195

96.0

1.201

113.3

1.081

104.8
1.112

101.9
1.199

94.5

1.218

101.2

0.974

102.5

1.112

91.0

1.239

81.7

1.239

1.012

1.039

1.027

1.015

1.133

1.062

1.151

1.175

THE FLUIDITY OF SOLUTIONS

187

always higher than that of the chlorides and that of the chlorides
is always higher than the fluidity of the corresponding sulphate.
The ratio of nitrate to chloride is 1.02 and of nitrate to sulphate
1.14. We may also compare the salts of different metals joined
to the same acid radical and thus get a ratio in terms of one
metal taken for reference, as potassium. Considering the com¬
plex effects due to dissociation, hydration and perhaps other
causes, the presence of even imperfect relationships of this kind
is remarkable.

CHAPTER VI

FLUIDITY AND DIFFUSION

According to Stokes (1851) a sphere of radius r, impelled
through a fluid under a force F, will attain the velocity v

v

Ftp

67rr

(62)

This formula is of fundamental importance in the study of the
settling of suspensions, diffusion, Brownian movement, the rate
of crystallization of solutions, migration velocities and transfer¬
ence numbers of the ions and in the conductivities of solutions.

Settling of Suspensions. —In the case of a falling sphere, the
force becomes

F = | tt^ 3 (p 2 “ Pi)

where p 2 and p i are the densities of the sphere and the medium
respectively, so

V = | g(p 2 — pi)r 2 <p (63)

This formula enables one to calculate the speed of settling of
suspensions. It has been utilized in determining the viscosity
of very viscous liquids, e.g ., Tammann (1898) and Ladenburg
(1907), for determining the radii of the particles in gold suspen¬
sions, Pauli (1913), for measuring the charge on the electron in
air, Millikan (1910).

The Diffusion Constant. —Sutherland (1905), Einstein (1905)
and Smoluchowski (1906) have derived the relation between the
diffusion coefficient 8 and the fluidity,

N 6t rr

where T is the absolute temperature, R is the gas constant
(83.2 X 10 6 c.g.s. units) and N is the number of molecules in a
gram molecule (70 X 10 22 ). The diffusion coefficient is defined
as the quantity of solute diffusing per second through a unit
cube when the difference in concentration between the two ends
of the cube is unity. But Stokes’ Law was derived for particles

188

fluidity and diffusion

189

which are spheres and having a radius large in comparison with
the molecules of the solvent. If the particles are so small that
the free path a of the molecules of the suspending medium is ap¬
preciable in comparison with the radius of the particles, Suther¬
land (1905), Cunningham (1910) and Millikan (1910) have shown
that Stokes’ formula becomes

5 =

RT

If'*"

67rr

(64)

where A is a constant and equal to about 0.815.

The following table from Thovert (1904) indicates that the
product of the diffusion constant and the time of efflux is approxi¬
mately constant for a considerable number of substances.

Table XLVIII.— The Relation between Diffusion and Viscosity

(Thovert)

Substance

8 X 10 s

T, time of
efflux

8 X T X 10 4

Ether.

3.10

315

98

Carbon disulfide.

2.44

405

99

Chloroform.

1.50

660

99

Mixture ethyl alcohol and ether.

1.51

660

100

Benzene.

1.24

790

98

Methyl alcohol.

1.16

820

95

Mixture ethyl alcohol and
benzene.

1.03

950

98

Water.

0.72

1,330

96

Ethyl alcohol.

0.59

1,620

96

Turpentine.

0.48

2,020

97

Amyl alcohol.

0.155

5,900

92

Glycerol solution.

0.0104

94,000

98

On the other hand, Oeholm (1913) finds that 8ri is not exactly
constant for a series of alcohols as compared with water when
glycerol is the diffusing substance. Oeholm thinks that associa¬
tion and hydration will account for the variations, at least in part.

Bell and Cameron have applied Poiseuille’s formula to diffusion
through capillary spaces and find that the distance y which a
liquid moves in a given time t is given by the formula y n = kt,

190

FLUIDITY AND PLASTICITY

where n and k are constants, and by derivation n = 2. The
formula is important in dealing with diffusion through porous
materials such as soils. But in this type of diffusion, it has
often been noticed that there is a separation of the components
of the diffusing substances. This subject will come up for con¬
sideration later.

Brownian Movement.—Einstein (1906) has shown that the
mean square of the projections l of the displacement of the
particle in time t on the axis of displacement is

P = 28t

Substituting into this equation the value of the diffusion, given
above

RT <pt~
N Zttv

(65)

This is the equation used by Perrin in his brilliant investigation
of the Brownian movement.

The Velocity of Crystallization.—As a crystal forms in a
solution, the molecules of the solute are drawn to the growing
face of the crystal. The solution bathing the face of the crystal
has therefore a lower concentration of solute than the main
body of liquid and a process of diffusion must be set up to restore
the equilibrium. The rate of crystallization must therefore
depend upon the fluidity of the solution. Even in an under¬
cooled liquid, where there is no opportunity for a change in con¬
centration, the viscosity of the liquid retards the proper orienta¬
tion of the molecules, and crystallization does not take place
instantaneously. H. A. Wilson (1900) has demonstrated that
the velocity is directly proportional to the fluidity of the liquid
at the face of the crystal, according to the formula,

v = a(t — to)<p

( 66 )

where v is the velocity of solidification in millimeters per second,
a is a constant and t 0 is the temperature at which the velocity of
solidification is zero, i.e,, the solidifying point, found by extrapo¬
lation. This point differs somewhat from the melting-point
of the substance, being 37°C for salol instead of 42°C. This
signifies that the above relation does not hold when the amount of
undercooling, and hence the velocity of solidification, is very
small. Since, Tammann has proved that purifying a substance

FLUIDITY AND DIFFUSION

191

always diminishes this region of small velocity, Wilson very
properly attributes this effect to impurity.

Wilson experimented with salol, benzoic anhydride, benzo-
phenone, and azobenzene confined in long glass tubes of varying
diameter. A thermocouple was used to get the temperature of
the solidifying surface, which was of course different in tubes
of various diameters. How well the observed and calculated
velocities of solidification agree can be seen in the following
table for salol.

Table XLIX- —The Velocity of Crystallization of Undercooled

Salol from Wilson, Melting-point, 42.0, 2 0 = 37.0, a == 0.065,6

Temperature,

degrees

t - t Q

<P

\

v , calculated v,

observed

35

2

8.77

1.15

1.25

33

4

8.19

2.14

2.5

31

6

7.31

2.90

3.2

29

8

6.49

3.40

3.7

27

10

5.84

3.82

3.9

25

12

5.16

4.05

4.0

21

16

3.90

4.08

4.1

19

18

3.51

4.13

4.1

15

22

2.77

4.00

4.1

Since for all of these liquids the fluidity is as a first approxi¬
mation a linear function of the temperature, for salol (p =
(t — 9.8) 2.9, Eq. (66) may be written

V = at 2 — fit + 7

where £ = a(9.8 + * 0 ) and 7 = 9.8 at. Thus, whereas it is
possible to express the velocity of solidification as a function of
the temperature only, it is much simpler to express it as a func¬
tion of the fluidity as was done in Eq. (66).

Migration Velocity, Conductivity and Transference Numbers*
That the movement of the ions under the action of electrical
attraction should be dependent upon the fluidity of the solution
seems a natural inference from what precedes, and a large num¬
ber of researches have been devoted to the elucidation of the
exact relationship. Since the measurement of electrolytic dis¬
sociation depends upon this relationship, there can be no question

192

FLUIDITY AND PLASTICITY

about its importance. One has only to compare the migration
velocities of a series of ions with the fluidities of their chlorides
in normal solution, as shown in curves I and II of Fig. 71 after
Bredig, (1894), to see that there is a definite relationship between
the two. In seeking an exact quantitative relationship we are
met again by the awkward fact that water is associated and elec¬
trolytes in it perhaps always form hydrates, so that an apparently
simple aqueous solution is not simple in fact. The study of molten
salts, of liquid metals and alloys, and of non-aqueous solutions for
this reason take on a particular importance, but aqueous solutions
have naturally received the greatest attention. The method
of investigation is usually to change the fluidity of the liquid
by altering the temperature, concentration or pressure and to
observe the corresponding change in conductivity.

As early as in 1851 Wiedemann investigated the viscosity and
conductivity of various salt solutions of varying concentration.
Wiedemann calculated the value of the ratio m^/A, where m is
the percentage of salt and A the conductivity. He found that
the ratio varies within narrow limits for each salt, e.g., for copper
sulfate the value varies from 22.8 to 24.2 when the concentration
is increased from 31.17 to 187.02.

Gour6 de Villemont6e in his monograph on Resistance Electri -
que et Fluidite has used the results of Bouty and Bender to prove
that the ratio m<p/ A varies with the temperature in a manner
which is the same for all salts. (Cf. Table L.) We have seen
that over a small range of temperature

ip = <Po(l + 00

so similarly

A = Ao(l 4“ of)

where a and 0 are arbitrary constants and <po and A 0 are the
fluidity and conductivity at 0°.

Grossman (1883) recalculated the results of Grotrian (1876)
and found that the ratio ^ is a constant independent of the tem¬
perature, and the temperature coefficients are the same to within
1 per cent. Bousfield and Lowry (1902) using parabolic formulas,
cf . Eq. (53a), in place of the simpler linear ones given above,
found that the constants in the two formulas were the same
within experimental error.

FLUIDITY AND DIFFUSION

193

Table L.—The Temperature Coefficients of Fluidity (0) and
Conductivity ( a )

Normality

0

a

0/«

Potassium chloride

3.0

0.0294

0.0230

1.3

2.0

0.0332

0.0259

1.3

1.0

0.0372

0.0291

1.3

0.5 |

0.0404

0.0302

1.3

Sodium

chloride

3.0

0.0390

0.0279

1.40

2.0

0.0394

0.0290

1.37

1.0

0.0410

0.0292

1.40

Table LI.— Fluidity and Conductivity of Fused Salts and Salt
Mixtures, after Foussereau

Temperature

degrees

<p

A

<p/A

Sodium

nitrate

305

0.377

0.459

0.821

320

0.439

0.526

0.799

329

0.454

0.555

0.818

340

0.498

0.599

0.832

355

0.561

0.662

0.848

Potassium nitrate

334

0.545

0.631

0.863

340

0.572

0.661

0.866

358

0.660

0.790

0.835

1 g mol Sodium nitrate -f

1 g mol Potassium nitrate

232

0.248

0.463

0.534

242

0.264

0.502

0.526

266

0.310

0.616

0.504

287

0.361

0.724

0.499

304

0.418

0.791

0.528

313

0.436

0.840

0.519

332

0.532

0.971

0.548

348

0.584

1.123

0.520

359

0.624

1.176

0.530

13

194

FLUIDITY AND PLASTICITY

Foussereau (1885) has examined the changes in fluidity and
conductivity of pure water with the temperature and proved
that the conductivity is directly proportional to the fluidity.
He has also examined several fused salts and salt mixtures and
obtained a similar result. We reproduce in Table LI his results
for sodium nitrate, potassium nitrate and an equimolecular mix-
ture of the two salts. It is to be observed that not only is the ratio
different for the different salts but the conductivity is relatively
much higher for the mixture than for either of the individual salts.

Vollmer (1894) studied solutions of various salts in methyl
and ethyl alcohols and found the temperature coefficients of

Table LII.—The Fluidity and Conductivity of Tetraethylammonium
Iodide at Infinite Dilution in Various Solvents at 0° and 25°C
(after Walden)

Solvent

/°

a°;

<p! a oo

a 25 °

<p/ A. CO

Acetone.

252.0

177.0

1.41

316.0

225.0

1.41

Propionitrile.

185.0

129.0

1.43

242.0

165.0

1.47

Methyl alcohol.

118.0

90.0

1.31

172.0

124.0

1.39

Ethyl mustard oil.

118.0

82.0

1.44

162.0

106.0

1.53

Acetylacetone.

87.0

57.0

1.52

128.0

82.0

1.56

Ethyl alcohol.

55.9

37.0

1.51

92.6

1 60.0

1.54

Benzonitrile.

51.6

35.5

1.45

80.0

56.5

1.42

Nitrobenzene.

32.6

25.0

1.30

55.0

40.0

1.37

fluidity and conductivity very nearly identical. Walden (1906)
has gone further and proved that ip/A m is a constant even when the
solvent is varied widely. He used tetraethylammonium iodide
in some forty different organic solvents and found <p/A K = 1.43,
which is independent both of the nature of the solvent and of the
temperature. A portion of his data will serve to show the nature
of the concordance.

These researches all point to the conclusion that Stokes’ Law,
with a possible correction as already suggested, holds for the dif¬
fusion of molecules and ions, so that if a given particle has the
same size in different solvents and at different temperatures,
the velocity imparted by a constant force will be proportional
to the fluidity of the medium.

FLUIDITY AND DIFFUSION

195

In ordinary electrolytic solutions, the dissociation is incom¬
plete, hence it is necessary to introduce into our formula the
dissociation factor a in order that we may always be dealing
with an equivalent number of ions, thus

^ = const. (67)

A v A<»

where <p v and A v are the fluidity and conductivity at the volume
v. This formula is of use in obtaining the percentage of dissocia¬
tion by the conductivity method, as indicated by Sutherland
(1902) Bousfield (1905) and Pissarevski and Lempke (1905).
Working with mixtures of alcohol and water, Doroshevskii and
Rozhdestvenskii (1909) found that the ratio Dp /A was a constant
over a considerable range of concentrations of alcohol, D being
the dielectric constant of the mixture. There is of course a rough
proportionality between the dielectric constant and the dis¬
sociating power of the solvent.

Hartley, Thomas, and Applebey (1908) have applied the
Eq. (67) to solutions of lithium nitrate in mixtures of nicotine and
water. These mixtures resemble ethyl alcohol and water in
exhibiting a pronounced minimum in fluidity. The coefficient
of ionization, calculated by formula (67), shows a maximum.
The molecular conductivities of solutions at infinite dilution of
the salt show a minimum, closely resembling the fluidity curve,
whereas the molecular conductivities of an eighth normal solution
show a point of inflection, due to the small ionization in the pure
solvent.

Heber Green (1908) has started considerable discussion
by the discovery that in water-sucrose mixtures, the conductivity
varies, not directly as the fluidity, but

Aqo = K<p m (68)

where m = 0.70 for lithium chloride and potassium chloride and
0.55 for hydrochloric acid. In the case of lithium chloride no
single value for m can be found which will give entirely satis¬
factory results. As a matter of fact, Washburn (1911) has found
that for the first six sucrose concentrations, a value of m of 0.94
gives better concordance than Green’s 0.70. Johnston (1909)
has determined the values of m for a number of other solvents
using the data of Dutoit and Duperthius (1908) for sodium

196

FLUIDITY AND PLASTICITY

iodide solutions, and he finds that in no case does the value of m
depart from unity by more than 0.2.

Johnston has calculated the value of m for many cations
and anions using different temperatures from 0 to 156°, but
found that no single value could be assigned for the hydrogen
and hydroxyl ions. The following table will show the nature
of his results.

Table LIII.—The Relation between the Conductances and the
Fluidities of the Individual Ions at Different Temperatures C
(after Johnston)

Ion

a°;

m

1

<p! a co

0°

<p/ A 00
100°

<p/ A 00
156°

K.

40.4

0.887

1.39

1.71

1.81

NH 4 .

40.2

0.891

1.40

1.71

1.80

Cl.

41.1

0.88

1.37

1.70

1.81

N0s.

40.4

0.807

1.39

1.98

2.19

Na.

26.0

0.97

2.16

2.27

2.31

MCa.

30.0

1.008

1.88

1.84

1.84

C 2 H 3 O 2 .

20.3

1.008

2.77

2.73

2.73

mso 4 .

41.0

0.944

1.36

1.51

1.55

H.

OH.

240.0

105.0

0.234

0.535

0.550

0.806

0.741

0.971

The slightly hydrated ions K, NH 4 , Cl, and N0 3 have*a
high conductivity and a small value of m, corresponding to^n
increasing ratio of <pf the presumably highly hydrated ions
Na, HCa, C2H3O2, and 1/2S0 4 have a low conductivity, a high
value of m and a nearly constant ratio of <p /Hydrogen
and hydroxyl are most like the unhydrated group of ions in that
they have a very high conductance and a low but rapidly
increasing value of <p/

The explanation of these curious facts is not at hand, but
apparently we must assume that the conductivity does vary
directly in proportion to the fluidity and seek to explain the
inconstancy of the <p/k& ratio in the changing solvation of the
ions. The phenomenon is as if the unhydrated ions increased in
volume with the temperature, whereas the hydrated ions do not.

FLUIDITY AND DIFFUSION

197

- In the same way the effect of the addition of sucrose to lithium
chloride solutions would be explained by an increase in one or both
of the ionic‘volumes due to uniting with the sucrose molecules.
That such a hypothesis is not improbable, it is well to add that
C. H. Gill has found that sucrose does form crystalline com¬
pounds with the halides of sodium and ammonium. He did not
find that lithium chloride forms such a compound but there may
be a sufficient tendency to unite in solution to explain the effect
which seems to be peculiar to sucrose. Glycerol, although
highly viscous like sucrose, gives values of m which are unity,
according to the determinations of Massoulier (1900) as calcu¬
lated by Green.

In conclusion, it may be added that there is no connection
between the conductivity and the fluidity of a colloidal solution
of gelatine, as demonstrated by Griffiths (1896) (see also
Ltideking (1889)). The reason for this peculiarity lies in the
heterogeneous character of colloidal solutions as will be more
fully discussed later. Schweidler (1895) has also shown that
there is no relation between conductivity and fluidity in mercury
and certain amalgams.

The Transference Number.—The transport number n A is
expressed by the equation

A*

— . I A

A A + A K

If the equivalent conductances of the different ions change with
the fluidity at different rates, the transport number must be also
a function of the fluidity. We have the two equations

and

a*=aoob (^~y K

Voo/

whence, according to Washburn (1911)

where Ncoa is the transport number of the anion at infinite dilu¬
tion and m is the exponent of Eq. (62) for the salt.

CHAPTER VII

COLLOIDAL SOLUTIONS

If it is highly important to discover the relation between
fluidity and conductivity, it is vastly more important to have a
solution of the numerous problems in connection with the
viscosity of colloidal solutions. Indeed it has been said that
the viscometer is to colloid chemistry what the galvanometer
is to the subject of electricity, and Graham referred to the vis¬
cometer as a colloidoscope. Since 1 per cent of colloid like agar
agar may give water the properties of a stiff solid, the advantage
of employing this property in recognizing the colloid state is
clearly apparent.

A pure liquid, at a given temperature and pressure, can
have but a single fluidity, but in our study of liquid mixtures we
have seen that a mixture of liquids may have an indefinite number
of fluidities dependent upon the method of mixing, in other words,
upon the structure of the liquid. Since colloidal solutions are
always heterogeneous, they always possess structure, and there¬
fore we have this variable always entering into our consideration,
whereas heretofore we have given it but scant attention. There
is, however, every gradation from a pure liquid, to an incom¬
pletely mixed solution, an emulsion, suspension or typical gel.

The Two Types of Colloid Structure. —The structures which
may occur are of two kinds, which must be clearly differentiated
from each other, because they give rise to phenomena which are
in some respects exactly opposite, and this is true in spite of the
fact that the two structures may in certain cases merge into each
other.

In the one case typified by gelatine, the structure requires
time to form and the fluidity at a given moment depends upon
the previous history of the solution. When moreover the
solution is agitated by shaking or stirring or when it is heated,
the structure is damaged and the fluidity is affected. This
structure is similar in results to that which would be produced

i no

COLLOIDAL SOLUTIONS

199

if an undercooled solution crystallized out needle-shaped crystals
throughout the solution so that flow of the resulting mass was
stopped except by breaking the crystalline structure. Such a
structure is a matter of slow growth, it may be partially destroyed
by purely mechanical means, and it arises from forces which are
of a polar nature. In view of this analogy we may speak of this
type of structure as polar, whereas the second type is non-polar.

In the second type of colloidal solution, typified by clay
suspensions those forces are absent which bring about the
setting of the gel. We have in the typical case merely particles
of suspended solid which affect to some extent the fluidity of
the solution, but as we shall see the amount of lowering of the
fluidity is very much less than when the structure is polar in
character. If the distribution of the particles is uniform, the
fluidity of the solution will be independent of time, agitation,
and previous treatment.

Suspensions.—For the simplest conceivable case of a solid
suspended in a liquid, we can imagine lamellae of solid parallel
to the direction of shear as discussed on page 104. If the
alternating lamellae are sufficiently numerous, the flow will
take place without separation of the components, even though the
fluidity of the one component is zero. The fluidity of the
suspension is

cj> = cup i (69)

where a is the volume percentage of the medium whose fluidity
is <pi that is, the fluidity of the medium in the limiting case will
be decreased in exactly the ratio which the volume of the solid
bears to the total volume of the suspension.

If the lamellae have an irregular surface, this law becomes
invalid. If, for example, the lamellae are pierced by a number of
fine pores, the fluid will fill these pores, yet the stream lines will
not pass through the pores or be appreciably distorted by their
presence. The fluidity is then

(j> = (a —* d)(pi (70)

where d is the fraction of the total volume which is pore-space.

The ordinary suspension consists of discrete particles, and
for the simplest case we may consider a sphere suspended in a
fluid of its own specific gravity. The shearing of the fluid,

200

FLUIDITY AND PLASTICITY

which causes any cubical figure of the fluid to assume the form
of a rhombohedron, will cause the sphere to rotate, thereby
assisting the flow. The stream lines are curved on account of the
presence of the sphere, but the sphere itself moves in a linear
direction and with the velocity of the stratum of fluid which
would, if continuous, pass through the center of the sphere.
Spheres in the same stratum do not approach each other since
they all have the same velocity.

Spheres in different strata move with unequal velocity,
hence collisions must take place, depending upon the radii of
the spheres, their number per unit volume, and also upon any
attraction or repulsion which may exist between them. The

a bed

Fig. 72. —Two spheres before, during, and after collision. The initial rota¬
tion of the individual spheres is lost on collision and this results in the dissipation
of energy as heat. In the place of this individual rotation there develops a
rotation of the system. It should be noted that this latter rotation causes the
centers of the spheres to move in a transverse direction, indicated by the dis¬
tances from the dotted lines.

surfaces of two spheres which are approaching each other must be
moving in opposite directions, which are at right angles to the line
joining their centers, Fig. 72. The viscous resistance to this
shearing action which is set up as they approach will rapidly
dissipate as heat their energy of rotation. In other words, their
energy of rotation is converted into heat by the “collision” of
the particles.

The contact of two particles, which are large in comparison
with molecular dimensions, brings the laws of ordinary friction
into play. The spheres cannot rotate unless the torque exceeds a
certain definite value, which will become very important when
we come to consider plastic flow. This value depends upon
the pressure existing at their point of contact normal to the
surfaces and this pressure in turn depends not only on the rate
of shear but on the attraction or repulsion which may exist

COLLOIDAL SOLUTIONS

201

between the particles. So when two spheres come into contact,
Fig. 72 b, they must remain in contact for a definite period
unless the spheres are small enough to exhibit Brownian move¬
ment. If the spheres were without attraction or repulsion for
each other, they would become separated as soon as their centers
have come to be in the same vertical plane.

The spheres cannot rotate as individuals during the period
of contact until the torque exceeds a certain minimum value.
The result is that during the time of contact the group of spheres
begin to rotate as a whole, and they pass out of the strata to which
they formerly belonged, Fig. 72c, and into layers of different
velocities. During this period of acceleration, the liquid will
flow around the spheres and through interstices between them.
Thus other spheres tend to collide with those already in contact
with each other, after which the combined mass tends to rotate as
a whole. When equilibrium is reached these clots will have a
certain average size, depending upon the number, size, and spe¬
cific attraction of the particles.

For the present purpose, the important thing to observe
is that in the collisions of the particles we have a new source
of loss of energy, and if these clots increase in size and number
there must come a point when the clots come in contact across
the entire width of the passage. At this point viscous flow
of the material as a whole stops and plastic flow begins.

For a given substance and volume concentration, the number
of collisions will be proportional to the number of particles, which
varies inversely as the cube root of the radius. But if the
angular velocity is independent of the radius, the energy of
rotation will be proportional to the square of the radius, hence
the loss of energy, due to collisions will be inversely proportional
to the radius. This conclusion, if correct, is very important in
indicating that very finely divided particles give comparatively
viscous liquids or at higher concentrations plastic solids.

Bingham and Durham (1911) have studied suspensions of
infusorial earth, china clay and graphite suspended in water,
as well as infusorial earth suspended in alcohol as already referred
to on page 54. For each temperature, the fluidity falls off rapidly
and linearly with the concentration of solid, so that at no very high
concentration by volume the fluidity of zero would be reached, as

202

FLUIDITY AND PLASTICITY

shown in Fig. 73, for English china clay and water. This concentra¬
tion of zero fluidity is independent of the temperature and is the
concentration which serves to demarcate viscous from plastic flow.

Fig. 73.—The fluidity of aqueous suspensions of clay in water according
to measurements of Durham.

We are not to conceive of a suspension of zero fluidity or infinite
viscosity as incapable of being deformed, but it would not be per¬
manently deformed by a very small shearing force. It remains
an important question which we are unable to answer positively

COLLOIDAL SOLUTIONS

203

as yet, whether the viscosity of a suspension is independent of the
instrument in which the measurement is made or not. It seems
a necessary conclusion that the concentration of zero fluidity
must be determined in a long, narrow capillary. The fluidities of
suspensions follow the empirical formula

4> = (l — ™)<?i (71)

in which 6 is the volume concentration of the solid and c is the
particular value of b at which the fluidity of the suspension
becomes zero. The value of c can vary only from 0 to 1, the
value increasing with the size of the particles. This equation
closely resembles Eq. (70) and becomes identical with Eq. (69)
when c = 1.

In Table LIY the fluidities of graphite suspensions are compared

Table LIV. —The Fluidities of Suspensions of Graphite in Water
at Different Temperatures, (after Bingham and Durham)

C = 5.4 PER CENT

Temperature,

degrees

Volume

percentage,

graphite

Fluidity-

observed

Fluidity

calculated

Volume

percentage,

graphite

Fluidity

observed

Fluidity

calculated

30

0.396

116.8

115.7

1.048

100.9

100.7

35

0.395

129.8

128.3

1.046 1

113.4

111.7

45 '

0.394 !

156.3

154.8

1.042

135.0

134.8

55

0.392

184.9

183.0

1.037

161.7

159.5

65

0.390

215.5

213.1

1.032

192.1

185.7

with the values calculated by formula (71). The two agree
extremely well, which may be due to the fact that the graphite
suspensions (aquadag) are very stable, obviating trouble due to
settling out and clogging the capillary. That the subdivision
of the graphite is carried very far is indicated by the very low
value of the concentration of zero fluidity, c = 5.4 volume per
cent.

Some of the suspensions of sulfur by Oden (1912) are plotted
in Fig. 74 using volume percentages, taking 1.90 as the specific
gravity of sulfur. These values indicate a zero of fluidity at
about a 25 volume per cent suspension. Some of the values are
not on the curves, particularly at the high concentrations; but the

204

FLUIDITY AND PLASTICITY

measurement of the fluidity of suspensions is rendered difficult
by the fact that partial clogging of the capillary gives too low
fluidities, and settling out of the solid gives too high fluidities.
In reference to the discordant observation at 5°C, Oden remarks
that the suspension was strongly flocculated.

O 5 10 15 eo 25

Per Cenf

Fig. 74.—Fluidities of suspensions of sulfur in water at various volume per¬
centages, at 5°, 20°, and 30°C. (After Oden.)

It is interesting to find that Trinidad Lake asphalt, treated
with benzene gives suspensions which according to measurements
of Clifford Richardson (1916) indicate a zero fluidity at 24.6
volume per cent. The fluidities of the suspensions agree well
with our formula, which is surprising, since each solution was
centrifuged to remove that portion of the suspended matter
which would not remain in suspension at that particular con¬
centration.

COLLOIDAL SOLUTIONS

205

The curves of infusorial earth in water, page 55, are convex up¬
ward at the lower temperatures and convex downward at the
higher temperatures. The explanation of this behavior is not
known. Plotting the fluidities and concentrations of “ night
blue” studied by Biltz and Vegesack (1910) we find that all of
those curves are convex upward, the zero of fluidity being at

Table LY.—Fluidities of Suspensions of Trinidad Lake Asphalt
in Benzene at about 20° (after C. Richardson)

Per cent asphalt
by weight

Per cent colloid
in asphalt

Fluidity observed

Fluidity calcu¬
lated c = 34.5
weight per cent

0

153.0

i

2.54

153.0

149

2

2.01

146.0

144

5

2.09

132.0

131

10

2.73

104.0 j

109

20

3.13

61.0 |

64

30

4.19

24.0

20

40

6.51

11.0 !

13

50

10.69

3.1

about 9.2 weight per cent. Allowing the suspensions to stand
for several days causes a marked decrease in the fluidity as does
also the purification of the material.

Woudstra (1908) investigated colloidal silver solutions. In a
solution containing only 0.0046 per cent silver by volume,
the fluidity at 26° was lowered 4.3 per cent so that it seems
possible that a solution containing less than 1 per cent of silver
would have zero fluidity! The data are too scanty to permit
an exact estimation of the zero fluidity concentration and the
fluidity-volume concentration curve is highly convex upward.
With the elapse of time and under the influence of electrolytes
colloidal silver solutions coagulate and there is a simultaneous
increase in the fluidity. This is in accordance with our other
knowledge of the effect of size of particle but it is in marked
contrast to the effect of “setting” on the fluidity of the polar
type of colloids.

206

FLUIDITY AND PLASTICITY

Einstein 1 and Hatschek 2 have both considered theoretically
the case of suspensions of spherical particles at low concentrations.
They both arrive at the formula

H = vi (1 + kb)
or

, = vi
^ 1 + kb

(72)

where 6 is the fraction of solid present by volume and k is a
constant for which Hatschek deduced the value of 4.5 and Ein¬
stein of 1. The formula is hyperbolic in form while the formula
obtained from available experimental material is linear. Their
curve is concave upward, and if it held for high concentrations
the pure solid would have a fluidity of 18 per cent (Hatschek)
to 50 per cent (Einstein) of the fluidity of the continuous
medium, which is absurd.

Hatschek states, “It is obvious that the liquid at the upper
pole of each spherical particle moves with a somewhat greater
velocity than at the lower pole, which is equivalent to a transla-
tory movement of the particles with a velocity equal to half the
difference of the two velocities prevailing at the two poles.”
He thus neglects entirely the rotation of the spheres and
assumes that they are moving faster than the stratum of fluid
which would pass through their centers. That these two motions
are equivalent is at least not self-evident. His formula is ob¬
tained by the employment of Stokes* formula for a sphere moving
through a viscous medium without rotation.

The view is commonly held that dilute suspensions have a
viscosity which is very little different from that of the dispersion
medium, but that as the concentration is increased the viscosity
suddenly increases. Thus Ostwald in his Kolloid Chemie states,
“The curves and tables show that at certain concentrations there
is a very sudden increase in viscosity. For silver and glycogen
hydrosols these concentrations are respectively about 3.5 and
30 per cent.” If the fluidity is in fact linear as we have indicated
is the case, the viscosity curve is hyperbolic. There will naturally
be a rather sudden increase in viscosity but it has no significance*
The question arises, “Does the glycogen fluidity-concentmtion

1 Ann. der Physik 19, 289 (1906)?

2 Kolloid-Zeitschr ., 7, 801 (1901); 8, 34 (1911); Trans. Faraday Soc. (1913).

COLLO / DA L HOL (FT ION N

207

curve show a sudden drop in fluidity at about 30 per cent?” The
glycogen suspensions were; studied by Botazzi and d’Errieo (1906)
using two different viscometers, one from 0 to 20 pen* cent and
the second from 20 per cent on. On plotting the fluidities we
find that the values for each viscometer lie on a straight line, but
the two lines do not coincide. For the first viscometer, the fluid¬
ity of water is 144.0 and the weight concentration of zero fluidity
is 27.5, while for the second viscometer it is necessary to assume
a fluidity for water of 77.6 and a zero fluidity at 41 per cent
concentration. Using formula (64) the calculated values agree
well with the observed except at 45 weight per cent which is
beyond the concentration corresponding to zero fluidity, as shown
in Table LVL Bottazzi and d’Krrioo give their viscosities as
times of flow, which of course; arc not proportional to the; vis¬
cosities, as is so often assumed, so this may perhaps explain the
discrepancy between the; two viscometers. But more work
mods to he done on this suhje;ct to definitely establish whether
the viscosity of a suspension is independent of the; dimensions
of the* instrument or not. At any rate there is no evidence; that
the* fluidity of concentrated supensions is abnormally low. In
fact these experiments lead to the opposite conclusion.

Tahuv. LVL— The Fhxntnrtm or QhYcemm Huspknhions at 37°(J (a mat

Bottazzi ash r/Kuiuco)

Per cent glycogen j

|

Fluidity calcu¬

by weight

Fluidity observed^

lated by formula

h

1

s

; i

(71)

0

i 144.0

144

j

1

138,0

130

Viscometer No. 1

5

114,0

118

vi 144.0

10 1

80.0

02

r - 27.5

15 ;

00.0

<10

20

i 40.0

40

20

t 40.0

40

; . '

25

32 0

34

1 Viscometer No. 2

20 1

20,0

21

1

-4

35 1

12.0

11

c * 41.0

40 [

5.0

2

45 i

2.3

208

FLUIDITY AND PLASTICITY

Botazzi and d’Errico obtained the viscosity of glycogen
solutions both on raising the temperature and on lowering the
temperature to the point of measurement. The difference was
hardly more than the experimental error, which shows that the
fluidity of a suspension is not dependent on its past history.
This is in marked contrast to the behavior of polar colloids. On
the other hand, non-polar colloids are very susceptible to the
effect of electrolytes, even the merest traces often causing a
change in fluidity. As a matter of fact many suspensoids show
a slow increase in fluidity on standing, due to. the gradual increase
in the size of the particles on precipitation, as shown by Woud-
stra’s experiments with colloidal silver suspensions.

Generally speaking, dilute acids and salts with an acid reaction
coagulate suspensions and lower the fluidity, whereas dilute bases
and salts with a basic reaction have a deflocculating action.
Neutral salts may act in either way or be without effect. This is
shown in the following table.

Table LVII.—The Effect of Electrolytes on the Fluidity of
Suspensions (after Bingham and Durham, (1911))

Dispersoid

Concentra¬
tion disper¬
soid

Fluidity of
suspension

Substance

added

Concentra¬
tion of elec¬
trolyte

Fluidity
with elec¬
trolyte

Infusorial earth....

6.46

62.1

KC1

1:80,000

53.2

Infusorial earth....

6.46

53.2

NaOH

1:20,000

58.3

Graphite.

0.396

116.8

KC1

1:20,000

116.9

Graphite.

0.396

116.8

HC 2 H 3 0 2

1:20,000

64.5

China clay.

2.63

41.5

KC1

1:40,000

65.8

The decrease in the fluidity due to acids is attributed to the
increase in cohesion between the particles, which results in coagu¬
lation. It is a matter of common experience that acids cause
the particles to cohere together and it has already been explained
on page 200 how increased cohesion decreases the fluidity.

We need not here discuss the reason why the cohesion of the
particles is so much greater in acid solutions, although the
subject is one of great interest in the theory of emulsification
with its important application in the detergent action of soaps.

Crystalline Liquids. —Reinitzer in 1888 first discovered that

COLLOIDAL SOLUTIONS

209

cholesterolbenzoate melts at 145.5° to an opalescent liquid which
at 178° became suddenly clear and isotropic. The optical
properties of this and other substances of similar behavior was
carefully studied by Lehmann. Schenck (1898), Eichwald (1905),
Buhner (1906), Bose (1907) and Dickenscheid (1908) have studied

Temperature

Fig. 75. —Fluidity-temperature curve (continuous) and specific volume-
temperature curve (dashed) of p-azoxyanisole. (After Eichwald (1905) and
Btihner (1906).)

the viscosities of these substances and shown that these so-called
“ crystalline liquids” have a higher fluidity than isotropic liquids.
The specific volume of crystalline liquids is smaller than that of
isotropic liquids of corresponding temperature. In other words,
when an anisotropic liquid is heated to the clarifying point, there

14

210

FLUIDITY AND PLASTICITY

is a sudden increase in volume and decrease in fluidity as shown
in Fig. 75 for p-azoxyanisole from the measurements of Eichwald
and Buhner. As the temperature is raised, the fluidity increases
in a nearly linear manner, passes through a sharp maximum,
and suddenly falls to the clarifying point, where there is a dis¬
continuity in the curve. As the temperature is raised still
further, the fluidity again increases in a linear manner.

This behavior resembles that of molten sulfur which increases
in fluidity up to 150°, where the fluidity is 11.4 according to the
measurements of Rotinjanz (1908). It then suddenly falls off
to 0.0018 at 180° after which the fluidity gradually increases up
to 1.14 at 440°.

Drawing a parallelism between anistropic liquids and molten
sulfur, in no way explains the phenomenon, for the behavior of
sulfur is unexplained. Bose regards anistropic liquids merely as
emulsions of very long life. But an emulsion has invariably a
lower fluidity than a homogeneous solution at the same tempera¬
ture, and according to the theory this must always be the case,
so that the emulsion theory seems to be excluded. The phenome¬
non cannot be accounted for on the basis of the observed vol¬
ume change, because the volume of the isotropic liquid is greater,
which would lead to an increase in the fluidity. We apparently
have but one explanation left, viz., that as the anistropic liquid is
heated to the clarifying point a new molecular arrangement is
formed which has a much larger limiting volume, so that although
the molecular volume is increased the free volume is lessened.
The same explanation would apply to sulfur.

Emulsions and Emulsion Colloids. —In our discussion of the
critical solution temperature, it was made clear that the separa-
, tion of the components of a mixture in the form of an emulsion
is attended by an increase in the viscosity. It seems probable
that this increase is due to the viscosities in emulsions being
additive, for it follows of necessity that when the viscosities
are additive the viscosity will be greater than in a homogeneous
mixture of the same composition. As in the case of suspensions,
there is considerable evidence that decreasing the size of particle
of the disperse phase brings about a corresponding decrease in the
fluidity. Martici (1907) experimented with oil-soap emulsions
and found that the fluidity becomes less as the drops become

COLLOTDAL SOLUTION'S

211

smaller. iBuglia (1908) has found that the fluidity of milk is
lessened when the milk is “homogenized” by being squirted
Against an agate plate, thereby increasing the number of fat
globules. The apparent decrease in fluidity with emulsification
finds excelled practical examples in the manufacture of solid
lubricants and of certain household products such as mayonnaise,
“ whipped cream,” and beaten egg albumen. In engine grease
less than 1 per cen t of water emulsified by means of a solution of
soap with, mineral oil produces a salve-like grease. Such bodies
have the properties of solids and may also be considered in
connection, with the plasticity of solids.

The question inevitably arises, “ How is it possible that water
with a high fluidity can decrease the fluidity of a heavy oil,
or air decrease the fluidity of albumen, so that the resulting
product, emcrulsion or foam as the case may be, has the rigidity
of a solid ? To answer this question it is necessary to return to
the consideration of our simple case of lamellae of different liquids
at right angles to the direction of shear. The theory that
viscosities of emulsions are additive, will account for the fluidity
being less than the fluidity of the homogeneous mixture but it will
in. no way account for the case we have here where the fluidity
of the emulsion is less than the fluidity of either component.

As the shear progresses, it is to be noted, Fig. 34, that the
lumellse are greatly elongated. But in immiscible liquids
this thinning out of the layers is opposed by the surface tension
which tends to keep the surface area a minimum. If therefore
the shearing force is less than the maximum force arising from
the surface tension, continuous deformation will not result.
There will be a certain amount of temporary deformation but
this too will disappear as soon as the shearing force is removed.
In other words, the substance shows not only rigidity but also
elasticity; if the shearing force is greater than “the elastic limit,”
continuous deformation wjfll take place, but since we are dealing
with immiscible liquids, the lamellae will not be thinned out indefi¬
nitely , but torn into portions which will gather into drops under the
influence of surface tension. Thus in an emulsion, shear tends
to make the droplets continually smaller, and consequently to
raise the -viscosity. This corresponds to the “cold working”
of metals. This effect is opposed by the spontaneous coalescence

212

FLUIDITY AND PLASTICITY

of the particles on standing, analogous to the “annealing” of
metals, so it appears that an equilibrium results and the maxi¬
mum in viscosity in emulsions may depend upon the rate of shear.

As the lamellae of the simple case, which we have taken for
consideration, are broken up, the viscosities are no longer
strictly additive. The droplets become smaller and smaller,
the surface tension becomes more and more effective, the droplets
become true spheres with an inappreciable amount of flow within
the spheres, so that finally the distinction between emulsion and
suspension disappears.

We pass finally to that class of polar colloids typified by
gelatine, soap and rubber. In some ways they are in sharp con¬
trast with the type which we have just been considering, because
their viscosity increases tremendously on standing and decreases
as a result of shear, but they are alike in the more fundamental
respect of exhibiting the properties of rigidity and elasticity.

It is assumed that the process of gelatinization is the result
of polar forces producing a network of crystals or crystal-like
material interlacing throughout liquid, without necessarily
taking up more than a small portion of the space. The solid
network performs the function of the lamellae at right angles to
the direction of shear in our simple case. The cohesion of the
solid opposes the shear and gives rise to the rigidity of the gel.
The ability of the solid to be deformed without fracture deter¬
mines its elasticity. This property of elasticity is enormously
developed in rubber, and we have seen that it is noticeable in
foams and emulsions. Barus (1893) has noted the considerable
degree of elasticity in marine glue which may be regarded as
a very viscous liquid. It also is of importance in suspensions,
as for example in the manufacture of pencils, the “leads’’ expand
considerably, as they are forced out of the die previous to baking.

If gelatinization is analogous to crystallization, we should
expect the viscosity to increase on standing and that it would
be hastened by “seeding” the solution with a more viscous
colloids. We can readily see that shearing the material would
result in the destruction of the polar structure of the material
and consequently in a decrease in the viscosity. We refer the
reader to the rich material furnished by Garrett (1903).

When a hydrogel is exposed to dry air, it loses moisture

COLLOIDAL SOLUTIONS

213

and the structure gradually collapses. But showing the proper¬
ties of a true solid, it remains under tension, and when placed
again in water, it swells to approximately its former size, but not
indefinitely, as shown by Bancroft.

Increase in concentration of the internal phase very naturally
increases the viscosity of the colloidal solution. The addition
of non-electrolytes generally affects the viscosity in the way that
we would expect from the change produced in the fluidity of
the external phase. Since the colloid may unite with the water
to form hydrates or with the non-electrolyte, we should expect
exceptions to the quantitative application of this rule. Electro¬
lytes have a similar effect on the viscosity of emulsion colloids,
potassium nitrate, ammonium nitrate, and potassium chloride
which increase the fluidity of water also increase the fluidity of
gelatine solution according to the measurements of Schroeder
(1903). Sodium sulphate, ammonium sulphate, magnesium sul¬
phate and lithium chloride depress the fluidity. Acids and
alkalies however first lower the fluidity and then raise it. For a
more adequate account of this complicated subject the reader is
referred to the original papers, Schroeder, Pauli, etc.

It has often been a cause for wonder that a gel which has con¬
siderable rigidity offers hardly more resistance to diffusion than
does pure water. We merely cite the names of Graham (1862),
Tietzen-Henning (1888), Voightlander (1889), and Henry and
Calugareanu (1901), giving a single observation from Voight¬
lander to the effect that a 1 per cent solution of sodium chloride
in a 1,2, and 3 per cent solution of agar gave a diffusion constant
of 1.04, 1.03, and 1.03 respectively. Similarly Liideking (1889),
Whetham (1896), Levi (1900) Garrett (1903) and Hardy (1907)
have found that the conductivity of solutions remains constant
during gelatinization.

To understand these peculiarities, it is necessary to consider
the phenomenon of seepage of a fluid through a porous material.
Suppose, for example, that we consider a single pore; we must
assume that since it is a tube of capillary dimensions, the flow
must follow the law of Poiseuille and be proportional to the fourth
power of the radius of the pore. The question arises, “What will
be the effect upon the volume of flow of substituting for the single
pore a number of smaller pores whose total pore opening is the

214

FLUIDITY AND PLASTICITY

same as that of the single pore?” It is easy to calculate from
Poiseuille’s law that for a given area of pore opening the volume
of flow will be directly proportional to the square of the radius of
the individual pores, which are assumed to be alike. If the small
pores have a diameter which is only 0.0001 that of the large one,
the flow which takes place through the large pore in 1 minute
will require about 12 years through the multitude of pores having
the same total area. The underlying principle on which the
explanation is based is the fact that each layer in viscous flow is
carried along by the layer immediately below it, the velocities of
the layers increasing in arithmetical progression. The laws of
viscous flow are therefore capable of explaining why fluids do
not readily flow through jellies and other finely-divided materials.

It is well known that compact clay is almost impervious to
both water and oils, and therefore they are often associated, the
clay forming an impervious stratum through which the oil or water
do not penetrate. The subject of pore openings is therefore
fundamentally important to the subject of the circulation of
water through soils as well as of their retention of water. The
use of compact clay in the cores of dams finds an explanation on
this basis.

When it comes to a single particle diffusing through a liquid
impelled by electrical attraction or other force, the above con¬
siderations no longer hold and the walls of the pores offer no
serious resistance, the particle moving through the medium as if
it alone were present, withqut the surrounding network.

CHAPTER VIII

THE PLASTICITY OF SOLIDS

Only by the behavior of materials under shearing stresses are
we enabled to distinguish between a fluid and a solid. If a
body is continuously deformed by a very small shearing stress,
it is a liquid; whereas if the deformation stops increasing after
a time, the substance is a solid. This distinction is theoretically
sharp like the distinction between a liquid and a gas at the critical
temperature, but just as a liquid may be made to pass into a gas
insensibly, so a solid may grade insensibly into a liquid. Glass
and pitch are familiar examples of very viscous liquids. Paint,
clay slip, and thin mud in a similar manner must be classed as
soft solids. According to the experiments of Bingham and
Durham (1911) the concentration in which the fluidity becomes
zero under a very small shearing force serves to demarcate the
two states of matter.

This simple distinction is not always sharply drawn nor is its
significance thoroughly appreciated; and for this reason much
labor has been ill-spent in the attempt to measure the viscosity
of solids, on the assumption that solids are only very viscous
liquids and therefore that plasticity and the fluidity of solids
are synonymous terms. The results are unintelligible because
the viscosity as so determined in various instruments is widely
different.

The views of Clerk Maxwell expressed in his “ Theory of Heat”
are especially noteworthy and are quoted at length:

“If the form of the body is found to be permanently altered when the
stress exceeds a certain value, the body is said to be soft or plastic and
the state of the body when the alteration is just going to take place is
called the limit of perfect elasticity. If the stress, when it is maintained
constant, causes a strain or displacement in the body which increases
continually with the time, the substance is said to be viscous.

“When this continuous alteration of form is only produced by stresses

216 FLUIDITY AND PLASTICITY

exceeding a certain value, tlie substance is called a solid, however soft
it may be. When the very smallest, stress, if continued long enough,
will cause a constantly increasing change of form, the body must be
regarded as a viscous fluid, however hard it may be.

“Thus a tallow candle is much softer than a stick of sealing wax;
but if the candle and the stick of sealing wax are laid horizontally
between two supports, the sealing wax will in a few weeks in summer
bend under its own weight, while the candle remains straight. The
candle is therefore a soft (or plastic) solid, and the sealing wax is a very
viscous liquid.

“What is required to alter the form of a soft solid is sufficient force,
and this, when applied, produces its effect at once. (This is, of course,
only relatively true, because plastic deformation is a function of the
time, as will appear later.) In the case of a viscous fluid, it is time which
is required, and if enough time is given the very smallest force will
produce a sensible effect, such as would be produced by a very large
force if suddenly applied.

“Thus a block of pitch may be so hard that you can not make a dent
in it with your knuckles; and yet it will, in the course of time, flatten
itself out by its own weight and glide down hill like a stream of water.”

The italics and parentheses are ours. Butcher (1876) has
expressed views quite similar to those of Maxwell.

We may now define plasticity as a property of solids in virtue

of which they hold their shape
permanently under the action of
small shearing stresses but they
are readily deformed, worked or
molded, under somewhat larger
stresses. Plasticity is thus a com¬
plex property, made up of two inde¬
pendent factors, which we must
evaluate separately.

Reverting to our fundamental
conception of flow between two
parallel planes separated by a
distance dr and subjected to a shearing force F , we have found
that in a viscous fluid

dv = <pFdr

so that if we were to plot the volume of flow through a tube or
the rate of shear against the shearing stress, wc would obtain

Fig. 76.—Typical flow-shear
diagram for a series of viscous
liquids.

?r

f

foi

tth,

-tlx,

plf

wt

m

plo

nol

]

IRe

vis

1

um

the plasticity of solids

217

for a series of fluids a family of straight lines passing out from
the origin illustrated in Fig. 76.

'In a plastic solid, a certain portion of the shearing force is
used up in overcoming the internal friction of the material. If
the stress is j ust equal to the friction or yield value, the material
may be said to be at its elastic limit. If the stress is greater than
the friction f, the excess, F — f, will be used up in producing
Plastic flow according to the formula

dv = m (F - /) dr (73)

where n is a constant which we will call the coefficient of mobility

JB'jg. 77. —Flow-shear diagram of a plastic solid.

in analogy to the fluidity >of liquids and gases. If we were to
plot the volume of flow against the shearing stress we would
again obtain a straight line for a given material but it would
not pass through the origin, ABC Fig. 77.

It is easy now to see why the “viscosity” of plastic substances,
as measured In the usual way for liquids, is not a constant.
Referring to the figure, if we take two determinations of the
flow A and B 9 we see that they correspond to entirely different
viscosity curves OB and OE .

When the stress is not equal to the yield value, the material
undergoes elastic deformation and an opposing force arises

218

FLUIDITY AND PLASTICITY

which would restore the body to its original shape if it were
perfectly elastic, as soon as the stress was removed. On the
application of the stress, the restoring force is first zero, then
gradually increases to a maximum, when at last the flow causes
the strain to disappear as fast as it is produced.

The elasticity e of a solid may be calculated, according to
Morris-Airey (1905), from the fundamental formula

ds = eFdr (74)

where ds is the distance which one plane of the material is sheared
in reference to another plane which is separated from it by a
distance dr, each being subject to the shearing force F . Morris-
Airey has applied this formula to tubes of circular cross-section
filled with gelatine and obtained the rigidity 1 £ which is the
reciprocal of the elasticity

, irgWP
* SVl

(75)

where V is the volume of the temporary deformation. It is
assumed that the solid is incompressible. The analogy of this
formula with that of Poisuille is striking.

The Methods for Measuring the Friction and
Mobility

To determine the two quantities, friction and mobility, which
go to make up the plasticity of a material, i.e., to locate the curve
in Fig. 77, it is necessary to make at least two measurements of
the flow, using different stresses. We may use the tube method
(Bingham (1916)), the torsion method (Perrott (1919)), or we
may observe the flow in a rod under traction or torsion, the
flow of a cylinder under axial compression, the rate of bending
of a horizontal beam of the material under its own weight, or the
flow of a freely descending stream of material, (Trouton and
Andrews (1904)). Still other methods have been suggested
such as the rate of decay of vibrations in solid bodies, (Kelvin
(1865) and others).

The friction is most easily obtained by the graphical method,

PR

plotting the rates of flow V/t, against the shear, F = -2f and

1 The assumption which is sometimes made that the rigidity is the re¬
ciprocal of the mobility is incorrect.

THE PLASTICITY OF SOLIDS

219

extrapolating the curve to the axis; the value of the intercept
will evid en tly be the friction. We may also use the algebraical
method. In either case at least two measurements of the rate
of flow V^x/ti = V\ and V^/h = ^2 are necessary corresponding to

the shears F t and F 2 , respectively. Assuming that the mobility
is independent of the rate of flow, Eq. (73) integrated in Eq. (89)
gives us

/ = V -F 1 ~ v iF2

Vi — Vi

( 76 )

220

FLUIDITY AND PLASTICITY

The following table, taken from the work of Bingham and
Green on paints, proves the validity of the general law of plastic
flow expressed in Eq. (73). The friction, when expressed in
terms of shear—and not in terms of pressure—is nearly constant
and not a function of the dimensions of the capillary. It is a
fact, however, that the rate of flow is not directly proportional to
the shear, when the shear is too small, but when the shear is suffi¬
ciently high the relation becomes linear, as is proved by plotting

Fig. 79.—The relation of fluidity and friction to volume concentration of solid
in clay suspensions.

the values in the table, Fig. 78. The table also indicates that the
mobility is a constant independent of the rate of flow or of the
dimensions of the capillary. The reason for the rate of flow-
shear curve not being linear as the rate of flow is decreased will
be considered when we come to discuss the theory of plastic
flow.

By measuring the fluidity of suspensions containing increasing
amounts of solid in suspension, Bingham and Durham found
it possible to obtain a concentration which would possess zero
fluidity when the shear was very small. Conversely, by measur¬
ing the friction of suspensions containing decreasing amounts of
solid, it is possible to find a concentration which would have

THE PLASTICITY OF SOLIDS

221

zero friction, Fig. 79. Evidently these two concentrations are
identical, and the concentration of zero fluidity or of zero friction
is a fundamental constant of the material giving important
information in regard to its nature, it being intimately related to
the size of the particles and to the adhesion between them.

The flow of a given material is defined completely by a knowl¬
edge of the friction and mobility, but when the concentration of
the suspension is changed, a knowledge of the concentration of
zero fluidity is necessary in order to estimate the effect produced

Fig. 80.—[Relation between mobility and volume concentration of solid in clay

suspensions.

upon the friction and mobility. It therefore seems probable that
the concentration of zero fluidity is a variable which is inde¬
pendent of both the friction and the mobility.

Finally, we may add that the mobility of suspensions de¬
creases very rapidly with increasing concentration of solid
as indicated by measurements of the author which are plotted in
Fig. 80. Clay suspensions were used having a concentration of
zero fluidity of 19 per cent by volume. The mobility starts at a
very large but undetermined value and quickly falls to a very
small value in a concentration of about 50 per cent by volume.
The friction on the other hand, starts at zero in the 19 per
cent mixture and rises steadily and in an apparently linear man¬
ner as the concentration is increased as seen in Fig. 79.

222

FLUIDITY AND PLASTICITY

Table LVIII. —Friction and Mobility of a Paint as Measured by
Bingham and Green 1

Num¬
ber of
obser¬
vation

I

V—t centi¬
meters per
second

Pressure

grams

per

square
centi¬
meter !

„ FR
F “ 21
dynes
per

square

centi¬

meter

/

dynes

per

square

centi¬

meter

F-f

Remarks

Obser¬
vations
used in
calcu¬
lations

1

0.0005836

670.8

1030.7

98.2

938.7

0.260

Capillary S

1 and 2

2

0.0004557

537.8

826.3

84.6

734.3

0.260

r = 0.014486

cm

2 and 3

3

0.0003344

409.3

628.9

75.9

536.9

0.261

l = 4.620 cm

3 and 4

4

0.0002133

277.5

426.4

[66.0]

334.4

0.267

4 and 5

5

6

0.0001661

0.0001019

225.6

152.9

346.6

234.9

[57.7]

254.6

0.273

5 and 6

7

0.002424

670.2

1458.6

101.0

1366.6

0.253

Capillary VI

8

0.001912

538.5

1171.9

85.4

1079.9,

0.254

r = 0.020805

9

0.001418

409.5

891.2

87.5

799.2

0.255

l = 4.684 cm

9 and 10

10

0.0008987

274.3

596.9

165.6]

504.9

0.256

10 and 11

11

12

0.0004164

0.0002880

143.3

106.7

311.8

232.2

[53.7]

219.8

0.272

11 and 12

13

0.004638

671.7

1723.0

81.6

1631.0

0.246

Capillary III

13 and 14

14

0.003678

539.1

1382.9

93.2

1290.9

0.246

r = 0.02450

14 and 15

• 15

0.002726

409.0

1049.2

85.1

957.2

0.246

l ~ 4.681 cm

15 and 16

16

0.001758

275.6

706.9

[75.1]

614.9

0.247

16 and 17

17

18

0.0008267

0.0005856

145.1

110.0

372.2

282.2

[63.5]

280.2

0.255

17 and 18

The average friction used in calculating the mobility is 92.0 dynes per square centimeter, .
which gives an average mobility of 0.257. When the rate of flow V/t is too small, the
friction becomes smaller, as seen in the table and the last two values for each capillary may
well be neglected.

The Capillary Tube Method. —Unless the conditions of
flow are carefully chosen, the friction constant does not manifest
itself, or at any rate the amount of shear is not a linear function
of the shearing stress. This departure from linearity is very
often shown at the low rates of shear as indicated in Fig. 76
by the curve FG.

This peculiarity is not fully understood at present and the
worker will do well to avoid anxiety in regard to it by choosing
the conditions as nearly ideal as possible so that the flow will
be a linear function of the shearing stress.

Nevertheless the cause of the above peculiarity must be
investigated in detail if we are to understand fully the nature

1 Proc. Am. Soc. for Test. Mats. (1919).

THE elasticity of solids

223

lastic flow and it has already had the attention of Bucking-
(1921)- In plastic material confined between two parallel
es of indefinite extent which are being sheared over each
r, the shearing stress F will be identical at every point. But
low through a capillary tube according to Buckingham
is ixot the case; the shear increases continually from the
er of the capillary outward and only at a certain distance r Q
the shearing force become sufficient to overcome the friction,
■efore the material at the center of the capillary moves as a
plug with the velocity v 0 , and beyond the radius r 0 the mate-
moves in telescoping layers. This results in the flow not
? a linear function of the pressure.

it there are other possible causes of the peculiarity which
be mentioned here. The plastic material next to the
may have a lower concentration of solid than elsewhere
ting in apparent slippage. Or the shearing stress may cause
iquid to flow between the particles of solid, seepage.
Lckiiigham suggests that the friction between the particles
ig flow may not be the same as the static friction. It
s further possible that the friction will need further definition
. the individual particles of the plastic material arc of very
ent slizes. We shall at first assume that slippage and
<ge are both absent and that the particles of solid are uni-
y small.

e total force producing the flow through a capillary tube
R 2 and since there is no acceleration, this is opposed by a
in the opposite direction 2 ttRIF. If p is the pressure
i is used up during the flow in overcoming the friction, the
m f is defined by the equation

- R

f = 2l p W

o follows that

/ - §P (78)

P = 2l P

ce tiie speed decreases as the radius increases, Eq. (73)
les

dv = - ^ “ /) dr (80)

224

FLUIDITY AND PLASTICITY

where v is the velocity parallel to the axis at the radius r. The
speed of the material in the variable region is obtained by
integrating Eq. (80) from r = R to r = r or

■f*.

*

Ur

Pr 2
41
P

■fr

41

(P 2 - r 2 ) - /(P - r)

(81)

(82)

The speed of the solid plug is obtained by making r = r 0 in
Eq. (81), and is after simplifying

(PR* If
Uo = n~4T + p

The volume of flow per unit of time is 7 /t and
7 = S'*
t “Jo

or using Eqs. (81) and (82)

7
t

But from Eqs. (78) and (82)

4 Pf(PR 2 . If

+ ^ -/«)

27irvdr

nr Q 2 vo + 27 r I rv r dr.

/r#

■r

(83)

xr 0 2 7 o = itjx

P 2

41 + P

~/p)

and from Eq. (81)

r rR

2lT | TV r dr = 2 TTJjL

■f

= 2* +

/'

41

(P 2 r - r 3 ) - /(Pr - r 2 ) dr

+'(*!’-¥)]• <*>
and introducing the value of r 0 from Eq. (78), we have
/P 4 P P 3 / RHf 2Rl*f 5 l 3 / 4
2P + P 2 3 P 3 '
Introducing these values of the separate terms of Eq. (83) and
simplifying, Eq. (83) becomes
7

(R'P P 3 /

“ Mia " T ■

(85)

7 _, /P 4 P P 3 / . 21 3 / 4 \
1 *** \ 3f 3 + 3P 3 /

or

V

■t

81

^ 4 rp - _L

L 3 \ P/ 3P 3 \ P / J
■ E

and now introducing the value of p given by Eq. (77),
7 x^P 4
t 81'

( p

4 „ ,

3 P ^ 3P 3 /

( 86 )

TUB PLASTICITY OF SOLIDS

225

For large values of the applied pressure, the last term of Eq.
( 86 ) becomes very small and the curve becomes very nearly
linear and coincident with its asymptote

V

t

7 rfjR A

r

(87)

The curve rises above the asymptote as the applied pressure
becomes very small, but it crosses the pressure (or shearing stress)
axis when P = p (or F =/). On differentiating Eq. ( 86 ) in
respect to the pressure one finds that the slope of the curve
vanishes when P = p, hence the curve is tangent to the axis.

The intercept of the asymptote is thus 4/3 of the true friction
which would be obtained by other methods as, for example,
plastic material confined between parallel planes which are being
sheared over each other. If in practice conditions may be
controlled so that all of the observed points lie on a straight line,
it will mean that the flow is taking place practically throughout
the capillary in telescoping layers, the term p/3P 3 being negligible.

Were we to assume that the material throughout the capillary
flows in telescoping layers.for all shearing stresses above/, we
will obtain

r‘ J dv

7TM (P - y)
21

V = 7 ipR 3
t 4

(*-/)

7 T JJL (P — p)R A
81

(89)

which differs from Eq. (88) in having / in place of 4/3/. It is
highly desirable that some one measure the friction both by the
capillary tube method and other methods using a given material,
to make sure that they give identical values for the friction.

Not being able to reproduce satisfactorily the data of Bingham
and Green, Buckingham has attempted to allow for slippage. If
there is a thin layer of viscous liquid of thickness e separating the
plastic material from the wall, it will increase the velocity of the
plastic material by the amount epF, hence the increase in the
volume of flow per unit of time over that given by Eq. (88) is
ir R 2 e<pF approximately. But at present it is not certain that there
is slippage after the flow is established by increasing the shearing

226

FLUIDITY AND PLASTICITY

stress somewhat above the friction, so we have no idea as to how
the value of e may vary with the rate of shear, and the equation
becomes unmanageable. .Fortunately by using the higher rates
of shear we can apparently always obtain the simple linear
relationship. If later experiments prove that this is not the case
it will be time to use the more complex formulas.

The Traction Method. —Trouton has discovered that the rate
of flow in a rod of material subjected to traction is not propor¬
tional to the tractive force T, but analogously to Eq. (73)

dv = \(T — t)dr (90)

where X is the coefficient of plastic traction, and Hs a tractive
friction constant. The value of t may be found by plotting
the elongation dv/dr against the tractive force and extrapolating
the curve to the axis. Trouton has obtained values of X for #
pitch of 2.3 X 10~ 10 and for shoemakers’ wax of 1.7 X 10 -7 . To
obtain the relation between the coefficient of plastic traction and
the mobility, we note that the tractional force applied to a rod
may be resolved into two equal shearing stresses at right angles
to each other and at 45° to the direction of traction. The value
of either shearing stress is one-third of that of the tractive stress,
hence the friction is one-third of the tractive friction and the
mobility is one-third of the plastic traction coefficient as shown in
Table LX.

The Torsion Method. —Trouton applied a constant torque
to the ends of a cylinder or tube of substance and observed the
relative motion of the ends. He found that rods which were
carefully made could be twisted almost indefinitely, provided
that they were maintained in a horizontal position. The motion
was fastest when the stress was first applied but the angle of
twist per unit length U soon became a linear function of the time.
Conversely when the stress was removed, the bar started to
twist in the opposite direction. He made the experiment of
removing weights at such a rate that the rod would not move in
either direction, and found that the weights remaining were
inversely proportional to the time elasped. This kind of elastic
recovery was found to be present in glass and sodium stearate.
Trouton does not seem to regard his materials as solids but he
makes it very clear that the angular velocity is not directly pro-

THE PLASTICITY OF SOLIDS

227

portional to the torque, and there is a very considerable magni¬
tude to the value which can be assigned to the friction in his
experiments with pitch.

Trouton assumes from symmetry that any two planes in the
material, lying at right angles to the axis of the cylinder, move
over each other, about the common axis, remaining plane all the
while.

Let 5x be the distance apart of the two planes, and 80 ) be the
relative angular velocity of the planes, then

„ . 2tt C R , ,

V 5x Jo

where F is the torque applied and / is the force used up in Over¬
coming the friction, obtained by extrapolation.

4 Thus for a solid cylinder we have

it UR 4

(91)

M 2 (F-f)

and for a tube of material this becomes

TiUiRt* - RS)

(92)

M 2(F — f)

where Ri and J2 2 are the external and internal radii respectively.

Trouton proved the validity of the fourth-power law by using
two cylinders of pitch whose radii were 0.36 and 0.67 cm and
obtained mobilities of 1.01 X 10"” 11 and 0.99 X 10 -11 respectively
which is excellent agreement.

The Sagging Beam Method.—The rate of sagging U of a rod
at its center is found to be

TJ = _JL gmL *

384 X/

(93)

where m is the mass of the rod between the supports, L is its
length and I is the moment of inertia of the cross-section of the
rod, and g is the gravitation constant. This does not take
account of the friction.

In order to prove that the rate of sagging of a beam varies as
the fourth power of the length, Trouton measured the times T
which beams of different lengths required to sag a certain dis¬
tance. Table LIX shows that TL 4 is very nearly constant.

228

FLUIDITY AND PLASTICITY

Table LIX.— -Experiments on the Sagging of a Rod of Pitch at 15°,
Demonstrating that the Time T Required to Sag a Given Dis¬
tance Varies Inversely as the Fourth Power of the Distance

L BETWEEN THE SUPPORTS (AFTER TROUTON)

L T TIJ

33

14.6

1.7 X 10 7

30

18.5

1.5

27

30.4

1.6

24

47.0

1.6

That the different methods agree with each other is shown
in Table LX.

Table LX.—A Comparison of the Coefficients of Plastic Traction
and Mobility as Determined by Various Methods (after Trouton) *

Substance

A

Method

u

Method

j“A

Pitch I.

. 2.3X10- 11

Traction....

7.1 X10~ u

Torsion..

3.07

Pitch II.

. 2.8X10" 11

Traction....

l.OXIO-io

Torsion...

3.60

Pitrh TT

. 3.OX 10~ lx

Sagging.

3.30

Pitch and tar I.

. 7.8XIO- 9

Traction....

2.4X10" 10

Torsion..

3.07

Pitch and tar II.

. 1.5 X10“ 10

Traction....

4.5X10-1°

Torsion...

3.04

Shoemaker’s wax.

. 1.9 X10“ 7

Traction....

5.0 X10“ 7

Torsion..

2.95

Pitch and tar 3:1 III.

. 1.3X10“ 5

Sagging.

3.8 X10“®

Efflux....

3.25

Pitch and tar 3:1 IV.

. l.lXlO" 8

Descending

column.

3.6X10- 8

Efflux....

2.91

The Theory of Plastic Flow

A plastic solid is made up of particles which touch each other
at certain points. The spaces between the particles may be
empty or it may be filled with gas, liquid, or amorphous solid.
Flow necessitates the sliding of these particles the one over the
other according to the ordinary laws of friction, so long as the
particles are large enough so that their Brownian movement is
negligible. It is by no means necessary that the particles be
touching at the maximum number of points, corresponding to
“ close-packing.” As a matter of fact, close-packing of the
particles prevents flow from taking place. It is merely necessary
that the particles touching each other form arches capable of
carrying the load, as already indicated on page 201. It is evident

THE PLASTICITY OF SOLIDS

229

that as aggregates of particles are formed in the process of
collisions, and the size of these aggregates increases as the
concentration of solid increases, there must come a time when
such aggregates or clots will touch each other and form an arch or
' bridge across the space through which the flow is taking place.
At that concentration the friction will have a finite value, and the
material may be said to have a structure just as was the case of
the jelly or foam already considered.

The pore space may vary between very wide limits, but
if the suspended particles are assumed to be uniform spheres, it
can easily be calculated that cubical close-packing, would leave
a pore space of 1 — 7r/6 or 47.64 per cent by volume, irrespective
of the size of the particles. It is possible to get the particles
still closer together until with tetrahedral close-packing, which
w we have in a pile of cannon-balls, the pore space is 1 — 7r /3y/2 or
25.96 per cent by volume, but in this case the particles are
interlocked and no true flow is possible but rupture , with dis¬
integration of the particles. When the pore space is roughly 50
per cent, the mobility is zero, and it is only as the pore space
is in excess of this figure that the mobility has a finite value.
This excess pore space thus plays a role which is analogous to
the free volume of liquids.

As there is a minimum in the allowable pore space in a plastic
solid, so there is a maximum, for as the pore space increases the
substance finally ceases to become a solid. This concentration
of zero friction was found for a certain English china clay to be
19.5 per cent by volume when suspended in water containing
one-tenth of 1 per cent of potassium carbonate. If the particles
of clay were spheres of uniform size, suspensions of this material
would show plasticity in concentrations of solid from 19.5 to
47.64, i.e., over a range of roughly 30 per cent. Colloidal
graphite exhibits zero fluidity when there is only 5.4 per cent
in suspension, hence it has a plasticity range of concentrations
of over 40 per cent. On the other hand, suspensions of many
coarse materials have a plasticity range which is much con¬
stricted, which for practical purposes, is sometimes a serious
disadvantage.

There is abundant evidence that as the diameter of the
particles is decreased, the opportunity for the particles touching

230

FLUIDITY AND PLASTICITY

is increased, which enhances the friction, but this effect reaches a
limit eventually when the particles are so small that their
Brownian movement becomes appreciable and strains in the
material are not permanent.

If, as we have intimated, the friction is subject to the laws
of ordinary external friction, the friction should be closely
dependent upon the adhesion of the particles to each other !

but independent upon the nature of the medium so long as it is
inert. In confirmation of this we note that whereas the china
clay referred to above showed zero friction when the volume ;

concentration was 19.5 per cent, the same clay thoroughly shaken •

down in a measuring flask in the dry state showed a pore space of
18.4 per cent, the pore space in this case being filled with air.

The two values are in very close agreement. Infusorial earth |

exhibited zero fluidity in water when present to the extent of
12.9 per cent by volume, whereas in ethyl alcohol the corre- \

sponding concentration was 12.1 percent. Finally it has been
observed that the temperature and therefore the fluidity of the
medium is without effect upon the friction.

Adhesion between the particles may be influenced in a marked f

degree by the addition of small amounts of substance^ of the j

most diverse character. Generally speaking, substances which
yield hydrogen ions increase the adhesion, i.e., promote floccula¬
tion, while substances which yield hydroxyl ions decrease the
adhesion and promote deflocculation. Colloids also have a
noteworthy effect. In flocculation, structure is produced and
therefore the friction is enhanced. In a given instance, using
50 per cent china clay in water, the friction was lowered from
78 to 59.5 by adding merely one-tenth of 1 per cent of potassium
carbonate, which of course yields hydroxyl ions.

The mobility is dependent upon the fluidity of the medium.

This in turn is influenced by the temperature, hence we may
expect that the mobility of a solid will be dependent upon the
temperature. Thus in a 50 per cent clay suspension the mobility
at 25° was found to be 5.11 and at 40°, 7.88. The ratio between
these mobilities is 1.54 which is very close to the ratio of the
fluidities of water at these two temperatures

= L 6 M = i 49

25 ° 11L7

THE PLASTICITY OF SOLIDS

231

The result of defloceulation is to greatly increase the mobility.
Thus one-tenth of 1 per cent of potassium carbonate raised the
mobility from 1.17 to 5.11 which is an increase of over 330 per
cent, a truly remarkable effect.

Seepage and Slippage

When the shearing force is just a little less than the friction,
there is generally a certain amount of flow which is due to two
different causes. In the first place, under ordinary conditions
of flow the pressure tends to cause the medium to seep through
the material. With this filtration phenomenon there is a local
change in concentration and therefore a change in the char¬
acter of the flow. Seepage is unimportant when the medium is
viscous and the suspended particles are small as in paint.

The second difficulty is due to slippage, which comes from lack
of sufficient adhesion between the material and the shearing
surface. The shearing surface is wet with the liquid medium and
the smooth surface affords little opportunity for the attachment
or interlocking of the particles. The result is that there is a
layer of liquid between the .shearing surface and the main body of
the suspension and flow takes place in this layer according to
the laws of viscous rather than plastic flow. Green (1920)
has observed this phenomenon in paint under the microscope,
the material moving as a solid rod until the shear reaches a
certain value when it begins to move in telescoping layers.
This slippage causes the rate of flow-shear curve to be no longer
linear when the rate of flow is small and the curve passes through
the origin.

Difficulties due to seepage and slippage can be overcome
by using sufficiently high pressures, so that the viscous flow
factor will become negligible. In this case there should be a
linear relation between shear and rate of flow.

Hydraulic Flow and the Plastic State

So far as known to the author, no one has yet used rates
of flow high enough to bring about eddy currents, which are so
troublesome in the case of liquids. But there is the same
necessity for using long narrow tubes for measuring the flow,

232

FLUIDITY AND PLASTICITY

rather than orifices or very short tubes, for the flow of a plastic
material through an orifice gives no idea of the mobility of the
material, just as the flow of a liquid through an orifice is largely
independent of the viscosity of the liquid. Flow through an

Fig. 81.—Hydraulic flow of a plastic material after experiments of Simonis.

orifice does, however, lead to a knowledge of the friction constant
of the plastic substance, as proved by the experiments of Simonis
(1905).

Simonis used 40 g of Zettlitz earth with 100 g of water
to which were added successive portions of a dilute solution of

THE PLASTICITY OF SOLIDS

233

sodium hydroxide containing 1.795 g per liter. The pressure
seems to have been measured as centimeters of water head, and
the volume of flow in milliliters per 600 sec. He measured the
flow of 16 mixtures and pure water, designated by the numbers on
the curves in Fig. 81. The amounts of sodium hydroxide solu¬
tion added are noted in the second column of Table LXI.

The curves are nearly linear except when the volume of flow
is small. The curvature is probably due to seepage. ' The hori¬
zontal distance of the different curves from the curve No. 10
is evidently a relative measure of the friction constant. The
values of the friction constant / as obtained graphically are given
in the table. We have found that it is possible to calculate this
relative friction constant /' by means of the formula

/' = 154 - 14.1c (94)

where c represents the number of milliters of sodium hydroxide
added. It appears, therefore, from a comparison of the values of
/ and f that Simonis’ experiments confirm our conclusion that
the friction is a linear function of the concentration. We note
that the friction constant continually decreases as water is added
until 11 ml have been added after which further additions are
without effect upon the rate of flow. On adding 11 ml, the
material reaches the concentration of zero fluidity or zero friction,
and the curve 10 should pass through the origin. That the curves
10 to 17 all coincide with curve 10 accords with what we should
expect of liquids flowing through an orifice.

The fact that all of the curves are sensibly parallel constitutes
the remarkable difference between flow through a capillary and
flow through an orifice. It does not signify that the plastic
mixtures all have the same mobility any more than it signifies
that all of the liquid mixtures have the same fluidity. It means
simply that the rate of flow through an orifice is independent
of the fluidity or mobility. If in the equations for the flow of a
viscous or a plastic substance through a capillary we make the
length of the capillary zero, we obtain the identical equation

f = kTR\pL (95)

z \ mp

where h is a constant. This is the characteristic and familiar
equation for the flow of liquids through an orifice in which the

234

FLUIDITY AND PLASTICITY

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the plasticity of solids

235

dity or mobility does not appear. The table shows that the
her rates of flow may be calculated quite accurately by means
■he formula

j- = 23.4(P-/') -h 168.7 (96)

Historical

large amount of work has been devoted to the flow of solids.
:hods of measuring plasticity, consistency, and hardness
e aimed to give a single numerical value to a property which
>und to be complex. Plasticity itself has hardly been meas-
1, but rather some property instead which is supposed to be
ted to it, such as the amount of water required to bring a
to a given consistency, the tensile strength on drying, the
irptive capacity for certain dyes such as malachite green, the
unt of shrinkage on drying, etc. It is no doubt true that
e properties are dependent in large measure upon the fineness
~ain which also essentially affects the plasticity, but a knowl-
! of these properties leaves the subject of plastic flow in a
llous state.

any investigators have investigated the so-called “ viscosity
>lids,” assuming that solids obey the ordinary laws of viscous
, and Tammann has identified fluidity with plasticity. Heyd-
er (1897) has measured the viscosity of menthol in both the
d and the solid condition. Weinberg (1913) Dudetzkii
4) and Pochettino (1914) have measured the viscosity of
i or asphalt- Segel (1903) worked with sealing-wax and
is (1893) with marine glue. Barus made the important
rvation that if the rod of material coming out of the capillary
in his measurements was cut off neatly with a knife, the
ders thus formed were in a strained condition. They
taneously change their shape, the advancing end becoming
wed in and the following end being bulged out. This proof
rain is very similar to that observed by Trouton.
esca (1868) did valuable work in forcing metals through
:es and proving that they may be made to flow in a linear
ler much as liquids do. It gives good reason for the pre-
ition that it is practicable to measure the friction and mobil-

236

FLUIDITY AND PLASTICITY

ity of metals and alloys. The work of Andrade on the differ¬
ent types of flow in metals may be referred to.

Werigen, Lewkojeff, and Tammann (1903) measured the rate
of outflow of various metals and arranged the metals in a plastic¬
ity series as follows: potassium, sodium, lead, thallium, tin,
bismuth, cadmium, zinc, antimony. They observed that with
equal pressures and openings, the efflux increases by about 100
per cent for every rise of 10° in temperature. This is shown by
the following table:

Table LXII.—The Relative Efflux of Metallic Lead through a Small
Orifice at Various Temperatures (after Werigen, Lewkojeff
and Tammann)

Temperature,

degrees

Efflux (relative)

Temperature,

degrees

Efflux (relative)

0.5 1

0.8

60.3

42.4

10.4

1.2

70.0

84.3

20.5

2.3

79.3

157.5

30.4

4.7

89.6

211.5

50.7

22.9

When a wire, which is stretched by a weight, is subjected
to torsional vibrations, the amplitudes of the vibrations form a
series in geometrical progression, and therefore the logarithmic
decrement of the amplitude is a constant. A part of the energy
of vibration is given to the surrounding atmosphere and a part
is transmitted to the support, but a portion of the energy is
dissipated within the wire itself. It is generally agreed that this
loss is due to the lack of perfect elasticity in the wire. In other
words, the wire when subjected to shearing stress suffers per¬
manent deformation even though the stress is not equal to the
elastic limit. This deformation causes a shift in the position
of rest, so that as the pendulum passes from its new position of
rest to its old position of rest, it does so at the expense of its own
momentum and there is thus a loss of energy. This flow is
entirely analogous to the flow of various plastic materials such
as clay slip and paint, which we have already considered, when
the shearing stress is less than the friction.

THE PLASTICITY OF SOLIDS

237

Since the flow is of the nature of local slippage rather than true
plastic flow, strains accumulate and they remain after the stress
is removed. The result is the same as that observed by Trouton
in pitch, in that the substance tends to creep slowly back toward
its old position of rest during a period of time which in pitch is
comparatively short but may be observed in metals for hours or
even days. The elastic u after effect” has been the subject of
exhaustive investigation by Weber (1835), Warburg (1869),
Kohlrausch (1863-76), Boltzmann (1876), G. Wiedemann (1879),
Pisati (1879), Streintz (1879), Rakkuk (1888), Wiechert (1889)
and others.

Kupffer (1860) was inclined to attribute this partial flow of
the metal to what he would denominate the fluidity of solids
in analogy to the fluidity of liquids. He says, “II paratt que
les molecules des corps solides possedent la propriety non seule-
ment de s’6carter les unes des autres en produisant une resistance
proportioned aux hearts, mais aussi de glisser les unes sur les
autres, sans produire aucune effort. Cette propriety est poss6d£e
a un haut degr4 par les fluides; je le nommerais volontiers la
fluidity des corps solides; le coefficient \p pourrait 6tre appel£
coefficient de fluidity la malleability des metaux paratt en de-
pendre et peut-etre aussi leur duret4.” According to the present
views we would say that this partial flow was evidence of low
friction or high mobility.

In harmony with this view, it has been found that the logarith¬
mic decrement of the amplitude of vibration is low in hard metals
like steel and high in soft metals like lead. The logarithmic
decrement also increases as the temperature is raised but in this
respect iron and steel are exceptional below 100°C according to
Kupffer, Pisati, and Horton (1905). It will be recalled that
sulfur presents a similar exception in the case of liquids.

According to this view, the elastic limit is reached when the
shearing stress is equal to the friction constant, for at this value
of the stress the material begins to yield. But since the deforma¬
tion takes place with exceeding slowness at this particular stress,
a wire may be loaded considerably beyond the elastic limit before
the flow becomes appreciable. The yield point naturally depends
to some extent upon the rate with which the load is put on.

Just as Trouton found that a given shearing stress produced a

238

FLUIDITY AND PLASTICITY

more rapid rate of flow at first than later when the strains were
developed to their maximum amount, so it is common experience
that metals become harder with working, but that they may be
softened again by annealing. In the process of annealing, the
plasticity is increased by raising the temperature and thus the
strains relieve themselves more quickly than otherwise would be
the case.

An entirely different view from that given above has been
presented by Lord Kelvin and it has had many followers. Noting
that the logarithmic decrement of the vibration is greater in lead
and zinc than it is in steel, he reasoned as follows:

“ Hence, there is in elastic solids a molecular friction which may be
properly called viscosity of solids, because as being an internal resistance
to change of shape depending on the rapidity of the change, it must be
classed with fluid molecular friction, which by general consent is called
viscosity of fluids ”

However, he further stated:

“ But at the same time it ought to be remarked that the word viscosity,
as used hitherto by the best writers, when solids or heterogeneous semi¬
solid-semi-fluid masses are referred to, has not been distinctly applied to
molecular friction, especially not to molecular friction of a highly elastic
solid within its limits of high elasticity, but has rather been employed to
designate a property of slow continual yielding through very great, or
altogether unlimited, extent of change of shape, under the action of
continued stress /*

It has thus come about that the logarithmic decrement has
been taken as a measure of the viscosity of a metal, so that
according to this nomenclature lead has a higher viscosity than
steel and the viscosity of lead increases as the temperature is
raised, which point of view is just the opposite of that used
by Kupffer and to which we are generally familiar in discussing
the viscosity of fluids. Since, however, several investigators have
followed Lord Kelvin in his nomenclature, there is danger of
considerable confusion. If we hereafter refer to the friction and
mobility of solids, the term “viscosity of solids” becomes
unnecessary; and we may confidently expect that the friction
constant of lead will be found to be lower than that of steel and
that it will decrease with the temperature.

THE PLASTICITY OF SOLIDS

239

In conclusion, we note again, cf. page 58, that Reiger (1906)
and Glaser (1907) have carefully investigated the question as to
whether the laws of Poiseuille may be applied to soft solids, using
as their material suspensions of colophony in turpentine. They
concluded that with a tube having a radius of 0.49 cm the vis¬
cosity was independent of the pressure between the limits of
136 and 2,172 g per square centimeter; and in a similar way it
was independent of the length of the tube for lengths varying
between 2.4 and 20.6 cm. They found that with a pressure of
1,965 g per square centimeter, if they varied the radius of the
tube from 1.52 to 0.34 cm, the viscosity remained constant but
for tubes of smaller radii the viscosity rapidly increased until
finally the material seemed to have infinite viscosity. This
inferior limit is unlike anything observed in the flow of liquids, for
the smaller the radius of the tube, the better are the laws of
Poiseuille obeyed, and in large tubes the flow is largely inde¬
pendent of the viscosity of the fluid. It seems probable that the
use of such very large tubes has prevented Reiger and Glaser
from discovering the friction constant just as, in the period before
Poiseuille’s study of flow in capillaries, the use of large tubes
prevented the discovery of the laws of viscous flow. In large
tubes the shearing stress is very large in comparison with the
friction which may possibly explain the fact that the “ viscosity”
was found to be independent of the pressure or length of the tube.

We note that the inferior limit of the radius of the tube is
increased as the percentage of solid in the mixture is increased.
This is what we should expect since this procedure raises the
friction constant. With an 80 per cent of colophony the lower
limit of the radius was found to be 0.100 cm, with an 85 per cent
mixture it was 0.576 cm., and with a 90 per cent mixture it was
1.019 cm. We give below a r€sum6 of the data of Glaser for the
90 per cent suspension of colophony in turpentine, the pressure
throughout being 2,040 g. per square centimeter.

The subject of the plasticity of ice takes on exceptional
interest and importance in connection with the flow of glaciers
and it has been the object of research by many investigators,
among whom we mention Pfaff (1875), McConnel (1886),
Miigge (1895), Hess (1902), Weinberg (1905) and Deeley and
Parr (1914). It is a, noteworthy fact that the precipitous moun-

240

FLUIDITY AND PLASTICITY

Table LXIII. —The Effect of Varying the Radius of the Capillary
on the “ Viscosity of a Solid’' (after Glaser)

Temperature,
degrees C

Radius,

centimeters

Length,

centimeters

Time,

seconds

Volume,

cubic

centimeters

“Viscosity,”

absolute

12.2

1.525

1

25.1 i

16,200

0.331

4.59X109

12.3

1.241

15.9

43,200

11.20

4.54X109

12.3

1.019

15.9

173,000

2.060

4.59X10®

12.3

0.746

16.0

258,000

0.756

5.62X10®

12.3

0.576

15.1

171,000

0.129

7.91X109

12.3

0.364

15.8

350,000

0.0866

25.2X10®

tain peaks maintain their sharp outlines through geological ages
whereas ice flows steadily in spite of apparent hardness. This
indicates that the friction constant of ice is incomparably lower
than that of most silicate rocks. Whereas the glacier scrapes its
bed to some extent (slipping), there is an abundance of evidence
that there is differential flow in the glacier mass, so that although
regelation introduces a new factor into the problem, the flow is
essentially plastic in its nature.

CHAPTER IX

THE VISCOSITY OF GASES

In 1846, the same year in which Poiseuille published his
principal paper on the laws of viscous flow in liquids, Thomas
Graham published the first of a series of papers on the “ trans¬
piration” of gases through tubes of small diameter, which
have great historic interest. Graham sharply differentiated the
flow of gases through an aperture (effusion) and flow through a
long narrow tube (transpiration); he noted that the resistance of
a tube of a given diameter was directly proportional to its
length. Also “ dense cold air is transpired most rapidly,” and his
experiments led him to a relation between the time of transpira¬
tion and the density of the gas. Graham studied the effect of
different pressures and concluded that “for equal volumes of air
of different densities, the times of transpiration are inversely
as the densities,” as exemplified in the following table:

Table LXIV. —The Effect of Pressure upon the Transpiration
of Air (from Graham)

Pressure, atmospheres

Observed time of trans¬
piration for equal vol¬
umes (relative)

Calculated time

1.0

1.0

1.0

1.25

0.795

0.800

1.5

0.673

0.667

1.75

0.589

0.571

2.0

0.524

0.5

When Clausius proposed the kinetic theory in 1857, all of
the properties of gases took on increased interest, and Maxwell
in 1861 published a paper in which he discussed the three kinds
of diffusion: (1) Diffusion of heat or conductivity, (2) Diffusion of
matter, and (3) Diffusion of motion or viscosity. The third or
16 241

242

FLUIDITY AND PLASTICITY

viscosity is the simplest to obtain and it may be used to calculate
the other two, so viscosity played an exceedingly important part
in the years that followed in the establishment of the kinetic
theory on a firm basis. Maxwell defined the unit of viscosity;
and the theory of viscosity and its measurement was rapidly
advanced by Maxwell, 0. E. Meyer and many others. After
many vicissitudes, the conclusion was reached that viscosity
is a fundamental property independent of the particular method
used in its measurement. Thus, for instance, Millikan (1913)
brought together the results for air at 23° by five different
methods and found them to agree to within less than 0.1 per cent
as given in Table LXY.

Table LXY.— The Viscosity of Air at 23°C (from Millikan)

0.00018258 Tomlinson.Damping of the swinging of a

pendulum. (1886)

0.00018229 Hogg.Damping of an oscillating

cylinder.'. (1905)

0.00018232 Grindley and Gibson.Flow through a large tube. (1908)

0.00018257 Gilchrist.Method of constant deviation.. (1913)

0.00018227 Rapp.Transpiration method. (1913)

0.00018240 Average value

Between 12 and 30° the viscosity of air is given by the following
formula with an accuracy of nearly 0.1 per cent according to
Millikan:

t u = 0.00018240 - 0.000000493 (23° - t)

The reader may, however, be referred to the more recent paper of
Vogel (1914).

The Theory of the Viscosity of Gases

The theory of gaseous viscosity has been so often stated
that it need be stated here only in the simplest terms. The
viscosity of a gas is given by the tangential force required per
unit area to maintain a unit velocity in a plane of indefinite
extent at a unit distance from another parallel plane supposed
to be at rest, the space between the planes being occupied by the
gas. It is assumed that if the shearing force is equal to the vis¬
cosity, the velocity v at any point will be numerically equal to its

THE VISCOSITY OF GASES

243

distance $ from the plane which is at rest. If, with Joule, we
think of one-third of the molecules as moving in a direction which
is at right angles to the shear, then these molecules are the only
ones concerned in the transfer of momentum which is the cause
of viscosity in gases. Through a unit area of a plane separating
any two layers of fluid there will pass per second in either direc¬
tion 1 /QNV molecules, N being the number of molecules in a
unit volume and V their average velocity as calculated from the
kinetic energy. The molecule in passing through the given plane
comes from a distance which is equal to the molecular mean free
path L, and therefore from a plane in which the velocity is not v
but v — L in one direction and v + L in the other direction.
The molecule which diffuses into a more slowly moving layer
loses momentum represented by m(v — L), and similarly a
molecule diffusing into the more rapidly moving layer gains
momentum represented by m(v + L), so that the total loss
of momentum is

v = ±NVm[(v - L) - (v + L)]

=-NVmL

or since Nm = p

u = \ P VL (97)

If the speed of the molecules £2 is the mean value as calculated
according to Maxwell’s law of distribution, the formula for the
viscosity becomes, according to 0. E. Meyer (1889),

v = 0.30967 QL (98)

Since the length of the mean free path varies inversely as the
pressure, whereas the density varies directly as the pressure,
it was seen at once that the viscosity of gases should be inde¬
pendent of the pressure. This surprising result was confirmed
by 0. E. Meyer (1866) calculating out the measurements of
Graham, also by the measurements of Maxwell (1866) and 0. E.
Meyer (1865), and it did much to establish the kinetic theory.
With the acceptance of the kinetic theory it can be seen that vis¬
cosity measurements give a very convenient and simple method
for the determination of the mean free path.

244

FLUIDITY AND PLASTICITY

Table LXYI.— Evidence prom Maxwell (1866) that the Viscosity
of Air is Independent of the Pressure

Temperature, degrees Pressure in mercurial Logarithmic decrement

Centigrade

inches

of oscillating disks

12.8

0.50

0.15378

12.8

5.52

0.15379

13.3

29.00

0.15398

Warburg and Babo (1882) were the first to prove that the
viscosity of a gas fluctuates widely with the pressure in the
neighborhood of the critical temperature, using carbon dioxide
as their experimental substance. We have already commented
upon the data for this substance recently obtained by Phillips.

Kundt and Warburg (1875) measured the viscosity of carbon
dioxide by the disk method at pressures as low as 0.1 mm of
mercury and they found that the logarithmic decrement of the
amplitude of vibrations became noticeably smaller when the
pressure became less than about 1.5 mm, the distance between
the disks being from 1 to 3 mm. At atmospheric pressure the
molecular mean free path of carbon dioxide at 0° is 0.0000065
cm, and at 2 mm the mean free path is therefore approximately
0.02 mm. Since a considerable portion of the molecules depart
widely from the mean velocity, we should expect the viscosity to
decrease long before the molecular mean free path became equal
to the distance between the boundary surfaces. Kundt and
Warburg believed that the decrease in viscosity due to the in¬
creasing length of the mean free path should not occur so long as
the thickness of gas was 14 times the mean free path and they
therefore assumed that at high exhaustions there is “slipping”
at the boundary. No one has yet explained why a molecule of a
rarefied gas is any less likely to give up its translational velocity
than a molecule of gas at ordinary pressures. Whether the
decrease in the viscosity is due to the increase in the free path or
not, the hypothesis of slipping seems improbable, and there may
be some other explanation for the results observed. For example,
in the case of the experiments of Kundt and Warburg with hydro¬
gen, the decrease in viscosity at moderately low pressures is,
according to Crookes, “ probably due to the presence of a trace of

TUB VISCOSITY OF GASES

245

foreign gas rnost likely water,” which seems to have been sus¬
pected by jCundt and Warburg themselves.

Crookes (1881) measured the logarithmic decrement of
a mica disk swinging in a glass bulb and supported by a glass
fiber, using pressures as low as could be measured, by means of a
McLeod gage. The gases employed were air, oxygen, nitrogen,
carbon dioxide, carbon monoxide, and hydrogen at 15°C. In
the case of hydrogen the logarithmic decrement was found to be
almost perfectly constant from atmospheric pressure down to
0.25 mm. A-t about this pressure the viscosity of all gases
decreases rather suddenly. With other gases there is a slow
decrease with the pressure even from atmospheric pressure,
except in a sample of air which contained some water vapor,
in which case the logarithmic decrement was at first that of
air, but at about 50 mm it decreased rapidly to that of pure
hydrogen. In an absolute vacuum we must assume that the
fluidity is infinite, hence Maxwell's law must break down at very
low pressures.

According to the data of Phillips, Fig. 54, we should expect
that Maxwell's law would break down at low temperatures
or at very high temperatures. There is a curious dearth of data
with which to test out this point. However, a hydrocarbon
vapor, “kerosoline, ” was measured by Crookes and the viscosity
was found to decrease rapidly from the highest pressure obtained
of 82.5 mm down to 8 mm. Lothar Meyer found in experiment¬
ing with benzene that the viscosity of the saturated vapor was
smaller the higher the back pressure at the exit end of the capil¬
lary tube. At high temperatures we are led to expect that just
the opposite conduct will be observed, viz., that the viscosity will
decrease as the pressure is increased , see Fig. 54, but there is so
far as known to the author no data to support this conclusion.

Viscosity of Gases and Temperature

From the formula

v .= 1/3 P VL

it is evident that the effect of an increase in temperature will be
to increase the mean velocity, but it is not known what effect
the temperature may have upon the mean free path, although it