(Frontispiece'). FLUIDITY AID PLASTICITY BY EUGENE C. BINGHAM, PH.D PROFE8SOB OF CHEMI8TBY AT LAFAYETTE COLLEGE, EASTON, FIEST 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 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. EUGENE C. BINGHAM. EASTON, PA. Feb. 11, 1922. A {£' LIBRARY M PAGE PREFACE........................... 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 IV. Is THE VISCOSITY 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 DlFFUSIONAL 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 XL 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 PAET I ^ISCOMETRY 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 (1860), 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- I bach was the first to attempt to make this correction but Neumann (1858) and Jacobson (1860) were the first to put the I correction into satisfactory form. Thus both the method of ; measurement and the formula used in calculation of absolute vis- I 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 i rediscovered, and there is an evident confusion in the minds of I many as to the conditions necessary for exact measurement. | 1 ! The so-called "transpiration77 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 | | j developed by Moritz (1847), Stokes (1849), 0. E. Meyer (1860), : Helmholtz (1860) and Maxwell (I860). The well-known method ; of the falling sphere was worked out by Stokes (1849). j During the period to which we have just referred, Graham v (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, 0. 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. I To chemists, on the other hand, impressed by the relations i. . between physical properties and chemical composition, so forcibly^ ; brought to their attention by the work of Kopp, the viscosity of I 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. METHODS OF MEASUREMENT 3 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 arid 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 AA'D PLASTICITY of The study of elasticity has been the dominant idea in of researches. Very little work has been done upon the viscosity of matter in the states of aggregation taken as a whole. If it has. been shown that our knowledge of viscosity consists of unrelated groups, it is equally apparent that such a is artificial and that nothing could be more important for our understanding of viscosity, than to bring these into an inter-related whole. We shall therefore not an attempt to follow the chronological method, where it with the consideration of the subject as a whole. the groups of researches to which we have alluded out rather clearly. The methods of measurement in use will be first considered, after which we shall study the viscosities 01 solutions, solids, and gases respectively. Deformation, Plastic, Viscous, and Turbulent Flow.— If a perfectly elastic solid be subjected to a shearing stress a strain is developed which entirely disappears when the is removed. The total work done is zero, the process is and viscosity can play no part in the movement. is not a of flow but of elastic deformation. If a body is imperfectly elastic as regards its form be subjected to it will be found that a part, at least, of the will remain long after the stress is removed. In this has been done in overcoming some kind of internal We may distinguish the kinds of flow under three It is characteristic of viscous or linear flow that the of deformation is directly proportional to the deforming the ratio of the latter to the former gives a measure erf It has been questioned at times whether this ratio is but it appears that only one qualification is In very viscous substances time may be necessary for the to' reach a steady state, aside from any period of because with substances like pitch the viscous develops slowly, so that the above ratio gradually when the load is first put on, but even in this case the reaches a value which is independent of the amount of tie As, however, the deforming force is steadily in- 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 rj Fs m >?=T (1) The dimensions of viscosity are [MLrlT~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 we have = -• 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 ttie 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 rilled 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. Venant (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). (/) A body moving freely under the action of gravity, e.g., falling sphere of platinum, mercury, or water, a f ailing body of other shape than a sphere, a rising bubble of air. Stokes (1845), Pisati (1877), Schottner (1879), de Keen (1889), 0. Jones (1894), Duff (1896), J. Thomson (1898), Tarnmann (1898), Schaum (1899), Allen (1900), Ladenburg (1906), Valenta (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), Prfbram and Handl (1878), Slotte (1881), Stephan (1882), Foussereau (1885), Couette (1890), Bruckner (1891), Thorpe and Rodger (1893), Hosking (1900), Bingham and White (1912). (6) Efflux through a vertical tube of small diameter. Stephan (1882), METHODS OF MEASUREMENT 7 igler (1885), Arrhenius (1887), Ostwald (1893), Gartenmeister (1890) sydweiller (1895), Friedlander (1901), Mclntosh and Steele (1906)' wikine (1910). (c) Efflux through a bent capillary. Grfineisen. (1905). (£) Bending of beams and torsion of rods of viscous substance. Trouton 906), Trauton and Andrews (1904). (e) Itate 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). 00 Decay of waves upon a free surface. Stokes (1851), Watson (1902). (c) Decay of vibrations in a viscous substance. Guye and Mintz (1908). <<*) Rate of crystallization. Wilson (1900). ITomenclature.—A. great variety of names have been given to struments devised for measuring viscosity, among which we ay cite viscorneter, viscosirneter, glischrorneter, microrheom- 3r, stalagnometer, and viscostagnometer. All but the first "0 are but little used and their introduction seems an unneces- :y complication. Viscorneter and viseosimeter are about ually used in England and America,, but such a standard work Watt's Dictionary uses only viscometer. Viseosimeter in its 5rin.au equivalent Viskosimeter is entirely satisfactory, but English viseosimeter is apt to be mispronounced viscos- eter. Furthermore viseosimeter does not so easily relate elf in one's mind to viscornetry which is the only word recog- :ed in the standard dictionaries to denote the measurement viscosity. Professor Brander Matthews kindly informs rne it the formation of the word viscometer is quite as free from jection as that of viseosimeter, and viscometer is in harmony bh modern spelling reform. Hence viseometer should be opted as the name for all instruments used for measuring vis- sity. The different forms are distinguished by the names of jir inventors. CHAPTER II THE LAW OF POISEUILLE Experimental Verification.—Prior to 1842 it had not been established as a fact that the movement of the blood through the capillaries has its origin solely in the contractions of the heart. There were theories current that the capillaries themselves caused the flow of blood or that the corpuscles were instrumental in producing it. Poiseuille reasoned that if the lengths and diameters of the capillaries are different in the various warm- blooded animals and if the pressure and temperature of the blood vary in different parts of the body, light might be thrown upon the problem by investigating the effects upon the rate of flow in capillary tubes of changes in (1) pressure, (2) length of capil- lary, (3) diameter of capillary, and (4) temperature. The results of Poiseuille's experiments were of a more funda- mental character than he anticipated for they proved that the conditions of capillary flow are much simpler than those in the wide tubes which had previously been employed, and by his experiments the laws of viscous flow became established. Not only did Poiseuille perform experiments which resulted in the law which bears his name, and therefore have affected all subse- quent work, but he measured the efflux times of water by the absolute method taking elaborate precautions to insure accuracy, and using capillaries of various lengths and diameters which are equivalent to separate instruments—in all over forty in number. Thus one is justified in studying his work in considerable detail, not only for its historic interest, but on account of its bearing upon questions which will arise later. In the Appendix his measurements are reproduced in full. In Fig. 1 is shown the most essential part of the apparatus of Poiseuille. It consists of a horizontal glass capillary d joined to the bulb, whose volume between the marks c and e was accu- rately determined. The bulb is connected above with a tube which leads to (1) a 60-1 reservoir for keeping the pressure of the air within the apparatus constant, (2) a manometer, filled with THE LAW OP POISEUILLE 9 FIG. 1.—Poiseuille's viscomo- ter. 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. 10 FLUIDITY AND PLASTICITY TABLE I.—CAPILLARY A' Pressure in Observed time millimeters of for transpiration Calculated Per cent mercury at of 13. 34085 cc time difference 10°C of water 97.764 10,361.0 147.832 6,851.0 6,851.91 0.01 193.632 5,233.0 5,231.22 0.03 387.675 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.™ Pressure in Observed time millimeters of for transpiration Calculated Per cent mercury at of 13. 34085 cc time difference 10°C of water 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 POISEVILLE 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 K is 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 0.1135 0.1117 2,052.98 75.050 0.1140 0.1120 1,526.20 1,539.0 0.85 49.375 0.1142 0.1122 998.74 1,004.0 0.53 23.575 . 0.1145 0.1123 475.18 476.8 0.34 9.000 3.900 0.11441 0.1124 j 0.11451 0.1125J 199.39 110.20 181.4 86.4 -9.05 -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"pjl where I 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 Volume calculated Per cent, difference M 0.0013949 0.0014648 0.001465 -f<3.02 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 0.0029384: 0.00139494 = 0.028826 :x, or x = 0.001465. The agreement is very satisfactory, hence the THE LAW OF POI SEVILLE 13 formula becomes V = K pd* 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 I and d in centimeters. He experimented with alc,ohol 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, r>d* Q = 1,836,724,000(1 + 0.0336793Z7 + 0.0002209936r2) — 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 I = 10.05 cm d = 0.0141125 cm p = 776 mm of mercury. Time of flow 1,000 sec. WEIGHT OF EFFLUX WEIGHT OF EFFLUX CALCULATED BY TEMPERATURE OBSERVED FORMULA 0.6 5.0 10.0 15.0 20.0 25.0 30.1 35.1 40.1 45.0 5.74376 6.60962 7.64649 8.74996 9.91530 11.14584 12.45631 13.80695 15.21866 16.67396 5.73955 6.60381 7.64435 8.74705 9.91191 11.13892 12.45423 13.80710 15.22184 16.66860 Since the values calculated are weights and not volumes, the values of Q are not proportional to the fluidity. This formula pd* remains empirical, but the expression V = K —•r- 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- I 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- jl' ment of the flow in long narrow tubes by Stokes (1849). I Imagine a horizontal capillary whose bore is a true cylinder to I connect two reservoirs L (left) and R (right) there being a differ- , j; ence of pressure between the two reservoirs, at the level of the | capillary, amounting to p grams per square centimeter. If the pres- il sure in L is the greater the direction of flow through the capillary ! I will be from left to right. The total effective pressure p is used ,' 7 up in doing various forms of work, several of which can be differ- and 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 •AMPLIFICATION OF THE LAW OF POI SEVILLE 19 method used by Couette—and as we have seen, also by Neumann, I^inkener, 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 ctnd 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 817 VI , mpF2 P = ^K5 + ir^gRH ' ^ we observe that since for a given tube and liquid only p and t 15 9^ ^P > 14 ^ k_ )-JL X3 Zc j r^ ^o 2 q n — '• .— — • 0-— ,-. — & Jrf c^5 x 8 0 ^ *— •Q*" *** r--- ^ S 5 fj^* S /" J/1^ r i 01 23456789 10 II 4*10* "Pro. 2.—Finding the value of »n for the kinetic energy correction. 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 \/t as abscis- arid of pt as ordinates Knibbs obtained the curves shown in ;. 2, using the data for Poiseuille's tubes Av, Avn, Bv, and Cv. 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 m = b- 20 FLUIDITY AND PLASTICITY Using a combination of numerical and graphical methods following values were obtained. TABLE VI. — VALUES OF m DEDUCED BY KNIBBS FROM EXPERIMENTS Tube Length in centimeters Mean radius in centimeters Values of m Am .............. 2,55 0.00708 1.04 A™ ..... 1 57 0.00708 1.02 Av ........ 0.95 0.00708 1.15 AVI ......... 0.68 0.00708 1.08 Avn ............. 0.10 0.00708 1.12 B ...... 10 00 0.00567 1.23 BIV ........ 0 90 0.00567 1.14 w ........ 0 39 0.00567 1.03 Cv 0 60 0 00427 1 87* F1 20 00 0 03267 1 08 F11 9 97 0 03267 1.33 F111 5 04 0.03267 1.16 JiIY 2 60 0.03267 0.82* Fv - 1 07 0 03267 0 82* The mean is 1.14 or rejecting the values for Cy, F1V, and 1.13. Certain of the tubes, viz., A, A1, A11, B1, B11, B111, C1, C11, Cm, CT, D, D1, Dn, Dm, D17, E, E1, E11, and F no satisfactory indication of the value of m. Knibbs dedu.c*o the value of m from 34 series of experiments made by Jacobso and obtained an average value of 1.14. This seems like a remould able justification of the deduction of Boussinesq. But it shou! 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 ovoi emphasized since in some instances the amounts of the correctior] are much smaller than the discrepancies among the observe tions themselves. Knibbs thinks that the values do vary than can possibly be accounted for by the experimental error 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 viscous 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 correc Accoi- small* the re diame differc There To -be Poise \ the si: times the s nearly m — AMPLIFICATION OF THE LAW OF POI SEVILLE 23 the actual length of the capillary, which he represents by A. The corrected viscosity r\Q should therefore be calculated by the formula According to Couette the corrected viscosity is always a little smaller than that calculated by means of Eq. (8) and we obtain the relation I A =1 Since A may be presumed to be the same for tubes of equal diameter but of unequal lengths I and Z', one should obtain different viscosities t\ and rjf by applying Eq. (8) to the same fluid. There would thus be the relation A = (11) To test out his theory, Couette used experimental results of Poiseuille with tubes A1V and Av which gave poor agreement with the simple law, Eq. (5) cf. Table II, VII and VIII. The efflux times are given in column 1, the viscosities yp calculated from the simple Poiseuille formula (5), in column 2, the more nearly correct viscosities rj and 77', calculated from Eq. (8) taking m = 1.00, in column 3. TABLE VII. — VISCOSITY OF WATEH CALCULATED FROM POISEUILLF/S EXPERIMENTS WITH TUBE A™ For dimensions cf. Appendix D, Table I, p. 331 Time r,P Eq. (5) 77 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 Av For dimensions cf. Appendix D, Table I, p. 331 Time TjpEq. (5) -n' Eq. (8), m = 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 77 vary but little around the mean 0.01329, while the values of rip show a regular progression, thus demonstrating the importance of the kinetic energy correction. The first three values of tf 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 i\ and V, which are notably different in value, the corrected viscosity T\C as well as the value of A may be obtained by the use of Eq. (11). We get yc = 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 ( «p = 2.868) • 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 TI and T2 are the two capillaries connecting three reservoirs M, N, and P. The pressure in each reservoir is measured on the differential manometer H. 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 or 7 7 + A) 8Vrjc(h -• A) irgR*t pi — * AMPLIFICATION OF THE LAW OF POT SEVILLE 25 13 y thus eliminating the correction for the kinetic energy and the e?:nds of the tubes, Couette obtained, for the corrected viscosity (-37 J of water at 10°, 0.01309 which is in excellent agreement with t>lie value calculated above from Poiseuille's experiments. If o:n the other hand, the viscosity fo) is calculated by means of Eg.- (8) with m = 1.00 for one of Couette's tubes, the apparent •vriscosity 0?) is 0.01389. From the values of t7 and t\e the value o:f A may be calculated as above. It is 0.32 em and the dia.zneter FIG. 4.—Capillary-tube viscometer. Couette. of the tube Is 0.090 cm so that the fictitious length to be a*dded is &, 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 eqiz^l to LOG instead of the more probable 1.12. Turtliermore the range of data used in establishing his conclusion is rather limited. He*n.ce, Knibbs has made an extended study of the same sul>ject. If for A we substitute nR, Eq. (9) may be written nR\ 26 FLUIDITY AND PLASTICITY but since from Eq. (9a) we have that and therefore 87T nR\ •fT This is the equation of a straight line. If values of ~nrrT 5.—Finding the value of n for the "end correction." 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 I = co 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 n required. Figure 5, taken from Knibbs' work, illustrates the method as applied to the tubes used by Poi- seuille B to Bv and F to FIV. The values of n are found to be —5.2 and + 11.2 respectively. According to Knibbs "these results challenge the propriety of Couette's statement that A may be always regarded as positive and 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 (qf. 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 y xio» t R* X 10l° f\ D ......... 22 0.242840 0.013074* M .............. 37 0.002367 0.013090* C .............. 42 3.250400 0.013028* D1 ............... 44 0.233770 0.013020* B ................ 56 10.235000 0.013202 C1. 57 3 265900 0 013071* E.. 64 0.047160 0.013242* A ........... 70 24.941000 0.013145 Mean 0 013109 B1 ............... 75 10.276000 0.013134* F ................ 85 11,207.000000 0.013147 C11, 86 3.298000 0.013151* D11. 87 0.227870 0.013078* A1 ........ 93 25.059000 0.013109* BII ............... 115 10.303000 0.013070* A11 ............... 139 25.183000 0.013119* F1 163 11,187.000000 0.013065 Mean 0 013109 E1 ............ 174 0.048400 0.013588* QIH 175 3.339400 0.013092* Dm 219 0.224400 0.013045* B111 240 10.331000 0.013002* A111 ...... 277 25.231000 0.012946 Fir ............... 326 11,233.000000 0.013249 QIV 421 3.339400 0.012498* A"" . . 450 25.231000 0.013343 Mean .... 0.013096 M1 ....... ........ 558 0.002367 0.013181* giv 630 10.357000 0.012742 F111 646 11,290.000000 0.013967 DIV ........ 649 0.223310 0.012652* E1 ........... 706 0.048400 0.013222* cv ........... . 709 3.339400 0.012015 Av 742 25.231000 0.013515 AVI .......... 1,046 25.231000 0.013607 Mean. ......... 0.013113 piv 1,254 11,316.000000 0.014891 Bv 1,455 10.368000 0.012193 Fv ..... 3,034 11,316.000000 0.014851 Avn ........... 7,088 25.231000 0.016980 28 FLUIDITY AND PLASTICITY Bingham and White (1912) have confirmed the conclusion of Knibbs by a study of interrupted flow. A capillary I = 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 + 6A) 6A) = C ML Z + 6A 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 p\ and when broken t% and p% respectively, we obtain the relation I +6A hence, I -f A K TABLE X. — EXPEBIMENTS TO DETEEMINE THE "FICTITIOUS LENGTH" OF A CAPILLARY XTKDEE CONDITIONS OF INTEERUPTED FLOW Number of Pressure in capillaries 6 Time grams Cpt K A per cm2 1 179.7 87.46 0.0836 2 * 180.2 87.77 0.0837 1.00 1+0.009 a 182.4 87.32 0.0835 0.99 9-0.006 4 183.1 87.75 0.0836 1.00 0 0.000 6 185.0 88.25 0.0838 1.00 2+0.003 AMPLIFICATION OF THE LAW OF POISE UILLE Since the values of K are unity within the experimental error to the length is zero. In no single instance does of A amount to even one-half the diameter of the tube. however the value of m had been taken as unity, A would appeared to have positive value. Had A been found to have a definite value, it would boen 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) fo-ut 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 irnpoir- tan.ce in determining the development of turbulent flow, under coir- 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 oscilla-fc- ing disk of white metal immersed in water, and he noted ttta^t coating the disk with tallow or sprinkling it over with sandstone h_ad no effect upon the vibrations. This seemed to prove that "bite flizid in contact with the disk moved with it, and that the proper-ty b-oing measured was characteristic of the fluid and not of ttte jxa/ture of the surface. These observations were confirmed O. Meyer in 1861. After the Law of Poiseuille had been experimentally theoretically established, it was still unsatisfactory that ttie results of measurements of viscosity by the efflux method did n_o"fc agree with those by other methods. It was natural tx> sxippose that the discrepancy might be explained by the friction between the fluid and the solid boundary which had assumed by Javier (1823), cf. also Margnles (1881) and Hada- mard (1903). Helmholtz in his derivation of the Law of ]Pol- seiiille had taken into account the effect of slipping ajad obtained tlie formula, -which in our notation is V = + 4>J&« (13) wh.ere X depends upon the nature of the fluid as well as upon of the bounding surface. In treatises on hydrodynamics this Is written 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 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 a'Vc'd or if the films B and D are more viscous than the mercury the section may be better represented by a"6"c"d. But from Eq. (1) dv oc 0 35 40 f 45 50 55 60 65 10 Temperature Degrees Reaumur -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 ("RheMand 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: AMPLIFICATION OF THE LAW OF POISEVILLE 37 Tube Radius, inches Length, inches Narrow , 0.053844 18.092 Mean. . . 0.077394 41.650 Wide ............... 11.391400 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 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.77 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 regime, 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 pRI

may be remarked here that hydraulicians have usually p .1 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 BT / 21*20 PIG. 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 s while g and g1 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 regime 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 regime, and the measurements were very difficult to ob- tain with precision. He proved that pRItp — 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 RADII R Temperature V • v E 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 tarns of the outer cylinder per minute is taken as proportional to the mean velocity, a being a constant. TABLE X"VII.—LAW OF DENSITIES Substance *7 P al Ckppl Water .......... 0 01096 1 0000 56 5 100 Mi ........ 0 00018 0 0012 &on ^ 3Oft 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 I = 6 ft. R = 0.38 in. measuring the flow of water over a range of temperatures from AMPLIFICATION OF THE LAW OF POISEUILLE 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 (V) 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. Por 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%. 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 Z7, 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) dp irgR4* P " But -7^— n is constant and therefore -57- = -^-j — ~ loij Jr dl I 50 FLUIDITY AMD PLASTICITY and we obtain z (l6a) CM?7 p or „ - 17 ~ where Fi and ¥2 are the volumes corresponding to pressure Pi and P2. Since p = Pi — Pz we observe that the ratio between the P1 _|_ p2 values of the viscosity calculated by Eqs. (17) and (5) is —^-p— |i| [ where P may have any value between PI and ?2 depending upon if I the value of V which is employed. If V be taken as \W 2Pt T, _ 2P2 „ (17) becomes identical with Eq. (5) and becomes unnecessary. The derivation of the law for gases was made by 0. E. Meyer (1866) aad 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 PVn as constant. Equation (16) becomes on integration l + i H-i ! '-Pi " Stn When n = <» this becomes identical with Eq* (4), for incom- jj! 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 = CP/CV = 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 l + PF! (18) . Pi" +Pi» 'P, + . . .P,= In the extreme case where n = 1, if p is not greater than P2/10 AMPLIFICATION OF THE LAW OF POI8EUILLE 51 V will not differ from — —;- — 2 by much over 0.2 per cent. Zt This means that working at atmospheric pressure, with a hydro- static pressure of over 100 cm of water, one may take the volume Vi + 72 of flow as - o — 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 thn kinetic energy as the fluid leaves the tube. This is irp^R2!^. The total energy supplied in producing the flow is 7rK2I2(Pi —Pz)g and the difference between the two is the energy converted into heat 7rR2h[(Pi — P%)g — qj[£\. The loss of head in dynes per cm2 in imparting kinetic energy to the fluid is therefore With this correction, but neglecting the slipping, we obtain = 77 _ Z 2P2 Substituting V for F2 and remembering that pzVs 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 P2. 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 filled with finely porous material like Fuller's earth, Gilpin1 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 " Fourth K«§girne." "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 he 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, I = 10.5 crn he found TABLE XVIII.-—THE VISCOSITY OF AN 85 PER CENT COLOPHONIUM—15 PER CEKT TURPENTINE MIXTURE AT 11.3° AND UNDER A CONSTANT PRESSURE OF 2,040 CM WATER IN TUBES OP VARIOUS DIAMETERS R I t y 77 X 107 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 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 0.158 15.1 173,500 0.023 19.90 0.50 0.117 15.4 3 weeks 0.000 00 0.00 Am. Chem. J., 40, 495 (1908); 44, 251 (1910); 50, 59 (1913). AMPLIFICATION OF THE LAW OF POISEUILLE 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 f 0.5 1.0 Fadius of Tube 1.5 Fia. 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. l! II by working in the same way with. 80 arid 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 smch rapidly increasing values, the viscosities are inconvenient to plot, we have changed his viscosities to fluidities. All of Ms 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 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, we have calculated the mean velocity by means of Eq. (6). It is of the same ordqr 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 jjiass, but rather of the nature of a thin rnud. 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 or* solid. If this view is 13 __ _ ____ 3 4- 5 ^ T S 9 10 1! Percerrfoge Volume of Earth 19. - The fluidity of suspension of infusorial earth In water. . used the zero of fluidity would be changed because there would be enough pressure to more than overcome the plastic resistance in the 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, I = 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 (RONCEBAY) P centimeters, water Time of flow of 10 ml in air at 17° Time of flow immersed 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 (cf. 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 :Q_7u . Other sources AMPLIFICATION OF THE LAW OF POISEUILLE 57 ^ tfj, of error such as surface tension, slipping at the boundary, f necking in of the lines of flow at the entrance of the capillary, | eddy currents inside of the capillary, resistance to flow outside f of the capillary, peculiar shapes to the ends of the capillary f| affecting the magnitude of the kinetic energy correction have f 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 p same Law of Poiseuille but the volume of flow is approximately 1 the mean of the volume at the entrance and at the exit of the f capillary Fl + V'2. I Plastic solids in their flow do not obey the Law of Poiseuille f and their study is deferred until Chapter VIII. Many attempts 14 have been made to measure the viscosity of soft solids. The | fluidity of such a substance is not a constant quantity but f 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 |f distinguished from a true fluid, just as Reynolds' criterion ff enables one to distinguish between viscous and turbulent flow. 4J 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 t if IS TUE VISCOSITY A DMVINITE 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 Am AT 0°C. Method

boiling-point to be made while the liquid is being fractionated To avoid contamination by dust and moisture in filling the vi* eometer, Thorpe and Rodger used a special apparatus, Fig. ii? The liquid was placed in the bottle H and forced over into tb* right limb of the viseometer M by means of the pressure of *-* mercury head A. The viseometer was held in a frame an* I supported on the vertical rod by means of the setscrew X FIG. 26.—Apparatus of Thorpe and Rodger for obtain- ing dust-free liquid. TEE VISCOMETEE 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 Zr (or 2) was turned to air and the cock Z (or Z') to pressure. As the liquid rose in the left limb, it finally overran in to the trap T/ Fig. 22. At the instant that the meniscus in the right limb reached the point k2j the cock Z was turned to air. Thus the working volume was adjusted. A measurement of the fluidity is made by turning the cock Zf 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 V. As the meniscus passes the point ra' the time recorded is begun. Keeping the temperature constant the time is taken as the meniscus passes the point W2. The preS- FIG. 27.—Pilling device of Thorpe . ,, , i <• j and Rodger. sure is then read as before, and before the meniscus reaches the point &' 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 Fig. 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 bath, 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 ti be the time of flow from left to right and U the corresponding time from right to left. Let p0 be the pressure as corrected, except for the average resultant head of liquid in the viscometer. Suppose this latter correction to amount to x em of the liquid as the liquid flows from left to right. In this case the total pressure becomes equal to p0 4- P% and when the liquid flows from right to left, it becomes equal to pQ — px. Since Eq. (8) when used for a given viscometer may be written in the form , = Opt - C'p/t (22) THE VISCOMETER 75 •where C and C" are constants, which caa be calculated, we obtain r, •+ C'pfi, cu wheace, 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 viscorneter, many investigators have followed the example of Pribram and Handl in disregarding these dimensions, and calibrating the instrument with same standard liquid. The most important instrument of this class, is that of Ostwuld, Fig. 28. It consists essentially of a U-tube with a capillary in the middle of one lirnb 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 T? that of liquid to be measured, we have from Eq. (22) FIG. 28.— The Ostwald viscometer. '/O Vy//OHI ^ P(J/ m and if -r\ is very nearly equal to T?O or if t and fa are very large, this may be written 1= 170 (23) 76 FLUIDITY AND PLASTICITY The pressure in this instrument must proportional to the densities so that be PIG. 29.—Viscom- eter suitable for the relative me a sure- merit of not too Tiscous liquids. = which is the formula suggested by Ostwald. The formula is true for dilute solutions when water is taken as the standard, for fj is then nearly equal to T?O. 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 calculation of this correction by assuming pQ = p. After obtaining the value of the hydrostatic head x in this way, the true value of C may be calculated from an observation upon the time of flow of any liquid whose viscosity is accurately known. In the use of any relative instru- ment, it is important that two stand- ards be employed so as to obtain a check upon the method. For this purpose a single liquid may be used at widely different temperatures or two or more liquids may be used of widely different viscosities. While this test is very simple and its importance is obvious, it does not appear to have been frequently employed. Viscosity Measurements of Liquids above the Boiling-point.— If the viscosity of liquids is to be 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, SecfionV-Yl FIG. 30.—Plastometer. For use or with plastic 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 tlie efflux into air, because the effect of surface tension would "be PIG. 31.—-Viscometer for gases after Schultze. negligible at these high pressures. A viscometer designed f or" very viscous substances is shown in Pig. 30. The use of this form of apparatus is. described in detail in connection with plastic flow (qf. Appendix Br p. 320). Tfee Viscosity Measurement of Gases.—A very satisfactory apparatus for the measurement of the viscosity of gases by the THE VISCOMETER 79 f efflux method has been worked out through the labors of Graham 5 (1846-1861), O. E. Meyer (1866-1873), Puluj (1876), E. Wiede- I mann (1876), Breitenbach (1899), and Schultze (1901). We may 1 describe briefly the form used by Schultze as illustrating tihe | modifications which are necessary in the apparatus used for \ liquids, In Fig. 31 the glass capillary, Z = 52.54 cm, R = \ 0.007572 cm, is contained in the upper chamber of the hath /, ;• which is maintained at constant temperature by water, water | vapor, or aniline vapor. A condenser is shown at b and SS is a i shield to protect the rest of the apparatus from the radiation. On either side of the bath the apparatus is exactly similar, so that I only the right side is shown in. the figure. The gas is contained I in the bulbs P and Q (and J?r and Q' on tie left side) surrounded 'I by a separate bath. The lower bulbs are each connected with two stop cocks B and C (or B' and C'); from B (or 5') a rubber 1 tube leads to the mercury reservoir G (or (?'), aad from C (or C') ? there is a glass tuhe 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 eock A (and A'} \ \ and a connection with a manometer K (and #')- By means of stop cocks at E and E1 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 O /w r> JL o = pi$i " hence TT = Rl v\ S Fig. 32.—Diagram to illus- trate additive viscosities. But since Si/o is the fraction by volume of the substance A present in the mixture, which we may designate a, and similarly s2/S = b, etc., JT = am + fci,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 EL Fluidities Additive—Fluid Mixtures.—If the larnelke are arranged parallel to the direction of shear, as shown in Fig. 33, we have a constant shearing stress, so that (24a) v 2, are the partial velocities as indicated in the where 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 viscorneter of the Coulomb VISCOSITY AND FLUIDITY or disk type, we actually measure the velocity v, BS in the figure, and we very naturally assume that P = ffv R 2. It is more usual, however, to calculate the viscosity from the volume of flow, as in the Poiseuille type of instrument. let vf, £S' in the figure, be the effective velocity which the surface JBS would have, were the series of lamellae replaced by S' S 5" FIG. 33.—Diagram to illustrate additive fluidities. 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, as for a homo- geneous fluid, be H', then •flV ZH'U P = E (24b) It is to be noted that had the less viscous substance been in contact with the surface AE, the effective velocity of flow would have been represented by the distance BS'7. We shall take the former of these for our definition of the viscosity of a mixture, 86 FLUIDITY AND PLASTICITY I 1 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% + . . . we obtain from Eqs. (24a) and (24b) that or since — ~, b = rt, etc. the fluidity of the mixture is 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 U = If there were n pairs of alternate lamellae of the two substances A and£ U = -JjfiviTi + n(n -f R n(n - (26). Since n = , Tl + 7*2 ui and v%, we get U and if & = Tr/ we obtain from Eq. (24b) JO. 6 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 A CD, 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. N"o "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 \vhen 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 but 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 be. 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 or b'. For all intermediate values of a and b we desire to learn VISCOSITY AND FLUIDITY 89 whether

12 — 2 = 91 +0?2 — 2 — e 300 zoo 100 joio .00$ Temperature Centigrade. 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 p-ure 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 ffarst approximation of Meyer and Rosencranz (1877) 1 Landolt and Bernstein, Tabellen, 3d. ed., p. 41. 94 FLUIDITY AND PLASTICITY when put in the form ? = 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 observedT' 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 ^oro. 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 Triibungsgrad und Tem- peraturkoefficient der inneren Reibung zeigen eine starke Zu- ^riedlandler, 439, VISCOSITY AND FLUIDITY 95 nahme im kiitischen Gebiete und stehen rnit 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 Lehrnann. 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'5 were charged an increase in viscosity might result. But by experiment Fried- lander found that an electrical field was without noticeable effect upon aa opalescent liquid. Friedlander's values are expressed in relative units. Scarpa (19O3) and (1904) has measured the viscosity of solutions of phenol and water, expressing his results in absolute units. For a J jj given temperature, he plotted the viscosities against the varying concentration?, 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 U 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 Donnan'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 "crystallin" or "anistropie" 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 "crystallin" in the pure condition. The mixture therefore 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 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 considerer la fluidity comme une fonction lin^aire de la temperature, sauf pour les corps trds associes comme Peau, ...... "Pour les melanges, loin de la temperature critique la variation de la fluidity est encore Iin6aire. Mais plus on approche de la region critique, moins les formules lin£aires sont exactes. Elles ne peuvent m6me plus pr£tendre & un semblant d'exactitude, ainsi que le montrent bien les lignes de fluidity des melanges it concentrations voisines de la concentration critique: elles sont tout & fait courbes et concaves vers Paxe des temperatures. D'aiUeurs, d6j pour des concentrations 61oign£es 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 pr£s de la concentration critique. "Si Ton examine Failure des isothermes de fluidite, on voit que pour les melanges de corps normaux la loi d'additivite: est assez bien satisfaite a des temperatures superieures & la temperature critique de dissolution. J'ai porte en abscisses les d e< C( is 11 f 1 gl^s- B ? -• < 3 pt 3 |Kfl sonccntrations en poids, niais en prenant les concentr volume la loi d'additivite n'est pas mieux v&ifi6e. Ce 5* « 0 P lt>U \ i I __-. ^ cv ciojL EX^ ij^_ — • •^^ ' ^ ^Xs v , — - 28.45%^ "^ ^ ^N ^ **~~ ^— — — • "^ 85 \G3 ^ ^.^ ^ 5.0 315. '.-3- §c*- iS 4- .' *""^ i ____ . — — 4? '45 * 5? % r\f ^5> ^^ NS 3 .-^ i^. 0° ^ ^*- ^— N ^ o * u — £2 ©^ -o C*- C+- U- *— ' £ sri ^ -^-— . — - "58.75% >8^ ^ :i^ ^^ I , x' ^^ 7? % •71 5l X!l -A' ^ a^ »x. — 'r- —• — ^ ^ ^ ^> »rt £> »-* 3V 5 § » p, g 2. Ill 0? c^ w ~ FIG. H-. ^ ° tfc 3 Hj 1 \ ^*§ ^ ^ __ i utiLOL =: — ' -• ^ 1 1 .0° e5° 30° 2>5Q 0% e5°/o 50°/o 15°/o !OC Temperature Weight Concentration Aniline 37t — Fluidities of mixtures of cyclohexane and aniline at the critical-solution temperature. (Aj Drapier.) B." % p- a- sri 100 FLUIDITY AND PLASTICITY Commenting on the theory of v. Smoluchowski by way of explanation he remarks, "II est probable que de pareilles h<§t£ro- g6n&t6s produirairent une augmentation de la viscosit6 et pourraient done expliquer la courbure, toujours de merne sens, des courbes d'6gale concentration et par consequent les hearts & 101 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 cause1 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 (cf. 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, = 200) observed Difference CH2 Slope at (4> = 200) Absolute temperature (<£ - 200) calculated Per cent, difference Hexane .............. (255 I)1 \ (2 88) 254 6 0 2 Heptane ............. 276 . 1 / (21.0) 277 3 0 4 Octane ............... 299 . 1 } 23 0 2 44 300 0 0 3 Isohexane ............. (249.0) \ (2 79) 247 0 0 8 Isoheptane 269 2 / (20.2) 2 68 269 7 0 2 Methyl iodide . 290 2 19 0 1 92 287 4 1 0 Ethyl iodide . . 309 2 \ 1 80 310 1 0 3 Propyl iodide ......... 332.7 f 23.5 1 82 332 8 0 0 Isopropyl iodide ........ 324.5 \ 1 92 325 2 0 2 Isobutyl iodide 345 5 / 21.0 1 86 347 9 0 7 Allyl iodide. . . 330 5 1 82 328 8 0 5 Ethyl bromide. 268 7 2 22 273 5 1 8 Propyl bromide ... 296.6 27.9 . 2 08 296 2 0.1 Isopropyl bromide ... . 289.4 2 22 273 5 1.8 Isobutyl bromide ....... 315.0 25.6 2 08 311.3 1.1 Ethyl propyl ether (255 0) (2 70) 256 1 0 5 Dipropyl ether 279 0 (24.0) 2 62 278 8 0 1 Methylisosbutyl ether. . . Ethylisobutyl ether ..... (251.1) 270.1 (19.0) (2,75) 2.68 248.5 271.2 1.0 0.4 Values in parentheses are extrapolated. TABLE XXVI.—THE VALUE OF THE ISO-GKOUPING Substance Temperature observed, normal grouping Temperature observed, 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 . . . Propyl chloride. . . . Butyric acid 296.6 261.5 381.6 289.4 255.2 371.6 7.2 6.3 10.0 Methyl butyrate . . 304.2 295.8 8.4 j 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 nCH2 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 H2 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 observed, normal propyl Temperature observed, 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 OP THE OXYGEN ATOM Substance Temperature observed O».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," 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 (E^O)* at the fluidity of 200 the absolute temperature is 328.9, while the value calculated from the gas formula H20 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 TEMPEEATUEES AND SLOPES OF SOME ASSO- CIATED COMPOUNDS CORRESPONDING TO A FLUIDITY OF 200 C.G.S. UNITS Substance Absolute temperature for ( == 200) observed Absolute temperature for (<£ = 200) calculated Slope for (<£ = 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 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 ....... 305.2 165.3 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 result 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 p FLUIDITY AND THE CHEMICAL COMPOSITION 121 TABLE XXXI.—A COMPARISON OP THE VALUES OF ASSOCIATION AS DETEK- MINED BY DlPFEEENT INVESTIGATORS Substance R. & S.,1 16-46° R. & S., corrected by Traube Traube,2 15° Longi-nescu8 B. & H.,4 temperature of (<£ = 200) Water ..... . ........... f 3.55 1.79 3.06 4.67 2.31 Dimethyl ketone ......... \1.64 1 26 1 18 1 53 1 60 1 23 Diethyl ketone .......... 1 25 1.16 Methyl propyl ketone. . . Formic acid ........... 1.11 3 61 1.10 2 41 1.43 1 80 1.25 1 80 1.14 2 05 Acetic acid ......... f 3.62 2.32 1.56 1.75 1.77 Propionic acid ......... \2.13 1.77 1 45 1 46 1 55 1 57 Butyric acid 1 58 1 35 • 1 39 1 36 1 51 laobutyric acid. ... .... 1.45 1 28 1 31 1 51 [Benzene 1 01 1 05 1 18 > 1 17 1 08 <1 517 Methyl alcohol i3. 43 2.53 1.79 3.17 1.84 Ethyl alcohol ............ 2.32 2.74 1.80 1.67 2.11 1.83 Propyl alcohol . . 1.65 2 25 1 70 1 66 1 67 1 74 Isopropyl alcohol 2 86 2 00 1 53 1 75 Butyl alcohol ............ 1.94 1.47 1 62 Isobutyl alcohol 1 95 1 53 1 54 1 66 Active amyl alcohol ...... Allyl alcohol ............ 1.97 1.88 1.54 1.50 1.53 1.55 1.80 1.54 1.69 Methyl formate 1 06 07 (1 60) 1 12 1 25 Ethyl formate ........ 1.07 .08 1 . 39o° 1.19 Methyl acetate .......... 1.00 .04 1 . 48o° 1.09 1.17 Ethyl acetate . . . 0 99 04 1 25 1 00 1 12 Propyl acetate .... ...... 0.92 .00 1.31 1.00 1.11 Ethyl propionate ........ 0.92 1.00 1.27 0.94 1.08 Methyl butyrate 0.92 1 00 1 30o° 1.00 1.10 1 RAMSAY and SHIELDS, Zeitschr.f. phyaik. Chem., 12, 464 (1893); 15, 115 (1894). « TRAUBE, Ber. d. deutach. chem. GeselL, 30, 273 (1897). « J. chim. Phys., 1, 289 (1903). * BINGHAM and HAKRISON, 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 temperature (<£ == 200), observed Hydrocarbon residue, calculated Chlorine Association Propyl chloride . 261 5 127 3 134 2 1 105 Isopropyl chloride ..... Isobutyl chloride. 255.2 285 2 119.7 142 4 135.5 142 8 1.11 1 13 Allyl chloride 256 0 123 3 132 7 1 10 Ethylene chloride . 336 5 45 4 145 5 1 27 Ethylidene chloride.. . . Methylene chloride. . . . Chloroform 291.2 279.1 " 305 3 45.4 22.7 - 36 5 122.9 128.7 113 9 1.10 1.15 1 04 Carbon tetrachloride . . Carbon dichloride ..... 347.0 356.3 - 95.7 - 77.0 110.7 108.3 1.01 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 1.00i5o. 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* 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

+ C - - (53) 9 For the simplest case, which we have in mercury, the constants B and C are each zero and our equation, becomes that of Batschin- ski (41); for other substances at high temperatures the term B/

yl chloride .... 0.25540 0.24993 7,465.2 5,881.9 246.74 234.22 0.05 0 Ol 31 Isobutyl chloride ........... 0.28656 4,973.0 253.00 O.O2 32 Ally! chloride 0.26292 6,377 2 234 10 0 O3 33 34 Methyl ene chloride ........... Ethtylene chloride .......... 0.39806 0.44121 2,666.5 2,219.3 212.97 258.83 0.04 O.O8 35 Ethylidcne chloride 0 . 33277 4,651.6 247.99 O.O4 36 Chloroform ............ 0.40697 4,400.0 245.73 O.O7 37 Carbon. tetracliloride .......... 0.47337 1,807.5 262. 15 O.O8 38 Carbon dichloride 0.55768 3,081.6 259.13 0.20 39 C&rbon disulfide ..... 0.26901 16,751.0 282.23 0,17 4O Methylsulfide ............... 0.25514 8,387.6 230.57 O.O1 41 Ethylsulfi.de 0.28517 6 , 447 . 9 258.00 O.ll 42 Xhiopheiie ....... • 0.38204 2,967.2 254.95 O.O9 43 DirnethxyUcetone .............. 0.23871 8,906.0 247.64 O.O3 44 45 Methylethylketone ........... Diethylketone ...... 0.27275 0.28145 5 , 572 . 6 6,179.3 252.32 262.41 O.O8 0.16 46 0.29251 5 , 544 . 3 263.21 0.15 47 0.15205 18,364.0 265.13 O.O06 48 Formic acid ............... 0.52465 963.4 281.57 0.19 136 FLUIDITY AND PLASTICITY TABLE XXXIV.—(Continued) Substance A B C Mean per cent difference 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 6th. 6r . . 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 Met a 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 69 Trimethyl carbinol ........... Amyl alcohol, active .......... 0.29657 0.35020 260.86 376 . 48 305.73 311.50 0.33 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 82 Methyl butyrate ............. Methyl isobutyrate 0.30210 0 28615 4,315.3 5 073 2 265.89 264 44 0.15 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 Di propyl 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 137 observed values is 0.17 for the 87 substances and based on some 1,OOO duplicate observations. If we omit the alcohols, this difference falls to 0.09 for 70 substances. This is much better than Thorpe and Rodger obtained with Slotte's , since the percentage difference is nearly twice the , 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 Slotte's 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-Av + C- (54) 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 O.O7 per cent which is nearly within the limits of the experi- ixiental 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 XXXV. XXXV.—THE CONSTANTS IN THE EQUATION T = A

utyric acid .............. 0.23862 43,665.0 433 . 17 200 0.06 JPropionic acid anhydride ..... 0.23619 52,294.0 425 . 82 250 0.07 IBTityl alcohol ...... .......... 0.23605 4,802.0 349.71 40 0.04 Isotoutyl alcohol ............. 0.23700 2,993.7 340.66 30 0.09 -A.csti.ve amyl alcohol 0.24650 2,942.8 346 . 82 30 0.08 Inactive amyl alcohol ........ 0.24101 3,908.7 354.17 35 0.09 H>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, T? = .70(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 106 ......... 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 — tp)/ti]10Q, is plotted as ordinates FIG. 53, FIG. 50. ^ •*. *. V X \ ^^ gf X \ \ \jO 12 16 20 24 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 conciliation 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 MobiUtl 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 T ^ are mixed and a contraction takes place, there seemt- " ' decrease in fluidity. Alcohol and water, acetic acid '' ' and chloroform and ether are a few examples. On th* 4 when liquids mix with an expansion in volume, the mi% greater fluidity than we would expect from the linear ftf * ume concentration curve. Methyl iodide and carbon ' 4 j furnish an example of this sort (BinghameZ al.91913). * facts have suggested to various workers (Brillouin (I * ** 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 presmi* * In spite of this intimate relationship, it has been lit f *v gated. Slotte (1894) stated that the logarithms of th< 1 are proportional to the logarithms of the specific veil** from this observation he deduced his second Eq. (48). most important discovery was made by Batschinski in I r 4 found that the relation between the molecular volume * fluidity may be expressed in the following formula: 0 = 0(7- Q) where 0 and C are constants. The constant 0 may be* * ipl the limiting value which the molecular volume of any iir j have as its fluidity approaches zero, and it is theref'**' the "molecular limiting volume.7' Consequently the *i« V — Q may be called the " molecular free volume5" and ft* relation may be very simply expressed as follows: Th*~ varies directly as the free molecular volume. Sixty-six n * < 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 agreeni**iu1 good include the alcohols, water, some of the acids, the fit t« drides, and some of the halides. These substances arc* regarded as associated and it may well be that the u**, weight is not constant as the temperature is raised. Tit** of the very remarkable agreement obtained is shown t*i XXXVII containing data for benzene obtained by Thf*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 = (V — 81.76)/0.045,35 Temperature

Calculated Difference 0.0 110.8 1.1124 111 0 10.0 131.5 1.1242 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

-f" C -- : |=: T~ 7 f TN\9

7\ ?l — ^Id ~T~ We \& • ) where rjd is the diffusional viscosity and rjc is the collisional viscosity. According to Maxwell, as discussed in Chapter XIV, the viscosity of a gas varies as the absolute temperature, so rid = 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 1) — co FLUIDITY AND TEMPERATURE 153 TABLE XXXIX.—THE FLUIDITY OF CARBON DIOXIDE AS CALCULATED BY MEANS OF THE FORMULA (p = (v — are constants and v is the specific volurP- ^ per gram. We have then 77 = BT V — co or A+BT(v-u) ' It is truly remarkable that so simple an equation 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 secured by a happier choice of constants, but consider* ^S difficulties in these measurements, the percentage of between the calculated and the observed values is not Having established a fairly exact relationship between- and volume, and indirectly with temperature and the problem of associated substances again presses the foreground as it tends to do so often. A means found for bringing these substances into conformity others, but the solution is not yet forthcoming. Dr. Kendall inquires in regard to the foregoing: — formula of Batschinski of such great importance as treatment of it would lead the readers to believe? IB it; merely an interpolation formula? Would it not be mention something about the alternative formula (1918). The expotential formula of Arrhenius (1918) dLoos lead us to a definite mental picture, and like many gtxxot frankly empirical formula was omitted in this brief troa/tono. of the subject. The relation of Batschinski fills a need ^wi was felt in many minds, cf. p. 142. It leads us at oneo t definite mental picture which is neccessary in building *ut] consistent theory, so that we are now able to explain the rola/t of fluidity to volume, temperature and pressure et ce£. i: manner which is so natural, so unexpectedly simple and so ~fc>€ tifully in accord with observed facts that it is hard to soo "W more evidence is needed to carry conviction. CHAPTER IV ; 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 1 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 i which are directly dependent upon the fluidity. There are 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. I 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 F measured by AC1 or CE, but at the same time 1 ) Ib 50 75 10 ?ene Volume Concentra'tion Ether Efh« Fia. 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. 400 350 p / / / ^o: vx / / / ' 500 / A / / /10DC '•6 13 ^50 LL. // // zoo / / / vr-; VTT- Additive Fluid ,alcu1ated by \ ^t/idittj Cala by Formula ity fo1.°/0 jlated // B / o ( Dbserved Poirv ^5 0 25 50 75 IOC enzene Vriinm«»rftwr«wW«4^ «£Fiu^ ^"he 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 s whereas the specific volume should be 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 + bvz = (a — mX — (n — 6)02 = (a - m) (P! — »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 $ $ = k(v — w) — KAv = v — KAv = a) as calculated with the use of Eq. (61), using 40 as the value for K. The fluidities ( calculated, Eq. (61) calculated, Eq. (61)

calculated, Eq. (61) v calculated, Eq. (25) Calculated Eq. (56) 155! 1 104.4 30 118.9 136.2 136.8 139.2 136.7 149.0 150.5 153.5 150.3 163.4 163.4 165.7 163.5 178.0 Observed calculated, Eq. (61)

calculated, Eq. (61)

calculated, Eq. (61)

per cent . va — «: 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 o> 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, which measure the slope —^— of the fluidity-specific volume curves are verv much less v — co J ^ 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 ANX> RODGER) Temperature 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 0 168.3 168.6 193.1 193.0 192.3 207.5 207.5 207.6 213.2 212.5 213.3 222.7 223.1 223.0 228.3 228.7 228.4 232.8 233.1 Observed Calculated, Batschinaki Calculated, Gibson 10 186.6 186.2 211.4 211.3 211.1 225.7 225.6 225.7 225.9 230.5 230.0 242.1 241.6 242.0 248.1 248.0 248.0 253.0 252.1 Observed Calculated, Batschinski Calculated, Gibson 20 205.3 205.3 230.9 230.4 229.2 243.9 243.8 244.0 248.1 248.5 248.2 261.8 260.6 262.0 268.1 268.0 268.0 272.5 271.9 Observed Calculated, Batschinski Calculated, Gibson 30 224.8 224.7 250.6 250.2 250.8 263.2 262.8 263.2 267.4 267.0 268.0 280.1 280.0 281.0 288.2 288.2 288.0 293.5 292.2 Observed Calculated, Batschinski Calculated, Gibson 40 244.7 244.8 271.0 270.2 270.8 282.5 282.3 282.6 285.7 286.2 286.0 299.4 300.1 300.0 309.6 309.8 309.5 314.0 313.4 Observed Calculated, Batschinski Calculated, Gibson Value of u> from mixtures. . . . from solvents ---- 0.38081 0.4405 0.4420 0.4855 0.4908 0.5096 0.5171 0.5722 0.5758 0.6161 0.6143 0.6642 Value of k from mixtures.. . . from solvents 3.5271 3,040 3,224 2,633 2,982 2,465 2,852 2,341 2,561 2,322 2,370 2,123 1 Using the densities of Thorpe and Rodger we have recalculated these constants, instead of taking the values of Batschinaki. When there was a volume change on mixing, Gibson assumed that the specific volumes vi and #2 were not the same as for the THE 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 v = mo i the values of Vi and v% 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) Temperature Per cent carbon disulphide by volume 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 J] ;| HI POSITIVE CTJRVATION ANTD CHEMICAL COMBINATION 4 Before considering the meaning of positive curvature in detail, it is necessary to emphasize the fact that a minimum in the fluid- JJ 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 Kndlay (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, I; we will present three cases of increasing complexity, in all of *; which there is the same amount of chemical combination, it being I' assumed that in the feeble combination with which we are dealing I the two components A and B are always in equilibrium with a f small amount of the compound C so that }' A + B & C L, Case I.—The fluidity-temperature curves of two closely related • jfl substances are represented by the curves A and B in Fig. 63a. 4! If there were no combination between the components on mixing, w the curve for the 50 per cent mixture would lie half-way between ft, the curves A and B (dotted). Let it be assumed that this mix- jl ture does show the maximum amount of combination and that the curve is thereby lowered to 0.5S. Using the data plotted in Kg. 63a it becomes possible to plot the fluidity-volume concen- tration curves for the various temperatures £1, £2, *2> etc., as shown in Fig. 63&. 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 0.5B falls between the curves A and B, Fig. 64&. As a result the fluidity-volume concentration curves, Fig. 646, no longer exhibit a minimum although, by assumption, THE FLUIDITY OF SOLUTIONS 173 the kydration is the same as before both in relative composition and amount, However, it is clear that the deviation of the fluidity 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 rQinimum in the fluidity-concentration curves. Tejnp. A °/ B B a h FIG. 64.—Diagram illustrating how when two substances A and B are mixed whose fluidities are very different, the formation of a solvate produces ao 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 case 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 65&. 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 floes 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 Tins 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 Kodger. Chloroform, like carbon tetrachloride, is prohably 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- mum 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 (^HioO.CECla contains 38.30 per cent ethor by weight. Guthrie has noted that heat is evolved on mixing and that it in 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 CjHioO.CHClg. 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 CdHioO.CHCls should have a fluidity of 200 at the absolute temperature of -588.6°. But actually a mixture of this composition has a fluidity of 200 at 282.9° absolute (9.9°C). Pure ether and pure chloroform have fluidities of 200 at 216.5° and 305.3° absolute respectively, BO that if the mixture were wholly uneombined, the absolute temperature neeensary 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 538.6s •+ 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 Mass Law holds, we have . cHcy 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 _ 74.08X1.526 where the specific gravities of ether pe and chloroform pc are taken as 0.7366 and 1.526 respectively and their molecular weights, me and mc) 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 - 5.46Wme][(43.74 - 4.25)Pc/mc] ^ 9.71p/(me+mc) 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, mc = 4.696 me 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 WATEE 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 CVEI60.- 4H2O, 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 Pro. 06.— -Mixing solutions of ethyl alcohol in water (corresponding to the composition OsHaO.SHvO) with solutions of acetic acid in water (corresponding to the composition CaH^Os-HsO) 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 C2H60.4H20 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 eeixt mixture has a fluidity of 200 is 362.3°. Hence, if we let x represent the fraction by volume of the mixture combined as C2H0O.4H20, the rest remaining unchanged, we have 335.5 (1 - a?) + 758.4s = 362.3 and x = 0.0634. 12 ^ 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 (C2H402.H20) is mixed with ethyl alcohol solution (C2H60.3H20), 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). IV. 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 Hiibener (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," d2 (p/db2< 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 XLVL—SUBSTANCES WHICH APPEAR TO EXHIBIT THE SO-CALLED "NEGATIVE VISCOSITY" IN AQUEOUS SOLUTION" Substance Observer Bromic acid.............. Hydroforornic acid......... Hydrocyanic acid......... Hydriodie acid............ Hydrosulfuric acid......... Nitric acid............... Ammonium acetate........ Ammonium bromide....... Ammonium. cMoride....... Ammonium ehromate...... Ammonium iodide......... Ammonium nitrate........ Ammonium thiocyanate.... Caesimm chloride.......... Caesium nitrate........... Perrouis iodide............ Mercuric chloride......... Mercu.ric cyanide.......... Potassium bromide........ Potassium chlorate........ Potassium chloride........ Potassium cyanide......... Potassium ferricyanide..... Potassium iodide—...... Potassium nitrate......... Potassium thiocyanate..... Rubidium bromide........ Rubidium chloride......... Silver nitrate............. Sodiu.m iodide............ Tetramethylammonium iodide Thallium nitrate.......... Urea..................... Poiseuille (1847) Poiseuille (1847) Poiseuille (1847) Poiseuille (1847 Poiseuille (1847 Poiseuille 1847) Poiseuille (1847) Sprung (1876) Poiseuille (1847) Schlie (1869) Hubeaer 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 (1S47) Hubener (1873), Kanitz 1897) Poiseuille (1847) Poiseuille (1847) Sprung (1876), Gorke (1905) Davis, Hughes and Jones (1913) Wagaer (1890) Poiseuille (1847) Poiseuille (1847) Schlie (1869) Sehottner (1878) Miitzel (1891) Most workers have con.fi.ned 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 350 ZZ5 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 Z25 200, 175 P 100 75 50 25 ^* SN X 10 ZO 30 40 50 60 70 50 90 100 Fluidity of Solutions of Urea • FIG. 68. 200 100} \ \ "0 10 TLO 30 40 50 60 TO 80 90 JOO Fluidify of Solutions of Silver Nitrate 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 chloridej 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 24-0 40 50 60 Per Cent 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 Gorke.) Miihlenbem (1903), however, showed that the dissociation hypothesis was by itself insufficient as an explanation since salts like NaN03 and K2S04 are highly ionized and yet do not show negative curvature as does KN03. 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, Mclntosh, 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 THE FLUIDITY OF SOLUTIONS 185 10 20 30 40 50 60 70 80 90 100 Up 120 130 HO 150 160 170 180 190 "200 210 Atomic Weight- 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 which one further example may be cited. In ammonium nitrate solutions, the expansion is least in a 7-weight per cent solution arjtd 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 sulf ates 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 XLVII.—A COMPARISON OP THE FLUIDITIES OP VABIOTTS METALS AND ACID RADICALS IN NORMAL SOLUTION AT 25°C (AFTER WAGNEE) N03 C13 S04 N03 Cl NO3 SO4 K 114 7 113 3 101 2 1 012 1 133 K/H . 1 053 1 081 0 974 H ....... 108 9 104 8 102 5 1 039 1 062 K/Na .. 1 095 1 112 1 112 Na ..... 104 7 101 9 91 0 1 027 1 151 K/Zn 1 195 1 199 1 239 Zn ....... 96 0 94 5 81 7 1 015 1 175 K/Mg ...... 1 201 1 218 1 239 Mg 95 5 93 0 81 7 1 26 1 169 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 » = &• ^ 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 4. F = g 7T#r3(p2 — pi) where p2 and pi are the densities of the sphere and the medium respectively, so = -L( — V2 (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 d and the fluidity, RT v d = AT where T is the absolute temperature, R is the gas constant (83.2 X 106 c.g.s. units) and N is the number of molecules in a gram molecule (70 X 1022). 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) andMillikan (1910) have shown that Stokes' formula becomes 0 = RT N (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 XLVIIL—THE RELATION BETWEEN DIFFUSION AND VISCOSITY (THOVERT) Substance <5 X 105 T, time of efflux d X T X 104 Ether ........................ 3 10 315 98 Oa,rbon disulfidc . . . 2 44 405 99 Chloroform ...... .... 1 50 660 99 Mixture ethyl alcohol and ether . Kenzene .................. 1.51 1 24 660 790 100 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 A.m.yl alcohol ................. 0.155 5,900 92 Grlycerol solution . . . 0.0104 94,000 98 On the other hand, Oeholm (1913) finds that 871 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 yn = kt, 190 FLUIDITY AND PLASTICITY I: (65) 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 I of the displacement of the particle in time t on the axis of displacement is Z2 = 25t Substituting into this equation the value of the diffusion, given above Z2 = RT /. 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 9 A ,/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 -fig 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 VAEIOUS SOLVENTS AT 0° AND 25°C (AFTER WALDEN) Solvent *°° AO° ACO

/A«, is a constant even when the solvent is varied widely. He used tetraethylammonium iodide in some forty different organic solvents and found

is « = cupi (69) where a is the volume percentage of the medium whose fluidity is = (a - d)vi (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 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. 725, 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 I 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. Volume percentage Clay FIG. 73.—Tlie 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 (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 LIV the fluidities of graphite suspensions are compared TABLE LIV.—THE FLUIDITIES OF SUSPENSIONS OF GEAPHITE 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 .048 100.9 100.7 35 0.395 129.8 128.3 .046 113.4 111.7 45 0.394 156.3 154.8 .042 135.0 134.8 55 0.392 184.9 183.0 .037 161.7 159.5 65 0.390 215.5 213.1 .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. 130 I 1201-^ 110 100 90 80 TO 60 50 4-0 50 10 25 10 15 Per Cent* PIG. 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 LV.—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 calculated c = 34.5 weight per cent 0 153.0 1 2.54 353.0 149 2 2.01 146.0 144 5 2.09 132.0 131 10 2.73 104.0 109 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 Einstein1 and Hatschek2 have both considered theoretically the case of suspensions of spherical particles at low concentrations. They both arrive at the formula H = rn (1 + Jfcfe) or , _ «*l /79N * - r+Tb (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 jflmdtfa/-concentration 1 Ann. der Physik., 19, 289 (1906)? 2 Kollmd-Zeitschr., 7, 301 (1901); 8, 34 (1911); Trans. Faraday Soc. (1913). COLLOIDAL XOLUTIONX 207 curve show a nuddon drop in fluidity at about 30 per cent?77 The glyeogen suspensions were studied by Botazzi and iPErrieo (1906) using two different viKcorneters, one from 0 to 20 per cent and the Heecmd from 20 per cent on. On plotting the fluidities wo find that the values for eaeh viscometor lie on a straight lino, but the two linos do not coincide. For the first vinoornotor, the fluid- ity of wafer is 144.0 and the weight concentration of zoro fluidity in 27.5, while for the second viscomoter it is necessary to assume a iluidity for water of 77.0 and a zoro fluidity at 4.1 per cent concentration. Using formula (64) the calculated values agroo well with the obsorvcnl except at 45 weight per cent which in beyond the concentration corresponding to zero fluidity, an shown in Table LVI. Bottazzi and d'Krrico give their viscosities an times of flow, which of course are not proportional to the V!H~ coHitioH, as is HO often assumed, HO this may perhaps explain the discrepancy between the two viseomctors. But more1, work needs to be; done on this subject to definitely establish whether the viscosity of a suspension is independent of the*, dimensions of the instrument or not. At any rate then; in no evidence that the fluidity of concentrated HUpennionH Is abnormally low. In fact those experiments lead to the opposite conclusion. TAIILK LVf,—••• THK Px,uti>rnKH or OLVCOCIKK HUHPHNHIONK AT 37°C5 Per c«»nt glycc»g<*n 1 hy weight t> j Fluidity calo.u- Fluidity ohm^rvfuii latfid by fonnuln \ (7!) 0 144.0 144 1 138 0 139 Vi«eomc4c»r No. 1 5 114.0 118 yn - 144,0 10 KILO 92 t - 27.6 16 (WU) fill 20 40.0 40 20 40 0 40 25 32 0 »4 VincoincUir No. 2 30 20 0 21 v\ 77.0 35 12.0 11 c *•> 41.0 40 5.0 2 45 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 LVIL—THE EFFECT OF ELECTROLYTES ON THE FLUIDITY or SUSPENSIONS (AFTER BINGHAM AND DURHAM, (1911)) Dispersoid Concentration disper-soid Fluidity of suspension Substance added Concentration of electrolyte Fluidity with electrolyte Infusorial earth. . . . Infusorial earth — Graphite .......... 6.46 6.46 0.396 62.1 53.2 116 8 KC1 NaOH KC1 1:80,000 1:20,000 1 : 20 , 000 53.2 58.3 116.9 Graphite 0 396 116 8 HC2HsO2 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 9E 91 90 88 81 86 I ^7*1 0.&8 a CQ O.S1 0.86 115 120 125 130 135 140 145 150 155 Temperature FIG. 75.—Fluidity-temperature curve (continuous) and specific volume- temperature curve (dashed) of p-azoxyanisole. (After Eichwald (1905) and Buhner (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 COLLOIDAL SOLUTION'S 211 smaller. ZBtiglia (1908) has found that the fluidity of milk is lessened wfron the milk is "homogenized" by being squirted against an ^Sate plate, thereby increasing the number of fat globules. The apparent decrease in fluidity with errmlsification finds excelled practical examples in the manufacture of solid lubricants and of certain household products such as mayonnaise, han 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 la,mellse a,re greatly elongated. But in immiscible liquids this thinning out of the layers is opposed by the surface tension which tends "bo 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 wjQl take place, but since we are dealing with immiscible liquids, the lamella will not be thinned out indefi- nitely, bu/b torn into portions which will gather into drops under the influence of surface tension. Thus in an emulsion, shear tends to make tlie 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 Ludeking (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, without 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, the 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 forrn, 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 Shearing Stress FIG. 76.—Typical flow-shear diagram for a series of viscous liquids. distance dr and subjected to a shearing force F, we have found that in a viscous fluid dv = ic solid, a certain portion of the shearing force is **sed irp i** overcoming the internal friction of the material. If the stress is j "ust equal to the friction or yield value, the material **iay be said "to be at its elastic limit. If the stress is greater than the friction f* the excess, F - /, will be used up in producing Plastic flow according to the formula dv = M (F - /) dr (73) where M is a constant which we will call the coefficient of mobility 01*. Shearing Stress JJ*XG. 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 throiagh 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 B9 we see that they correspond to entirely different viscosity curves OD 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 rigidity1 £ which is the reciprocal of the elasticity 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 = ~^r 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 / r "J extrapolating the curve to the axis; the value of the intercept will eviclently be the friction. We may also use the algebraical method. In either case at least two measurements of the rate of flow V^/ti = v\ and Vz/tz = v2 are necessary corresponding to 0.025 0.020 0.015 O.OIO 0.005 / / JJ. // f 500 1500 2000 1000 Shearing Stress FIG. 78. - IFlow-shear curves of a certain paint, using capillaries of varying length and radius. the shears FI and Ft, respectively. Assuming that the mobility is independent of the rate of flow, Eq. (73) integrated in Eq. (89) gives us 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 200 25 JO 75 VOLUME ^PERCENTAGE CLAY 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 25 50 75 VOLUME: PERCENTAGE CLAY 100 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 Pig. 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 BlNGHAM AND GREEN1 FR y Pressure dynes Obser- Num- V-t centi- grams dynes per vations ber of observation meters per second per Square centi- per square centi- square centimeter F-f M Remarks used in calculations meter meter 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 2 and 3 cm 3 0.0003344 409.3 628.9 75.9 536.9 0.261 I = 4.620 cm 3 and 4 4 0.0002133 277.5 426.4 [66.0] 334.4 0.267 4 and 5 5 0.0001661 225.6 346.6 [57.7] 254 . G 0.273 5 and 6 6 0.0001019 152.9 234.9 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 1 = 4.684 cm 9 and 10 10 0.0008987 274.3 596.9 [65.6] 504.9 0.256 10 and 11 11 0.0004164 143.3 311.8 [53.7] 219.8 0.272 11 and 12 12 0.0002880 106.7 232.2 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 1 = 4.681 cm 15 and 16 16 0.001758 275.6 706.9 [75.1] 614.9 0.247 16 and 17 17 0.0008267 145.1 372.2 [63.5] 280.2 0.255 17 and 18 18 0.0005856 110.0 282.2 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 1Proc. Am. Soc. for Test. Mats. (1919). THE PLASTICITY OF SOLIDS 223 lastic flow and it has already had the attention of Buckiiig- (1921)- In plastic material confined between two parallel es of indefinite extent which are being sheared over each r? th_e shearing stress F will be identical at every point. But low tfctrough. a capillary tube according to Buckingham is n.ot the case; the shear increases continually from the sr of tfa.e capillary outward and only at a certain distance r0 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 v0, and beyond the radius TQ the mate- m.o\res in telescoping layers. This results in the flow not ? a linear function of the pressure. it tbiere are other possible causes of the peculiarity which be mentioned here. The plastic material next to the ma/y have a lower concentration of solid than elsewhere ting in apparent slippage. Or the shearing stress may cause Lquid to flow bet~ween the particles of solid, seepage. Lcklngham 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 are of very •ent sl^es. We shall at first assume that slippage and * Jo (P - 21 V t (P - 81 (p-n (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 erptive 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- j of these properties leaves the subject of plastic flow in a ilous state. "any investigators have investigated the so-called "viscosity >lids,37 assuming that solids obey the ordinary laws of viscous , and Tarnraann 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 i) 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 rmich as liquids do. It gives good reason for the pre- >tion 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 EFFLXJX OF METALLIC LEAD THROUGH A SMALL OBIFICE AT VABIOUS TEMPERATURES (AFTER WERIGEN, LEWKOJEFF AND TAMMANN) Temperature, degrees Efflux (relative) Temperature, degrees Efflux (relative) 0.5 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 "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 propri6t£ 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 propri6t6 est poss6d6e a un haut degr6 par les fluides; je le nommerais volontiers la fluidity des corps solides; le coefficient if/ pourrait £tre appel£ coefficient de fluidity; la mall6abilit6 des metaux parait en d6- pendre et peut-etre aussi leur duret£." 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.'J 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, Radius, Length, Time, Volume, "Viscosity," degrees C centimeters centimeters seconds centimeters absolute 12.2 1.525 25.1 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 4.59X109 12.3 0.746 16.0 258 , 000 0.756 5.62X109 12.3 0.576 15.1 171,000 0.129 7.91X103 12.3 0.364 15.8 350,000 0.0866 25.2X109 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/7 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) Observed time of trans- Pressure, atmospheres piration for equal vol- Calculated time umes (relative) 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 t'o agree to within less than 0.1 per cent as given in Table LXV. TABLE LXV.—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: rit = 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 s 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 +1/), so that the total loss of momentum is t; - L) - or since Nm = p 7? = (97) If the speed of the molecules 0 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), 77 = 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 LXVI.—EVIDENCE FKOM MAXWELL (1866) THAT THE VISCOSITY OF AIR is INDEPENDENT OF THE PEESSURE Temperature, degrees Centigrade Pressure in mercurial inches Logarithmic decrement of oscillating disks 12.8 12.8 13.3 0.50 5.52 29.00 0.15378 0.15379 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 THE VISCOSITY OF GASES 245 foreign gas rnost likely water/'which seems to have been sus- pected by jCxindt an(j 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 17.= 1/3PFL it is evident tttat the effect of an increase in temperature will -be to increase the mean velocity, but it is not known what effect the tempera.ttrre may have upon the mean free path, although it 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 Ms experiments that the viscosity varies directly as the first power of the absolute temperature. Barus (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. Holrnan (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 condensihle 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- rnann (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 = KT3A. (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 comnmnicates 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 VISCOSITY OF fVASA'.x 247 f orc*e» then each molecule of gas when it out of the of t,h^ molecular force of the solid must have the as when it went In, so that without of gas and solid there can t>e no of to the gas. If, then, molecules of gas and of gas must collide amongst themselves." I xi the theory of diffusional viscosity explained It plain that there would be viscous If the failed to collide with each other entirety. I£LXXtieric pressure. Ex:amirimg Sutherland's formula, we observe ike csorxsfcant C is small in comparison with the -th.e formula reduces to the simple theoretical formula 1? = KT* Tlio discovery, (cf. Vogel (1914)), that Sutherland's formula a/fe* low temperatures indicates tkat it does not account of the deviation from the simple formula. Quite in harmony with the above> it is found that the 248 FLUIDITY AND PLASTICITY TABLE LXVIL—CONCORDANCE BETWEEN SUTHEHLAND'S FORMULA AND HOLMANT'S DATA FOB CARBON DIOXIDE. C = 277, ij0 = 0.000,138,0 Temperature, degrees Centigrade i\ X 107 observed 17 X 107 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 Tcr Tcr = 1.12C (101) TABLE LXVIII.—THE RELATION- OF THE CONSTANT C IN SUTHERLAND'S EQUATION TO THE BOILING-POINT AND CRITICAL TEMPERATURE Substance TCr3 Critical temperature, absolute C Tcr/C Tb, Boiling temperature, absolute C/Tb Helium ............ . ..... 9.0 78.2 0.11 4.3 18.3 Hydrogen ................. 37.0 83.0 0.45 20.4 4.1 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 1,14 170.0 1.46 Carbon dioxide ............ 304.0 259.0 1.17 194.0 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 24§ which suggested to Vogel a similar relation to the boiling temperature 2\ C = L47I\ (102) This formula indicates that C increases rapidly than the temperature, and since T& is large for vapors, the less perfect agreement of formula is partially explained. This, however, is not of hydrogen and helium which present curious as in Table LXVIII. VISCOSITY AND CHEMICAL COMPOSITION If the mass of a particle in a rarefied gas Is by changing its chemical composition, the velocity will be rr* times the original velocity, so that the momentum of TABLE LXIX.—THE VISCOSITIES OF PERMANENT GASES AND VAPORS AT 0°C Substance Molecular weight i?, X 107 TV 1* X 107 Hydrogen ....... 2 0 850 31 0 Helium . ........... 4 0 1,871 5 21 Methane ............... 16.0 1,033 183. Neon .................. 20.2 2,981 Nitrogen ......... 28 0 1,678 Carbon monoxide 28 0 1^672 133. Oxygen ................. 32.0 1,920 154. Argon ........... 39 9 2,102 155.6 1,253 Nitrous oxide 44 0 1,362 Krypton. 82.9 2,334 210.5 1,806 Xenon . . 130.2 2,107 288. 2,266 Ethyl alcohol ........ 46.0 827 513. 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 n1/2-f old that of the smaller molecule. But the number of excursions of the molecules will be in proportion to rr**, 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 T* "lit? 0 100 200 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 cxf 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 yc and their atomic weights M at the critical temperatures. He finds them related together by the formula 77C2 = 3.93 X 10~10M THE VI8COMTY OF C/JXEK 251 depicted In Fig. 82, The critical constants of and *e not yet been determined. Rankine has further sam-« general formula applies to the but different being 10.23 X 1Q~10. He for iyf = XI !0~7 and for bromine ye = 2,874 X ID™"7 (ff. Fig. S2). v^re to use the molecular weights instead of the atomic ts? "the constant would be 5.12 X 10™ 18 which is that "the jraxe gases but still not identical with it. THE VISCOSITY OF GA.SEOUS MIXTURES SInco in a rarefied gas the viscosity is to the of molecules in a unit volume, i.e., to the viscosities will be additive when arc in percentages by volume; but since the viscosity of a gas is also independent of the weight of the the law loses its significance. In gaseous mixtures at ordinary pressures the 1 = ~ applies, it being merely necessary to find the mean -values of p, V, and L. This has by (1S68) and Puluj (1879), and one obtains the (cf. IVteyer^s Kinetic Theory of Gases, page 201 el seq.) Gratiam (1846) observed that mixtures of oxygen or- oxygen and carbon dioxide in all proportions of traxispiration which are the arithmetical of the two components. Thus for air, 0.0001678 X0.7919 - 0.0001920 X 0.2081 - Calculated ^ri^iosity of air........... 0.6001728 Observed Tiscosity of air........... 0. (1914). and others have noticed that when is 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 = 1 irCVaLiQi + NJkQ*)/N (104) Ni, 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 = Ni + N%. 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 = TrOL (105) o If we neglect the small difference between Oi and ti due to temperature difference the conductivity of heat k becomes k = l have been attempted. The connection of superficial fluidity -with emulsions be mentioned at this point although we cannot stop to it. We can merely refer the reader to the fascinating studies of Pla- teau, Quincke, and Lord Rayleigh upon the nature of ting films. The recent paper by Irving Langmuir (1919) on the theory of flotation is very suggestive. Maay of the examples which we would naturally cite as pies of the second ease given above may really be of the third instead. It is certain that in most emulsions a stance is necessary to stabilize it and it may give rigidity. 256 FLUIDITY AND PLASTICITY Hi i 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 work 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 ^ Am. Chem. J. 40, 495 (1908); 44, 251 (1910); 60, 59 (1913). SUPERFICIAL FLUIDITY 257 radius whereas the volume of flow varies as the of the radius, Eq. (6), we may expect to find the effects fluidity shown to the best advantage in very fine tubes. There "are a variety of causes which may cause the near the bouo-clary- to have a different fluidity. The most cause results from the selective adhesion of the of the fluid for the solid. If one of the components of the L* more strongly attracted than another, separation becomes ble, and the magnitude of the fluidity of the mixture as will theoretically he affected The adhesion between and liquid or liquid and liquid Is doubtless Just as specific a as is the better known cohesion or surface tension of and we are coming to understand the nature of adhesion tetter through, the efforts of Langmuir (1919) and Harkins (1920). We have seen that it is possible to greatly affect both the and the mobility of plastic substances by the addition of amounts of acid or alkali. Just what happens in such might be subject to dispute, but it is certain that of substances adsorbed on to the surface of a solid may change the character of the solid which is in contact with the liquid. Thus Henry Green (1920) has observed the of small amounts of gum arabic to a suspension may deereas© the yield value and increase the mobility, in of the high viscosity of gum arabic solutions. This is as being due to the decrease in adhesion between the pended particles. The well-known work of Schroeder (11)03) upon thte effects of electrolytes on the viscosity of gelatine of Handowsky (1910) upon serum albumin should also to. We Imve already proved on page 86 that if any in the fluid near the boundary becoming different from the re- mainder of the liquid, the resulting fluidity will be This theorem is therefore useful in explaining superficial fluidity. We will now prove that the components of a mixture under conditions will undergo partial separation. The conditions will be unade more general by using the non-homogeneous mixture considered on page 86. Considering the mixture as up of the two components A and B7 arranged in alternate plane layers, the total quantity of A flowing in a unit of time, regardless IT 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 : -L. ^ """ and similarly the rate of flow of component B is , H 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 = <» , (110) and there will be no separation at all. The separation may be calculated from the expression Ui __ a naz When n = 1, the component A will flow at only one-third of the rate of J5, 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 pip.e 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 ^Engineering Record, 57, 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 ~r 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 7 exerts a force — 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 and by integration «• - f •* The quantity of 0.5^ 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 ladius 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 LUBRICATION When- a solid substance Is subjected to a stress, ife undergoes plastic flow if the stress is greater the value of the material. In this process of shear, lateral arise and if the material is not supported laterally by pressure, rupture of the material will finally result. surfaces formed by rapture slide over each other 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 j oil, a layer of oxide, etc. So two surfaces formed by a rupture as, for example, two broken pieces of porcelain do adhere together firmly even when they seem to fit together very nicely. So also the resistance to movement between smooth, surfaces is far less than the resistance to low. If, liowever, sufficient force is brought to bear between sliding surfaces of similar material, there will occur, far the molting point of the substance, a welding together of surfaces into a more or less compact whole, there is present some substance which prevents such welding. Two surfaces of glass ordinarily touch each other at very few and thiey do not adhere strongly, but when the two are ground to an optical surface and cleaned, it Is difficult to separate the two surfaces without tearing them, after they been brought together. A motor bearing wMch has care- fully fitted by "lapping la'7 may be ruined completely by a slight turn with the hand after the surfaces have 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 in gage testing are made of hardened steel with are •exceptionally true. When these blocks are placed one on top of the otter, the adhesion between them Is so great that a 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- tioual resistance P varies directly as the load W or P = fW (111) 11 and the coefficient of friction f is defined as the ratio between the P !; friction and the load. | • 2. The force P0 required to maintain an indefinitely small rate i of shear, the so-called static friction, is greater than when the | rate of shear is appreciable. The dynamic friction is independent I of the velocity. I 3. The friction is independent of the area of the surfaces in LUBRICATION 283 **E>parent contact, within wide limits. The ^v^er, be large enough so that the surfaces Since it is impracticable to obtain a pair of "Unlxibricated surfaces, it is needless to say are irxexact. As already intimated, well-fitting and of similar material would probably seize of flow, which are very different from the have, however, both historic interest Just as the laws of solid friction are to the laws of plastic flow, so these laws are in to the laws of viscous flow which apply to With well-lubricated surfaces we have the where S is the area of surface in contact, dv is the velocity and dr Is the thickness of the oil film. According to 1 . The frictional resistance P is independent of the 2. The friction is directly proportional to the is "tbterefore zero when the velocity is zero. 3. The friction is also directly proportional to the of faces in contact. In view of the absolute antithesis between sets of laws, it is not surprising that the results of the study of as recorded in the literature are often contradictory. We liowever, state broadly that slow-moving, poorly faces follow approximately the laws of friction, rapid-moving and hence necessarily well-lubricated such as electric dynamos and motors, follows the of friction. Most bearings are imperfectly lubricated neither set of laws exactly. Petroff (1887) seems first to have applied the of friction to lubricated bearings testing out Ms views by Most important in its relation to the development of lubrication is the experimental work of C 1885-4), undertaken at the instance of the of Mechanical Engineers. His experiments were extreme care and under varied and well-chosen His results, as obtained under ordinary conditions of hibri- 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 the oil, which at the crown of the bearing was shown by actual measurement to be 625 Ib. 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: "The results, 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' 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 ia the direction perpen- LUBRICATION 2tw ciicular to the paper. As soon as a load Is on tk Iwvirltu "block, the liquid begins to be squeezed out i*>t*x»i : tlr surfaces. If this space is divided into the tvtiJ .,r*vi* indicated by the dotted lines, these lines, moving wnh *ht *r.«; I "will after a time occupy the positions of the curved Ikt^; ui*d the distances moved "by the particles are by th» dNtai, ^ "between tbe corresponding points on the two o*" ui v* ^ .4- ^J3*1 for the pointPj and the slopes of the curves iT dk at ^+L* air* •- tions of the forces in the fluid Just, as if the liius wero ^ reti VI A < " 8 FIG. 83.—The simplest case of lubrication. Two parallel, surfaces. elastic threads. The pressures exerted along CD are shown in the curve of pressures CFD, the being proportional to the vertical height above the line CED. -At the center of the block the pressure is a the liquid is squeezed out to the right left of this For -fctds section alone, there is a uniform "variation of ~A. R to CD, siich as would be true of all if the -A J3 and CD were not approaching. Case IL Surfaces Indined— —• If now the bearing block is tilted, we have the condition for continuous lubrication, for the is to sustain a load without the surfaces "Were we to assume that in this the velocity formly from U at AR to zero at CD, the quantity of ajxy cross-section Jf JV would be proportional to MN X CT/2, or simply to MN. But since the quantity of cross-section must be the same, there must be an to the right and left of the cross-section MW, at the is a' maximum, so the flow at any section MN is (MN - MWO 17/2 At the cross-section MN, the -velocity 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 GH 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 MzNz on either side, just as they are concave to a section MtNi on the right. The reason for this is that with a uniformly varying velocity more fluid would be brought in at the right of MiNi than would pass the section GH, hence the fluid must flow outward from MiNi, 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 M^Nz. 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- J sures may be eliminated. LUBRICATION 267 If the oil supply is limited, the oil not wet the bearing but form an oil pad in the of Off, the of course reaching a zero value at the the oil meets the bearing surface. If d is the of the oil film outside of the pad, the quantity up to the pad per second will be Ud, and the quantity the JlfuWi where the velocity varies uniformly Is If lX'l 17/2, and there is no accumulation of oil, these two be and Jf uZV, = 2ci also If 2AT2 = 2d Case IF. EevoMng Cylindrical also Cylin- drical.—In a very common example of we a cylindrical journal partly or wholly surrounded the or "brass" CD in Fig. 85. The oil is drawn up into the BD FIG. 85.—The lubricated journal and bearing. and creates a pressure which is a maximum at Q. The of nearest approach between journal and is at middle of the bearing 0 but at a point some 40° further on at G toward the so-called "off-side" of the 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 I** 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, Sommerf eld, Michell, Lasche to name but a few. LUBEICATION 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 2*]fl urveyors of the fatty oik claimed that oils pos*>>*•«! renter uoiliness," "body" or "lubricating value/' bo! *i.nce tese claimants seem never to hare ease by the e-femal measurement of "oiliness" and in- cialism requires vastly more oil for p,,*- ft>ly be met by the available supplies of fatty oik, the ctmi-option f "the property of oiliness has gradually a sort of will o* ko wisp vaguely referred to in treatises on lubrication, ami ively used by energetic salesmen in convincing a myer of the superiority of a given brand of oil all Pfcie theory predicted that so long as the viscosity to *roduce the necessary pressure required to the it of no moment what the chemical nature of the 3, provided only that the quantity of was Tltie practice has therefore been to use an oil is viscous than is really necessary and to a in in order to insure against any discontinuity in the oil There are, to be sure, many instances which be wliere an experienced engineer has a hot by substituting a fixed oil with which he was for the oil in use. However, in comparing two oils for lubrication, there are so many factors which the comparison such as the quantity of oil, the ixesrature of the oil film, the condition of the titia,t instances which might be cited are easily by the skeptical. Nevertheless, there is a for cztnts which will be less wasteful of power which at the s.«tni€ time give the maximum assurance that the not t>e injured in use. With the aeroplane in it is s«try to keep the motor going at aU of the l^exiod of flight, and an overheated may the oomplete wreckage of the machine in mid-air, so the of the best lubricant for severe conditions the of c * oiliness" becomes now vitally important. the evidence on this point is obtained from cutting Cuttiag Lubricants.—It is the well-nigh of mechanicians that in certain cutting oils are absolutely necessary and that mineral oik wii mot as a satisfactory substitute. Voluminous 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 Science and (London)). The surface tensions of mineral and of oik an* :i* it ii in different. These oils are however very differ* nt in < r.o IIL' respect t»£z., that the fixed oils ail have an active .lu ^Lu which, gives them a strong adhesion for metals, ho th.it *• oil is not readity squeezed out from between two :n«»tj* faces (£/. langmuir (1919), Harking (1920), Bin^iit*m • Lord Hayleigh (1918) has shown that a Saver ol la')ri- Fio. 86.—lEustration of the necessity for high in aa cl w to have the best lubricating quality. monomolecular thickness possesses truly in reducing the friction between solid bodies of the contamination probably serving to prevent the together of the surfaces. According to a film formed from paraffin oil can be readily by a stream of running water from platinum, etc,, but a f ornxed "by oleic acid cannot be thus removed. To get a clearer idea of the action of a cutting we mill follo-w tlie operation of an Armstrong parting tool in off disks from a rod of soft steel 1% in. in diameter, a a constant speed and feed and as lubricants a of lard oil or of mineral oil of the same viscosity. 30 were made with lard oil and at the end there was no M f I 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 6 is being cut away by the tool c, a chip / 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. Per 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 TO, 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, i.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 1R 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: " 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 i i; l The walls of the aeroplane motor, the crankshaft et cet. are made so ,1( light that the seizure of a siagle bearing will result in the wrecking of the LUBRICATION 275 Benzene (C6H6) represents a typical paraffin-base hydro- carbon, diallyl (C6Hi0) may be taken to represent an unsaturated non-cyclic hydrocarbon,, whereas benzene (C6Hg) and hexa- methylene (C6H12) represent types of cyclic hydrocarbons. All of these compounds have the same number of carbon atoms, but 10° 40° 50° 60° 10° 80° Temperature, Centigrade 90° 100° 110° Pro. 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 tfaie 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 tb.e 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 the paraffin is strikingly shown by introducing a paraffin residue (CH3) into the benzene ring, which results in toluene (C6H8) having a higher fluidity than benzene (C6H6). On the other hand, toluene has a much lower fluidity than the purely paraffin compound heptane (C7Hi4) which contains the 100 ?00 800 900 1000 300 400 500 600 700 Vapor Pressure in mm FIG. 89.—Fluidity-vapor-pressure curves of hydrocarbons of different homolo- gous series. (Of. 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 hexarnethylene 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.—AVEEAGE FLUIDITIES AND VAPOR PBESSUBES FOB CORRE- SPONDING TEMPERATURES Tem- Toluene1 Benzene2 Hexamethylene3 Hexane4 pera- ture, Vapor Vapor Vapor Vapor degrees 9 pres-

pres-

4 and NaaSt^.lOHsO-1 Further work along thinlino in zioodc.d to differentiate the effects of chemical com posi- tion, ttMwtitution, and juwociation, inniHiiring the iluiditioH over a ranfcfc of tcmpomturcH. AH in other linos of physical chemical investigation, the* importance of making doterininatioiiH at more than one temperature can hartlly be overestimated because «ul)8tunc«H must be compared under conditions which are truly coin parable*. Various colloidal solutions Rueb as those of rubber, glue, •vis- COHO, nitrocellulose,, dextrine*, gluten, et cet., offer problems of importance which can bo most appropriately solved by the vi»- tjornotcr. 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 HO! to the plasticity of the gel is practically a dosed book. To indicate how complex the phenomena may be, we may add that Carl Berquint of the Corn Products Refining Company han found in an investigation of corn dextrines, tapioca dextrine*, borax, gums and starches that as the percentage of doctrine mereiweB during the process of conversion, the mobility steadily risen whereas the* friction first falls, then rises, and again falls/1 The quick netting of a gurn Boenw to bo associated with a high friction. TliUH the addition of .25 per cent »odium 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 Hturch will set harder and have "better body" than an acid starch. Nitrocellulose Solutions.—The fluidities of nitrocellulose solu- tions a« calculated from the determinations of Baker (1913) would indicate that nitrocellulose solutions never become true solids as the percentage of nitrocellulose in increased, for the fluidities approach the ^ero 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 viecomoters in order to get the necessary range, and each one is calibrated from another, the possibility of error is considerable. *(?/. Dunfttan and Langton (1012). •Privately communicated. Cf. Herschel and Bergquisfc (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 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 it 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 I f 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 2H2 FLUIDITY AND PLASTICITY material entails IOHH physical and mental fatigue on the part- of the* plasterer, and ho in thereby led unwittingly to produce u better quality of work/1 Rmley pointB out that the method of Hlaking the lime has much to do with the development of pla«t,ieity, but that quite ae impor- tant IK the* Hotiree, and by informer, the chemical cornponition of the lime. A lime high in mapicBimn oxide is capable of develop- ing a high plantleity more readily than one which is low in the dolomttie oxide. The growing practice of buying Ohio finishing lime, already hydra ted, even when local lime may be purchased for about one-half the price is a reflection of the above facts and IB a demonstration of the indiiHtrial importance of plasticity. In handling roud-building and roofing materials, a knowledge of the principles of plastic flow might enable UB 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* as high as practicable. The "metal" of the rail- road is made up of coarse crushed stone of uniform size which glve« excellent drainage and a very high yield fxrint. Where liquid hydrocarbons are used as binder, a considerable amount of fine! material must be UBcd in order to raise the yield point sxiffi- cictntly 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 hut the mobility must also bo high so that the painter may spread it without undue fatigue. Other thingn being equal, these ends are, both achieved by the u»e of finely-divided materials, and at the time the covering power is augmented. Perrott (1919) hm a study of the plasticity of "long" and "short" carbon blacks. tip to 1914, Austrian ozokerite was thought to be essential in the wax uned in making electrotypes. Research has shown that a good impreftBion can be obtained and held with waxes which do not contain the Austrian material TeictHes and Belting.—If a cotton window cord is ran 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 it- If 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 the organs of a frog. When blood flows through the capillaries the 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. 1 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 ri the milk of a cow, goat, or horse but does not affect the fluidity of P woman's milk. According to Alexander1 human milk contains [ a protective colloid not present in cow's milk, hence coarse curds ^ are not formed on adding acids. I The action of ferments upon milk has been studied by Gutzeit u (1895) and Fuld (1902). The decline in the viscosity of a solu- I 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 Bayliss (1904). This is in accordance i 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 1 Zanda (1911) investigated the changes in viscosity during diges- I tion by pepsin. ) Ceramics and Glass Making.—The thorough mixing of glass t melts, the removal of bubbles of gases, and the pressure necessary ! to blow the glass at a given temperature all depend upon the i fluidity of the melt, hence the control of the fluidity of glass t melts is of importance. 1 j The Seger cone method of determining temperatures suggests f the possibility of measuring high temperatures by the viscometric method. Barus proposed to use the viscosity of a gas for this purpose. The manufacture of porcelain is concerned with the principles 1J. Soc. Chem. Ind. 28, 280 (1909). APPLICATIONS OF THE VISCOMETJRIC 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 cryptocrystalline, 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 hi 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 'soF pass into the plastic 'gel' at a perfectly definite concentration and temperature ? " ArisJZ (1915) Las 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

90 19.0 20.3 21.7 23.1 24.4 25.8 27.1 28.5 29.8 31.2 g 60.8 7 1.0 91 32.5 33.9 35.2 36.6 37.9 39.3 40.7 42.0 43.4 44.7 o 8 l!l 92 46.1 47.4 48.8 50.1 51.5 52.8 54.2 55,6 56.9 58.3 o 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 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 IV.—VALUES OP N. COEEECTION IN PRESSURES (GRAMS PER SQUARE CENTIMETER) FOR VARIOUS TEMPERATURES AND MERCURIAL HEIGHTS Temperature, degrees Centigrade Height of mercury, centimeters 10 20 30 40 50 60 70 80 90 100 5 0.4 0.7 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 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 - .0 -1.2 —1.5 -1.7 -2.0 -2.2 -2.4 31 -0.3 -0.5 -0.8 - .1 -1.4 -1.6 -1.9 -2.2 -2.4 -2.7 32 -0.3 -0.6 -0.9 - .2 -1.5 -1.8 -2.1 -2.4 -2.6 -2.9 33 -0.3 -0.6 -1.0 - .3 -1.6 -1.9 -2.2 -2.6 -2.9 -3.2 34 -0.3 -0.7 -1.0 - .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 Difference in temperature between mean temperature of emergent steam and bath. Corrections in degrees to be added to the observed temperature exposed 30° 40° 50° 60° 70° 80° 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 0', 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 0', 0", etc. are closed. By allowing air to pass very slowly through the valve D the air will be gradually forced down the tube Hf 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 Jv jiv jm an(j ju jri jn or(jer 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. APPENDIX A O I 2. 34- 5cm. \ FIG. 93.—Details of bath, frame, and clips for holding viscometer. 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- rected simply for the buoyancy of the air gives the correct 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 \ FIG. 94.—- A pycnometer for liquids. 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 The formula to be used in obtaining the density is: 309 WQ WQ where w' = weight of liquid at t°C, WQ = weight of water at fC, po = density of water at £°C. The liquid is introduced or removed from the pycnometer by I P \ i|:J !' ! ' ' ' I IwK ii !!vif TOSUC7JON 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 II li I 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 HO, Fig. 29, is graduated in millimeters. By filling the viscometer with mercury from A to (?, and weighing this mercury, the work- ing volume V can be actually determined. And by filling a given length of the capillary HG with mercury, the volume vf 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 s0 at temperature t0 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 s must be = i (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 (cf. 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 if. TABLE VIII.—LAFAYETTE COLLEGE VISCOSITY RECOHD Page I Substance Pure Water Remarks Calibration W. G. K. Date Observer .0.46 Viscometer No. JL, Pycnometer No. 2 C' Log C' = 8.37598-10 Log ~- =6.22122 ~~~Q Time Manometer , upper read- Temperature bath Limb Min- Sec- Time, seconds ing, lower reading Temperature Sum difference = Weight pyc. utes onds ho Start Finish 20 L 5 7.0 307.0 259.46 259.46 21.2 287.58 28.12 231.34 20 R 5 8.2 308.2 259.48 259.48 21.1 287 . 60 28.12 231.36 P hip K ±hi PK±L Po Cip C*2 P t\ in cp V V Remarks 0.9982 + .44 -.79 -.35 230.99 1.76 229.23 1.0050 99.50 water mano- -.44 -.79 -1.23 230.13 1.75 228.38 1.0052 99.48 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 I = 7.5 cm hence Cf = 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 X U Calculating now the hydrostatic head, using this value of C, we have from Eq. (5) , hi = 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 C = 0.1005 X 307 + 0.02377 X 0.998 231 X 307 X 307 = 1.428 X 10~7. 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 Cpt - C'p/t = CPT C'P P = 7>-^ (U) Ofc which is evidently the pressure consumed in overcoming viscous resistance solely. In this case 7^ = 1.76 hence P =229.2. The fluidity

__ ri where #3 and JR4 are the average radii of the two ends, e = n _, x? and 25 and 2C 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, I, hence the pressure (h + l)p added to pi, 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 TJ-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 plastoraeter occording to the latest design used by Mr. H. D. Bruce is reproduced in the frontispiece. TREATMENT or PLASTICITY DATA The data may be analyzed either algebraically or graphically. The formula for plastic flow through a capillary tube is 1 _ irRH mpV M~ 4 where ^ is the mobility, and / the friction or yield value. The 7?P/y 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 Kv /1Q\ A* = j—j (13) where v is the volume of flow per second and K is a constant whose R* value is 384.8 -y. If we substitute in Eq. (13) the values Fiy vi and F^ v<> from two observations of the flow, we find that = so that both ja and / are readily determined. Since however the weight of flow w = vp, a more convenient expression for the friction is - 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- ff^ cometer in Great Britian, the Saybolt Viscosimeter in the United J States, the Barbey Ixometre in France and a host of others. Most I of them seem to have been devised with the idea in mind that the I 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 2-^-f P * 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 rj/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" (cf. Ubbelohde (1907)), and Redwood (cf. 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. 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- Sectional View of Standard Oil Tube. A Oil Tube Thermometer. B Bafh Thermometer. C Electric. Heafer. Q Turntable Cover. £ Overflow Cup, F Turn fable Handles. 6 Steam Inlet or Outlet. H Steam U-Tvbe. •J Standard Oil Tube. K Stirring Paddles. L Bath Vessel. M Elecfric Heafer Receptacle. N Outlet Cork Stopper.. P Gas Burner. Q Strainer. R Receiving Ffask. \ S Base Block. 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 Ry 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 tube3 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 I7, and all oil entering the standard oil tube shall be strained through a 30-mcsh 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 eup of the stand- ard oil tube. "(&) The standard oil tube / 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. . . Length of outlet tube .......... 0.1750 1.215 0.1765 1.225 0.1780 1 235 Height of overflow rirn above bottom of outlet tube ..... 12.40 12 50 12 60 Diameter of container of standard 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-pomt (open 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. j[? It is difficult to adequately clean this type of instrument or to |l 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.rum.ent has considerable range, for wires of differing diameters caix 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 (cf. Herschel (1020)). For liquids of high viscosity, the falling sphere method is used industrially. If the containing vessel does not have a diameter a/t least 10 times that of the ball, a correction must be applied Slxeppard (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 fliiids 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 liave been accurately determined should be available for tlie 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 X10"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. "5 3 t w ^ K fe P *5 Designation of tube 0.955 1.575 2.555 5.11 7.58 ^ 10. 05 Length in centimeters a 0 d o 0. 01430 0. 01425 0.01420 ? 0. 1415 Major axis Open end Diameter of capillary in centimeters d o d p 0.01405 0. 01405 0. 01400 ? 0. 1395 Minor axis d o d o d P d p d o ^ 0. 1430 Major axis Bulb end d p d p d o d p d o ? 0. 1405 Minor axis d p d p d p d o 10° 4».^COCOtOtOi-lH-« OiOCnOCnOCnOCnO M Temperature of experiment O^MMQOOOOOi ° d o a P d p d 0 d o 13. 34085 Volume of bulb in cc at 10°C -* *a oo co ** o rf»- to rfx M CO 00 CO CO CO •sj £J |_l. (-a •^ oo (O rf>- to >f>. to Cn 00 rf^ 00 00 <® fr> -a co P^ --* c> to co i-4 to to oo co O M Cn CO »-« O» 00 !-• O fcO ^ bO Or Oi §O Cn W CO <£) c> O M CO CO *-* *-L OO Cn o to Cn 0 00 *• ^ CO ^ H- rf* tO tO 0 0 Cn >-* O (^ o a GO ^ o ~J t-« ^1 CO CO O> O> 0> O» Cn t5 rf^ rf^ 00 OOCO^OOOOCn^*>'b3tf^i--00 tOCnOiCn00tOcDrfxoeO*.i-»q^v-*ooo>co.o H-« fcS 01 bO CO -i CO O -» cn -i O Ot tt>- Cn s3s 00 M Cn Cn O M Cn I-* to oo co o 00 tO tO O> t-* H1 Cn <|COCOOi-'COCn-OCntOCnOCnOOtO * fi S" H 24.753 3,828.75 50. 001 1,923.75 >o 99.343 994.00 JLVI & Do. Do. Do. Do Do. Do. 148. 618 682.00 6 193.010 537.75 387. 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 e- 49.040 571.75 Av« O 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 10 ^ UD U3 388.256 4,103.5 J5 iO r-( rH rH r-l o 00 739.333 2,156.0 0 O O 0 O rH ^ 777. 863 2,060.0 CD O 6 0 6 «o 55. 286 21,430.0 o 97. 922 12,079.0 »0 rt* 2 148. 275 7,981.5 B1 s 0 O Do. Do. Do. Do. 193.947 6,100.0 t>: d o 387. 695 3,052.0 739.467 1,600.0 774. 891 1,526.5 99. 163 7,804.0 1C 2J N 149.679 5,165.0 fin CO 0 0 Do. Do. Do. Do. 193.441 3,997.0 •"*' o 0 387. 130 1,995.0 774. 796 999.0 49.091 7,471.0 >o CO CO 98.315 3,729.0 £111 in 2 Do. Do. Do. Do. 148.571 2,473.0 CO 0 o 193. 877 1,892.0 co' 0* 6 " 388. 100 946.0 774. 880 473.0 1 3 0 a ^ to Designation of tube 4.97 7.495 10.0325 10. 0325 0.39 0.9000 Length in centimeters 0. 0085 0.00850 0.00845 0.01145 0.01144 Major axis Open end Diameter of capillary in centimeters 0. 0084 d p 0.00845 S PS- 8 0 •- * 0.01125 0.01124 Minor axis d p d p 0.00860 0 0 0 8c$& B d p d p Major axis W cr 1 d 0 d p 0.00850 d o d o Minor axis d p d p 10° rf^)^COCObOh3i-«(-' CnOCnOCnOCnOOCnO d P d p Temperature of experiment d P d p 2.1057 2.1057 d P d p Volume of bulb in cc at 10°C -a oo H-« »-« -q 00 CO £>• «O )&. CJ CO CO CO M *4 CO J-« H-* <1 CO 00 CO rf^ CO Cn £•• OO ~4 CO CD 00 IO •-* H* •^ 00 CO (^ CO ^ K> £k ^ 00 • )-« rf»- H-» Cn *4 M tO £»• t-« (-1 •>! §CO i-» O5 Cn CO 00 OJCSCnCnOOi— '•OOOOOO -a.rfi. OOC5rfi-COi-'OOOOO*aOO INP O »-" O I-1 Cn tO oo to co co co -a co to rf^ o oo CD oo o Cn O H* O tO 00 -^ >£>• tO CO 00 i-i Cn Oi O >P>. rf^ tv3 tf^ -> to )-* t-« CO OS 00 U> CO N> tO £- H_i t_t !-> h-i h-i 1-1 t\5 tO tO L\5 i— W H-i *-* CO rf^ 0 H* CO >-* 00 rf*. K) tO CO 00 0 CD • 00 H-« 00 05 I-1 tO Cn i_* CO -^ CD ^ rf*-CD Cn 00 Cn O to CO Time of efflux of volume of bulb in seconds o o *-• co CD 10 O H-« G5 CO CO On *-« to o to )— CO i-i Cn CD 00 CO O> 00 O 00 O Oi O H-1 h5 CO CO H-i CO 10 O cooi-»to>f>-o>ooocotii.oo tli.rfi.rfxoobOi-itOCOOitOH-1 COtOCOO3-«4'cnOi rf».rfxCn£-Cn»^.COrf>>tf».»*>.C»fcOCDCOCntf>. M M CO M CO 00 >-» CO <| ><». CO «O O rf*- <1 CO t-i t-» • to >P» ->l rf^ 00 00 CO rt»- •^ CO H* >-» <| 00 CD rf»- CD >£>• rf>> OOO>CncDfcOC7i««JOQC>cn «O H-> tO g CO rf* W bO tO l-1 H* Or H« fcO CO i-« >-» O O <£> O >-* t-t t-1 Cn CO CO . en co l-« CO »-* tO rf^ Qi oo H* Cn O O> »*»• 00 -^ H4 H1 CO O to co >-» SCO 00 to zn -4 00 CO 00 CO CO 00 O i— to co -q rf*. to "co tn "oo "-3 00 Oi t4». 00 CO S Time of efflux of volume of bulb in seconds ^toCOOi*a«OH-«cnOOOiOOOOi^tf>-COil*-^JCOi-itoeOOcOOfcO-4loi-'eo<35 OOa>CntOCOOOCOO)OQO>Cni-'t-*~400O 00 O | Diameter of capillary in o o "o i JD "S centimeters X .2 * •!•» 1 0) r> ;d £> *o d d .2 d OJ o Open end Bulb end "S o f-, •9 jo 1 § .2 g II .»-> oS .5 (A rH £) O 98.917 10,149.00 N O o CO 147.857 6,789.00 £>" rH I i Do. Do. ^ Do. 193.485 5,178.00 N d d 386.847 2,589.50 773.985 1,293.00 50. 374 7,978.00 CO CO 97. 124 4,136.00 X>m IO CO o Do. Do. Do. Do. 148. 248 2,706.00 8 § 192.707 2,084.75 d d d 387.419 1,038.00 775.866 519.00 23.884 5,479.00 50.276 2,611.75 "5 w 97.440 1,373.00 DIV CO CO I 8 Do. Do. Do. Do. 147. 889 897. 50 d d d 193.459 697. 00 387. 062 349.00 772.117 176.00 0 CO 58.211 26,625.00 CO S 8 S 386.218 4,020.00 S CO 8 8 • 1 1 2 OJ 737. 829 2,103.00 d d d d d 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 45.00 774. 635 908. 75 g M 96. 693 5,903.50 CO Oi 147. 588 3,868.00 E1 S 8 1 Do. Do. 10° Do. 193. 100 2,955.00 d d d 386. 787 1,469.00 773. 880 736. 75 24. 301 5,661.00 49. 994 2,751.00 En 0 Do. Do Do. Do. Do. Do. 96. 123 1,426.00 * £ 126. 92 1,623.00 303. 08 678.00 a rt a & o d *o '•§ *O J-H 1 g x "* d d Open end Bulb end 2 S d § 3 ft IB ^ o '•& 0 d 5 -p d d ®o •2 a QJ CP " 0 J3t i rCj W) d »4 0 OB •3 -a ll 1 2 *s? *a s B •5 a II 1§ d ^ *o "ti ilo S 1°° *O *S 111 Q J S * S - s * s * H & S> - £ a ^ £ P 0 74.29 114.00 83.89 130.00 162. 89 63.00 329.39 39.00 FV 1 Do. Do. Do. Do. Do. Do. 653. 49 25.00 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 s o TH 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 e— 18.90 1,019.870 323.00 £i 0 Do. Do. Do. Do. 18.80 Do. 2,014.160 162.00 s 18.70 3,437.360 97.00 18.80 6,841.370 48.75 18.80 10,191.540 33.00 o § ?H 19.25 • 269. 220 597.00 £ii § i CO O CD O 19.30 Do. 518. 940 305. 50 s d d d d 19.50 1,019.670 155.00 19.50 2,014.360 79.75 CO CO 8 11.00 2,316.870 9,048.00 # 8 8 Q 11.00 0.5 3,837.000 5,438.00 CO* o" d 11. 10 6, 117. 600 3,460.00 10.80 3,850.160 388. 00 o 10.90 4,610.230 319.00 J o 11.00 1.0 5,370.130 267. 00 iO d d 11.00 6,127.360 235.00 7.50 6, 130. 080 261.00 CO o 11.00 54.987 8,590.00 o i— i CO i—t CO 11.00 210. 129 2,250.00 /C o "<# 0 o 11.00 1.0 419. 645 1,125.75 CO CO d d 11.00 835. 565 565. 00 12.00 1,576.000 286. 00 22 338 FLUIDITY AND PLASTICITY TABLE I.—(Continued) s 1 Diameter of capillary in o "o o •§ a centimeters 2 fl a % > S *o d *0 a .2 ? § Open end Bulb end 1 11 •a 1 11 1 bO '5 *3 "S a 4 -a 1-3 s „ 'cf'fl I- a .5 x e8 fl § | ft.g S 1H "3 ^ Ilo S | o ill Q ^ § * s * S * S * H a >• * fi S 3 ° 11.00 1.0 2,338.376 197. 50 CD O 11.00 3,095.540 154.00 8 CO CO 11.00 3,856.939 123.00 Z O o 11.00 4,616.534 106.25 8 O o 11.00 5,376.534 88.25 11.00 6,136.534 77.50 7.00 6,136.534 86.75 OS 0 O5 0 O5 o M S oo 8 8 8 8 10.00 i 775. 000 1,240.00 *"* o o 0 0 o jn § Do. Do. Do. Do. Do. Do. 775. 000 84.50 0 TABLE II.—VISCOSITY OF WATER IN CENTIPOISES AS DETERMINED BY DIFFERENT OBSERVERS Temperature Poiseuille Sprung Slotte Thorpe and Rodger Hosking Bing-ham and Wnite Average Calculated by formula A° C D' E o 1 7755 1 7900 1 7944 1 777 1.807 1 . 7766 1 7928 1 7960 1 7887 1 7921 5 1.5108 1.5137 1.5143 1.5142 1.5089 1.523 1.5083 1.522 1.5241 1.5155 1.5188 10 1.3045 1.3078 1.3088 1.3088 1.2995 1.313 1.3014 1.3105 1.3002 1.3061 1.3077 15 1 . 1385 1.1464 1 . 1465 1.1456 1.1334 1.143 1 . 1324 1.142 1.1373 1.1406 1 . 1404 20 1.0028 1.0073 1.0063 1.0087 0.9978 1.007 1.0005 1.006 1.0054 1.0046 1.0050 25 0.8900 0.8964 0.8966 0.8973 0.8947 0.895 0.8900 0.8926 0.8940 0.8941 0.8937 30 0.7958 0.8016 0.8011 0.8027 0.8183 0.802 0.7965 0.800 0.7991 0.8019 0.8007 35 0.7154 0.7194 0.7190 0.7207 0.7216 0.723 0.7190 0.724 0.7223 0.7205 0.7225 40 0.6466 0.6523 0.6508 0.6531 0.6558 0.656 0.6525 0.657 0.6557 0.6533 0.6560 45 0.5867 0.5934 0.5937 0.5932 0.6001 0.601 0.5959 0.600 0.5984 0.5958 0.5988 50 0.5512 0.552 0.5464 0.5500 0.5491 0.5497 0.5494 55 0.509 0.5044 0.508 0.5073 0 5072 0 5064 60 0.471 0.4676 0.469 0.4728 0.4701 0.4688 65 0.437 0.4343 0.436 0.4362 0.4359 0.4355 70 0.407 0.4048 0.406 0.4069 0 4062 0 4061 75 0.380 0.3782 0.380 0 3794 0 3794 0 3799 80 0.356 0.3547 0.356 0 3558 0 3556 0 3565 85 0.334 0.3336 0.335 0 . 3337 0.3341 0.3355 90 0.315 0.3140 0.316 0.3133 0.3146 0.3165 95 0.297 0.2970 0.300 0.2983 0.2981 0 . 2994 100 0.281 0.2814 0.284 0.2821 0.2838 1 to CO CO CO 340 FLUIDITY AND PLASTICITY TABLE III.—FLUIDITY AND VISCOSITY OF WATER CALCULATED BY FORMULA1 FOE EVERY DEGREE BETWEEN 0° AND 100°C Temperature, °C Fluidity Viscosity in cp Temperature, °C Fluidity Viscosity in cp Temperature, °C Fluidity Viscosity 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 20.20 100.00 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 2.1482{(t - 8.435) + VS078.4 + (t - 8.435)2} - 120. f Viscosity in the Variation of the Action of Invertase According to the Conductivity of Saccharose. 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Uber die Viskositat des menschlichen Blutes bei Schwitz- 11 proceduren. Deutch. Arch. f. klin. med. 80, 308 (19—). | i LORENTZ. Verlagen der Akademie Van Wittenschappen te Amsterdam | j 6,28(1897). If, LORENZ, R. & KALMUS, H. Die Bestimmung der inneren Reibung einiger * I 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. LUDWIGJ C. & STEPHAN, J. tJber den Druck, den das fliessende Wasser senkrecht zu seiner Stromrichtung austibt. Wien. Sitzungsber. (2A) 32, 25 (1858); 18 pp. Cp. Stephan. LUDWIK, P. tlber 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 qualite* dans les gommes adragantes au moyen d'un viscosimetre tres simple. Bull. soc. ind. Mulhouse 66, 64 (1896); 9 pp. LUNGE, G. & ZILCHERT, P. Untersuchung der Zahflussigkeit von Gummi und Traganthlosungen mittels des Lunge*schen Viscosimeters. Z. angew. Chem. 437 (1895); 3 pp. LUSSANA, F. Sulla viscosita del latte. Bologna (1905). LUSSANNA, S. & CINELLI, M. L'attrito interne e 1'attrito elettrolitico nelle soluzionL Atti d. R. 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A, The Ion Mobilities, Ion Conductances, and the Effect of Viscosity on the Conductances of Certain Salts. J. Am. Chem. Soc. 43, 1217 (1921); 10pp. 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 Sperimental! agrarie italiane 37, 383. MAGNUS, G. (1) tlber die Bewegung der Flussigkeiten. 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, 0,674 (1913); 1 p. MAIN, J. Note on some Experiments on the Viscosity of Ice. Proc. Boy. 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 4=5, 126 (1888); 7pp.; (2) Experiments on Fluid Viscosity. Proc. Roy. Soc. London 59, 38 (1896); 2 pp.; Phil. Trans. (A) 187,41 (1896); 16pp. MALUS, C. (1) Etude de la viscosite" du soufre aux temperatures sup6- rierures a la temperature du maximum de viscosite". Compt. rend. 130, 1708 (1900); 3 pp.; (2) Recherches sur la viscosite* 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). MABEY. 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. MAEGULES, M. 29, tlber die Bestimmung des Reibungs- und Gleitungs- coefficientea aus ebenen Bewegung einer Fliissigkeit. Wien. Sitz- ungsber. (2A) 83, 588 (1881); (2) Wien. Sitzungsber. (2A) 84, 491 (1881). MARIE, C. Surtension et viscosit6. Compt. rend. 147, 1400 (1908); 2pp. MAEIOBT. A Liquid Passing through Another without Mixing. M6m. soc. sci. phys. nat. Bordeau 2, 51 (1886). MAEIOTTE. 1, Traite* du mouvement des eaux. Paris (1700). MABKOWSKI, 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 MARKWELL, 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, Contribute alia conoscenza delle emulsioni. Arch. d. Fisiol. 4, 133 (1907). MARTIN, H. M. Lubrication. Proc. Phys. Soc. London, II, 32, 11 (1919); 4pp. 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). MASSOTJLIER, P. 197, (1) Relations entre la conductibilite* electrolytique et le frottement interne dans les solutions salines. Compt. rend. 130, 773 (1900); 2 pp.; (2) Relation qui existe entre le resistance elec- trique et la viscosit6 des solutions electrolytiques. Compt. rend. 143, 218 (1906); 2 pp. MASTROBTTONO. Viscosity of the Aqueous Humor. Arch. Ottalm.-Centr. Augenheilk. Erganzungsheft (1908). MATHIE^, £). 14, Sur le mouvement des liquides dans les tubes de tres- petit diametre. 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 156, 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 viscosite* dans les phenomenes osmotiques et dans les ^changes organiques. Compt. rend. Soc. Biol. 53, 1138 (1901); (2) fitudes viscosim<§triques sur la coagulation des albuminoides du plasma sanguin par la chaleur. Compt. rend. Soc. Biol. 54, 367 (1902); (3) Coefficients de viscosite* du serum et du plasma sanguina normaux. Compt. rend. Soc. Biol. 54, 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). MclNTOSH, 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). McEJBEHAN, 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 tiber 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. 15, 596 (1914). MELLOR, J. W. Clay & Pottery Industries. Griffin & Co., London (1914). MENNERET, M. 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Viscosity of Binary Liquid Mixtures. Z. Physik. Chem. 95, 349 (1920); 29 pp. METER, L. 245, (1) tlber 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); 5pp. MEYER, L. & SCHUMANN, O. tJber Transpiration von Dampfen. Part II. Wied. Ann. 13,1 (1881); 19 pp. MEYER, 0. E. 2, 6, 29, 50, 79, 127, 242, 243, 251, (1) tJber 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) "Qber die innere Reibung der Gase. Part L Pogg. Ann. 126, 177 (1865); 33 pp.; (6) Do., Pogg. Ann. 125, 401 (1865); 20 pp.; (7) Do., Pogg. Ann. 125, 564 (1865); 36 pp.; (8) t)ber die Reibung der Gase. Pogg. Ann. 127, 253 (1866); 29 pp.; (9) Do., Pogg. Ann. 127, 353 (1868); 30 pp.; (10) tJber 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) Hydraulische Untersuchungen. Pogg. Ann. Jubelb. 1 (1874); (6) Bemerkung zu der Abhandlung von Dr. Streintz uber die Dampfung der Torsionsschwingungen von Drahten. Pogg. Ann. 164, 354 (1875); 7 pp.; (17) Beobachtungen von A. Rosen- cranz iiber den Einfluss der Temperatur auf der innere Reibung von Flussigkeiten. Wied. Ann. 2, 387 (1877); 20 pp.; (18) tJber 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 (1891 J; 14 pp.; (21) De Gasorum Theoria. Diss. Uratislaviae (1866); 15 pp.; (22) tlber 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) tJber 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) tJber die Bewegung einer Pendelkugel in der Luft. Do., 336 (1872); 12pp. MEYER, 0. & ROSENCRANZ, A. 6, 93, 127, 134, Cp. Meyer, Wied. Ann, 2, 387 (1877). INDEX 399 MEYEE, O. & SPEINGMUHL, F. Uber die innere Reibung der Gase. VI. Pogg. Ann. 148, 526 (1873); 30 pp, METEE, P. fc Apparatus for determining the viscosity of liquids. Ger. Pat. 244,098, June 16 (1911). MICHAELIS, G. tTber die Theorie der elastischen 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. 25, 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, 253, (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., MAEQUINA, M. & SANTOS, G. -Viscosidad y conductibilidad ele"ctrica 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. 1,20,23 (1920); 12 pp. MONROE. 105, Cp. Kendall and Monroe. MONSTEOV, 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. MONTEMARTJNI, C. The Relations between the Water of Crystallization of certain Salts and the Viscosity of their Solutions. Atti. A. Ace. 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); 14pp. 400 INDEX MOOBE, 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 liber 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); 12pp. 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); 27 pp. MtrCHIN, G. Fluidity Measurements of Solutions. Z. Elektrochem. 19 819 (1914); 2 pp. MUELLER. Romberg Deutsch Med. Woch. 48 (1904). MUNZER, E. & BLOCK, F. (1) Die Bestimmung der Viskositat des Blutes mittels der Apparate von Determann und Hess nebst Beschreibung ernes eigenen Viskosimeters. Z. exp. Path. Ther. 11, 294 (1913); Med. Klinik, #9, #10, #11 (1909); (2) Experimented 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, 0. 239, Plasticity of Ice. Nach. G. Wiss. Gottingen. 173 (1895); 3pp. MUHLENBEIN, J. t)"ber die innere Reibung von Nichtelectrolyten. Diss. leipzig (1901); p. Schettler's Erben (1901). Cp. Wagner. MULLER, A. "Uber 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, Wber innere Reibung von Flussigkeiten. Wied. Ann. 43, 15 (1891); 28pp. NAPIERSKY. Versuche uber die Elasticitat der Metalle. Pogg. Ann. Ergsbd. 3, 351 (1853). NATANSON, L. (1) tJber die Gesetze der inneren Reibung. Z. physik. INDEX 401 Chem. 38, 690 (1901); 15 pp.; Cp. Phil. Mag. (6) 2, 342 (1901); (2) tlber die temporare Doppelbrechung des Lichtes in bewegten reibenden Flussigkeiten. Z. physik. Chem. 39, 355 (1902); 9 pp.; (3) "Ober die Fortpflanzung einer kleinen Bewegung in einer Fliissigkeit mit innerer Reibung. Z. physik. Chem. 40, 581 (1902); 16 pp.; (4) t)ber die Dissi- pationsfunction einer zahen Fltissigkeit. 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) tTber 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). NAVIBR. 1, 29, (1) Me*moire sur les lois du mouvement des fluides. Me*m. de FAcad. roy. des Sciences de 1'inst. de France 6, 389 (1823); 52 pp.; (2) Me*moire sur r<§coulement des fluides elastiques dans les vases et les tuyaux de conduit. Me*m. de 1'Acad. roy. des Sciences de Tlnst. 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) tJber elastische Nachwirkung. Pogg. Ann. 157, 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. Kriiger. NEUMANN, F. 2, 14, 17, (1) Vorlesungen iiber 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. & BAUME", 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. Hajmodynamics. 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). 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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 Visconieter. Chem, Hev. Fett.-Hartz-Ind. 18, 272 (1911); 3pp. ONFRAY '» II 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 -f, 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 Silvernitrate, 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, el 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 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. Vis co meter, 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 Gtimbel, 413.8 Gurney, 377.2 Lunge, 394.5, 413.6 MacMichael, 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 Yield value, 217, et seq., 237, 257 Zero fluidity concentration, 54, 201, 203, 205, 220 Zinc-cadmium, alloys, 424.9 Zinc sulphate, 394.3 'v?Tr-—-<^~ j