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BOOK 54 1.34.B27 c. 1 

BARRER # DIFFUSION IN AND THROUGH 



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Digitized by the Internet Archive 

in 2009 with funding from 

Boston Library Consortium IVIember Libraries 



http://www.archive.org/details/diffusioninthrouOObarr 



THE CAMBRroGE SERIES OF PHYSICAL CHEMISTRY 



GENERAL EDITOR 

E. K. RIDEAL 

Professor of Colloid Science in the 
University of Cambridge 



DIFFUSIOiSr IN AND 
THROUGH SOLIDS 



CAMBRIDGE 

UNIVERSITY PRESS 

LONDON: BENTLEY HOUSE 

NEW YORK, tI'ORONTO, BOMBAY 
CALCUTTA, MADRAS: MACMILLAN 
TOKYO : JMARUZEN COMPANY LTD 

All rights reserved 



DIFFUSION 

IN AND THROUGH 

SOLIDS 



BY 
RICHARD M. BARRER 

D. Sc. (N. Z.), Ph. D. (Cantab.), F. 1. 0. 

Head of the Chemistry Department, The Technical 

College, Bradford ; forrherly Research 

Fellow, Clare College, Cambridge 



NEW YORK: THE MACMILLAN COMPANY 
CAMBRIDGE, ENGLAND: AT TttE UNIVERSITY PRESS 

1941 






PRINTED IN GREAT BRITAIN 



CONTENTS 

PAGE 

Foreword. By Professor E. K. Rideal ix 

Author's Preface xi 

Chapter I. Solutions of the Diffusion 

Equation ' 1 

Differential Forms of Fick's Laws 1 

Steady State of Flow 4 
One Dimensional Diffusion in Infinite, Semi-infinite 

and Finite Solids ' 7 

Solutions of the Radial Diffusion Equation 28 

Diffusion in Cylindrical Media 31 

Diffusion Processes Coupled with Interface Reactions 37 

Instantaneous Sources 43 



Treatment of the Equation — = ^ I Z) ^ I 47 

A Note on the Derivation of Diffusion Constants from 

Solutions of the Diffusion Equation 50 

Some Cases for which there is no Solution of the Diffu- 
sion Equation 50 

Chapter II. Stationary and Non-Stationary 
States op Molecular Flow in Capillary 

Systems 53 

Types of Flow .53 

Permeability Constants, Units, and Dimensions 60 
Some Experimental Investigations of Gas Flow in 

Capillaries 61 

Flow of Gases through Porous Plates 65 

Permeabfiity of Refractories 69 

Flow in Consohdated and Unconsohdated Sands 73 

Flow through Miscellaneous Porous Systems 78 

Separation of Gas Mixtures and Isotopes 78 

Non-Stationary States of Capillary Gas Flow 82 



VI CONTENTS 



PAGE 



Chapter III. Gas Flow in and through 

Crystals and Glasses 91 

Structures of some Silicates and Glasses 91 

Diffusion of Water and Ammonia in Zeolites 96 

Diffusion in AlkaK Halide Crystals 108 
Diffusion of Helium through Single Crystals of Ionic 

Type 116 
Permeabihty of Glasses to Gases 117 
The Solubility of Gases in Silica and Diffusion Con- 
stants within it • 139 

Chapter IV. Gas Flow through Metals 144 

Introduction • 144 

The Solubility of Gases in Metals ^ 145 

The Solubility of Gases in Alloys 158 

The Measurement of Permeation Velocities 161 

The Influence of Temperature upon Permeability - 162 
The Influence of Pressure on the Permeability of 

Metals to Gases 169 

Permeation Velocities at High Pressures 175 

Some Mechanisms for the Process of Flow 178 
The Behaviour of Hydrogen Isotopes in DiiBfusion and 

Solution in Metals 183 

The Influence of Phase Changes upon PermeabiUty 191 

The Influence of Pre-Treatment upon Permeabflity 192 

Grain-Boundary and Lattice Permeation 197 

Flow of Nascent Hydrogen through Metals 200 

Chapter V. Diffusion of Gases and Non- 

/ Metals in Metals 207 

Introduction 207 

Measurement of Diffusion Constants in Metals 208 
A Comparison of Methods of Measuring Diffusion 

Constants 219 

The Diffusion Constants of Various Elements in Metals 221 



CONTENTS Vll 

PAGE 

Degassing of Metals 226 
Diffusion and Absorption of Gases in finely divided 

Metals . 230 

The Influence of Impurity upon Diffusion Constants 234 

Chapter VI. Diffusion of Ions in Ionic 
Crystals, and the Interdiffusion of 

Metals 239 

Introduction and Experimental 239 

Types of Diffusion Gradient 245 

The Structure of Real Crystals 247 

Some Equilibrium Types of Disorder in Crystals 248 

The Influence of Gas Pressure upon Conductivity 251 

The Energy of Disorder in Crystals 254 
The Influence of Temperature on Diffusion in Metals 

and Conductivity in Salts 257 

The Identity of the Current-carrying Ions 265 
The Relation between Conductivity and Diffusion 

Constants 268 
Diffusion Constants in Metals and Ionic Lattices 272 
Diffusion Anisotropy 276 
The Influence of Concentration upon Diffusion Con- 
stants in Alloys 279 
Summary of Factors influencing Diffusion Constants 283 
Models for Conductivity and Diffusion Processes in 

Crystals 291 

Chapter VII. Strijcttjre-SensitiveDifpusion 311 

Types of irreversible Fault in real Crystals 311 

Non-equilibrium Disorder in Crystals 313 

Structure -sensitive Conductivity Processes 321 

Structure-sensitive Diffusion Processes 327 

Chapter VIII. Migration in the Surface 

Layer OF Solids 337 

Introduction 337 



VUi CONTENTS 



PAGE 



Evidence of Mobility from Growth and Dissolution of 

Crystals 337 
Evidence of Mobility from the Condensation and 

Aggregation of Metal Films 339 
Measurements of Surface Migration in some Stable 

Films 347 

The Migration of other Film-forming Substances 368 

A Comparison of the Data 370 
The Variation of the Diffusion Constants with Surface 

Concentration 371 

Phase changes in Stable Monolayers 374 

Calculation of the Surface Diffusion Constant 375 
Applications of Surface Mobihty in Physico -Chemical 

Theory 377 

Chapter IX. Permeation, Solution and Dif- 
fusion of Gases in Organic Solids 382 

Permeability Spectrum 382 

Structures of Membrane -forming Substances 385 

Permeability Constants of Groups A and B of Fig. 132 391 

The Air Permeability of Group C of Fig. 132 406 

The Solution and Diffusion of Gases in Elastic Polymers 411 

Models for Diffusion in Rubber 422 

ChapterX. PermeationofVapoursthrough, 

AND Diffusion IN, Organic Solids . 430 

Water-Organic Membrane Diffusion Systems 430 
The Permeability Constants to Water of Various 

Membranes 438 

Sorption Kinetics in Organic Solids 443 
A Modified Diffusion Law for Sorption of Water by 

Rubbers 445 
The Passage of Vapours other than Water through 

Membranes 447 

Remarks on the Permeation and Diffusion Processes 448 

Author Index 454 

Subject Index 460 



FOREWORD 

The general theory of diffusion is based upon analogy to the 
flow of heat through solid media, as is exemplified in the 
classical treatments of Fourier and also of Lord Kelvin in the 
Encyclopedia. In the actual process of diffusion of molecules 
and ions through and in soKds a whole set of new phenomena 
is observed. For example, hmitations on the magnitude of 
the diffusion potential are much more frequently imposed on 
these material systems by such factors as solubility or com- 
pound formation than are observed in systems m which the 
flow of heat alone is concerned. Again the flow of matter 
through solids is frequently composite in character, the 
various modes of transport being dependent on micro- 
heterogeneity, as is exemplified by lattice diffusion and move- 
ment along crystal boundaries, canals and capillaries, or even 
on molecular discontinuity as is the case when migration 
depends on the existence of molecular or ionic "holes" or 
vacant lattice points in a crystal. In studying the movement 
of material particles through a sohd, we must consider the 
nature of the interaction between diffusing material and 
diffusion medium. We may in a somewhat general manner 
note that either only dispersive or Van der Waals interactions 
are involved, or that electronic switches have taken place 
leading to chemi-sorption or chemical combination. Migra- 
tion across a surface or through a solid medium may thus 
involve movement across an energy barrier from one position 
of minimum potential energy to another. At sufficiently high 
temperatures in the higher energy levels activated surface 
migration naturally merges into free migration, with a con- 
sequent change in the temperature dependence of the 
diffusion. 

The mechanism of the transfer of material across phase 
boundaries likewise presents a number of novel and interesting 
problems. Here we have to consider firstly the abnormal 



X FOREWORD 

distribution of dififusing material at the phase boundary. We 
note that the Donnan membrane equilibrium may apply to 
electrolytic systems diffusing through a solid membrane, the 
Gibbs relation for a non-electrolyte or the adsorption iso- 
therm for a gas permeating a solid. Diffusion must then take 
place into and out of the boundary layer from the homo- 
geneous phases On both sides. Energies of activation are 
involved which may differ considerably from those required 
for the diffusion process in the homogeneous phases. Dr Barrer 
has been interested in these problems for a number of years, 
from the theoretical as well as from the experimental point of 
view. I pointed out to him that they were the concern of 
many who would find a monograph on the subject invaluable 
in their work, and I experienced a deep sense of satisfaction 
when my colleague assented to the suggestion of writing a 
book. 

E. K. RIDEAL 



Laboratory of Colloid Science 

The University 

Cambridge 



12 Fehruary 1941 



AUTHOR'S PREFACE 

Diffusion processes are related to chemical kinetics on the one 
hand, and to sorption and solution equilibria on the other. 
There are available excellent surveys deahng with chemical 
kinetics and also with sorption equiUbria. No previous 
text has attempted to correlate and summarise diffusion data 
in condensed phases, save briefly and in relation to one or other 
of these fields. It is apparent that the study of diffusion touches 
upon numerous aspects of physico-chemical research. There 
are in general two states of flow by diffusion— the so-caUed 
stationary and non-stationary states. From the former one 
derives the permeability constant (quantity transferred/unit 
time/unit area of unit thickness under a standard concen- 
tration or pressure difference) and from the latter the diffusion 
constant. The permeabihty constant, P, and the diffusion 
constant, D, are related by 

J. — -tv - , 

ox 

when dCjdx is a standard concentration gradient. One there- 
fore has to do also with the solubihty of the diffusing substance 
in the solvent. The aim of this book has been to study the 
permeabihty of materials to solutes, and the diffusion con- 
stants of solutes within them, and not specifically the sorption 
equihbria partly controUing the permeability. However, 
wherever the permeabihty and diffusion constants are dis- 
cussed it becomes essential to outhne also the data on solubihty, 
and this has been done throughout the book. 

In treating the data, the author has tried to keep a balance 
between experimental methods and their mathematical and 
physical interpretation; the hsting of adequate numerical 
values of permeabihty and diffusion constants which may 
serve as reference material, and as starting points for further 
investigations; and outhnes of current theories of processes of 



Xll INTRODUCTION 

permeation, solution and diffusion in solids. Many problems 
remain partially or completely unsolved, but if this book 
directs attention to them, and produces further work in a very 
fruitful field of research, it will adequately repay the time and 
trouble involved in its preparation. 

In Chapter i is given a number of solutions of the diffusion 
equation suitable for treating the various diffusion problems 
that may arise. The solutions are as exphcit as possible, so that 
they may be employed at once, or with the aid of tables of 
Gauss or Bessel functions. The purpose of this chapter is to 
provide ready-made integrals of the diffusion equation, thus 
avoiding too long a search through a scattered hterature for 
suitable solutions. Not aU experimentahsts have the mathe- 
matical training to derive at will solutions to suit the boundary 
conditions of their particular problem. Chapter ii is devoted 
to a survey of the different types of gas flow in capillary systems, 
from the study of which interesting methods of fractionating 
gas mixtures have been evolved. The pecuhar permeabihty ©f 
some glasses to certain gases has been known for a con- 
siderable time, and Chapter ni considers gas flow through 
glasses, and in crystals. These diffusions are much more 
selective than those considered in capillary systems, but they 
are not specific as are the processes of diffusion of gases through 
and in metals, discussed in Chapters iv and v. The uptake 
and evolution of gases by metals is a subject of technical 
importance concerning which a great body of experimental 
evidence has been amassed. These chapters, however, show 
that many problems remain unsolved. 

Chapters vi and vii describe the phenomena of conductivity 
and diffusion of ions and atoms in ionic lattices and metals. 
The first of these chapters has to do with reversible diffusion 
phenomena, and the second with irreversible diffusion pro- 
cesses depending upon the past history of the system con- 
cerned. Many of the problems discussed in Chapter vii have 
to do with diffusion processes down grain boundaries or 
internal surfaces, so that it is but a step to the treatment of 
numerous and interesting surface diffusions, described in 



INTRODTJCTIOISr XUl 

Chapter vin. Finally, in Chapters ix and x the problems of 
gas and vapour flow through and diffusion in organic polymers 
are considered. In these chapters a large number of perme- 
abihty and diffusion constants are collected for systems of 
technical importance. As examples one may cite the passage 
of gases and vapours through rubbers, proteins and celluloses, 
and of water through leather, or insulating substances such 
as vulcanite, ebonite, rubber, or guttapercha. 

Among many colleagues I wish to thank Professor E. K. 
Rideal, and especially Dr W. J. C. Orr, of the Colloid Science 
Laboratory, Cambridge, who read the proofs and made many 
valuable suggestions. I am glad to take this opportunity of 
thanking the University Press for their painstaking work, and 
numerous authors for permission to reproduce diagrams. 

E.M.B. 
March 1941 



CHAPTER I 

SOLUTIONS OF THE DIFFUSION EQUATION 

Differential forms of Fick's laws 

No adequate compilation of those solutions of the diffusion 
equation applicable to the diffusion of matter has as yet been 
made. The author has become aware of the difficulty of 
obtaining suitable solutions in a number of studies of diffusion, 
and it is hoped that this chapter will provide a source of 
reference for such solutions. Equations oY chemical kinetics 
can be used readily in differential or integral form, but the 
equations of diffusion kinetics require for their treatment a 
special and often laborious technique. Many of the cases which 
can arise have not yet been solved rigorously, though the field 
is a rich one both from the mathematical and the experimental 
viewpoints. The present chapter ^ims at giving some solutions 
of the diffusion equation in a form in which they may be applied, 
together with cases of diffusion systems in which the boundary 
conditions are those of the diffusion equation solved. 

Two familiar differential forms of Fick's laws of diffusion 
are i . n y^ 

Equation (1) gives the rate of permeation, in the steady state 
of flow, through unit area of any medium, in terms of the 
concentration gradient across the medium, and a constant 
called the diffusion constant D. The second equation refers to 
the accumulation of matter at a given point in a medium as 
a function of time. That is, it refers to a non-stationary state 
of flow. This second equation may be derived from the first, 
by considering diffusion in the + x direction of a cylinder of 
unit cross-section. The accumulation of matter within an 
element-of volume dx bounded by two planes, 1 and 2, normal 



2 SOLUTIONS OF THE DIFFUSIOX EQUATIOISr 

to the axis of the cyhnder, dx apart, may be estimated as 
follows : 

The rate of accumulation is 

which gives -m=^-d^^' 

In two dimensions the above equation becomes 

and in thiee dimensions 

dc ^d^c ^dw ^d^c 

or ~ = DV^G (5) 

dt 

dC 
or ^:— — D (divgrad) C, (6) 

ot 

as it may alternatively be written. All the equations (3)-(6) 
assume an isotropic medium, but if the medium is not isotropic 
one may simply write 

and then make the substitution 

to bring the equation to the form of (4): 

This transformation involves then a change of co-ordinates, 
after which methods suitable for the solution of (4) appl}^ 
also to (8). 

In many cases the diffusion constant, D, depends itself upon 
the concentration in the 'medium, e.g. the interdiffusion of 



SOLUTIONS OF THE DIFFUSION EQUATION 3 

metals (Chap. VI), the diffusion of water in certain zeoHtes 
(Chap. Ill), the surface diffusion of caesium, sodium, potassium, 
or thorium on tungsten ( Chap . VIII ) , or the diffusion of organic 
vapours in media such as rubber which swell during the 
permeation process (Chap. X). It is now necessary to solve 
an equation of the form 



dc 

~di'' 



d_ 

dx 



D 



dx 



H 1 D — 

9y L ^y. 



d 


~dC^ 


+ ^^ 


^ ^ 


dz 


L s^J 



(9) 



Equation (4) may be expressed in spherical polar co- 
ordinates r, d, and ^, by means of the transformation 
equations 

X = r sin d cos (j), ] 

y = rsiwdrnKf), j- (10) 

Z ~ T COS d. 



Equation (4) then becomes 



dt 



dr \ dr / sin i 



sini 



dd 



/ sm^ 



d'-C 



d d(/>'\ 



■ (H) 



Equation (11), for a spherically symmetrical diffusion, re- 
duces to 



ac 
'dt 



= D 



-d^C 2dG' 
dr^ r dr 



(12) 



for then dC/dd = and B^C/a^^ _ q. 

If one desires to express equation (4) in cyhndrical co- 
ordinates r, 6 and z, the transformation equations are 



X = r cos d, ^ 
y — r sin d, ) 

so that the equation becomes 



(13) 



ac' 
dt 



DVd i dc\ a /iao\ a/ ac\i 



Equation (14) again takes simpler forms in certain cases. In 
problems concerning the sorption and desorption of gases in 



4 SOLUTIONS OF THE DIFFUSION EQUATION 

and from long metal wires, for example, where end-effects 



are small, the equation reduces to 

= ^[^1(^^11. (15) 



dt \_rdr\ dr)} 



because dC/dd = 0, and d^CJdz^ = 0. For examples of the use 
of the fuU equations (11) and (14) text-books on the conduction 
of heat may be consulted (e.g. Carslaw, Mathematical Theory 
of the Conduction of Heat, Macmillan, 1921; IngersoU and 
Zobell, Mathematical Theory of Heat Conduction, Boston, 
1913). The equations (1)-(15) serve to show most of the 
differential forms of the diffusion equation which may be 
encountered in studies of diffusion kinetics. 

Various solutions taken from different fields of diffusion 
kinetics may now be given. In order, the solutions given will 
be for 

A. The steady state of jflow. 

B. The non-stationary state of flow: 

(1) Diffusion in one dimension— the infinite, semi- 
infinite, and finite solid. 

(2) Radial diffusion. 

(3) Diffusion in cy finders. 

(4) Surface reaction and diffusion simultaneously. 

(5) Instantaneous point, surface, and volume sources. 

(6) Diffusion when D depends on the concentration. 
The solutions given cover a great variety of diffusion 

systems, and with the aid of appropriate tables for error and 
Bessel functions may be readily applied to practical problems. 

A. The steady state of flow 

The diffusion equation takes particularly simple forms when 
the term dC/dt = 0, i.e. as much solute leaves a given volume 
element as enters it per unit of time. The equations of flow then 
become: 

d^C 
Through a plate: p^-^=0. (16) 



THE STEADY STATE OF PLOW 

Through a hollow spherical shell: 
Id^rC) 



r dr^ 

Through a cylindrical tube: 

I djrdC/dr) 
r dr 



= 0. (17) 



= 0. (18) 



The plate. Consider the diffusion of a gas through a plate 
of thickness I. If the boundary conditions are 

C = C\ 8bt X = for all t, 

= 6*2 at X = Z for all t, 

the solution C = Ax+B (19) 

of equation (16) may be further elucidated by putting x = 0, 
and X = 1, and eHminating A and B. This procedure leads to 
the expression ^ ^, 

where C denotes the steady state concentration at any value 
of X. The flow through unit area of the plate is 



where Cg denotes the concentration of the gas which has 
diffused through the plate into a volume F at a time t (the 
condition C = C^Sitx = I for all t naturally makes it necessary 
that Cg<^ Cg). Then the amount of gas which has diffused in 
time t is n n 

VCg=.D^-^^t. (22) 

The hollow sphere. For the diffusion of a gas through a 
hollow spherical shell of internal and external radii 6 and 
a respectively, with boundary conditions 

C = Oi at r =Ai"or all t, 
(7 = Cg at r = a for all t. 



6 SOLUTIONS OF THE DIFFUSION EQUATION 

the solution of (17) is 

Ci-C a{r-b) 



(23) 



C1-C2 r{a-by' 

where C is the steady state concentration at any value of r. 
The outward flow of gas per unit time and per unit area of 
the shell is then 

Over the whole area of the shell the expression is 

S = -Wi>(^)^^^ = 4.I>(C,-Q^^, (26) 

and the quantity diffused in time t, Q = qdt, is 

Jo 

nh 
Q = 47rD{C,-C,)j^-^^t. ■ (26) 

The cylindrical tube. The solution of equation (18) is 
C = A\nr + B, 
and if the boundary conditions are 

C = C^ at r = b for all t, 
C = C2 Sit r = a for all t, 

where b and a are respectively the internal and external radii 
of the tube, one may as for (20) and (23) eliminate A and B 
and obtain ri ^ /^ ^ 1 n ^ 

C = — ^lnr + — =^ . (27) 

In 6 — In a hi 6 — In a 

This leads to an expression for Q, the quantity wliich diffuses 
through unit length of the wall of the cylinder in time t : 

2nDiC Cn ^ (28) 

In a — mo 

The equations given will be found useful in evaluating the 
permeabihty constants of various types of membrane. The 
diffusion constants may also be evaluated when the absolute 
values of C^ and Cg are Iviiown. The permeability constant 



THE STEADY STATE OF FLOW 7 

may conveniently be defined as the quantity of gas diffusing 
per unit of time through unit area of the outgoing surface of 
a plate, cylinder, or hollow sphere when unit concentration 
gradient exists at that surface. Other equivalent definitions 
may be found more practical, and will be used in later 
chapters (e.g. Chap. II). The permeability constant as defined 
above is numerically equal to the diffusion constant. The 
dimensions are, however, different. 

B. The nox-stationary state of plow 

(1) Solutions of the linear diffusion law 

Solutions of the Fick law for diffusion in the x-direction 
may be divided into those for infinite, semi-infinite, and finite 
solids. The infinite solid extends to infinity in -!- and —x 
directions, the semi-infinite solid extends from a bounding 
plane at a: = to x — +o3, and the finite solid is bounded 
by planes dit x = 0, x — I and sometimes x ={l + h). All these 
solutions are exemphfied by physical systems, which will be 
illustrated in the text. 

{a) The infinite solid. 

We have to solve the equation 

dt ~ dx^ ' 

given that C = C{x,t), when t>0, 

C =f(x), when ^ = 0, 

f{x) can be differentiated when ^ > 0. 

The solution may be written as 

C = <^{x)F{t), (29) 

which allows one to separate the variables in Fick's law. 
^{x) and F{t) take the forms 

<^{x) =A cos kx + B sin kx. 



(30) 



8 SOLUTIONS OF THE DIFFUSION EQUATION 

where A, B, a and k are real constants. Any sum of solutions 
is also a solution, and a new solution is therefore obtained 
when the equations (30) are integrated over all values of k: 

/*oo 

C = [g{k) cos kx + h{k) sin kx] e-^^^^dk. (31) 

In (31) g{k), h{k) are functions chosen to fulfil the condition 
C = f{x) where ^ = 0, and Fourier showed that these functions 
took the forms 



(32) 



1 r+°° 

g(k) = - f{^') cos {x'k) dx', 

TTJ -co 

h{k) = - r^f{x')sin{x'k)dx'. 



Thus when the equations (32) are substituted in (31) one has 
as a general solution 

C = - e-^'D^dk f{x')cosk{x'-x)dx'. (33) 

TTJo j-oo 

This equation, by an integration process, can be brought to 
the form 

Case 1 . Suppose we have diffusion of a solute across a sharp 
boundary at a; = from a solution into a solvent. The solute 
may be a salt dissolved in water; it may be radioactive lead 
dissolved in lead diffusing into pure lead; or it may be an ion 
in an ionic lattice diffusing into another lattice, e.g. silver 
diffusing from silver sulphide into copper sulphide, while 
copper passes in the opposite direction. For the equation (34) 
to hold we know that D must not be a function of C; and that 
the amounts of solution and solvent must be great enough, 
the diffusion slow enough, or the time short enough, so that 
no appreciable amount of solute diffuses from the far extremity 
of the solution, or reaches the far extremity of the solvent 
(Fig. 1). In these system^, at f = 0, C = Cq^ot x<0, C = 



THE KON-STATIONARY STATE OF FLOW 9 

for x> 0. For all positive values of x, the concentration at 
time t and at a point x becomes 






^0 

2 



(35) 
(36) 
(37) 



1 2 3 4 5 6 7 a 9 10 




-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 



Distance from Boundary 
Fig. 1. Diffusion across a boundary from a solution into a solvent. 

The second term is the Gaussian error function, erf (?/), and 
may be evaluated from mathematical tables. Equation (37) 
may be expanded as a series: 



C 



Go 



b-i 



2 ^{Dt) 3 . 1 ! {2 V(i)^)}3 ' 5 . 2 ! (2 ^{Dt)Y 



7.3\{2^{Dt)} 
The solution for :» < is correspondingly 



^mr^-'ll' 



(38) 



Co 



1 + 



+ 



2 L \J7r\2 ^{Dt) 3 . 1 ! (2 ^{Dt)Y 5 . 2 ! {2 ^{Dt)} 



■7.3!{2V(D0} 



,-^]. 



(39) 



10 SOLUTIONS OF THE DIFFUSION EQUATION 

. Case 2 . It is necessary to have a readily applicable solution of 
Fick's law for systems in which two diffusion media are present. 
In many instances, when metals interdiffuse for example, a 
number of sohd phases occur, as alloys, with sharp phase 
boundaries. This is true when molybdenum diffuses into 
iron(i) at a temperature of 1300° C, for there is a diffusion of 
molybdenum dissolved in a molybdenum -iron alloy formed in 
the outer layer of the iron, and a diffusion of molybdenum also 
in the purer iron at the core of the rod. This diffusion system 
is complicated further by a movement inwards of the alloy -iron 
phase boundary. The simpler case of a stationary phase 
boundary can be easily treated by the methods already 
outlined (2), 

The boundary conditions are 

the second solvent. 

t \/Dr=o 



D'6 



I \ 


[W.oo 


\^=7 I 




\^^o ^"■^-^-__ 



T ' -^ ■ -7 ' ■ 7 ■ 2 A — ^ 3 
Fig. 2. Diffusion in two phases I and II where h — ?> and B^ = D^. 

^-0 




Fig. 3. Diffusion in two phases I and II where k = \ and D-^ = D^. 



' THE NON-STATIONARY STATE OF FLOW- 11 

(ii) Ci = (7o for ^ = and ic < 0; C^ = ior t = and a: > 0. 



/dx^ 



x=0 

solvents -for all t. 

(iv) r {c,-c,)dx 



dx. 



(iii) DA^^^\ = DcX -P I at the interface between t&e 

\ ox 'x=0 \ ^^ / x=0 



Codt = the total amount diffused. 



(v) Cg/Ci = k, the partition coefficient, in the equihbrium 
state. 
The solutions of the problem for a; > and a; < are 

lVD^/,^Ar"'"''e-.V^], (40) 




Fig. 4. Diffusion ia two phases I and II where k — 1 and D^^ = iD.^. 




Ficr. 5. Diffusion in two phases I and II where A; = 3, Di = 4Z>2, and for the curves 
° (a) sJiD^t) = 1, V(-Oi*) = 2, while for the curves (6) V(A<) = 0-5, V(A*) = 1- 



12 SOLUTIONS OF THE DIFFTJSIOlSr EQUATION 

Some solutions of the equations (40) and (41) are illustrated 
graphically (3) in Figs. 2-5. 

(6) The semi-infinite solid. 

The medium now extends from x = to a?= +oo, and the 
general equation takes the form 

[x — x'Y'' 



'^=v(^)r"/(^''-p[ 



4.Dt 



dx'. (42) 



Case 1. At the plane a; = one may generate a supply of 
solute, in which the rate of supply is any function of time. 
Instances where the supply is constant will be of the greatest 
practical significance. Examples would be the sorption of 
gas at constant pressure in a large amount of a solid* (oxygen 
in silver, hydrogen in palladium, nitrogen in iron, ammonia 
in analcite) or a crystal in contact with a quiescent solvent 
hquid. 

When the supply at the plane a; = is constant the boundary 
conditions are 

C — Cq at X = Q for all t, 

C = at a; > and ^ = 0, 

C = C{x, t) at a; > and ^ > 0,^ 

and the solution giving the concentration C at a point x and 
a time t is 

/v' ^ 

where 



^ 2^{Dty ■ 

* Neglecting for the moment the possibility of a rate controlling process at 
the interface gas-aolid. 



THE NON-STATIONARY STATE OF FLOW 13 

(c) Finite solid. 

We have been concerned hitherto with so much solvent that 
its amount for practical computations of D can be reckoned 
as infinite. Such a supposition hmits the apphcabihty of the 
equations given, for it will be much more usual to work with 
small amounts of diffusion media. The solutions now to be 
given will concern themselves with problems such as the 
following. A solute diffuses from a solution bounded between 
the planes x =0, x = h into a solvent bounded between the 
planes x = h and x = I. The concentration-distance-time 
curves may be measured readily enough and have now to be 
interpreted so that the diffusion constants, D, may be 
evaluated. The new solutions of Fick's law will apply to 
numerous cases of the interdiffusion of metals, and salts, so 
long as D does not depend on the concentration, and whenever 
the amount of metal or salt is hmited. 

Another group of solutions will be given for the problems 
involving diffusion into, through, and out of a slab bounded 
by the planes x = and x = I. This group has to do with 
the diffusion of gases or Hquids through membranes, organic 
(rubber, cellulose, gelatin, plastics) or morganic (metals, 
crystals, and glasses). It is necessary in applying the solu- 
tions that the surface processes should be sufficiently rapid 
not to interfere with the internal diffusion process. This 
is true when gases diffuse through organic membranes, or 
inorganic glasses and crystals, but it is not true of gas-metal 
systems such as Ha-palladium, to which solutions of the 
diffusion equation must be apphed with caution. 

The general solution for diffusion within parallel boundaries 
which obeys the conditions 

^'^ ^^^0' (ii) at^ = 0, C'=/(x)for0^x^Z, 

(iii) C = Sit X = and x = I for all t, 
may be written as follows: 

C = 7 S e ^"""''^ ^*sm^- f{x')Qin-—dx'. (45) 

^ n=l ^ Jo f' 



14 SOLUTIONS OF THE DIFFUSION EQUATION 

Case 1 . Diffusion from one layer to another may be treated 
by regardmg the system as being a single layer with im- 
permeable boundaries in which the distribution at time ^ = 

is as follows : 

fix) = Cq for <,'c < Ji, 

f{x) ^ for h<x<l, {l>h). 

TKe solution for this particular case is 

= C.(^ + ?£^e-<"'«-^'cos'*f^m^V ' (46) 
\l 7T I n I I I 

For any numerical ratio oih/l, equation (46) may now readily 
be expanded as a series, each term of which may be given a 
numerical value. These numerical values have been worked 
out when |Z = h by Stefan (4) and Kawalki(5) whose tables 
may be consulted. 

Case 2. For some purposes it may be necessary to evaluate 
D from measurements as a function of time of the total solute 
which has diffused across the boundary. This quantity, Q, is 
given by the equation 

From equation (46) one has, at x = h, 

g^--6oyEe sm^. (48) 

Thence, substituting (48) in (47) and integrating, 

Equation (49) may be evaluated for various ratios of h/l. For 
instance, when h/l = |, equation (49) reduces to 

n ^^o^fi ^ ^ 1 / D{2 m+l)^7rH \-\ ,,., 
^ = ^L'-^sJo(2^r^TT?^^P( 1^ IJ' ^''^ 

which may again be exjjanded as a series and readily employed. 

The various possibilities which arise when a solute at a 

constant concentration diffuses into a slab may now be con- 



J 



THE ISTOlSr-STATIONARY STATE OF FLOW 15 

sidered. The slab, as before, is bounded by the planes a; = 
and X — I, and the boundary conditions are 

C = 6\ at a; = for all t, 

C = C^ Sbt X = I for all t, 

C=f{x)att = OfoTO<x<l. 

The general solution of this problem is (6) 

I 



X 2 °° CV COS Ti-TT — C'l . rnrx 
I 



C = C^+{C^- Ci) y + -i; -^ sm --- exp i 



2 ^ . nnx 



7T 1 

exp[ 



Dn^n'^ t 

72 



Z2 



^ „ . nTra;' , 
/ (a; ) sm — — dx . 
i 



(51) 



Ca-se 3. When the slab is initially free of solute, and the 
concentrations of solute at the faces are C^ (at x = 0) and 
6*2 (at X = 1), one has the example of diifusion of gas at 
constant pressure into a membrane of soUd free of gas. Here 
f{x') .— and the solution is 

(62) 
If instead of the membrane being initially free of gas there is 
an initial uniform concentration, Cq, of gas in the membrane, 
the solution is 

"J 



^ ^ / ^v ^ ^ ^ 2 ^ Co cos nn — C\ . nnx 
C = C'i + (6'2-Ci)y + -i;-^ -'sm^-exp 

V 7T \ To V 



X 2 ^ Co cos niT — Ci 



TT ^=o(2m+l) Z 



nnx 
sm ^— exp 

L 



V 



D(2m+lY-n^r\ 



(53) 



Again, if C^ = for all t, and Cq = at ^ = 0, the solution is 

~x 



c = a 



2^ (_l)ft _ riTTx { Dn^n^t)! 

7 + -S sm^— exp ^ 

_l n i n I { P 



1 



(54) 



Also, when C^ = Cg for all t, and Cq = at Z = 0, the solution 
becomes 



C = Ci 



4 ^ 1 



7r„i=o2»w+l 



sm 



{27/1+ I) nx 



exp - 



D{2m+\fn^t 



Z2 



(55) 



16 SOLUTIONS OF THE DIFFUSION EQUATION 

The reader may work out other examples from the general 
equation ( 5 1 ) , simply by giving O^ , C2 and / (a; ' ) their appropriate 
initial values. Special cases of equations (52)-(55) follow with 
equal ease. For instance, in equation (55) the concentration 
at the mid-plane of the slab (x = |Z) is given by putting 
X = II and so getting 

(a=.. = ^.[l-,J^(2;;^exp) -, }J. (56) 

All the equations (52)-(56) have to do with the flow of gas or 
other solute into a membrane. The use of these equations 
presents no difficulty. They may be expanded term by term, 
and all terms save the first two or three can be neglected, be- 
cause the series converges rapidly. In this form the equations 
become of special interest in the field of sorption kinetics. 

Case 4. The desorption of gas from a membrane is an 
equally important example of the apphcation of equation (51 ). 
Suppose at the time ^ = the membrane contains a uniform 
concentration Cq of solute throughout, from a:; = to a; = Z and 
Cj = Cg = 0. The solution in this case by substitution in (51) 
becomes 



^ 2Cn ~ . nnx f Dn^nH~\ C^ . nnx' , , 

4Cn ^ 1 . (2m+l)7rx F D(2m+lf7TH~] 

= —^2 Hy — rr\Si^^ r^ — ^-'^P — ^ — jt"^ — • 

7T ^=o(2w+l) I ^\_ l^ J 

(57) 

At the midpoint of the membrane (x — U) equation (57) 
reduces to 

Should the gas desorb from the slab not into a vacuum, as in 
(58), but into a gas atmosphere where at a: = and x = I the 
concentration is C^, the solution of (58) becomes 

C-C+(C C)^ y "^"^^"^'cxnT ^(2^ + l)'^'n 
C-C, + {C,-C,)^J^^ i2m+l) "^pL 1'' I 

(59) 



THE NON-STATIONARY STATE OF FLOW 



17 



Case 5. In the study of sorption kinetics one frequently 
requires an expression for the amount of gas (or other solute) 
left in a membrane at any time t, or alternatively, for the 
quantity which has diffused out of the membrane. The total 
amount of sorbed gas at the time t is given by the expression 

Q^ = C{x, t) dx 
Jo 



4Cp p » 1 

7T Jo,.=o(2m+l) 



. (2m+l)7ra; 
sm -^ zj—^ — exp 

V 



D{2m+1)^7TH 



]- 



When the integration is carried out one gets 

D{2m+lf7rH 



Qi 



8^-_ 1 

7T^ a (27 



^(2.^+l)^^^PL- 



P 



1 



(60) 



(61) 



Corresponding to equation (61), the amount of gas which has 
been desorbed is 

■■I 



Q2 = Qo-\ C{x,t)dx 



la 



^--. 2 



1 



,exp 



D{2m+lf7TH 



|] 



(62) 



7r%,-^o(2m+l)2^ ^i l^ 

The corresponding problem of the quantity of gas which has 
been sorbed in the membrane after a given time t appears 
from either of the two relationships 

when the boundary conditions in the most general practical 
case are those which led to equation (53). The solution when 
Cq = is simply equation (98), if u-^ and u^ are the constant 
concentrations at a: = and x = I respectively. The solution 
for the general case (concentrations C^ at x = 0, Cg at a; = ?, 
and C = Cq at i = 0) is 



Q = l 



— h '"^0 



1- 



r 7r\h{^m+lY^^^\ 



( D{2m+lf7TH 



P 



(63) 



18 SOLUTIONS OF THE DIFFUSION EQUATION 

Case 6. If one is measuring the rate of flow of a gas (or any 
other solute) through a membrane in which the gas dissolves, 
there will be an interval from the moment the gas comes into 
contact with the membrane until it emerges at a constant rate 
on the other side. By analysing stationary and non-stationary 
states of flow it is possible to measure the diffusion constant, 
the permeabihty constant, and the solubihty of the gas in 
the membrane. Once more one may employ equation (53) to 
determine the intercept {L), in terms of D, I, and C, which 
the pressure-time curve makes on the axis of time. 

The boundary conditions are 

C = Ci at ic = for all t, 
C = C^Sbt X = I for all t, 
C = Cq for < a; < Z and at i = 0, 

giving as solution 

^ ^ /^ ^x^ 2°° Co cos nn — C-, . rnrx F Dn^7TH~\ 

4C'o ^ 1 . (2m+l)7Tx r D(2m+1)^7TH~\ 

+ — ^ S 7^ — rT\^^^~ 1 exp ^ j^ . 

77- ,„=o(2m+l) I ^L ^ J 

(53) 
Equation (53) may be diiferentiated, and one obtains at a; = 
idC\ 0,-0. 2", „ ^, r Dn^TTHl 

t w=o L t J 

But if the gas flows through a membrane into a volume V, 
the flow of gas is given by 

Substitute (64) in (65) and integrate between the Umits and t, 
and so obtain 

4C„Z :^ 1 /. r D{2m+\f7TH~\\ ,^^, 



THE ISrON-STATIONAEY STATE OF FLOW 19 

Equation (53) as ^-»oo approaches the Kne 
D 



^'-IV 



'ir rUl 2?^^/C, cos 7^77 CA 



+i)d 



^TT^i) Jo(2m+l)2j (^^^ 

^f//. ^x C'oZ^ (7J2 72-1 

= ^[(^.-^i)^-^-^ + -^J. (67a) 

Had there been no time lag, Gg would be expressed by 

(^a = iyi<^2-Ci)t, (68) 

so that the intercept, L, on the time axis is given by 

{C^-Ci)\_6D^ SD 2DJ' ^^' 

In the case where Cq = at ^ = 

1 rC^P a PI 

and where Cq = and C^ is very smaU ( ~ 0) 

L- ^" 



(696) 



In equations (69), (69a) and (696), one is provided with an 
easy means of measuring D. 

(d) Finite solids with surface concentration a function of time. 

Case 1. In many experiments, e.g. the uptake of gases by 
sohds, the surface concentration is a function of time if 
sorption occurs at constant volume or variable pressure. The 
following considerations will apply to systems such as Og-Ag, 
Hg-Cu, Ng-Mo, NH3-, HgO-zeohte, where sorption occurs 
in a sheet of metal or crystal, provided there is no rate 
controlhng interface reaction. Then the boundary conditions 
are for the most general case: 

C = (f)-^{t) at a: = 0, 

C = ^2(0 8it X = I, 

C =f{x) whent ^ 0, 



20 SOLUTIONS OF THE DIFFUSION EQUATION 

and the solution becomes (7) 

'I 



rexpr:?^^]{55i(i')-(-l)«5i2(«')}d(j • (70) 



nDn f ^ 

+ ■ 



I 

When the slab is initially gas free and 0i(^') = 02(^') ^^^ ^9^^' 
centration becomes 

2 ^ -. (2m+l)7Tx(2m+l)D7T 
C = - H sm^ ^ J 



X exp ^ j^ — ^ '-\(j){t)dt , {tl) 



and Q, the amount sorbed in unit area at time t, is 

Q={'c{x,t) 

If in the interval ^' = to Ht is possible to represent (j>{t') by 

9{t') = C^{l+At' + B(t')% 
where Cq is the initial surface concentration and A and B are 



constants, one obtains 



2 ^ ' ' 71^ „,^o(2m+l)2 

VA 2Bt -IB /, A 2B\ H ,„, , 

|_a a a" \ oi cc- J J 

i)(2m + l)27r2 



where a = 



P 



It is possible, however, to solve the diffusion equation 
completely for this type of sorption problem, without recourse 
to empirical expressions for ^{t'), as the method of case (2) 
will show. 

Case 2. If one has a plate enclosed between the planes x = 
and X = 1 with the face x = impermeable, and an initial 



THE NON-STATIONARY STATE OF FLOW 21 

concentration Cq of solute in it ; and if another plate enclosed 
between the planes x = I and x = h + l is made of the same 
material with an initial concentration zero, then the solution 
of the problem of flow from one plate into the other is given 
by equation (46). 

Suppose, however, that the solute in the plate between 
X = I and X = h + l could be kept at the same concentration 
throughout by some process of stirring. The concentration in 
this plate would then be a function of time only. The diffusion 
equation in this instance is to be solved for the important case 
of desorption from a slab into a constant gaseous volume. 
Thus the desorption of water, ammonia, or another gas from 
a plate of a zeohtic crystal would be examples of diffusion 
systems of the kind postulated, save that the distribution 

coefficient k = -p:r7-. -. r^ is not unity. Another example 

C (m the gas) 

where A; = 1 is the diffusion of urea from a layer of aqueous 

gel into a stirred aqueous layer. 

The adaptation of the solution which will be obtained for 
the case where kj^l is very simply made, by supposing the 
actual plate between x = I and x = (l + h) replaced by a plate 
between x = l and x={l + hjk) in which the partition coefficient 
is now unity. 

The outhne of the solution for this interesting system may 

dC d^C 

now be given. The equation -^ — D ^^ has to be solved for 

the boundary conditions 

(i) dC/dt ^Ositx = 0,t>0, 

(ii) C = f{x) for 0<x<l and t ^ 0, 

(iii) contact between sohd and gas gives Sbt x — I 

C{1, t) = kCgit), 

where k is the distribution coefficient and Cg{t) denotes the 

concentration at time t in the space between x = I and x = l + h. 

First take k = 1 and write the solution as the sum of two 



22 SOLUTIONS OF THE DIFFUSION EQUATION 

where 

(iv) dCJdx = at a: = 0; 0^^(1,1) = 0; C^{x,t) = Cq{x) at 
t = 0, 

(v) dCJdx = at a; = 0; C^ih t) = Cg{t), t > 0; C^ix, t) = 
at ^ = 0. 

The solutions obtained by March and Weaver (8) then were 

^/ X ^ 4 ^ (-1)'' /2n+l7Tx\ 






C,{t) = Cg{<x>)-i:^e-fiiK . (74) 



where B^ and ^^ are constants which can be evaluated for the 
various ratios A, defined by 

h Volume from x = I to x = h Vg 
I Volume from x — to x = 1 V^' 

The quantities B^, /?^ contain the constant A (see below). 

In the case of urea diffusing from gel (volume Vg) into an 
aqueous layer (volume Vg), where A = VgjVg, equation (74) 
reduces to 

Equation (75) becomes 

C^{t) = S(^)±^Q(^)_[o.327e-4-ii'^'/''' + 0-0766e-24-i4^W=' 

ifFJF„=l. 

Should, however, the distribution coefficient k not be unity 

. Volume from a; = Z to a; = lijk h 
Volume from a; = to a; = Z Ik' 



THE NOJSr-STATIONARY STATE OP FLOW 23 

and this value of A must be used in evaluating B^ and y^^; and 
also C^(oo) in equation (74) becomes 

The constants /?^ and B^ are obtained in terms of D by means 
of the equations (77)-(81). The y^/s are found from the positive 
roots of the transcendental equation 

tang + Az^O, (77) 

which are easily obtained graphically by plotting the curves 
y = —Xz and y = tan 2. The solutions are the points of inter- 
section of the curves. 

One may now evaluate the constants B^, /?^. In equation (74) 

A = ^|' (78) 

when ^ = 77- (^9) 

The constants B^ may be evaluated from the relationships 

and the equations 

B'o = -0-099A + 0-644 -eo/ 

B[ = -0-146A + 0-894-61, 

B:. = _0-157A + 0-946 -62, 

^; = -0-159A + 0-958 -63, 
and for all B^ where ^ > 3 

B'^ = -0-162A + 0-972 -63.; 

The quantities e^ are then read from Fig. 6 for all values of 
A between and 5. Then in the final expression D disappears 
from the ratio E B^j^^. 

One now sees that it is easily possible to evaluate D, the 
diffusion constant, from desorption experiments at constant 
volume, provided 1 < A < 5, and that no rate controlling inter- 
face reaction intervenes. The advantage of this method over 
the method outlined in equations (72)-(72a) is that one does 



(81) 



24 SOLUTIONS OF THE DIFFUSION EQUATION 

not need to represent a ^(i) — t curve by means of an empirical 
equation, in order to obtain an expression for Q, the amount 
sorbed by the plate. Accordingly, the method of March and 
Weaver (8) has been treated fully as far as practical details 
for determining the constants in' the solution of the diffusion 
equation are concerned. . This method, however, apphes only 
to systems where the diffusing molecule exists in the same 
molecular state in both phases. For instance, it cannot be 
apphed to systems such as Hg-Pd, where in the gas phase one 
has molecules, and in the sohd phase, atoms. The earher 
method of equations (72)-(72a) is, however, still available. 



ua - - 
















" 


























































i 




































-^^===5 ;e,-}:: 
















^2 


^ ^^ 


















^ ^ 6" ^.1 
















/>.?>— — «.^. U. — m 


-^ "r^^ 


^, 














0-^—-- jf-A- 


_p.^,_ 


N. 














t/ - 




N A 














M / 




V, 


N. 
















s 


>.^^ 


















^^v 












h 






\^ 












it tt 






\ 


^ - 












i „ re. : : 






SS^ - 










u T- 








5v 










n -, ^ jl ^: 


" 


!^^ft. 




--^- 










0-1 ~—f- -^''■ 






■^^ 


Ss 










t -7 






*N 


s 


VS 








jL ^ 








"^s 


S^ 


1 






J-Z 








^s- 




^ 






t/ 








^> 


*, 


* s 


s 




V 










V 


v 


V 




:n 












\ 


>\ 




It 


1 












^\ 




r.r^t 














^J> 




OOL 


1 




_J 










^ 


A=l 


^ 3 






^ , 








5 



Fig. 6. 



Case, 3. It is easy to set up a diffusion cell so that one has 
a porous membrane separating two well-stirred solutions, or 
two gas volumes. The gas or solution on either side of the 
membrane contains the same constituents. This system is an 
important one biologically, as well as chemically. The mem- 
brane may be organic or inorganic if it separates two stirred 
solutions or two gas phases ; or it may also be a crystal if it 
separates two gas phases only. 



THE NON-STATIOlSrARY STATE OF FLOW 25 

The exact interpretation of the data to give diffusion 
constants may now be given (9). The general method of solution 
is similar to that of March and Weaver given in the previous 
section. One has a membrane between the planes a; = and 
ic = Z; and weU-stirred solutions of concentrations gC^, ^JJq are 
in contact with the planes x = and x = 1 respectively. The 
solution in contact with the plane x = extends from x = 
to X = —g; the solution in contact with x = I extends from 
X = I to X = l + h.Then the boundary conditions for which 
the equation 

. dt~~ dx'- 
is to be solved are 

(i) C = Cq{x) at ^ = for < x < Z. 

(ii) C{Q,t) =^ gCit); C {I, t) = j,C{t) for t>0. 

(iii) C = ^Cq at ^ = for < x < —g, and G = jfi^ at ^ = for 
I <x<l + h. 

dt g\dx).^^Q' dt h\dxJ^^Q 

For this system Barnes (9) found solutions in the form 

^C=C^~zi^e-^i^, (82) 

Si 

j,C=C^-i:^e-^ii, (83) 

where C^ denotes the uniform final concentration in the 
solution or gas phases. When the distribution coefficient 
between membrane and solution is unity, 

C^{g + Uh) = {gC,)g+j'^C,{x)dx+UCo)h. (84) 

Also, where ^ = 0, 

Ceo — ^C'q = 27-^- ; C'oo — hPo — ^~r^- (^5) 

Si hi 



26 SOLUTIONS OF THE DIFFUSION EQUATION 

There are three more relationships available for determining 
the ^'s, B's and ^'s which are 



/I cot^A _ ^_ 



l_cot^\ _^ 1 



A2 2.- / ^2;,-sin2v 



(86) 



and 



^[.^o-(-l)'^.^o-y£Q^)sm^^x] 



where 2^- = ^j-. From equations (86) one may form the 
equations 



B^ ^/ 1 1 A^- cot zjz/ 

and 22_(^^^^\^)2cot2-AiA2 = 0. (89) 

By finding the points of intersection of the graphs y = z^ — A^ Ag 
and y= (Ai + A2)zcotz, one can readily determine the 2's, 
for which real positive values only are taken. From the infinite 
set of linear equations (87), whose form is greatly simplified 
when Cq{x) = Cq (i.e. when there is a uniform concentration 
within the membrane at t = 0), and (88) and (85), one may 
determine B^ and A^ in terms of z^ and D. In the ratios 
ZA.JE,^, ZBf/E,^ which appear before the exponential terms of 
(82) and (83), the D cancels, since it is a simple multipher 
in ^^, B^ and E,^. As with all diffusion problems, at large values 
of t, all the exponential terms save the first become small, 
so that if one plots log (^,(7— Coo) against t one obtains a 
curve such as that in Fig. 7, in which the curve approaches 
asymptotically a straight line of slope — ^V In any case 
even at small values of t it will not be necessary to take many 
terms of the exponential series, in solving for A^^ and B^. 



THE NON-STATIONARY STATE OF FLOW 



27 



The complete solutions of (82) and (83) for the case when 
g — h (i.e. A^ = Ag = A, A; = 1, Cq{x) = = Cq) are 






, A A2 /^ A A2 



~ 4A 



,•2^2 



6A 



^l fTT" \ %^7T^ ' 






l2 



(t27r2 + 4A) I, (90) 



A A2 
~6^45 



n hCoY, A A2 /^ A A2\ f 2XDti 



~ . 4A 

y (-1)^— - 




Fig. 7. A typical diffusion rate curve of log (gC - Coo) against time. 
The units of ordinates and abcissae are arbitrary. 

which reduce as illustrated in Fig. 7 for small values of A and 
large ^ to 

2 



,0 = ^«[l + e-2'W], 



,C = ^°[l-e-2'^^n. 



(90a) 
(91a) 



Ai 


g 

Ik' 




h 


h 


^lk\ 



28 SOLUTIONS OF THE DIFFUSION EQUATION 

When the distribution coefficient of solute between the 

membrane and the solvent is not unity, but k — —pz-. , 

•^ Cm gas 

the problem can be treated identically by writing 



(92) 



and using for C^ the equation (93) instead of (84), 

CJg + kl + h) = {^C,)g+ CCo{x)dx+{,Co)h. (93) 

J 

If the diffusing solute exists in the membrane in atomic 
form, and in the gas phase (or in solution) in molecular form 
(e.g. Hg-Pd), then the treatment of Barnes must be further 
modified. 

(2) Solutions of the radial diffusion equation 

We may restrict ourselves to considerations of diffusions such 
that the spherical surfaces of constant concentration are con- 
centric; in this case the equation of diffusion is (12): 

?^ = i)/^ + ?^\ (12) 

dt \dr'^ r dr J ' 

Examples of diffusion in spheres may then be treatedin a very 
simple manner by making the substitution 

u — Cr, 

1 • 1 • du ^dH .. ., 

which gives -^ = ^ • ^ ' 

The methods used in the previous pages for all the linear cases 
may then be employed, and analogous solutions obtained. 
Some examples of these solutions of the diffusion equation 
will now be given. 

Suppose one has a sphere of radius a, and containing solute 
at an initial concentration C = f{r). The surface of the sphere 
is kept at a constant concentration Co. The substitution 



THE NON-STATIONARY STATE OF FLOW 29 

u = Cr in equation (12) leads to equation (94) whicH must be 
solved for the boundary conditions : 

ttj = at r = for all t, 

U2 = aC^ at r = a for all t, 

u = rf{r) at i = and for < r < a. 

These are the conditions for diffusion in a plate of thickness I 
and with surface concentrations at a; = and x = 1 oi and 
aCg. The solution is therefore (51) in the form (51a) 



u^r 2°°itpCOSW7r . mrr V Dn'^nH'l 
-^+-2-^ sm exp ^- 



2 °° . nnr 

+ - y sm exp 

a I a 



^]jV/(.>m'^'*'. (51a) 



Case 1 . The amount of absorption or desorption in spheres 
can easily be derived from (51a) for some important examples. 
For instance, when f{r) = Cq throughout the sphere at ^ = 0, 
equation (51 a) becomes 

n n 2a^(-l)'^ . ?^7^r ' r D^i^^q 

C' = C', + -i:^sm-^expL-^^J(0,-Oo). (95) 

Equation (95) corresponds to absorption with C'2>Co, and 
desorption with Cq > Og, and from it by means of the equation 

the quantity Q which has been absorbed or desorbed may be 
found. The necessary differentiation and integration leads 

to (96): (C2-(7o)a2/, 8^1 f ■Dn'-nH-W 

Case 2. In the problem of diffusion into or out of the wall 
of a hollow spherical shell of inside radius 6 and outside 
radius a, one may proceed analogously. The solution of 
duldt = Dd'^uldr^ is to be found for the boundary conditions: 

u = u^ = hC^ at r = 6 for all t, 

u — U2 = aC^ Sit r = a for all t, 

and C'c = for b<r<a and at t = 0. 



30 SOLUTIONS OF THE DIFFUSIOISr EQUATION" 

When one also makes the substitution r = b + x, the problem 
reduces to the case of flow into a plate of thickness I = (a — b), 
and with constant concentrations u-^^ and 2*2 ^^ the faces x = 
and X = I. Thus the solution is that given by equation (51), 
with appropriate substitutions: 

^x 2 °° Wo cos mr — u-, . nnx F Dn'^TTH~\ 
w = %+(^*2-%)y + -2-^ -^sm-y-expl p— J, 

(526) 

where % = 6C1 at a; = 0, 1*2 = ^^% ^^ x = {a — b) = I, x = r — b, 
u = rf{r) = Sbt t—0 {b<r<a). 

The quantity Q which has flowed into the slab at any time t 
is given by 

" 1 TT^ 



[' 



1 i ^,^±^.xpl-^^(^'^±g^11, (99) 



for sorption in tlie spherical sliell, on resubstituting aC^ for «2. 
6(7; for Ml and (a - 6) for i. Equation (99) for the case when 
C\ = Ci reduces to 

^-^^"^~L 7r^Jo(2^^Ti)2^''Pl " {a-bf IJ- 

(99a) 

It is to be noted that the equations (97)-(99a) refer to the 
quantity Q which has diffused into unit area of thickness I of 
the slab, i.e. into a volume I of the spherical shell, and that 
the total quantity Q^^^^^ = Q[^7T{a^ - b^)]. 



THE NON-STATIONARY STATE OF FLOW 31 

Case 3. The intercept L upon the ^-axis of the curve, where 
Cg (the concentration built up in the gas phase by permeation 
of gas through the hollow spherical shell of Case 2) is plotted 
against t, is given by the theory of equations (63)-(696). One 
has only to make the substitutions of Case 2 (equations (69 a) 
and (97)-(99a)) to obtain for the boundary conditions of 
these equations: 



U2,{a — bY %(a — 6)2~] 



'a€ 



■J (100) 



J V aG^{a-bf 66>-6)n 

-2-6CiL 6i) ^ 3i) J- ^^^^^ 

If Oi~0, which will be the case when diffusion occurs into a 
vacuum or near vacuum from the exterior of the shell to the 
interior, the lag L reduces to 



62) 



(102) 



which is just the same as for a plate. It must be remembered 
that in the equations (100)-(102) it is supposed that the 
initial concentration, Cq, of gas in the shell is zero. The 
equations (101) and (102) provide a very easy method of 
evaluating D, the diffusion constant, 

(3) Diffusion in cylindrical media 

Examples of diffusion problems in wires are fairly common. 
Of practical importance is the outgassing of metal filaments, 
and the converse process of sorption in wires. It is necessary 
to know how much gas remains in the filaments, or how much 
has diffused away under vacuum conditions. The diffusion of 
metals such as thorium into or out of tungsten filaments has 
a profound effect upon the thermionic and photoelectric 
emission of the filament, and this in its turn is of importance 
in various types of vpJve. The quantitative expression for 
the flow of thorium can be obtained by integration of the 
differential equation for flow in a cylinder, using appropriate 
boundary conditions. Sometimes a supply of a solute may be 



32 soLUTioisrs of the diffusion equation 

generated chemically or physically at the surface of a wire at 
a concentration which is a function of time. This provides a 
more complex example of diffusion into a cyhnder. For 
example, a supply of carbon may be generated by the de- 
composition of hydrocarbons at the surface of an iron wire and 
the carbon may then diffuse into the wire. Often analogous 
problems will arise in which gases are sorbed in, desorbed from, 
or diffuse through the walls of cylmdrical tubes. 
The equation for radial flow in a cyhnder is 



I- 



T dr \ dr 



(15) 



and by making the substitution C = ue~^'^"\ the equation (15) 
is transformed to 

d^u \du „ ^ , ^^. 

^-„ + -^ + a2w = 0, - 103) 

or'^ r or 

which is Bessel's equation of zero order. Solutions of Bessel's 
equation may be obtained in terms of the appropriate Bessel 
functions whose choice is governed by the boundary conditions. 

Case 1 . A circular cyhnder of radius r — a is, the diffusion 
medium, at its surface a constant concentration Cj is main- 
tained, and the medium is initially free of solute. In this 
example the solution may be given in terms of Bessel's function 
of the first kind and of zero order Jq{x) and its differential 
J'q{x). The solution is 

C = 0in:?S-#^le-^^, (104) 

where a^ is the nth root of the equation Jo(a„ a) = 0. The first 
four roots of jQ(a„a) = are 

2-405 5-520 8-654 11-7915 ,_^, 

ix = a, = , a, = '-, a. = . (105) 

a "a ^ a a 

These roots give four exponential terms in an infinite series, 
and it will be found as a rule that these terms are adequate to 
express the diffusion process. Indeed, for larger values of the 



THE NON-STATIONARY STATE OF FLOW 



33 



time, t, a single term will be sufficient. The functions Jq{x) 
and J'o{x) are given by the series 



^OK"^) — ^ {2"^) ^ -|^222 122232^ 



J'oi^) = -(1^) + 



(M! 

122 12223 



;k)' 



+ 



(106) 



and their values for any values of x are given in tables.* 

For most purposes it will be more important to know the 
mean concentration in the cylinder, or alternatively the 
quantity Q which has diffused into the cyhnder, per unit 
length. C, the mean concentration in the cylinder, is given by 



C 



I ra 2 /*« 

— — z 27TrCdr = —„ Crdr 



and 



Q = 27T \ Crdr, per unit length. 

Jo 



(107) 



The integration of (107) in which (104) has been substituted 
leads to the equations 



(J = C {1—^^ — ^~-Da2„/ 



a^ 1 a^ 



)• i 



Q = na^C,{l--,^-e 



-DaKd 



(108) 



where the first four values of a^ are given by equation (105). 
Equations (108) follow easily from the relation 



/; 



'rJ^ic^n^) = —J'q{cl^(i), 



and by substitution of (105) lead to a value for the mean 
concentration C in the filament: 






C 



"H (2-405)2 ^^P\ 02 I 



xexp 



•405)2 
D{6-520)H 



(8-654)' 



,exp - 



(5-520)2 
Z>(8-654)2^ 



1-4 

(108a) 



* References to suitable tables are given at the end of this Chapter. 

BD 3 



34 SOLUTIONS OF THE DIFFUSION EQUATION 

Case 2. If the cylindrical diffusion medium is of radius a and 
the boundary conditions are 

Cj = at r = a for all t, 

C = f{r) for a < ?• < at ^ = 0, 
the general solution is 

When/(r) = Cq, the solution is 

C = _?£o-glg_D.,.,jM^. (109a), 

By using the relations 

C = -.rCrdr and Q = na^CQ- 271^ Crdr, {107 a) 

C' Jo jo 

where Q is now the quantity of solute which has diffused out 
of the cylindrical medium per unit length, one obtains the 
solutions for O and Q just as for Case 1, equations (108): 






(110) 



Since the a's are given by (105) it is an easy matter to write 
the first four terms of equations (110), and so to employ them 
in practice. 

Case 3. If in the cylindrical medium of radius a the boundary 
conditions are 

C — Cj^ a,t r = a for all t, 

C = f{r) for a<r<0 and ^ = 0, 
the general solution is 



c = cJi+^i~^^!^.-^^n 

\ a I a„ Jo{oc,,a) 

rf{r)Jo{oc,j)dr 



^'a4'""""'-"- [j'.Mr ■^°'""'-'- <"" 



THE NON'-STATIONARY STATE OP FLOW 35 

l{f{r) — Cq for a<r <0 and at i = (which would correspond 

to the sorption equilibrium of a gas in a wire before admitting 

another dose of gas at constant concentration Ci), equation 

(111) reduces to 

2 00 ] T (nf r\ 

In turn, equation (112) gives for the mean concentration C 
in the cyhnder, or. the quantity Q of gas which has diffused 
into or out of the cyhnder per unit length, 

-^ = = Oi-^-^^^i;ie-^-«^ (113) 

na^ ^ a^ Oil 

where the values of a^, cc^, cl^ and oL/^ are given by (105), and 
permit four exponential terms of the series to be used. 

Case 4. If the surface concentration of solute is C = At, 
where J. is a constant, and there is an initial solute concen- 
tration Cq in the cyhnder of radius a at ^ = 0, the general 
solution is (9a) 

C = -^ri(a^-r2-l>^) + -i:-3#^e-^--^^^ 

+ — °i-i#4e-^V^, (114) 



and so one finds for Q and C the expression 

Q 



c 



«^ 4 « 1 „ ,n 20o ^ 1 r, -, 
8D aWiOc^ J a" a| 

(115) 



Case 5. If one has a hollow cyhnder of external and internal 
radii b and a respectively, the initial concentration in the 
cylinder is /(r) for a<r<b, and the concentrations at the 
surfaces at r = 6 and r = a are zero for all time t, the general 
solution for the concentration as a function of time (lO) is 



772 » Ji(a,,a) 



C = ^^oc^l 



^^Q-DccnH UQicc^r) rf{r) UQ{a,^r) dr. 



2f ^-Jl{^,^b)-Jl{a,ay 

(116) 
3-2 



36 SOLTTIOXS OF THE DIFFUSION EQUATION , 

In this equation the a^'s are the roots of the equation 

Jo{aa)HP{ab)-jQ{oib)HP{oui) = 0, 

in which H^'\(xr) denotes a Bessel function of the third kind, 
for Uo{ocr) = jQ{ar)H'^\xb)-jQ{ocb)H'^Q\ar), and with the 
above meaning of a„, f7Q(a„a) = C/o(a,^6) = 0, satisfying the 
bomidary condition that at r — a and r = b the concentration 
is zero. 

When/(/-) = Cq, equation (116) reduces* to 

r_4K_.|^n-| 

L 7TXI\ Jo(a„«)/J 

From this equation by means of the relation 

Q = 7T{b^-a^)CQ-27T( Crdr 

one may find Q, the amount of material which has diffused 
out of rniit length of the waUs of the cyhndrical tube, as a 
function of time, 

(118) 
The roots of the equation ?7Q(a^6) = = t'J)(a,ja) are com- 
puted from the expression (ii) 

7177 (p-1) r 100(^3-1) 1 lip- If 



a„ = 



(P-1) 



(p-1) r 100(^3_i) 1 -| (^_i; 

8p{n7T) ^ l3{8p)^ ip-iy {Spf_\ {nnr 



r 32(1073) (p°-l) 50(p3-i) 2_-|(p-l)5 

L 5{8p)Hp-l) ^3p{Sp)^p-l) (Sp)3j {nTT)-^ "^ 

* This integration makes use of the relationships 

fb rr ~i& 

J a L^n -a 

r^ cro(a„rn ^_-2i 
L cr Jr=6 n ' 

r dlUa„ry i ^ 2i J,ia„b) 
L cr Jr=a n Jo(a.„a)' 
the first of which will be found in Watson doa) and the others in Carslaw(io). 



THE NOJSr-STATIONARY STATE OF FLOW 37 

where p = bja. The values of jQ{oc^b) and /^(a^a) may then 
be found from tables of Bessel functions. 

Case 6. If one has a hollow cy Under of internal and external 
radii a and b respectively, and there is a concentration G^ at 
r = a and C*2 at r = 6, and a concentration Cq in the wall of 
the cyUnder, one may treat equation (116) in the same way 
as was done in equations (63)-(69 6) to find the rate of approach 
to the steady state of flow through the waU of the cyhnder. 
The intercept L made on the time axis by the asymptote to 
the curve of the quantity difiused plotted against the time is 
now given by 

when Ofl = 0, C^ = the most important experimental case 
arises, and the equatiori above reduces to 

•^,^i-L f;.K")y in^. (120) 

where a^ is as before the Tith positive root of 
C^K^) = - ?7o(a,,a), 

and one has once more a useful method of measuring the 
diffusion constant (cf. equations (69), (69a), (696) for plates, 
and equation (101) for a hollow spherical shell). 

(4) Diffusion processes coupled with interface reactions 

Sometimes there are slow processes at the surface of the 
diffusion medium which may sensibly alter the rate at which 
the diffusing substance leaves or enters the medium. For 
instance, under certain conditions the diffusion of hydrogen in 
palladium is fast compared with its rate of entry into the solid 
from the sorption layer. In this case it will be necessary to 
include a term to allow for the leaving or entering of hydrogen 
at the surface. In general two cases will have to be considered: 



38 SOLUTIONS OF THE DIFFUSION EQUATION 

(a) the solute is dissociated on entering the diffusion medium, 
and (6) the molecular condition is the same inside as outside 
the diffusion medium. The first condition wiU be encountered 
in systems such as Og-silver, Hg-paliadium, Ng-iron or 
molybdenum; and the second when ammonia, water or gases 
diffuse into an alkali halide lattice, or a zeoUte lattice. It 
will be possible to attempt a solution for the conditions 
of (6). 

Case 1 . The diffusion of ammonia gas occurs from a sphere 
of a zeoUte (anal cite). The initial ammonia concentration 
in the sphere is /(r); the radius of the sphere is a, and the 
concentration at r = a is maintained very low (by condensing 
evolved ammonia in hquid air). One has thus to solve the 
equation g^ ^^^^j 2 ac'\- , ^ 

when C=/(r)at« = 0, (6) 

dC 
and i)-7^-^-^C = at r = a. (c) 

or 

The substitution u = Cr 'gives 



^t-^d^^ (0<r<a), id) 



Yt 

^+{h— l/a) u = at r = a when h — k/D, (e) 

u = rf{r) at ^ = 0. (/) 

The problem is analogous to the process of cooling of a sphere 
with radiation at its surface (12), the radiation corresponding to 
the surface desorption. Proceeding therefore along analogous 
lines to those adopted in the problem of the cooling of the 

earth (13), one obtains 

C = — 2 -0- , \r -u \A »'// sma,,r'rf/ sma,,re-'0««^', 

(121) 

where a,^ is the wth root of 

aa 
tan aa = 



\—ah' 



THE NON-STATIONARY STATE OF FLOW 39 

When/(r) = Cq the equation reduces to 

1 1 af; af;^ a'^ + ah{ah — 1 ) r 

The quantity Q wliich has been desorbed from the sphere 
is then 

Q - f Tra^Oo - 47r ( "Or^ cZr = ^^ - 477 ( "Or^ fZr 
jo jo 

When one plots hi ^^ — yr against t, the curve for large values 

of t approaches a hne of slope — Daf, for which the intercept 
on the axis of Hs . 



hi 



Slll'^ 



(x-^a 



_a^ <x\cc\a^ + ah{ah— 1) 

These two equations can be solved for D and k with the aid 
of the equation 

tan(aa) = 7, 

I— ah 

i.e. tan2; + A2 = 0, where A = —p: — — and aa = z. The roots 

a{h—l) 

of this equation (8) are given by 

1 3A-1 



, 1575A3 + 1575A2 + 483A + 45 



315A7[(ti + i)7r]7 

39690A^ + 52920A3 + 24696A2 + 3834A + 9 
^ 2835A9[(ri + i)7r]9 "•• ^ ' 

In Fig. 8 are shown the first four roots of the equation 
tanz + Az = for A between 1 and 5(8), i.e. when ak/D Ues 
between 2 and 1-2. The radius a may of course be varied. 

When the desorption process at the surface depends on the 
square of the concentration, the problem cannot be solved 
along analogous lines. 



40 SOLUTIONS OF THE DIFFUSION EQUATION 

Case 2. If desorption into a vacuum occurs at the surfaces 
X = and ic = Z of a slab, the conditions of Case 1 become (14) 



(i) 

(ii) 

(iii) 
(iv) 



dC_ 
dt 



D 






dC 



— ^ — h ^c = at a; = 0, 

ox 



dC 



+ -;r- + he ^ ai,t X = I, where h = k/D, 

ox 

C =f{x) at i = 0. 



21 



2 



1-9 



1-6 



1-7 






'Z'^ 



Fig. 8. This figure gives the first four roots, z^, Sj, Zg? ^3> of the equation 
tanz+Az; =0, for values of A ranging from 1 to 5. For the curve z^ the 
vertical scale reads as shown. For z^, the values read from the curve should 
be increased by 3: for z^, add 6: for Zg, add 9. 

The expression e-^*^'(^ cos ax + B sin ax) 

may be used to develop the solution, for it satisfies (i), and 

also (ii) and (iii), when 

~aB + hA = 

and a{B cos al — A sin al) + h{B sin al + A cos al) = 0, 

2ah 



which in turn give tanaZ = 



a' 



7^2' 



A_B 
oc h 



(124a) 
(1246) 



THE NON-STATIONARY STATE OP FLOW 41 

The roots of the first expression are the points of intersection 

of the curves ii = - — - — i and y — t — • 
tanai h, oi 

The second curve is a hyperbola; the roots of the equation 
(124 a) are all real, not repeated, and only positive values are 
to be taken. 

The solution of the problem is 

r = 2 V r-T)'-n^t' ^^ ^^^ cc.^x + h shi g^ X 
nil {al + h^)l + 2h 

X f {x'){a^ COS oc^x' + h sin (x^x')dx, (125) 
Jo 

which, i£f{x') = Cq, reads 

fa^smaj-hcosaj ^ hi 
L a« ^nJ ' 

and gives for Q, the quantity desorbed per unit area, 

Q = Qq— Wdx 

ir 9r V ^ [«^ sin ccj-h cos a^ l + hf ,^ 

Thus, at large values of t one finds in plotting In— ^^^ — 

against t that the curve approaches the line of slope — -Oaf, 
and the intercept upon the i-axis is 

r2 1 [aisina^Z — /icosaiZ + ^jn 
L^^ {al + h^)l + 2h ^J' ■ 

The method outhned is inapphcable when the desorption 
velocity depends on the square of the concentration at the 
surface. 

Case 3. Barrer(i4a) treated the problem of diffusion through 
a slab of thickness I, when in addition to pure diffusion two 
other rate-controlhng surface processes are important: 

(i) The passage of the sorbed atom from the surface to the 
interior of the plate. Velocity constant k^. 



42 SOLUTIONS OF THE DIFFUSION EQUATION 

(ii) The passage of a dissolved atom from the plate to the 
surface. Velocity constant Ajg. 

It was assumed in order to treat the problem that the 
sorption and desorption, with velocity constants k^ and k^ 
respectively, in the gas and the surface layer occurred so 
rapidly that the surface concentrations were defined by the 
adsorption isotherm. The distribution of Fig. 9a was sub- 
divided into the two distributions of Figs. 96 and c. In Fig. 96 
the gas desorbs into a vacuum; in Fig. 9c one has the steady 
state of flow through the plate. The complete solution for 
Fig. 9 a was then (employing the notation of Fig. 9) 

V = u + w. 

The solution u^{Ax + B) (128) 

for the stationary state of Fig. 9 c between x = and x = I 
was given when the constants A and B took the values 



A =T 



k^ k^ vJ 1 - ^ j - ^^1 h Vii 1 - ^ j 



;i29) 



B^ 



(130) 

One may work out the different simple types of behaviour 
that may be encountered. In the equations (129) and (130) v^ 
denotes the saturation concentration at the surface, and Ug 
the saturation concentration in the metal. It was observed 
that very great concentration discontinuities may occur at 



THE NON-STATIONARY STATE OF FLOW 43 

the interfaces when diffusion is rapid compared with rates of 
transport from the surface to the interior of the plate and 
vice versa. 








X'O 3C-1. 2C.O 2e- 1 Distance ^'<^ ^'l 

Fig. 9. 

The complete solution for the non-stationary state of flow 
of Fig. 9a is given by 

V = u + w, ■ 

where u is defined by equations (128)-(130), and w is given by 

1 {cl^'h?)l^-^Ji 

rx 
X {vq — u) (a^cos oc^^x + h sin a^x)dx, (131) 
jo 

where a^ is the nth positive root of tan ocl — -^ — 7^, and h — y.. 



(5) Instantaneous sources 

A number of problems on diffusion into soMds require solutions 
of the diffusion equation for conditions described below. 
A definite quantity Q of matter is deposited at a point on the 



44 SOLUTIONS OF THE DIFFUSION EQUATION 

surface of a solid, and left to diffuse into it, or over its surface. 
Important examples of these diffusion systems are : 

(i) Diffusion of caesium along tungsten filaments (15). 

(ii) Diffusion of thorium around a tungsten strip (16). 

(iii) Diffusions of sodium and potassium into, and over, 
tungsten strip (17,18). 

It wiU be seen from the above examples, and in Chap. VIII, 
that most of our quantitative data on surface diffusions are 
based on treatments of Tick's law for instantaneous sources. 

The solution for an instantaneous plane source in an infinite 
solid may be derived from equation (34), in which is given 
the general solution 

^=2V(k)r>''-p[-^>^' (=**) 

dC d^C 

to the equation -—- — D ^r-^ , 

ot ox^ 

for the boundary conditions 

C = f{x) when t = 0, 
C = f{x, t) when t > 0. 

Suppose in the infinite soHd of equation (34) there exists 
initially a concentration C — Cq between the planes x = —^h 
and X = + Ih. Equation (34) becomes for the new conditions 

and if h tends to zero while CqH remains constant and equal 
to Q, one finds 

In Fig. 10 are given (i9) a set of curves showing the progress 
of diffusion according to (133) from a thin layer into a solvent. 
The curves of Fig. 10 may be compared with those of Fig. 128, 
Chap. VIII, for sodium diffusing into tungsten. 



THE NON-STATIONARY STATE OF FLOW 45 

When the instantaneous source, quantity Q, is deposited 
at the plane a; = of a semi -infinite sohd, the solution corre- 
sponding to (133) is 

Q 



G = 



. Q-xViDt^ 



(134) 



The variation of C with time at a: = is given by the equation 

obtained by putting re = in (134), Equation (135) gives a 
very simple method of obtaining D. 




-10 -9 -8-7-6-5-4-3-2-10 12 3 4 5 6 7 



9 10 



Distance •prom Boundary 

Fig. 10. Diffusion from a thin layer into a solvent. 

The solution in two dimensions of the problem treated in 
the equations (132)-(135) may be deduced in an exactly 
analogous manner. The equation corresponding to (132) 
becomes 



^=i^ci:>[- 



(,_..).^+(,-,-)^ j^^,^^, ^^3gj 



and leads to equation (137) for a point source, diffusing over 
an infinite plane surface: 



C 



Q 



^nDt 



exp 



r {x^+y') i. Q r r^i . 



if r denotes the radial distance in the plane from the point 
source. 



46 SOLUTIONS OF THE DIFFUSION EQUATION 

Similarly, for the point source diffusing in an infinite solid 
the solution for C in three dimensions is 

It will be noted that in the 2- and 3 -dimensional cases the 
value of C at r or i? = is 

and <' = sSw^' ("») 

respectively. Equation (139) should have ready application 
in evaluating D in the case of surface diffusion. For 
instance, one might by a molecular ray method deposit a 
metal film on a crystal at a point, and heat the crystal. The 
spreading of the metal could be followed photoelectrically 
(Chap. VIII). 

Equations (138) and (140) are not yet of particular interest 
in the case of the diffusion of matter, because the conditions 
of the solution are not capable of easy experimental reahsation. 
One might also have an instantaneous spherical surface source, 
for which the boundary conditions are that a tliin layer of 
matter exists in the shell 

a<r<a+h 
at a concentration C. Then 

and when, as in (132), we let h tend to zero while Q remains 
constant, the solution of the general equation 



a 




(141) 






THE NON-STATIONARY STATE OF FLOW 47 

Should one have a spherical volume source of radius a, 
where f Tra^Co = Q, one can obtain the solution by integrating 
equation (141) between the hmits and a, so obtaining (143): 






r C{r+a)l2V(.Dt) C(r-a)l2V(Dt) T 

e-y-dy- e-y~dy 



4a37rt\""^JL ^l ^Dt f ^""^ \ Wt j J " ^^^'^^ 

The equations (141)-(143) are more significant in considering 
flow of heat than of matter, and give the form of temperature 
waves spreading from areas or volumes through an infinite 
surrounding medium. - Equation (142) would apply to the 
diffusion of solute from a sphere of aqueous gel into a sur- 
rounding medium of solute -free gel. It might also apply to 
the diffusion of a metal solute from a sphere of a solvent metal 
embedded in a mass of the solute metal. 



(o) Treatment of the equation -^r~ = ^^ 1)^:— 

01 dx\ dx J . 

Very frequently the diffusion coefficient depends upon the 
concentration, and it is therefore necessary to solve the 

equation ^:- = i:—\D -^r- ) . The interdiffusion of metal-metal 

ot ox \ ox J 

pairs, the diffusion of ammonia or water in certain zeohtes, 
and the diffusion of vapours in media which swell during 
sorption are typical examples where one must use the above 
equation. The variation in D with concentration can be very 
great indeed — a thousandfold or more in some metal-metal 
systems (20). 

The most favoured experimental procedure in following the 
interdiffusion of metal pairs is to place a slab of each metal 
in contact, and heat them. The diffusion can be followed by 
X-ray or chemical analysis of thin layers near the origin, and 
the slabs may be regarded as of infinite thickness if the diffusion 



48 SOLUTIONS OF THE DIFFUSION EQUATION 

is slow. The conditions for which a solution is sought are 
now (21, 22) 



... dC d I dC\ 



(ii) C = Cq for X = to x = — 00 at f = 0, 

(iii) = for a; = Otoa: = +ooat^ = 0. 

BoItzmaiui(2i) showed that the solution would contain C as 
a function of a single variable A = xj^jt. If this variable is 
substituted in the diffusion equation it becomes 

When some of the diffusing material has passed across the 
plane x = 0, conservation of mass requires that 

xdC ^ \ {-x)dC; or xdC = 0. (145) 

J C(x=o) Jo jo 

The interdiffusion of two metals in contact is more compHcated 
because each metal acts as both solute and solvent, and after 
a time the original plane x = will no longer satisfy the 
condition above. It is necessary to choose a new plane at 
x' = so that 

/•Co /•C(,- = o) 

x'dC = {-x')dC. (146) 

J C(x'=o) J 

The zero plane may be chosen when the concentration- 
distance curve is plotted at a given time. Integration with 
respect to A then gives 

D,c^n)=-l^r^''^^dG, where ^'''''mC = 0. (147) 

At any time t — a constant, the equation is 

D,c^c,)=~^r^''°^dC, where ^^'''xdC^O (148) 
I Maiy J c=c^ Jc=o 

and D is easily evaluated, as a function of the concentration, 
by graphical integration. Boltzmann's original assumption 
used in treating this problem, that C is a function of A = xj^t 



THE NON-STATIONARY STATE OF PLOW 49 

only, can be tested by plotting x against ^^ for a constant 
value of (7. If for each value of C the origin is chosen so that 

XdC — 0, a series of straight lines must be found all 

J c=o 

of which pass through the origin. Matano (20) showed that this 
was true of data obtained on the metal pair Ni-Cu. 

One can also obtain a solution of the problem with the 
following boundary conditions: 

(i) C = at a; = for all t, 

(ii) C = Cq at < a; < 00 and at ^ = 0. 

If one writes C" = dC/dA in equation (144), it may be re- 
written in the form 

SO that ln{DC') = -( ^ + lna\ 

Jo '^J-' I 



(150) 



(7 = aJ^'^exp[-| ^1-1-6, (151) 



DC = ae^v[- jl^] 
Further integration gives 

2Z)J 



for which the boundary conditions (i) and (ii) above give 
6 = (from (i)) and 



■C'^dA r C^AdXl 



If there is an initial concentration at a; = of Cj , the integrated 
equation gives 

r^dx r r^xdr\ 

C'o-cJoD'^PL Jo 2d] ^^^3^ 



D 



^"4-JoWj 



Equations such as (151) and (153) are not of the same practical 
importance as are (147) or (148), 



50 SOLUTIONS OF THE DIFFUSION EQUATION 

(7) ^ note on the derivation of diffusion constants 
from solutions of the diffusion equation 
The solutions which we have given can be expressed, as pre- 
ceding examples indicate, as a converging series of exponentials : 

^~%=f^{A) e-^\(b)t +y^(^ ) ^-F,m +f^(A ) e-^3(«< + . . . . 



C'oo~C'q 



If therefore log (C — Cq)I{C^ — Cq) is plotted against t a curve 
of the type illustrated in Fig. 7 is obtained. This curve 
approaches asymptotically to the hne 

and so the slope, and the intercept on the axis of log C, gives 
fi{A) and F^{b). The theory then gives D, and all the other 
terms. This procedure is available for all converging series of 
exponentials. 

Special methods are available in other instances such as 
the graphical integration method apphed to equation (148). 
Solutions involving semi-infinite or infinite soHds all give 



^-^(; 



and so one can use the relationship between x and t for a fixed 
value of C, to measure D. Other methods, or examples of the 
apphcation of these methods, will be indicated at various 
points throughout the text (e.g. pp. 97 and 352 et seq). 

(8) Some cases for which there is no solution of the 

diffusion equation 

dC d^C 

Nearly all types of system in which the law -x- = D -^-^ applies 

have now given an appro^Driate solution of the diffusion 
equation. There are one or two outstanding and experi- 
mentally important cases: 

(i) Sorption or desorption occurs from a diffusion medium 
into a gas phase where the gas exists as molecules, whilst it 
exists in the medium as atoms (e.g. Hg-Pd). The desorption- 



THE NON-STATIOlsrARY STATE OF FLOW 51 

sorption processes occur into a constant volume system, so 
that the surface concentration is a function of time. This 
problem has been solved for the case when the m<slecular 
state is the same in gas and difiPusion medium (p. 21); b;;t it 
would now be necessary to have a gas phase volume altering 
with time also, owing to the existence of an equilibrium 

TT sOTT 

in the gas phase. 2^— 

(ii) There is the analogous problem of flow through a 
membrane from one constant volume system into another 
which has been solved for the case when the molecular state 
is the same in both gas phases and in the medium. When, 
however, the gas exists in the medium in the atomic state, 
consideration of an equihbrium such as . 

H2^2H 

would require, if the same solution were to hold, that the, 
volume of the two gas phases should become functions 
of time. 

(iii) There exist also more complex instances of (i) and (ii) 
above, when surface reactions occur simultaneously with 
diffusion. These problems may be solved by the methods 
outhned on pp. 37 to 43 as soon as the problems (i) and (ii) 
can be solved. 

REFERENCES 

For tables of Bessel functions the following may be iised : 

Watson, G. Theory of Bessel Functions, Cambridge Univ. Press (1932). 
Gray, A. , Mathews, G. and MacRobert, T. Treatise on Bessel Functions, 
Macmillan and Co. (1922). 

Values of error functions may be obtained in : 
Janke, E, and Emde, F. Funktionentafeln mit Formeln und Kurven, 

Teubner, Berlin (1938). 
Pierce, B. A Short Table of Integrals, Ginn and Co., Boston (1929). 

Further examples of diffusion problems on solids may be worked 
out by considering analogous problems in heat flow, as given in 
standard texts on the theory of heat conductivity. Suitable texts are : 
Carslaw, H. Conduction of Heat in Solids, Macmillan and Co. (1921). 
IngersoU, L. and Zobel, O. Mathematical Theory of Heat Conduction, 
Ginn and Co. (1913). 

4-2 



52 SOLUTION&/OF THE DIFFUSION EQUATION 

-(1) Jost, W. T^e. Chem. Reakt. in Festen Stoffen, p. 203, Steinkopf 
(1937J 

(2) The na^hod for problems of this type involving spheres and rods 

jg'outlined in Carslaw, H., Theory of Heat Conduction, pp. 206 et 
seq., MacmUlan and Co. (1921). 

(3) Jost, W. Diffusion ii. Chem. Reakt. in Festen Stoffen, p. 17 (1937). 

(4) Stefan, J. S.B. Akad. Wiss. Wien, ii. 79, 161 (1879). 

(5) Kawalki, W. Ann. Phys., Lpz., 52, 166 (1894). 

(6) Carslaw, H. Theory oj Heat Conduction, p. 67 (1921). 

(7) Theory o/ Heat Conduction, p. 68 (1921). 

(8) March, H. and Weaver, W. Phys. Rev. 31, 1081 (1928). 

(9) Barnes, C. Physics, 5, 4 (1934). 

(9a) Carslaw, H. Theory of Heat Conduction, p. 211 (1921). 

(10) ■ Theory of Heat Conduction, p. 127 (1921). 

(10 a) Watson, G. Theory of Bessel Functions, Cambridge (1932). 

(11) Gray, A., Mathews, G. and MacRobert, T. Treatise on Bessel 

Functions, Macmillan and Co. (1922). 

(12) Ingersoll, L. and Zobel, O. Mathematical Theory of Heat Con- 

duction, p. 136, Ginn and Co. (1913). 

(13) Carslaw, H. Theory of Heat Conduction, p. 136 (1921). 

(14) Theory of Heat Conduction, p. 74 (1921). 

(14a) Barrer, R. Phil. Mag. 28, 148 (1939). 

(15) Langmuir, I. and Taylor, J. B. Phys. Rev. 40, 463 (1932). 

(16) .Becker, J. A. Trans. Faraday Soc. 28, 148 (1932). 

(17) Bosworth, R. C. Proc. Roy. Soc. 150A, 58 (1935). 

(18) Proc. Roy. Soc. 154A, 112 (1936). 

(19) WUliams, J. and Cady, L. Chem. Rev. 14, 177 (1934). 

(20) Matano, C. Jap. J. Phys. 8, 109 (1930-3). 

(21) Boltzmann, L. Ann. Phys., Lpz., 53, 959 (1894). 

(22) Wiener, O. Ann. Phys., Lpz., 49, 105 (1893). 



CHAPTER II 

STATIONARY AND NON-STATIONARY STATES 
OF MOLECULAR FLOW IN CAPILLARY 

SYSTEMS 

Types of flow 

A number of types of flow have been found to occur in capillary 
systems, each of which is observed in the appropriate region 
of pressure difference, absolute pressure, and pore size. Each 
kind of flow may be characterised by different permeabihty 
constants, a fact which has often led to confusion in expressing 
the data. The nature of gas flow in single capillaries teaches a 
great deal concerning the more complex permeation processes 
through porous plates, refractories, and consolidated and 
unconsohdated sands, so that before discussing flow in these 
systems the different types of capillary flow wiU be discussed. 
The researches of Warburg (i), Knudsen(2), Gaede(3), 
Smoluchowski(4), Buckingham (5), and others (6, 7) have demon- 
strated the properties of the following types of flow: 
(1) Molecular effusion; (2) Molecular streaming, or Klnudsen 
flow; (3) Poiseuille or stream-hne flow; (4) Turbulent flow; 
(5) Orifice flow. These may now be considered in turn. 

(1) Molecular effusion 
If one has an orifice of area ^ in a thin plane waU such that 
its diameter is small compared with the mean free path of the 
gas the number of molecules N effusing up to it in unit time 
is given by the well-known kinetic theory equations 



N = lAN^Cw \ 



= lANr 



° \ V TTin) 
AN,p 



(1) 



^{27tMRT) ' 

In these expressions, Nq is the Avogadro number, C the 
concentration, w the mean square velocity, k the Boltzmann 



54 MOLECULAR FLOW IN CAPILLARY SYSTEMS 

constant, m and M the mass of a molecule and of a gram- 
molecule respectively. This equation may also be written 

since Px'^i — i^J^o) ^^^ where v-j^ is the volume of gas in 
gram-molecular volumes effusing per second at a pressure j^i- 
One sees that the equations for molecular effusion may be 
used to measure molecular weights, temperatures, or vapour 
pressures. The emission through such an orifice obeys the 
cosine law, so that by using two orifices in series, a molecular 
beam may be defined, whose intensity is given by the effusion, 
equation coupled with Lambert's cosine law. 

(2) Molecular streaming, or Knudsenflow 

When the orifice is of considerable length molecules will coUide 
with its walls on their way through. If the colHsions are elastic 
the flow through a smooth tube will be identical with the 
effusion velocity through the hole in the thin plane wall, since 
no molecules will be turned back in the original dkection by 
collision with the wall. For such a tube therefore the equations 
of flow are, as before, 



Such a flow is independent of the lertgth of the tube, which 
Knudsen(2) showed was contrary to experiment. Knudsen 
therefore supposed that of each N molecules striking the wall 
a fraction / was emitted with random velocity distribution, 
and a fraction (1 — /) was specularly reflected. Some molecules 
are then returned in the direction from which they came, and 
more return the greater the length of the tube. The number 
of molecules striking unit area in unit time at the inlet and 
outlet sides of the tube are respectively INqCj^w and ^-iVoCoW. 
The excess flow of molecules from the inlet to the outlet along 
the axis x can be shown to be hB{dN/dx){S), in the stationary 
state of flow, when jB is a constant depending upon the shape 



TYPES OF FLOW 55 

of the tube. iV is a linear function ofx, in the stationary state, 
and the nett rate of flow in mol./sec. is given by 



dt ''^^{27tMR)[^T, ^tJl\ f /' ^^^ 

where L is the length of the tube. The derivation of the factor 
{'2—f)lf will be found in numerous places (6, 2, 3, 4), and will not 
be considered here. For tubes of circular cross-section the 
constant B takes the value ^r^n, where r denotes the radius 
of the capillary. 

(3) Poiseuille or stream-line flow 

When an incompressible fluid flows down a tube without 
turbulence, the volume of fluid passing through a cylindrical 
tube in unit time(io) is 

^\^2iy~L'^ (PoiseuiUe's law) (3) 

(where d = the pore diameter, 

7} = the viscosity of the fluid, 
and L = the tube length). 
On the basis of the kinetic theory (lO) the flow in mol./sec. is 

where dC/dx denotes the concentration gradient in the direction 
of flow X, and the other terms have already been defined 
(except A, the mean free path). These equations are vahd for 
an incompressible fluid throughout the pore length; but for 
a compressible fluid obeying the gas law the equation must 
be written ^^ ^^ 

* = I28lj ,^^''^' '^^ 

which for an isothermal process, with tj — a constant, becomes 

dn d'hr 1 'p\—p\ 



dt USL7]ET 2 



(6) 



56 MOLECULAE FLOW IN CAPILLARY SYSTEMS 

Equation (6) may take the form 

dn dhr 1 _, 

where p — l{Pi+P2) i^ *^® mean pressm'e in the tube. Also 
dn dhr p 



dt USLtjM 



{P1-P2), ^ (8) 



since -^7^ = p, the mean density of the gas. 

A complete equation of flow. 

Two corrections may be apphed to Poiseuille's formula: 

(i) Only part of the pressure difference is used in over- 
coming friction; a fraction which must be subtracted from 
the original pressure difference produces kinetic energy of 
motion(io,ii,i2). 

(ii) In the boundary layer of gas, of thickness A, equal to 

the mean free path, there may be specular reflection at the 

surface, those molecules specularly reflected having the 

streaming velocity component of the flowing gas, A fraction 

/ only is emitted in random directions. The coefficient of 

2-f 
sUppage is then — ^Aii.c). 



The equation of flow then becomes 



dn_J_/ 8(2-/) yl\r 7r# pj-pl M /dnV' 
'dt~87JL['^ 7 d)\_16RT 2~ 7r\dtj_ 



(9) 



in which expression the last term gives the correction due 

to (i). It transpires, however, that this is only a small term; 

and also when the pores are small, or the pressures are low, 

2-f A 
that '—j^ 77 ^ ^ • Thus the equation becomes 

dn _ 1 dnW I J2-/\ 

. dt~ L 32 ^{2nMRT)^^'- ^^^\ f )' ^^^ 

which is the equation given for Knudsen's molecular streaming 
save that the numerical constants are somewhat different. 



TYPES OF FLOW 57 

(4) Turbulent flow 

The type of flow known as viscous, stream-line or Poiseuilie 
flow changes when a certain hmiting mass velocity W is 
reached. We will define a quantity [K] (13), called the " Reynold's 
number", by 

m-'-^, (11) 

where tj, denotes the ratio r— ; and is the so-called 

periphery 

hydrauhc radius, which for circular tubes is one-quarter of 

the diameter d. p and rj have their usual significance as fluid 

density and viscosity respectively. Then for such circular 

tubes it is found that Poiseuille's equation no longer applies 

when [i?] is greater than 580. In tubes of such a length that 

any nozzle effect may be neglected, the differential equation 

of flo wis (14, 15, 16) 

dx 2rj, ' ^ ' 

wherein y5 is a constant. For smooth tubes of glass or steel(i7) 

/? = 0-056^, (13) 

but is higher, and follows a different law for rough tubes. For 
some tubes /3 may be taken as independent of [R]. 

There is a formal connection between stream-fine and 
viscous-flow formulae which may be brought out as follows. 
An alternative form of the above equation of flow is 

fn^r.d^ 1 Idjp^) 

[_ 16^ RT_\ dx ' ^ ' 

since the mass velocity W and the volume v of gas transfusing 
in unit time are related by the expression v = ilnd^) W. When 
d{p^)/dx is constant down the length L of the tube, and since 



58 MOLECULAR FLOW IN CAPILLARY SYSTEMS 

for a gas at constant temperature pv or pv is also constant, 
one may write 



or {vp) 



whilst the equation for Poiseuille flow gives 

In the general case of isothermal high-pressure flow, whatever 
the Reynold's number, we may thus say 

{vjp)'>'cc{pl-jpl), (18) 

and the value of n then determines whether stream-Hne or 
turbulent flow is occurring. Never under any conditions have 
values of n been found such that 2<n<\. Later in this chapter 
will be given examples of gas flow in porous sohds which are 
partially turbulent (p. 73). The great importance of gas flow 
or vapour flow in turbine design or in aeronautics can easily 
be understood, and farther information on turbulent flow will 
be found in references given earlier(i4, I5,i6,i7). Equations for 
an adiabatic turbulent flow have been given by Stodola(i5). 

(5) Orifice flow 

In designing permeameters for studying gas flow through 
textiles, Buckingham (5) discussed another type of flow of a fluid 
through frictionless jets, the flow being supposed to occur 
adiabatically. 

Let V, K, E denote the volume, kinetic energy and internal 
energy of unit mass of fluid. For adiabatic flow 

K^-K = E-E^+pv-p^Vj^. (19) 

Let ^1 be a cross-section of the jet at its nozzle, and ^ be a 
cross -section farther upstream where the velocity is neghgible. 
Then iC at yl is negligible and 

Ki = E-E^+pv-p^i\. (20) 



TYPES OF FLOW 59 



If the gas is ideal, 



pv = RT, 

Gy = constant, 

E = TC^ + constant, 



(21) 



(Cp and Cy are, respectively, the specific heats at constant 
pressure and volume), which combined with the original 
equation give ^ = C (T — T) 

If S denotes the mean speed of unit mass over the cross - 
section ^j, ^ ^ 1^2 

i.e. S^ = ^{2C^{T-T^)}. (22) 

As we have assumed the flow to be frictionless, these con- 
siderations are limited to short well-formed nozzles. Then one 
may apply the adiabatic gas laws 

pv^ — constant, 
T = ^(7-i)/y X constant, 

whence -i = h^ = r(r-i)/y 



I (23) 



T \p 

or . S^^^{2TC^{l-r(y-^^ly)}. (24) 

The volume passing the cross-section J.^ in unit time is 
Vi = Aj^Si, which under the initial conditions of pressure is 

V — Ii(^/vi), and thus 

V = yli- V{2Tq,(l -r(^-i)/y)}. (25) 

When A^ is a circular cross-section of diameter d, and since 

vjv-^ — T^^y, one gets 

V^-^^d^^{TC^r^ly{l-r^y-'^^ly)]. (26) 

The conditions under which this type of flow is most accurately 
obeyed are known to require short smooth nozzles, or sharp 
perforations in a plate. The extent to which the law breaks 
down as the nozzles increase in length has not been studied 



60 MOLECULAR FLOW IN CAPILLARY SYSTEMS 

experimentally, nor have its possible uses in determining y, 
the ratio of specific heats, or C^ , the specific heat at constant 
pressure. Reference to orifice flow will be made again in 
considering the permeabifity of various paper and fibre -board 
membranes, when it will be seen that it does not in general 
occur with such membranes (-Chap. IX). 



Permeability constants, units, and dimensions 

By considering the flux of gas through porous membranes, as 
the volume V measured at a pressure p, passing through the 
membrane in time t, in relation to the equation governing the 
flow (19, 20), one can define permeabifity constants of various 
types, and having various dimensions. The equation of flow 
for an incompressible fluid in a pore system may be written: 

^ = AP^{Vi-P2l (27) 

where A is the cross -section of a pore. For a compressible 
fluid it is ^_ 

^ = ^P„^Ny,-p,). (28) 

In these expressions we can call i^ and Pq "permeabifities". 
As such they will be functions of the membrane tlrickness, 
pore type and size, and of the nature of the gas. By multiplying 
each of these permeabifities by /, the membrane thickness, 
one gets "permeability coefficients" Pj^l and P^jZ which are 
dependent only upon pore type and size, and the nature of 
the gas. FinaUy, multiplying each permeabifity coefficient by 
the viscosity i], one obtains the "specific permeabifity" Pjli] 
or Po^, which should depend only on the pore type and size. 
In considering the equation of flow of rarefied gases another 
set of permeabifity constants is obtained. The Knudsen 
equation, for a capillary system, may be written 

^-f = AP,j{p,-p,), (29) 



CONSTANTS, UNITS AND DIMENSIONS 61 

sothatij^ = P(j|(Pj^ +i?2)j ^'^dhas been called byManegold(i8, 19) 

the "molecular permeability". By analogy with the previous 

cases {PqI,Pj^I), one has also a "molecular permeabihty 

coefficient" Ptv^^; and finally a " specific molecular per- 

meabihty" Pjyjl^{M/IiT), which is dependent only upon 

pore size and type. The equation of molecular effusion through 

an orifice gives a "molecular effusion coefficient" P,^, and the 

equation of orifice flow an " adiabatic effusion coefficient " P^. 

These relationships may be summarised as in Table 1. 

The various constants defined above have, it will be seen, 

different dimensions. It is important therefore to specify 

accurately what constants are being employed. When, as is 

usually done, the permeabihty constant is formally defined 

as the number of unit volumes passing in unit time through 

unit cube having unit" pressure difference between its faces, 

its dimensions are cm.^sec.^g."^, which are those of the 

"permeability coefficient" of Table 1, for PoiseuiUe flow. The 

dimensions of the diffusion constant D defined by Fick's law 

dC d^C 

-IT- = D^:—r are cm.^sec."^, which are those of the molecular 

ot ox^ 

permeabihty coefficient of Table 1 . 

The numerical measure of the permeabihty constant is 
dependent upon the units in which it is expressed. The 
hterature shows a lack of uniformity with respect to units. 
It is specially important to state the thickness of the membrane 
used. In some systems it may also be an advantage to give 
other membrane properties such as porosity, defined by the 

^^*i° Pore volume 

Total volume " 

Some experimental investigations op gas plow 
in capillakies 

Warburg (1) first estabhshed the vahdity of Poiseuille's law 
in the region of high pressures. These measurements were 
made before the development of high vacuum technique, but 
at the lower pressures it was even then necessary to assume 



62 



MOLECULAR FLOW IN CAPILLARY SYSTEMS 



P3 



.2 3 


T 7_ 
bb ti 

6 cj 
(U (a 

a a a 

o o o 


d d 
a; ii 

00 05 

"a a "a 


d 


d 

tH 

a 

C3 


f 

ft 

O 


||i g 
^° a ^ 

o 5 g^ * 

i=^l6 M .a 

O CD 

3 g P O 3 

fl S « j3 « 

® o ® 5 ® 
S+3 § C § >, 


CD a a? ® 

.S ° P^ ft 
p. eg a> CD 

Kfe .a .a 

^H CD h 03 lUi 

O 5 O t>D o 

ft « ftt^H ft 

g^ ggg 

fl^ o fl « fl 

45 O CD _, iC 

T3 -a £ "O a -a 


03 ® 
g^ 

OJ O 

§ &^ 

'^-!: 

g-S ^ 

CD TS "3 
T3 C S 

C eS a 
1— 1 


03* 
g 

t+-i o 
O ft 

p 

II 

ll 

1— 1 


03 

1 

05 

c 
o 

1 

CD 

a 

Ph 


'o 'o ;^ a 

" " a a 

^ ^ ^ ii! O O 

C^ d (D CD -i^ -p 
CD CD rt fl O O 

a a 1 s g,§, 

(i, f^ 0^ a, Oh 0. 


1 ? ^ 

^ 3 r CD 

1 alg- 

5 ^^°J§ 


o 
u 

g 

03 

ID 

M 

o 

^^ 

2 !=) 

a. 2 


o 

o 

.2 

03 

4) 
O 

If 
«3.2 

■ — ■ o 


O 

o 
a 


+ 

1 Sin 

.2 II '^ 

CD 3 ^ 
CO ^ ^. — . 


g? 

fi ^ 
^ II 

1-^ 


o 

1 _ 

00 05 -^ 


g.r-g 

c» ^^ iL 

1 :^« 

c^J -St, c3 

« * h ! 

l"-2 



EXPERIMENTAL INVESTIGATIONS OF GAS FLOW 63 



a slip between the wall of the capillary and the adjacent gas 
layer due to specular reflection (compare with equation (9)). 
Warburg expressed his results by the formula 

4r 



G 



= lap + apyUp-^^-p^l 



(30) 



where his G expressed the quantity of gas flowing per unit of 
time (measured as the product of pressure in bars and volume), 

a IS a constant = --— [r being the capillary radius), p is the 

mean pressure = ^{p-^^ +P2)> a-nd ^ is the " coefficient of sKp ".* 

.J- 
-M 



0-03 
0-02 
0-01 




Fig. 11. The gas flow as a function of pressure. 

Knudsen (2) expressed his results by the empirical formula 

1 + CiP 



G 



=[ 



ap-\-h 



:](.. 



-i?2): 



(31) 



\+G^p_ 

where h was called the "coefficient of molecular streaming" 
and Ci and G^ were two quantities depending upon shp. At 
high pressure the first term in the square bracket alone is 
significant, but at the lowest pressure the second term in the 
bracket was predominant, and so reduced to 6. On the 
basis of the kinetic theory, values may be assigned to 6, G^ 
and Cg, and to t,. Fig. 11(6) gives the quantity of gas flowing 
as a function of pressure, curve I referring to the simple 
Poiseuille law G = CLpiPi — p^)^ curve II to the calculated rate 
of flow according to equation (31), and curve III a typical 
run by Knudsen. The experimental curve shows a minimum 



2 -f 
g= -~ A, as defined on p. 55. 



64 MOLECULAR FLOW IN CAPILLARY SYSTEMS 

when G is some 5 % less than O dut p = 0. This minimum^ 
which is characteristic of most of the results obtained (2, 6, 20), 
being true even for mixtures (2i) of gases, does not appear to 
have been explained. The minimum occurred in this case (6) 
when the mean free path was about five times as large as the 
capillary radius. Similarly, Fig. 12(6) gives another series of 
curves, calculated and experimental, by several authors. 
Curve I is for simple Poiseuille flow for hydrogen, curve II 
for Poiseuille flow with sUp, curve III for hydrogen employing 




750 P 

Fig. 12. Gas flow as a function of pressui-e. 



Knudsen's empirical equation for Gaede's(3) apparatus and 
results (the calculations being made by Klose (6)), while curve IV 
gives the experimental measurements of Gaede. The point 
marked " /Sm " on the axis of O was calculated by Klose from 
Smoluchowski's formula for a tube of rectangular cross- 
section. Fig. 12 shows the degree of divergence in theory and 
experiment of the various authorities. Clausing's(7) experi- 
ments, which are not cited in Frgs. 1 1 or 12, are probably more 
accurate than any series. In the region of molecular streaming 
or Knudsen flow he found Knudsen's formula satisfactory, as 
did Klose, who, however, still noted small deviations which 
he ascribed to an experimental error. 



EXPERIMENTAL INVESTIGATIONS OF GAS FLOW 65 

One may summarise the situation as follows: 

(i) For high pressures theory and experiment both lead 
to the Poiseuille law wherein slip may be neglected. 

(ii) For the lowest pressures theory and experiment are 
agreed on the type of molecular streaming proposed by 
Knudsen. 

(iii) In the region of intermediate pressures theory and 
experiment do not yet satisfactorily agree, nor is it certain 
to what extent the deviations are due to experimental errors 
and to what extent to inadequacies in the theory, 

Adzumi(20) has recently tested the Knudsen equation fof 
the mixtures Hg-CgHg, Ha-CgHg, finding a flow O = h{f-^—p<^ 
which was that given by the ideal mixture law b = n-J}-^ + n^b^, 
the TV's, being mol. fractions. In the mixtures at the lowest 
pressures the constituents diffused by molecular streaming 
independently, but when Poiseuille flow became predominant 
at high pressure the separation of constituents by flow became 
neghgible. 

Flow of gases through porous plates 

Since a porous plate consists of a medley of pores of different 
sizes, flow through such a plate need not be of one type. 
Instead, one would anticipate that flow would be a mixture 
of the various kinds, reducing to Knudsen flow at the lowest 
pressures. Effusion through an orifice or molecular streaming 
down capillaries should occur for all gases at rates inversely 
proportional to the square root of the molecular weight, the 
condition for effusion or molecular streaming being a suitably 
low pressure. It is interesting to see under what conditions 
the law is fulfilled. Ramsay and Collie (22), by diffusing hehum 
through unglazed clay and comparing the data with the corre- 
sponding diffusion of hydrogen, found the atomic weight of 
helium to be rather low (3-74 compared with 4-00); oxygen, 
argon and acetylene however obeyed the Knudsen flow formula . 
Donnan(23) measured the effusion of argon, carbon dioxide, 
carbon monoxide and oxygen through an orifice, finding small 



66 MOLECULAR FLOW IN CAPILLARY SYSTEMS 

deviations from the l/sJM law, which they attributed in part 
to orifice flow as described by Buckingham (5) and others. These 
measurements were made however at pressures approximating 
to atmospheric. 

Recently, Sameshima (24) has measured the rates of flow of 
various simple gases through a compact unglazed earthenware 
plate. The rates of flow definitely did not obey the Knudsen 
formula t = k^jM, where t denotes the time required for the 
efiusion at constant pressure of a volume F of a gas of molecular 
weight M, and where ^ is a constant. On the other hand, the 
law t = k^M was accurately obeyed when the gases effused 
through a platinum plate with a single orifice. For the 
earthenware plate Sameshima found a formula t = k7j^M^^^~''^^ 
to apply. If the wall was very thin n approached zero, and 
the simple behaviour of the perforated platinuni plate was 
found. If the wall was thick n approached unity and the 
equation became t = k7j {tj denotes viscosity). 

Adzumi(25) has elaborated a semi-empirical theory for flow 
processes through porous plates, based on the assumption 
that the plate is perforated by numerous fine holes, the 
diameters of which can vary down their lengths, so that 
efi"ectively each capillary is a number of capillaries of very 
numerous diameters arranged in series; and each such com- 
posite capillary is in parallel with all the other capillaries. At 
high pressures the Poiseuille law G^ = cip{pj^—p^) apphes 
(p. 55), and at low pressures the law G^ = ^'{'p-^—p^) (P- 54). 
J'or medium pressures, or for a porous plate, the two laws 
may be considered to overlap: 

G = G, + yG,,==K{p^-p^), (32) 

where y is a term due to the coeflicient of slip, and where 

K — A yV + jB-j. If one has n caj)illaries in parallel, 
G = K'{p^-p)^), 
K' = K^ + K., + ... 

= Ap-Zj + 7BZj. (33) 



FLOW OF GASES THROUGH POROUS PLATES 67 

When one has two capillaries of radii r j^ and r^ in series, 

^1 = ^Iv + 7^1m> 
Org = tr^v + y^2m5 

1 _ 1 1 

where 2^1 and p^ are the pressures at the inlet and outlet sides 
of the capillary, and Pq the pressure at the junction of its two 
halves of radii r^^ and r^. 

The last equation takes the form 

where Zl^ proves on inserting numerical values for a typical 
plate to be less than 1 % of -^r;^ . This gives 

which may in turn be written as 

where A^ again proves to be less than 1 % of iC". One obtains 
therefore afe the final constant K" for m capillaries in series 

^"-^P^+yB^, (38) 






while for the series-parallel arrangement of capillaries, pre- 
sumed to comprise the whole pore system of the plate, 

= ApE + yBF. (39) 

Fig. 13 gives the quantities of gas flowing through a porous 
plate (expressed as c.c. x atm.) as a function of pressure for 

5-2 



68 



MOLECULAR FLOW IN CAPILLARY SYSTEMS 



a number of gases. It will be seen that, as the theory requires, 
the rate of flow is proportional to the pressure, for a constant 
pressure difference, but with an intercept (= yBF) on the 
JK^-axis. 



0-02 




Fig. 13. Gas flow through a porous plate, where K is defined by equation (39). 



The treatment of Adzumi allows one to estimate a mean 
pore size, and the number of pores per unit volume. Should 
all the n pores be of radius r, 

E = n — ; F = Thy (Z = thickness). 

V V 

If iV = the number of pores/unit volume. 



^0 = Nr^; 



F, = Nr\ 



and one may readily compute E^ and Fq from E and F and so 
find N and r. This is done for a few permeable plates in Table 2, 



FLOW OF GASES THROUGH POROUS PLATES 



69 



For further examples of flow through porous plates one may 
study the permeabihty of various refractories, the properties 
of which, in relation to the dijffusion problem, are given in 
the next section. 



Table 2. The mean radii of pcyres, and the 
number of pores per unit volume 





Thick- 












Porous plate 


ness 
cm. 


Area 
cm.2 


E^ X 10" 


i^o X 10' 


rxlO^ 
. cm. 


N 


Unglazed earthen- 


0-15 


0-28 


0-97 


1-83 


5-3 


1-2 xl06 


ware 














Compact porous pot 


0-27 


2-22 


0-054 


0-195 


2-77 


2-4 xl05 


Rough porous pot 


0-41 


2-22 


0-154 


0-318 


4-84 


2-8 xl05 


Mantle of Daniell 


0-2 


0-27 


0-80 


4-98 


1-61 


9-6 xlO' 


ceU, I 














Mantle of Daniell 


0-2 


0-27 


1-09 


6-26 


1-74 


1-19 x 108 


ceU, II 















Permeability of refractories 

Diffusion through refractories is of technical importance. 
According to the fineness of the pore structure and the 
pressure the flow should conform predominantly to the 
equations for Poiseuille flow or Knudsen flow. In general 
the numerical value of the permeabihty depends on a variety 
of factors (26) which include : 

(1) The material of which the refractory is made (sihca, 
fireclay, alumina, chrome, or magnesite). 

(2) The physical constitution of the refractory, the grain 
size of the grog, the percentage of grog to bond, and the 
method of manufacture. 

(3) The nature of the diffusing gas. 

(4) The testing temperature, and the thermal expansibihty 
of the sohd. 

(5) The pressure difference maintained between the ingoing 
and outgoing surfaces. 

(6) The thickness of the specimen being tested. 



70 



MOLECULAR FLOW IN CAPILLARY SYSTEMS 



As a rule the physical constitution produces greater changes 
in the permeability than the chemical nature of the refractory. 
Thus one finds a very great range in permeabihty in different 
samples of the same chemical nature, as is illustrated in the 
following data (Table 3) due to Kanz(27). In agreement with 

Table 3. The range of 'permeability in 
some refractories 



Type of substance 


Permeability range 
(c.c./sec./cm.7cm./sec./cm. of 
water difference in pressure) 


Fireclay product 
SUica product 
Magnesite product 
Chromite product 
Insulating product 


0-00278-0-476 
0-0158 -0-132 
0-0275 -0-265 
0-147 -1-42 
0-00597-0-539 

















/ 




/ 














/ 






/" 












^ 


/ 




/ 




s< 










/ 






/ 




t 

> 

a 
o 








J 


^ 




o / 

/Pr 


XWCT 


I_ 


_Pr. 


)DUCT 


M 


/ 




/ 






/ 




/ 


^ 








Pm 






/ 




/ 














/ 




/ 














/ 


/ 




/ 














/ 


. / 

















Reciprocal of thickness 
Fig. 14. Gas flow through refractories as a function of thickness. 

both Poiseuille's and Knudsen's equations the velocity of 
transfusion is inversely proportional to L, the thickness of the 
specimen. Fig. 14, originally due to Clews and Green (28), shows 



PERMEABILITY OP REFRACTORIES 71 

this relationship for two refractories. The product I shows a 
gradual change in texture with thickness, so that the curve 
does not pass through the origin. Usually it appears that the 
velocity of flow is proportional to the pressure difference 
causing it, provided in the case of Poiseuille flow one measures 
the permeation velocity in c.c./unit time at the mean pressure 
2(^2+ Pi) obtaining inside the material. This proportionality, 
required by the equation already obtained, 

dn dhr 1 p 

was observed by Bansen{29) for a mortar joint, by Kanz (27) 
for a specimen refractory, and by Clews and Green (28), 

Temperature and permeability in refractories 

One may look for interesting results by considering the 
influence of temperature upon the rate of permeation of 
refractories. The effects of temperature help to sort out the 
different types of flow, Poiseuille, Knudsen, and activated 
diffusion. In a refractory should occur large pores and tubes, 
small pores, and pores of molecular or sub-molecular dimen- 
sions. In the large pores one would anticipate Poiseuille flow; 
the permeability Pq is proportional to 1/^/ {tj — viscosity); 
and the viscosity increases as temperature increases. Thus 
Poiseuille flow in capillary systems is marked by a diminution 
in flow rate with rising temperature. When Knudsen flow is 
occurring the permeability P-^j increases according to a aJT 
relationship. Finally, in a very compact sohd where activated 
diffusion predominates the permeabihty constant P contains 
a term q-^IRt ^ 

For systems in which Poiseuille flow is important Preston (26) 
has pointed out that for a given refractory, at all temperatures, 
the quantity Polrj is constant. Then if PqIti be the specific 
permeabihty (p. 62) at a temperature T° C, and PqI be the 
permeabihty coefficient at a standard temperature, the ratio 
Pq7j/Pq should also be a constant. The data of Clews and 
Green (28) are summarised by Preston (26) in Table 4, where it 



72 



MOLECULAR FLOW IN CAPILLARY SYSTEMS 



is seen that the product 7j{PqIP°q) shows small trends with 
temperature, but tends to be constant. 

The role played by alternative processes of Poiseuille, 
Knudsen, or activated flow is perhaps to be seen in the data 

Table 4. The influence of temperature upon 
permeability constants 



Temp. 


Viscesity 
ofNg 


, Fireclay product 


Silica product 










°C. 


(poises 
xl04) 


PgIPg" 


V{Pg/Pg°) 


Pg/Po° 


V(Pg/Pg°) 


10 


1-71 


0-997 


1-71 


0-992 


1-70 


100 


2-13 


0-841 


1-79 


0-860 


1-81 


150 


2-33 


0-785 


1-83 


0-828 


1-93 


200 


2-52 


0-740 


1-86 


0-759 


1-91 


250 


2-70 


0-710 


1-92 


0-670 


1-81 


300 


2-87 


0-687 


1-97 


0-599 


1-72 


350 


3-03 


0-662 


2-01 


0-546 


1-65 


400 


3-19 


0-639 


2-04 


0-517 


1-65 


450 


3-34 


0-613 


2-05 


0-493 


1-65 


500 


3-48 


0-592 


2-06 


0-473 


1-65 



Table 5. A comparison of theoretical and experimental 
permeation rate ratios 



Temp. 


Permeability COg 


Permeability Hg 


Permeability SOj 


°C. 


Permeability air 


Permeability air 


Permeability air 






Theoretical ratio 






0-81 


3-79 


0-67 


17 


0-80 


3-79 


0-65 


100 


0-80 


3-82 


0-98 


145 


— 


— 


110 


190 


— 


— 


0-95 


200 


0-79 


3-55 


0-96 


300 


0-79 


3-07 


Ml 


400 


0-80 


2-92 


113 


500 


0-83 


2-98 


— 


600 


0-84 


— 


— 


700 


0-84 


— 


^ — 



collected by Bremond(30) on the diffusion of gases through 
unglazed porcelains , between 17 and 700° C. This author 
discovered that for air, carbon dioxide, hydrogen, and sulphur 
dioxide the permeabilities at fii-et decreased and then increased 



PERMEABILITY OF REFRACTORIES 73 

with temperature. The temperatures of mmimum permeation 
velocity were: 

Gas Air , CO2 Ha SOg 

Temp. ° C. 250 240 340 250 

The theory of Knudsen flow requires that the ratios of the 
rates of flow should be in the inverse ratios of the square roots 
of the mofecular weights. Here again the ratios diverge from 
the requirements of the theory at high temperatures (Table 5). 
These ratios, as well as the increase in diffusion velocity, may 
imply a certain amount of activated diffusion at the highest 
temperatures. 

Flow in consolidated and itnconsolidated sands 

A number of experiments have been carried out upon fluid flow 
through sands, sandstones, or columns of beads. The experi- 
menters (3i) have sought to correlate the phenomena of gas or 
liquid flow with permeation rates of petroleum and its vapours 
through oil-bearing sands, and with the possibility of displacing 
oils from these sands by another fluid. Thus their measure- 
ments, whose technical application may become considerable, 
were carried out at high pressures, and as one would expect 
cover the regions of Poiseuille flow and of turbulent flow. The 
quantitative application of Poiseuille 's law to each of the pores 
in sand or a sandstone must have corrections for features such 
as the following (32, 33, 34): 

(1) Deviations of the cross-section of an average pore from 
circular shape. 

(2) The increased length of a sinuous path through the 
medium, as compared with its apparent length, and so the 
greater pressure gradient across the specimen needed to 
maintain a flow rate equal to that for a straight path. 

(3) The energy consumption due to alternate expansions 
and contractions of cross-section. 

There is little that can be done to allow for (1). Schlichter(35) 
calculated that in an assembly of spheres the actual path was 



74 MOLECULAR FLOW IN CAPILLARY SYSTEMS 

1'2-1'5 times the apparent path; while Chilton and Colbourn(36) 
claimed that 80% of resistance to flow should be due to 
alternate contractions and expansions of the cross-section, 
and only 20 % due to the viscosity. However, equations 
having the form of Poiseuille's law must and do apply, and 
one has as examples Adzumi's data for flow at moderately 
small pressures through porous plates (p. 66), and the data 
cited in the previous pages (69-73) for the permeabihty of 
refractories. In papers dealing with the permeabihties of un- 
consolidated and consolidated sands it has become customary 
to plot the logarithm of the so-called "friction factor", /, 
against the logarithm of Reynold's number [B] = dWpl4:7j{i.c.). 
The curve is rectilinear at low pressures, but undergoes a 
smooth continuous transition to another rectihnear portion of 
smaller slope at the onset of turbulent flow (37, 38) (Fig. 15, p. 75) . 
The friction factor is usually deflned by Fanning's (39) equation 
for a capillary 

where g = the acceleration due to gravity, d — the diameter 
of a capillary, p = the density of the gas at the temperature 
of the experiment, and the mean pressure ^{pi+P2) of flow, 
W = the velocity of flow in c.c./sec. Fanning's equation is 
derivable from an equation of turbulent flow 

iPv)'=j^{pl-pl), (41) 

similar to that described on p. 57. A denotes the area of 
cross-section of the capillary; and v the volume passing 
through per second measured at pressure p. Stream-line flow 
obeys a law wherein (pv) appears only to the power unity 
(p. 58), and so one sees how the curve log (/) versus log [E] 
is a straight line of slope — 1 while stream -hne, viscous or 
Poiseuille flow is occurring; and —0-5 when turbulent flow 
has set in. In Fig. 15, taken from the work of Fancher and 
Lewis(3i), log (/) versus log [R] plots have been given for a 
great variety of porous solids. The pressure at which the 
experiments were carried out varied from 10 atm. per square 



I0«10' 




















NOMEMCLATURE 































-+ + 


! 





■ — 






2 FEET PER SEC.PER SEC. 
METEPOfAVEOAGE GRA.N.FEFT 
E55URE OBOP LBS PER SI}. FT, 
NGTH or CUBE. FEET 


sample: no. 


SAND 


POROSITY 













J 


1 


, 






1 

2 

3 

4 
b 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
?0 
21 
22 
23 




y° 






V 






- + + 




— 


' — 




TT (/»P)=PR 


BRADFORD 

BRADFORD 

3 "O VENANGO 

CERAMIC A 

ROBINSON 

CERAMIC B 

WOODBINE 

WILCOX 

3"0 VENANGO 

ROBINSON 

ROBINSON 

3RDVENANGO 

WILCOX 

WA R R E N 

3B0 VENANGO 

ROBINSON 

CERAMIC C 

3"PVENANG0 

WOODBINE 


12-5 
12-3 
18-9 
37-0 
20-3 
37-8 
19-7 
15-9 
11-9 
19-5 
18-4 
22-3 
16-3 
19-2 
21-4 
20-5 
33-2 
219 
23-8 
26-9 
27-7 
22-1 
28-8 






% 
















L=l_E 






\ 


, 




















•^ 


















flATC Of FLOW rEET 


PER SEC 

DUS 

TFfiLPRESENT 
"T DATA 

jF*HCMEH.Lnwi5 

OF MINE5 






"^ 


■ SYMB 




I-Oiio' 








1 










- - ^-AB 




























































































































s, 








X 
















HOIL 






\ 








'-s 
















+ CA 






^ 








\^, 




















H 


\ 


\ 






"^ 


















I-0«IO^ 




\ 




^ 9 






1 


^J 










































































3 






































































^ ^ 




















5 










\ 


\ 


















"C 












\^ 


0^ 










--- 












'■"'v 


1 




N^ 


:s^ 


^ 










24 


u 


>JC0N50LI DATED 


38-5 




' 








FLINT 








\ 






|b 


^> 


u. 


< 




^^ 










25 
26 
27 


OTTAWA 
20-30 OTTAWA 
LEAD SHOT 


30-9 
34.5 
34-5 






"■ 




























- 










— 


1 









' — 






























































































































v> < 


^ \ 


/■ 


^ 


1 


























Q 
























\ 


\ 


















































°( 






\ 




S,'^ 


-^u 
















































4l 






\^ 


\ 1 


^\ 








































1-OxlO^ 










\ 






^n 


»1^^^ 

P^^ 


1 










' 






































































































































































































































































J: 


% 




• i S'"'"""^ 




^^^1 














































^ 


^ 


, 


>■>." 




. 
















































i^. 


N \ 




i^jv^ 










































5AHD 1 R 


. ^ 


k 


\ 




ijT^^ 


*~K 






























t-o.io' 


















K. 


\ 


^ 


>- 


.'X 


s 


■o 

X 




























































































































































































































"^ 
























































K 




, 




'V-- 


















































\ 




N 




^^^'- 








■>"-. 










































\ 




o 


\ 


'^V 


?o 
















































LEAO^ 
SHOT 


\ 




^ 


. \^ 


\J 






























10x10' 


























-^1 


N 


3 




tj 







































— 






— 








■^ — 


1 




- V" 


|-5 




— 














^ 


























































































■*■ 


° o 


























































■ o 




« 


























































■>. 


V 


























































' -t 


'», 
















^ 






10X102 


































' 1- 


r^ 


































= 


— 






=: 








:zz 




~ 


I i: 


' V^ 




b= 




i: 




^ 




M 


p:: 


:: 




































































































L+ 


I- 


^ ^ 


























































■ - 


rr-!^* 


























































^*^*~ 




























1 
































+- 




t, 


4-t 


i< 




























































0-( 


)0! 









■01 








01 


c 


W| 


e. 


10 










100 










100 










000 



Fig. 15. The curves log (friction factor) versus log (Rejoiold's number) in the flow of 
various simple fluids through porous materials. 



76 



MOLECULAR FLOW IN CAPILLARY SYSTEMS 



inch downwards. Fancher and Lewis's notation and the sub- 
stances used are explained in the figure. It is interesting that 
the unconsoUdated sands all give curves which fall near a 
single mean curve, so that they may be said to exhibit a 
typical behaviour. 

Several types of the apparatus (40,4i) devised had arrange- 
ments of pressure gauges at intervals down the length of the 
column ofsand through which permeation was occurring. These 
workers then were able to measure the pressure gradient along 
the length of the column. It was found that dp^/dx was 
constant, and thus equal to {pl—pD/L. The curves pv {or pv), 
against (p'f—pl), obtained by Muskat and Botset(40) for un- 
consoUdated and consolidated sands, emphasise still further 
that the law of flow is of the form 

{pl-pl) = k{pvr, (42) 

where l<n<2 (see pp. 57 and 58) according to the amount 
of turbulent or stream-line flow occurring. The chance of 
turbulent flow occurring decreased with decrease in grain 
size, whilst of course the permeability decreased as turbulent 
flow set in, i.e. as n increased from 1 towards 2. 

►Some typical permeability constants for the stream-line, 
flow of gases and liquids through media such as have just been 
considered are presented in Tables 6-8. The data recorded in 

Table 6. Data of Green and Ampti'i2) for flow through 
unconsolidated media 



Substance 


Mean 

radius 

cm. 

xlO-2 


Porosity 

/o 


Specific permeability 1 
c.c./sec./cm.2/mm. thick/cm. 

Hg/unit viscosity, when, 
meq.(28),2p/{p,+p.,)=l 


Specific per- 
meability by 
gas flow 
xlO-8 


Specific per- 
meability by 
liquid flow 
xlO-« 


Glass spheres 
Quartz sand 


4-69 
3-55 
2-49 
1-59 
1-25 
413 
1-45 
0-93 


3G-4 
37-3 
361 
36-3 
36-6 
34-7 
34-7 
37-7 


372 

250 

113 
47-6 
28-6 

157-5 
22-3 
101 


357 
. 252 
115 
4G-5 
28-3 
154 
22-3 
9-75 



Table 7. Data of Muskat, Botset and co-workers (4i) ; andFancher 
and Lewis (31). Stream-line flow through unconsolidated media 



Substance 


Mean 

radius of 

particles 

cm. 

X 10-2 


Porosity 
% 


Per- 
meability 

co- 
efficient 
PgI 


Specific 
permeability 


PgIv 

xl0-« 


PlIv 

xlO-« 


Glass beads 





33-8 


1350 


248 


— 


\ 40-45 mesh sand 


— 


— 


1025 


188 


188 


60-65 mefih sand 


— 


440 


233 


42-6 


— 


i 80-100 mesh sand 


— 


— 


183 


33-5 


29-6 


1 Heterogeneous sand 
j Lead shot 


5 


. 42-0 
34-5 


271 
3190 


49-6 

584 


— 


1 Ottawa sand, unsieved 


36-7 


30-9 


1175 


215 


— 


Ottawa sand, sieved 


35-7 


34-5 


1117 


204 


— 


! Flint sand, Pennsyl- 


18-75 


38-5 


308 


56-3 


— 


i vania 













Table 8. Data of Fancher and Lewis (-H). Stream-line flow 
through consolidated media. The specific permeabilities 
are measured with air, water, and crude petroleum; 
^p/iPi+Th) — ^- '^^^^ permeability coefficients are those 
calculated from Pfjlf) for air 



Sandstone 


Mean 

particle 

radius 

cin. 

xlO-» 


Porosity 
/o 


Per- 
meability 
coefficient 

PgI 

for air 


Specific 
per- 
meability 


PqIv 

xlO-» 


Woodbine, Texas 

Venango, Pennsylvania 
Woodbine, Texas 
Wilcox, Oklahoma 
Venango, Pennsylvania 
Wilcox, Oklahoma 
Venango, Pennsylvania 
Woodbine, Texas 
1 Warren, Pennsylvania 
Robinson, Illinois 

Venango, Pennsylvania 
Ceramic C (5% binding 

agent) 
Ceramic B (10% binding 
agent) 
1 Ceramic A (20 % binding 
1 agent) 
{ Bradford, Pennsylvania 


8-11 
8-35 
6-39 
12-53 
4-51 
6-99 
4-54 
6-97 
419 
3-13 
3-26 
2-74 
2-74 
2-71 
4-90 
1-43 

1-43 

1-43 

2-79 
2-82 


22-1 
26-9 

28-8 

11-9 

22-7 

16-3 

21-9 

15-9 

21-4 

23-8 • 

19-2 

20-6 

18-4 

19-5 

16-9 

33-2 

37-8 

37-0 

12-3 

12-5 


24-2 

18-2 

17-3 
8-79 
6-35 
4-08 
3-32 
2-57 
1-78 
1-73 
1-03 
0-925 
0-498 
0-466 
0-347 
0-273 

0-065 

0-0380 

0-0209 
0-0194 


4-62 

3-33 

3-16 

1-61 

1-16 

0-75 

0-607 

0-47 

0-326 

0-317 

0-189 

0-170 

0091 

0-0855 

0-0636 

0-0502 

0-0119 

00070 

000384 
000356 



78 MOLECULAR FLOW IN CAPILLARY SYSTEMS 

these tables show how particle size modifies permeability, at 
nearly constant porosity (Table 6) and how the nature of the 
particle (e.g. its smoothness) also controls the gas flow. In 
general, particle size controls the flow rate more than porosity; 
and the rate of flow is less through consoHdated than through 
unconsoHdated media. The tables also show a good agreement 
between specific permeabilities determined with liquids and 
with gases (Tables 7 and 6). 

Flow through miscellaneous porous systems 

In the literature will be found measurements on liquid or gas 
flow through a great variety of substances. The permeabihty of 
papers, leathers, textiles, and flbreboards is treated elsewhere 
(Chap. IX). The permeabihty of various kinds of wood (43), 
of a sihcic acid gel (44), of building and heat -insulating sub- 
stances (45, 46), and of various soils (42) illustrate the systems 
studied. Manegold(i9) summarised a number of measurements 
on these systems, employing the permeabilities or specific 
permeabilities of Table 1 . 

Separation of gas mixtures and isotopes 

Phenomena connected with effusion and molecular streaming 
through porous plates, and in tubes, have had some interesting 
applications in separating the components of gas mixtures 
and, in particular, isotopes. The original method for the 
separation of isotopes developed by Hertz (47) required a 
battery of mercury-in-glass diffusion pumps connected in 
series. Each pump circulates the gas mixture through a clay 
tube; a greater part of the lighter than of the heavier gas 
diffuses through the wall in transit down the tube, and enters 
a counter-current rich in the hght fraction on the other side. 
These fighter fractions tend to enter another higher separating 
unit, and the heavier fraction a lower unit. The method has 
served to effect a complete separation of the isotopes of 
hydrogen (Hg and D2)(48), and a partial separation of the 
isotopes of neon, carbon and nitrogen (49, 50). A schematic 



SEPARATION OF GAS MIXTURES AND ISOTOPES 79 

representation of a Hertz porous wall separation unit is 
shown in Fig. 16(49). 

Recently, however, the type of diffusion unit with a porous 
wall has been abandoned by Hertz and his collaborators (oi), 
and instead batteries of diffusion pumps are joined together 
directly, separation occurring by effusion together with 
mercury vapour through jets. The theory of the separation of 



7st Separation r 2ncl Separation i 3rd Separation i 4th Separation 
t^nit I unit ; unit i unit 




Fig. 16. A Hertz porous wall diffusion apparatus. 



isotopes by this method has been given by Barwich(52). For 
example, one might be employing a battery of pumps to 
separate the isotopes of carbon, C^g and C^^, in gaseous form 
as methane (53) . Then according to Barwich one set of conditions 
gives 



hxg = ^(i?-.-.o)[(^^-^J+ln| 



(43) 



where D^, D^ denote the diffusion constants of the isotopes, 
V is the speed of the mercury vapour jet, R, s, and Xq are 
geometrical factors of the apparatus, and q is the separation 
coefficient defined by 



Ci3 (exit) Ci2 (entrance) 
Ci2 (exit ) Cjg (entrance ) * 



(44) 



80 MOLECULAR FLOW IN CAPILLARY SYSTEMS 

This equation holds only when the pressure of diffusing gas 
is high. When the pressure and temperature are such that the 
densities of the diffusing gases are small compared with the 
density of the mercury vapour, one has D = \wA {w = the 
molecular velocity;^ = themean free path), and the expression 
for the separation of the isotopes becomes 

.\nq = const. p-^s^W^ + r^^f y/m^ - {r^ + Tj^g)^ V^j], (45) 
where r^, r^, r^g are the molecular radii of the two isotopes 
and of mercury atoms respectively (rj^rg), and yo^g is the 
density of the mercury vapour. 

A diagram of the diffusion unit (54) used to separate isotopes 
bythismethodisgiveninFig. 17. Itisnot necessary, according 
to a recent communication (53), to connect the pumps by a 
capillary tube. With a battery of fifty-one pumps it was found 
that a 32 % separation of C^a and Cjg could be effected from 
300 c.c. of methane at 1-8 mm. pressure. The advantage of this 
newer arrangement of Hertz's original method is that there 
is no accumulation of impurities within a porous wall, and that 
higher separation factors are possible. 

Another method of separating hquid and gaseous mixtures, 
which was first developed by Clusius and Dickel(55), is based 
upon Chapman's early study of thermo-diffusion in gases (56). 
This method promises to be very efficient in separating 
isotopic mixtures, and is simple to operate. If a hot wire passes 
axially down a tube containing a mixture of gases, the gases 
under the combined influence of radial and axial diffusion and 
of convection undergo a partial separation into the two 
components (57, 58) , Let the separation unit consist of a hot and 
a cold surface, plane and parallel, between which is confined the 
solution being disproportionated. The temperature gradient 
exists across this solution, in the a;-direction, and convection 
occurs in the 2-direction. Then the separation velocity is 
governed by the differential equation (57) 

where v{x) denotes the convection velocity as a function of x, 



SEPARATION OP GAS MIXTURES AND ISOTOPES 81 

and T the temperature. For a separating unit of length I 
there is a characteristic time 

which gives the order of magnitude of the time required for 
the separation equiUbrium to be approached. In a normal 



_-CM. 




Fig. 17. Mercuiy diffusion pumps. "A" is the Hertz pump and "B" is a unit 
with a modified mercury jet. The pumps are connected in series as shown. 
(Scherr (54).) 

apparatus this would be 1000 days for hquids, and < 1 day 
for gases. The equihbrium state is defined by the equation 



^3^0^ 



82 



MOLECULAR FLOW IN CAPILLARY SYSTEMS 



It was found in a liquid mixture D20(32%) + ^20(68°/^) that 
a separation of up to 4-8 % was established in two days — ■ 
about Yo of the equihbrium separation in the stationary state. 
The method was employed successfully to disproportionate 
the isotopes Zug^, Zugg, Zngg, as aqueous solutions of zinc sul- 
phate (57). The isotopes were estimated spectroscopically. The 
method has also been used (59) in the separation of isotopes 
of CI and of Hg. The method is of quite general apphcabihty 
in disproportionating or sedimenting such mixtures as 



ZnCL, ZnSO. in H20(57), 



?2-Hexane in CCl4(57), 
CgHg in C6H5C1(57), 



NaCl, Na2S04 in H20(57), 
Chlorophyll in CCl4(57). 
Table 9 illustrates its efficiency in separating gas mixtures (55). 

Table 9. The Glusius-Dickel method of separating 
gas mixtures 



Total pressure approximately 1 atm. 


Original gas 
mixture 

0/ 
/o 


Length of 

separating 

column 

cm. 


Temp, 
diff. 

°C. 


Composition at 


"Heavy" 
end 


"Light" 
end 


25 Bra 75 He 
4OCO2 60 Ha 

20 Og 80 N2 (air) 

Normal Ne 

(at. wt. =20-18) 
23H="C1 77H35C1 


65 

100 

(a) 100 

{b) 290 

260 . 

290 


-300 
-600 
-600 
-600 
-600 

-600 


100% Bra (Uq.) 
100% CO2 

42% Oa 

85% Oa 

At. wt. 20-68 

40% HS'Cl 
60% H35C1 


100% He 



NON-STATIOlSrARY STATES OF CAPILLARY GAS FLOW 

Stationary streaming proceeds at the same rate whether the 
molecules have a long hfe in the adsorbed phase or not, 
according at any instant to the equation 

dn IB 1 (v. v.\l2-f) ' 



dt 



(Pi i>2\(2-/) 
2L^{27rME)\^T, ^Tj f 



already discussed. In the stationary state every molecule 
leaving the wall is replaced by one striking the wall. Clausing (7), 



STATES OE CAPILLARY GAS ELOW 83 

however, has shown how non-stationary streaming may be 
used to determine the hfetime of molecules in the adsorbed 
phase, and heats of adsorption. His experiments and theoretical 
treatment are of interest because they obviate any necessity 
for direct measurement of the very minute quantities ad- 
sorbed on certain surfaces such as glass and crystals. The 
essential apparatus consists of two large flasks connected 
by a fine capillary. In one flask, for all times t>0 the 
pressure is p^^; while in the capillary and in the second flask 
the pressure P2 — 0- At time t = the gas enters the capiUary. 
A certain time must elapse before the first molecule enters 
the second flask, and this time is a function of the lifetime 
of the molecules in the adsorbed state, being greater the 
greater this lifetime; 

In obtaining a solution for the diffusion into the capiUary, 
Clausing assumed that surface diffusion need not be considered 
in comparison with bulk diffusion. The diffusion equation is 
then once more * _„^ _^ 

The initial condition is, inside the capillary, 

for ^ = 0, C = 0. 
The boundary conditions are 

for a; = 0, C = C^ for all t, 

for X = L, G -^ i) for all t. 
The solution of the problem then is 

C = C'J — ^ e-^'^ sin nt -^^ \ - ^e-^l"" sin 27t 



\_ L n[ \ L J '' \ L 

(49) 



if a = L^/tt'-D. 
When X = L 



(^) ^ -^{i-2(e-^.--e-^-^.'- + e-«^/-)}. (50) 

* In equations (iS) to (58) C denotes the number of molecules per unit length 
of the capillary, including those adsorbed on the waU. It is given by equation (55). 



84 MOLECULAR FLOW IN CAPILLARY SYSTEMS 

Thus in time t there flows into the second vessel a number of 
molecules N^ given by 

'■-I'A-'iL'- 

SO that 

N, = ^[t-2a{{l-e-fl-)-i{l-e-^l-) + l{l-e-^(l'^)- ...)]. 

^ (51) 

The value of D, Clausing was able to deduce from kinetic 
theory and from his own consideration of the mean life, Tg, of 
an adsorbed molecule. This value of D is 



so that „ = l^g)'g+^.), (63) 

and ^^ = _|„|a-=. (64) 

It was also shown from simple considerations that 

C, = lrN,w(^^ + T,y (55) 

In equations (52)-(55) r denotes the jDore radius, w the root 
mean square velocity of a molecule, and N-^^ the number of 
molecules in the first vessel per c.c. ; Cj is of course the same 
for stationary as for non-stationary flow. By treating 
stationary flow as a diffusion, one obtains 

_^_iN,w = -D^^^P. (56) 

Thence, by combining (51), (52), (55) and (56), one obtains 



STATES OF CAPILLARY GAS FLOW 85 

in which 



K^) = 



1 — ?^|(i_e-<M)_i(i_e-4</a) + i(l_e-9//«)_ ...} 

V 



For large values of t 

JV, = P,(l-^{l-i + l-...})=P,(l-^f)=P, (38) 

That is, the diffusion rate tends for longer times to become 
that for stationary streaming, as would be anticipated. The 
function ^{t/oc) then approaches the asymptote '^{tjcc) = 1, 
corresponding to stationary flow. 

In Clausing 's apparatus the flow through two capillaries 
with diameters in the ratio 1 : 10 was measured. The pressure 
on the ingoing side was virtually constant, and on the outgoing 
side was measured by two ionisation gauges. The current, 
which was then amphfied, was proportional to the pressure, 
and was registered by a hnear galvanometer with a vibration 
period of -^ sec. Thus the deflection was found as a function 
of time, and so, from the caUbration curve for the instruments, 
the value of p and therefore of N^ = P(X^{t/oc) was deduced. 
The record of the diffusion process was obtained photo- 
graphically on a moving photographic plate. Employing the 
foregoing theory to interpret his photographs, Clausing 
obtained the lifetimes of molecules of argon, nitrogen, and 
neon on glass. He also obtained the lifetime of argon, Tg, as 
a function of temperature, his results being expressible as 
linear logTg versus l/T curves {T in °K.) (Fig. 18). According 
to the theory of Frenkel(60), 

Tg-Toe^^/'^^, (59) 

where Tq denotes the vibration period of the adsorbed mole- 
cule on the sohd, and AH is the heat of adsorption. Thus 
the slope of the curve logXg versus l/T gives the heats of 
sorption, which are collected for argon on glass in Table 10. 
The heats of sorption obtained by Kalberer and Mark(6i 



86 



MOLECULAR FLOW IN CAPILLARY SYSTEMS 



for argon on dehydrated silica gel are fairly constant at 
2500 cal./atom, a figure somewhat lower than the above value 
for glass. Clausmg remarks that the results are not always 
easy to reproduce. For argon on glass between 78° and 90° K., 
the approximate value for r^ is 

T^f^l-7 X 10-1*63800/^2' sec. (59a) 




w^ 



!I50 



1200 



1250 



mo 



Fig. 18. The linear relationship between log t^ and l/T ( T in ° K.). ®, cleaned 
with chromic acid; Qj Aj cleaned with h3'drofluorie acid. 



At 78° K., rg^75 X 10-^ sec; at 90° K. t,^3-1 x lO-^ sec. For 
nitrogen t^ was of the same order of magnitude, whilst for 
neon Tg is less than 2 x 10"' sec. These lifetimes accord well 
with our knowledge of the relative adsorbability of these 
gases. 

The estabhshment of sorption equiUbria in porous soUds is 
another very important case of non-stationary flow into 
capillaries. These capillaries may range in size from molecular 
dimensions, when activated diffusion occurs, to capillaries 



STATES OF CAPILLARY GAS FLOW 



87 



of macroscopic dimensions. Any theoretical treatment is 
rendered more difficult by the occm'rence of an unknown 
distribution of pore sizes and channel sizes. It is true of a 
limited number of adsorbents, however, that the pores are 
nearly aU of one size, for example in dehydrated chabasite 
or analcite crystals (62), from which heat and evacuation 
removes water without destroying or fundamentally changing 
the crystal skeleton. A formal treatment based on Tick's law 
without a kinetic picture of what is happening has been apphed 

Table 10. The adsorption heat for argon on glass 



Cleaned with hydrofluoric acid 


Cleaned with chromic acid 


AH 
(cal./atom) 


Remarks 


AH 

(cal./atom) 


Remarks 


2450 
2430 
3300 
3930 


The first three results 
not quite trustworthy, 
the last result free 
from objection 


3320 
3430 
4050 
3440 
3810 
3880 


The first two results are 
not so trustworthy, but 
the last four are free 
from objection 


Mean of 
last four" 
results : 
3800 



to heulandite and analcite systems by Tisehus(63); and a 
kinetic theory interpretation of the diffusion has been outlined 
by Hey (64) (Chap. III). Other attemx^ts have been made, based 
upon the kinetic theory of gases, to derive formulae for the 
rate of flow of gases into capillary sohds(65,66). Owing to 
the complexity of the problem such attempts are not very 
successful. Often sorption kinetics, after some time has elapsed, 
tend towards a pseudo-unimolecular law. When oxygen, 
nitrogen, and hydrogen were sorbed by charcoal (67) at low 
temperatures, the " unimolecular " velocity constants then 

l:M:3-2. Molecular flow 
N2 : Hg = 1 : 1-07 : 4-0. Finite sorption velo- 
cities may in some cases have their origin in slow dissipation 
of the heat of sorption, or in the slow displacement of surface 



stood in the ratio Og : Ng : Hg 
would require Og . -^^ 2 • -^^2 



08 MOLECULAR FLOW IN CAPILLARY SYSTEMS 

impurity. In the experiments described, however, the first 
possibiHty was avoided by using minimal amounts of gas, and 
the second by the most thorough heat -evacuation treatment. 
Other authors (68, 69, 70) have observed finite sorption rates 
in charcoals, and a number of the equations used to express 
the velocity of sorption in various adsorbents have been 
summarised by Swan and Urquhart(7i), 



(1 
(2 
(3 
(4 
(5 
(6 

(7 
(8 

(9 
(10 

(11 

(12 
(13 
(14 

(15 

(16 

(17 
(18 
(19 

(20 
(21 
(22 
(23 
(24 
(25 
(26 
(27 
(28 
(29 



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90 MOLECULAR FLOW IN CAPILLARY SYSTEMS 

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CHAPTER III 

GAS FLOW IN AND THROUGH CRYSTALS 

AND GLASSES 

Structures of some silicates and glasses 

In this chapter are described the phenomena of gas flow in 
glasses and in crystals. While the structure of alkah halides 
and similar more simple crystals is well known, it is not so 
with the large families of derivatives of siHca. Nevertheless, 
considerable advance has been made towards an under- 
standing of structural relationships in some at least of these 
famihes of compounds. Since many siHcates are of importance 
to the present chapter, a few of the known structural relation- 
ships will be given for substances such as silica, silicate glasses, 
zeolites, mica, clays, feldspars and ultramarines, the first four 
of which are of great importance in discussing the permeability 
data. 

Certain units occur throughout aU these substances, for 
instance, a silicon atom (or Si+ + + + ion) surrounded by four 
oxygen atoms (or 0"" ~ ions). Sometimes one finds an analogous 
aluminium tetrahedron, whose resultant negative charge is 
balanced by cations, and which, as in the aluminosihcates, 
may replace the silicon tetrahedron. These tetrahedra are 
then hnked together to form chains and closed rings. Four 
tetrahedra can form a 4-ring; one can also find analogous 
6 -rings, and in certain irregular or acrystaUine siKcates (such 
as glass) also 5- and 7 -rings. 

If the structure is composed entirely, or nearly entirely, of 
linked silicon-oxygen tetrahedra, one can obtain the three 
crystalhne silica structures, showing a- and /^-modifications. 
These are quartz, cristobalite and tridymite. Also the irregular 
netting of the tetrahedra leads to fused siHca glass. Of the 
crystalline forms, quartz is the most dense; and y^-cristobalite, 
next to fused sihca, is the most open. In /^-cristobalite, zigzag 



92 GAS FLOW IN CRYSTALS AND GLASSES 

chains are cross -linked forming wide 6-rings, about 6 A. across, 
but in quartz one has spiral chains also cross -linked to give a 
ring structure, the widest aperture in the lattice being about 
4 A. (from ion centre to ion centre). 

As ah-eady observed, aluminium tetrahedra may replace a 
part of the siHca tetrahedra in the three-dimensional network, 
which becomes anionic, since Si++++ is replaced by A1+++. 
Cations now enter the lattice to restore electrostatic balance, 
with a corresponding distortion of the parent structure. In 
this way nepheline is built up from tridymite. Sometimes more 
cations are incorporated than are needed for electrostatic 
balance, and the nett positive charge is then in its turn 
balanced by the incorporation of anions SO4", CI', S". This 
type of structure is typical of the ultramarines, of which 
lazurite, whose simplest empirical formula is Na8Al6Si6024 
(S", SO4"), is an example. 

These substances in which extra cations or anions are 
incorporated in the network are called interstitial compounds. 
Another group of interstitial compounds, of the greatest 
interest from the viewpoint of gas flow, is the zeohtes, in which 
the very open anionic frameworks, electrostatically balanced 
by cations, contain also the neutral molecule water. The 
water may sometimes be replaced with varying lattice changes 
by the substances NH3, HgS, NgO, H2, 'N^, Ar, He, 1^, Hg, or 
CH3OH, so giving rise to very diverse types of diffusion 
system, and heterogeneous equihbrium. In the zeolites, the 
ultramarines and the clays, interstitial cations may undergo 
replacement and diffusion, although these properties are most 
readily observed in zeolites. 

W. H. Taylor (1) has classified the zeohtes, according to their 
structural properties, into 

(1) rigid frameworks, 

(2) semi-rigid frameworks, 

(3) platy frameworks. 

In the first class are chabasite and analcite, in which the 
water-containing interstices do not collapse markedly upon 



GAS FLOW IN CRYSTALS AND GLASSES 93 

dehydration at moderate temperatures (i, 2). In these structures 
the frameworks are thus to a large extent independent of tlie 
neutral molecules occupying interstices inside the frame- 
work. In the second class are the fibrous zeoHtes natroHte, 
scolecite, mesohte and edingtonite, all of which are formed 
by cross-hnking chains of (Si-Al)-O tetrahedra in different 
ways (Fig. 19). The fibrous zeohtes undergo reorganisation 
around the water-bearing interstices on dehydration, without 
any appreciable shrinkage of the chain length. The third class 
of zeohtes is exemphfied by heulandite, of which the great 
shrinkage in one dimension on dehydration and the X-ray 
diffraction patterns are compatible with a platy structure of 
two-dimensional lattice layers, separated by water molecules 
and cations. These cations serve to bind the anionic laminae 
together. In Fig. 20 a possible laminated lattice is shown (i). 
Mica is another case of a laminated sihcate crystal, while 
laminar crystals of sihca itself have recently been reported (3). 

Clays are aluminosihcate structures often having close simi- 
larities to the zeolites. Montmorillonite, for example, may be 
regarded as the counterpart of the zeohtes heulandite and 
stilbite. Like heulandite it swells greatly upon hydration (4), 
as is indicated by Fig. 21. The clays also show base -exchange 
properties hke the zeoHtes. In general they contain larger and 
less definite channels. 

Glasses are anhydrous sihcates,* in which A1+ + + or B+++ 
may replace some Si+ + + +, and are based upon the irregular 
network of sihca-glass. Electrostatic balance is restored by 
the various basic oxides of types MgO, MO, MgOg which they 
contain. By varying the proportions of constituents the most 
diverse properties may be conferred. 

Thus one finds in sihcate chemistry many examples of 
complex structures resolved by the classical chemical and 
X-ray methods. While diffusion phenomena have been httle 
studied in many of the structures (feldspars, ultramarines, and 

* Experiments upon the diffusion of deuterium through silica glass showed, 
however, that some hydrogen, probably as hydroxyl groups, still remained in the 
silica, since the deuterium content of the diffusing gas decreased (4a). 




I I I I I — I 

1 2 3 'i sa 

Fig. 19. The structures of some fibrous zeolites. The structure of one tetrahedron 
chain is shown in [d'), in which the large circles represent silicon or 
aluminium atoms, the small circles oxygen atoms, and the heights of the 
atoms are given in A. The same chain is represented diagrammatically 
in {d), where the numbers show the heights of sUicon and aluminiiuu 
atoms as multiples of c/8 (the c-axis is 6-6 A., so that 3c/8 = 2-5 A., 
5c78 = 4-l A.). The linked chains are shown in (a), (6), (c) as arranged in 
edingtonite, thomsonite and the natrolite group, respectively, and in each 
the unit cell is indicated by dotted Imes. 




Fig. 20. A lattice of two-dimensional laminae. Large circles represent oxygen 
atoms, small circles silicon or aluminium atoms. The upper diagram repre- 
sents a portion of an infinite sheet of tetrahedra in which all vertices are 
supposed to point upward. If these vertices lie on a reflection plane, a 
second similar tetrahedron sheet, in which all vertices point down, is linked 
to the first. The lower diagram represents the appearance of the linked 
sheets, which form a tetrahedron framework of finite thickness, when viewed 
in the direction indicated by the arrow in the upper diagram. 



MOLECULES OF WATER PER UNIT CELL 
2 'i 6 8 10 12 15 _ 20 J5 30 



% 



1 1 1 < 1 1 i 


1 


I ■- I 


r 1 


- Oven Dry 

- fWOV Air dry 


-* 


3_ 


-o- 


1 1 




/ 




/ 








-/♦ 








- 








10 20 30 UO 


50 


SO 70 30 90 


100 m 



20 

18 

ijioi '^S 
SPACING 

12 
10 



WATER -%0F miTED MATERIAL 
Fig. 21. The one-dimensional swelling of montmoriUonite on hj^dration. 



96 GAS FLOW IN CRYSTALS AND GLASSES 

clays), in others (glasses and zeolites) considerable data have 
been collected. It must be in the hght of the main structural 
features outlined that these data are examined. The mobility 
of ions has been observed in ultramarines, clays, zeolites and 
glasses — in the three former groups by base exchange experi- 
ments, and in the latter because of its function as a hydrogen 
electrode, or even as a sodium electrode. The mobihty of 
sodium (and potassium) ions in glass may be demonstrated in 
an experiment used to prepare pure sodium in vacuo. The cell 



Cu 



NaNOg 
(molten) 



Na 
glass 



Conducting I Glowing 

evacuated space | tungsten filament 



is constructed by dipping an evacuated soda glass bulb into 
molten sodium nitrate, into which dips a copper electrode. 
Inside the evacuated bulb is a hot tungsten filament, which, 
by thermionic emission, renders the evacuated space con- 
ducting. On making the tungsten negative with respect to 
the copper, metallic sodium appears inside the glass bulb. 



The diffusion of water and ammonia in zeolites 

Introductory 

Having briefly reviewed zeohtic structure and properties, 
one is in a position to consider diffusion within these 
crystals. The most complete and satisfactory studies are 
those of A. Tisehus (5, 6), whose methods and data will now 
be considered. In crystals such as NaCl, lattice diffusion is 
possible only at elevated temperatures because no channels 
are available for the diffusion. Very often, as when thorium 
diffuses in tungsten, diffusion occurs down the only available 
channels, the grain boundaries and faults in the tungsten 
crystal. In interstitial compounds such as zeoHtes there are, 
when the crystal is dehydrated, numerous channels through 
the lattice left by the evacuated water, and so diffusion by 
spreading through interstices, as opposed to high temperature 
processes of diffusion by place exchange, may occur. 



DIFFTJSION OF WATER AND AMMONIA 97 

The experimental and theoretical interpretation 
of diffusion data 

Tiselius(5,6) found that the double refraction of light by 
these crystals was dependent upon the extent of hydration 
of the lattice. The crystals were cut for this purpose in the 
form of thin plates, and the rays were examined in a polarisa- 
tion microscope. As a first approximation, Tiselius then 
considered Fick's law to hold. To interpret his data he used 

dC d'^C 

the solution of the diffusion equation ^ = D^^ for the 

diffusion of a substance along the co-ordinate x into a medium 
of infinite length through a surface normal to its length. The 
solution is (Chap. I): 



e^rilo--'^^' 





where /? = x/2.yJ{Dt); C^ = the concentration at plane. a: = 0, 
assumed constant; C^ = the concentration at plane x — x 
at time t; Cq = the concentration, constant for all x, at the 
beginning of diffusion. 

For this solution of Fick's equation the boundary conditions 

C = Cp for X — and all times t, 
C = Cq for x>0 and ^ = 0, 

C = Cq for a; = 00 and all t. 

The equation provides three methods of obtaining D, the 
diffusion constant, which are: 

(i) Measure a; as a fmiction of t for a constant known G; 
(ii) Measure C as a function of t for a given x; 
(iii) Measure C as a function of x for a given t. 

All three methods were employed successfully in Tisehus's 
experiments. 

The laminated lattice of heulandite has the laminae parallel 
to the 010 plane. The diffusion of water normal to this plane 
was found to be i)Qio< 7 x lO^^^cm.^sec."^ at 20° C; but 
diffusion of water parallel to the 010 plane was rapid, though 



98 



GAS FLOW IN CRYSTALS AND GLASSES 



proceeding at different rates across 201 planes and 001 planes. 
Thus one is introduced to the phenomenon of diffusion 
anisotropy, exhibited also in "platz-wechsel", or place- 
change, diffusion of ions in salts (4). 

The diffusion of water in the 010 'plane, normal 
to the 201 plane 

The first of the three methods of determining the diffusion 
constant, based on Tick's law, depends upon the Unear 
relation between x^ and t, at constant C. Fig. 22 gives the 
x^ — t curve at a constant C^ value of 13-87 % of water in the 




woo 2000 

Seconds 
Fig. 22. x^ as a function of t at constant C. 



mo 



lattice. Here Cq was 13-45 % and Gp was 19-67 %, the experi- 
ment being performed at 20° C. Thus 

D^Q^ = 3-4x 10-'^cm.2sec.-i 

When repeated at different values of C^ , there was evidence 
that D varied according to the water content of the lattice. 

When C was plotted as a function of ^ at a constant x, the 
curve of Fig. 23 was found; and when C was plotted as 
a function of x at constant t, the curve of Fig. 24 was 
obtained. 

In this latter figure the dotted curve gives the value of a 
C — x curve computed using D = 3-9 x 10~^cm.2sec.~^, and 



DIFFUSION OF WATER AND AMMONIA 



99 



assuming D to be independent of (7. In Tables 11 and 12 are 

collected the values of D according to these last two methods, 

dC d^C 

as functions of C, using the simple Fick law -w- = ^^-^ ^^ 

their evaluation. 




woo 7500 

t in seconds 



2000 



Fig. 23. C as a function of t at constant 



20 r- 



3; 



50 100 150 

t in seconds 

Fig. 24. C as a function of x at constant t. 



200 



When the difference between Cp and Cq was small the values 

dC d'^C 

of D were nearly constant, and the simple equation ^:- — D -^r-z 

ot ox^ 

more closely obeyed, as is indicated by Table 13. 

dC d^C 

Hitherto the simplest equation -^ = D^-^ has been used. 



7-2 



100 GAS FLOW IN CRYSTALS AND GLASSES 

Table 11. Diffusion constants from the relation between G 

as a function of t at constant x 

C^ = 19-67%, C'o = 12-30%, x = 9-832 x 10-^ cm., T = 20° C. 



c^ 


t 


Cj,-Cx 


/» 


Z>xlO' 


% 


sec. 


C^-Co 


cm.^ sec.""^ 


13-0 


81 


0-905 


1-181 


2-4 


13-5 


93 


0-837 


0-986 


30 


140 


112 


0-770 


0-849 


3-4 


14-5 


140 


0-701 


0-734 


3-6 


15-0 


169 


0-634 


0-639 


3-9 


15-5 


212 


0-566 


0-553 


4-2 


16-0 


283 


0-498 


0-475 


4-2 


16-5 


377 


0-430 


0-402 


4-4 


170 


577 


0-363 


0-334 


4-4 


17-5 


1060 


0-294 


0-267 


3-6 


18-0 


1755 


0-227 


0-204 


3-7 



Table 12. Diffusion constants from G as a function of x 
at constant t 

Cj, = 19-67%, Co = 8-3%, t = 1200 sec, T = 20° C. 



c. 


X in units 


CjJ-Ca; 


DxlO' 


% 


2-809 X 10-3 cm. 


c^-c. 


cm.2 sec.-i 


9 


108 


0-938 


11 


10 


107 


0-850 


1-8 


11 


105 


0-762 


2-6 


12 


99-7 


0-675 


3-4 


13 


93-0 


0-587 


4-2 


14 


83-0 


0-499 


5-0 


15 


70-8 


0-411 


5-6 


16 


56-5 


0-323 


6-0 


17 


38-0 


0-235 


5-3 


18 


21-0 


0-147 


4-2 


19 


7-2 


0-058 


3-3 



Table 13. Diffusion constants when the crystal was nearly 
saturated with water 

Cj, = 19-67%, Co = 16-20%, T = 20° C, x = 11-353 x 10"^ cm. 



c« 


t 


Cj, - Cj. 


/? 


Z>xlO' 


% 


sec. 


^J)-^0 


cm.2 sec.-^ 


16-5 


60 


0-915 


1-218 


40 


17-0 


119 


0-770 


0-894 


4-0 


17-5 


218 


0-625 


0-627 


4-1 


18-0 


398 


0-482 


0-457 


4-2 


18-5 


984 


0-342 


0-313 


3-8 


19-0 


1520 


0-266 


0-240 


41 



DIFFUSION OF WATER AND AMMONIA 



101 



However, there is considerable evidence that D may vary 
with C(7), in which case the equation to be solved is 

dt 



dx\ dx 



for which solutions (5, 6) have been given (Chap. I). Of use in 
evaluating D is the expression 

1 dx C ^=^» 
Dr-r = TTTT^ xdC, 

^-^- 2tdC] c=c. 

when C is measured as a function of x for a given time interval. 
The integral can be evaluated graphically, and the derived 
value of D refers to that value of at a; = x^. Employing this 
method, Tisehus succeeded in obtaining a number of diffusion 
constants for various values of G, which, as Table 14 shows, do 
indeed depend noticeably on C. 

Table 14. Diffusion constants evaluated from ^- = ^ 1 D -^ I 





Temp. 20° C, 


C^ = 19-67%. 






DxlO' 


DxlO' 


DxlO' 


c 


(cm.^ sec.~^) 


(cm.^ sec.~^) 


(cm.2 sec.~^) 


0/ 

/o 


(exp. with 


(exp. with 


(exp. with 




Co =8-3%) 


Co -13-20%) 


(7„ = 16-20%) 


10 


0-04* 








11 


0-2t 


— 


— 


12 


0-7 


— 


— 


13 


1-3 


— 


— 


14 


2-0 


21 


— 


15 


2-7 


2-6 


— 


16 


30 


3-5 


— 


17 


4-0 


4-2 


4-0 


18 


4-0 


4-1 


4-2 


19 


3-3 


3-5 


4-1 



* Very uncertain. 



t UncertaiQ. 



These results are numerically somewhat different from 
those obtained earher with the simple Fick treatment. They 
show in each case a maximum which might be explained as 
follows. The heat of sorption is greatest at small Gq\ therefore 
the water molecule is most strongly anchored and is not very 
mobile. When the lattice has its full complement of water, 
mobility must also be shght, since few water molecules will 



102 



GAS FLOW IN CRYSTALS AND GLASSES 



have vacant lattice cells to migrate into, and therefore for 
intermediate values of Gq the greatest mobihty is possible. 

Dijfusion anisotropy in heulandite 

The permeability perpendicular to the plane of laminae in 
heulandite is negligible; parallel to this plane it is > 10^ times 
as rapid, at room temperature. But different directions in the 
010 plane have different diffusion constants. This is shown by 
the photomicrographs in Fig. 25 in which the dark band gives 





Fig. 25. Progress of diffusion in heulandite, showing diffusion anisotropy. 

a measure of the rates of advance of diffusion normal to the 
201 and 001 faces, the most rapid advance being found normal 
,to the 201 face. The relative rates of advance of the dark 
bands are given in Table 5. 

Table 15. Dijfusion anisotropy 



c% 


^201 • -^^OOl 


13-21 


3-6 


14-43 


3-4 


15-50 


3-5 


16-25 


3-2 


17-36 


3-4 



The ratio 



'■'201 
"^001 



201 
001 



= 11-6, and is constant at a series of 



values of Cq. When the diffusion velocities normal to the faces 
201 and 201 were measured, it was found that they were equal. 
Thus for the particular sample of heulandite studied 

^ooi:-C>ooi:i)2oi= 1:11-6: 11-6. 



DIFFUSION OF WATER AND AMMONIA 



103 



The diffusion anisotropy phenomenon was studied in a 
number of other heulandites, and proved substantially the 
same in all of them. Always the minimum diffusion rate 
parallel to the 010 plane was normal to the 001 plane, and the 
following anisotropy ratios were found: 



Heulandite from 

Dalur, Faroerne 
Teigarhorn, Iceland 
Rodefiord, Iceland 
Sulitelma, Norway 



Daoi at 20^ C. 
and C = 15-0% 

2-7 X 10-' 
3-7 
3-5 
2-3 



-^201 

13-6 

15 to 20 
15 to 20 
15 



The energy of activation for the diffusion of water 
in heulandite 

Tisehus measured the diffusion constants at a number of 
temperatures, finding the values given below: 



Temp. 


Aoi X 10' 


Dooi X 10' 


°C. 


cm.^ sec.~^ 


cm.^ sec.~i 


20-0 


2-7 


0-23 


33-8 


41 


0-45 


46-1 


4-8 


0-66 


60-0 


7-6 


1-45 


75-0 


111 


2-8 



From these figures he computed the energy of activation 
for diffusion normal to the 201 and 001 faces respectively to 
be 5400 and 9140cal./mol. The temperature dependence of 
the diffusion constants he found did not depend appreciably 
upon the amount of water in the lattice, although we have 
seen that their absolute magnitudes do. 



Diffusion processes in analcite-ammonia 

These experiments were made upon a most complex zeolite, 
heulandite, and were later extended to analcite, which has 
only one type of lattice hole and a cubic or pseudo -cubic 
symmetry (7). Once more the optical experiments were per- 
formed upon narrow plates cut parallel to cube or diagonal 
faces. The material was thoroughly outgassed and the sorption 
processes with ammonia were observed. In Table 16, which 



104 



GAS FLOW IN CRYSTALS AND GLASSES 



gives the absolute diffusion constants, A /"denotes the difference 
of the refractive indices of the ordinary and extraordinary 
rays, which was used as a measure of the amount sorbed. 

Table 16. Diffusion 0/NH3 through 0-92 nun. plate 
cut parallel to cubic surface 

Temp. = 302° C, pressure = 761-2 mm., volume of NH3 sorbed = 27-5 c.c./g. 



t 


v< 


ArxW 


DxlO^ ! 


sec. 




cm.2 sec. 1 











1-4 


900 


30 


40 


10 


3,600 


60 


66 


1-4 


8,100 


90 


119 


1-2 


14,400 


120 


151 


1-2 


22,500 


150 


183 


— 



The diffusion rate was the same for a plate cut parallel 
to the diagonal surface, so that no anisotropy occurs. The 
temperature dependence of the diffusion velocity was found 
using powdered analcite, and the equation 



IDf 

AQcx:{C^^-Co)J-.. 



where AQ denotes the amount sorbed. This equation supposes 
that D does not depend upon AQ. If one writes 
AQ 



C^-C. 



= e=f{D,t) 



only, the curve d versus the time, t, or ^Jt, should be the same 
at all pressures. It was shown that this was so for two 
rate curves at 302° C, one with analcite outgassed and 
p — 758-6 mm., the other with the analcite saturated at 
Pq = 578-6, a further dose of ammonia being admitted at 
1385-4 mm. Then since 6 =f{D,t) only, 



Dr, 



and a simple method is available to determine the temperature 
coefficients and activation energies. The temperature de- 
pendence of the diffusion coefficient gave the activation energy 
for diffusion as 11,480 cal./mol. It was also noted that the 



DIFFUSIOISr OF WATER AND AMMONIA 



105 



ratio of the diffusion constants at two different temperatures 
was not dependent upon the amounts of ammonia sorbed. 

Other observations on zeolite systems 

In a study (2) of sorption by various zeolites, platy, fibrous, 
and rigid three-dimensional networks, several points of interest 
were estabHshed. Ammonia sorption was most rapid in the 
lattices of the three dimensional networks (chabasite, analcite) . 
It was least rapid in suitably outgassed fibrous and platy 
zeolites (natrolite, scolecite, and heulandite). Heulandite, a 



4.2-0 





12 3 5 

I (in days) V' (^ i" "''■n-) 

rig. 26. Sorption velocities. NHg on natrolite, showing tendency to autocatalytic 

sorption rate curves. (NH3 sorbed before commencing expt. = 12-11 c.c. 

at N.T.p. Inset shows NHg on heulandite, sorption rate following the 

parabolic diffusion law.) 

laminated crystal like mica, outgassed at low temperatures 
(130° C.) to prevent lattice collapse, sorbed ammonia at first 
rapidly and then more and more slowly, suggesting a rising 
activation energy with charge. When heated to 330° C, how- 
ever, heulandite sorbed ammonia much more slowly. Some 
profound change in the bonding between laminae must have 
been effected by the high temperatures, associated with the 
more complete removal of water. For heulandite crystals it 
will also be seen (Fig. 26) that the sorption rate obeyed the 
parabolic diffusion law initially, although this law quickly 
broke down. In natrolite, a fibrous zeolite, sorption in its early. 



106 GAS FLOW IN CRYSTALS AND GLASSES 

stages was autocatalytic (Fig. 26) and a more or less well- 
defined ammoniate resulted. The experiments bring out an 
important point: activated diffusion in zeolites may be a 
diffusion of interfaces in certain cases (natrolite) and not 
a diffusion of molecules down a concentration gradient (chaba- 
site, analcite, heulandite). 

Another point of interest was the difference in the velocity 
of ammonia sorption by analcite observed by Barrer(2) and 
TiseHus(6). The former found sorption rapid at temperatures 
as low as 200° C. ; the latter found that sorption equihbrium 
could not be estabhshed at 270° C, so slow was the ammonia 
uptake. These experiments all serve to show the complexity 
of behaviour of this very interesting series of diffusion systems. 

A kinetic theory of diffusion in zeolites 
These systems provide a very complete investigation of 
the diffusion of vapours in zeolites, and the types of 
phenomena hkely to be encountered. They also provide an 
interesting application of the methods outhned in Chap. I. 
Kinetic theory may also yield equations of acceptable form 
for velocities of sorption and for equihbrium data. Such 
equations were derived by M. Hey (7, 8), who expressed 
Tiselius's data in terms of them and concluded that they were 
satisfactorily able to represent Tisehus's experunents. Hey 
derived his equations on the assumptions that water or other 
vapours may occupy all or part of definite lattice positions, 
and that the water if given an energy of activation E may 
become mobile. A plate of zeolite is mounted with a free 
surface of unit area exposed to the vapour, which diffuses 
normal to this surface. Water molecules impinging on the 
surface with an energy E^ normal to the surface may, if they 
strike within certain areas, enter that lattice hole nearest the 
surface. Conversely, molecules in the first lattice hole by 
acquiring an energy E.^^ may re-evaporate, or by acquiring an 
energy E may diffuse into the lattice. E^ is usually regarded 
as being greater than E. In the diffusion problem two extreme 
cases arise: 



DIFFUSION OF WATER AND AMMONIA 107 

(1) Hydration of the zeolite is governed by the sorption 
and desorption rates at the surface. 

(2) Hydration is governed by the diffusion rate into the 
crystal. 

In the latter case ^ dJ(2E) „,„„ 
2) ^ ^^ ,^; e-EiRT 

where d denotes the distance between two successive points 
of equihbrium of a water molecule in the lattice, ;^ is a 
correction term for anharmonicity of vibration of water in 
the crystal, and the other symbols have their usual significance. 
When ^ is a function of x, the vacant fraction of water posi- 
tions, one may write E = Eq{\ -\-f{x)]. The equations involve 
simphfjdng assumptions, but were apphed to Tisehus's data for 
water-heulandite diffusion systems. Hey found for this system 
that for diffusion across the 201 face, from Tisehus's data, 
^2Qj = 4-79(1 + 0-13a;) x 10^ cal./mol. from which dix = 3-66 A., 
a reasonable value since x is approximately unity. The diffusion 
data across the 001 face gave a value of 

Eqq^ = 7-3(1 + 0-08a;) x 10^ cal./mol. and d/x = 19 A., 
by no means so likely a value. One difficulty is that all the 
water in heulandite is not held with the same energy; Hey 
considered three groups of water lattice positions to occur. The 
data above apply only to the most volatile group (13-11 % 
hydration) . Table 1 7 gives the calculated and observed diffusion 
constants computed using the data given above. 

Similar agreement was found for diffusion across the 001 
plane. It may be concluded that an equation of the type 
deduced by Hey can represent the experimental results when 
the constants are suitably chosen. 

Tisehus's data for ammonia-analcite systems however 
allow the simphfied form 

_ ^V(2£^o) EJRT 

^-ttx^m' 
to be applied, since E does not appear to vary with x. Hey 
calculated Eq = (1-35 ± 0-05) x 10* cal./mol. of ammonia from 
Tisehus's data, while from X-ray examination d = 5-93 A. 



108 



GAS FLOW IN CRYSTALS AND GLASSES 



Thus i)302»c ^ (3.5 4. 2) X 10-8 cm.2 sec.-i, which constant 
Hey (8) showed for various simple cases was one-third the 
diffusion constant along a particular set of channels, as 
measured by Tiselius. Thus from X-ray data and the calcu- 
lated value of Eq the diffusion constant was (1-2 + 0-7) x 10"^ 
cm. 2 sec. -^ compared with the directly measured value 
1-2 X 10-8 cm.2 sec.-i, both at 302° C. 

Table 17, Diffusion data for heulandite, calculated 

and observed " 



% H,0 


19 


18 


17 


16 


15 


14 


X 


010 


0-255 


0-41 


0-56 


0-71 


0-865 


io'xZ)i°;'^- 


3-3, 


4-0, 


4-0, 


3-5, 


2-6, 


2-1. 


cin.2 sec.-^ (obs.) 


3-5, 
4-1 


4-1, 
4-2 


4-0, 
4-2 


3-6 


2-7 


2-0 


io'xZ)i°;«- 


4-47 


4-27 


3-80 


3-32 


2-70 


2-14* 


cm.2 sec.-^ (calc.) 














Temp. °C. 


20-0 


33-8 


46-1 


60 


75 


— 


10' X D201, for 


2-7 


4-1 


4-8 


7-6 


111 





a; =0-71, in cm.^ 














sec.-i (obs.) 














10' X D201, for 


2-70 


4-10 


5-60 


7-90 


110 


— 


x = 0-71, in cm.2 














sec.-i (calc.) 














0/ TT 


18-5 


17-5 


16-5 


15-5 


14-5 


13-5 


X 


0-18 


0-33 


0-48 


0-64 


0-79 


0-94 


DZ""- fobs. 
Z)2o;c. Icalc. 


1-7 


2-0 


1-8 


1-8 


1-8 


1-7 


1-93 


1-93 


1-94 


1-98 


203 


2-09 



Diffusion in alkali halide crystals 
Introductory 
Experiments designed to clarify photochemical processes in 
alkali metal and silver hahdes have led to some interesting 
studies on the diffusion of metal vapours (9, lO), halogen 
vapours (11) and hydrogen (12) in alkali halides. It is possible 
to prepare transparent single crystals of the halides of the 
greatest chemical purity. It was found that the haUdes can 
act as solid solvents for a number of substances, and that 
the properties of these mixtures or solutions permit one to 
study photochemical processes and diffusion. Thus, potassium 



DIFFUSIOiSr IN ALKALI HALIDE CRYSTALS 



109 



vapour dissolves in potassium halides giving a number of 
"colour centres " or "Farbzentren " which give rise to a definite 
absorption band in the crystal (Fig. 27) (13). In a similar way, 
a solution of potassium hydride in potassium bromide gives 
rise to another typical absorption band (Fig. 27). Exposing 
a crystal containing colour centres to hydrogen produced 
a solid solution of potassium hydride in potassium bromide, 
whereas exposing the hydride-bromide solution to hght re- 
formed the potassium colour centres. What is true for one 



200 



Wove - length , fm/() 
J 00 H/O 



600 SOO 




6 S 4 J 2 eVoi'tf 

Fig. 27. Optical absorption bands of KH-KBr solid in which about half 
the KH has been decomposed photochemicaUy giving "Farbzentren". 

hahde may in general be taken as true for others (RbBr(i3), 
KI(13), KC1(13)). The studies on hahdes containing potassium 
and potassium hydride were extended to cover solubility and 
diffusion of halogens and hydrogen, and it is with these 
systems as with the mobihty of Farbzentren that the data now 
to be given are concerned. 

The number of colour centres per unit volume of halide 
may be measured optically, after impregnating the crystal 
with alkali vapour. It can be shown that 
'N = constant x K^ x H 

= constant x area of the band " ^" of Fig. 27, 
where K^ is the absorption constant at the maximum in 
the band, and H is the breadth of the band at K = ^K^. 



110 GAS FLOW IN CRYSTALS AND GLASSES 

The concentration of colour centres may also be determined 
from the conductivity, since each potassium atom can give 
rise to an electron (e.g. K in KI). 

The corresponding investigations upon solutions of the 
halogens in hahde crystals were also attempted using the 
conductivity of the crystal as a means of estimating the 
halogen excess (ii). For potassium bromide and iodide a rise 
in conductivity followed their exposure to Brg and Ig vapour 
respectively, but in CI2-KCI systems no rise in conductivity 
was observed. The diffusion of halogens in hahde crystals was 
also followed by saturating the crystal with thalhum vapour, 
which imparted to it a brown tint. Then, as the halogen sub- 
sequently diffused inwards, the brown tint disappeared as the 
boundary advanced, due to the formation of colourless or 
white thalhum hahde. The progress of the colour boundary 
served to measure the diffusion rate. 

In Hg-KBr systems the concentration of hydrogen was very 
sUght, and was best determined by subsequent heating of the 
crystal containing hydrogen in potassium vapour. In this way 
potassium hydride was formed, the concentration of which in 
the mixed crystal KBr + KH can be determined optically, the 
concentration in atoms per c.c. being 

(KH) = 5-1 X 1016 A^^ 

{K^ = absorption constant of optical absorption maximum). 
The diffusion of hydrogen was followed by the same method 
as was employed for the halogens, using as indicator metal 
potassium and following the movement of the boundary 
K-KH (blue to colourless) through the crystal. 

The experimental results on solubility and dijfusion 

Potassium dissolved in both potassium bromide and potassium 
chloride exothermally, the heats of the processes being 
given by 

'^^(K vap.^K in KBr) = " 5-8 k.cal./atom of K, 
^^(K vap.^K in Kci) = " ^'^ k.cal./atom of K. 



DIFFUSION IN ALKALI HALIDE CRYSTALS 111 

At high temperatures the potassium was dispersed in atomic 
form, but on slow coohng to lower temperatures it could be 
condensed into colloidal aggregates. The halogens and 
hydrogen dissolved endothermically in haUde lattices, the 
heats of solution being 

^^(i^Tap.^i.inKi) = 18-5k.cal./g.mol. I2, 

^^(Br.vap.->BranKBr) = 27-6 k.Cal./g.mol. Bt^, 
^^(H.gas^H.inKBr) = 16 k.Cal./g.mol. Hg. 

The main feature of interest in the results is that the solubiHty 
in each case is proportional to the external gas pressure, so 
that the halogens and hydrogen dissolve in molecular form. 
The results on diffusion show too that thermal diffusion occurs 
mainly in molecular form, there being httle dissociation, save 
perhaps for the halogens at the lowest pressures when simul- 
taneous atomic and molecular diffusion takes place (ii). 

It is doubtful, when solution occurs as molecules (e.g. Hg, 
Bra, CI2, in KBr, KCl), whether the molecules are homo- 
geneously dispersed. The "solute" is most likely to be 
dispersed along faults and ghde planes in the crystal. The mere 
introduction of large foreign molecules into a perfect lattice 
would distort the lattice locally, and create a fault. Hilsch 
and Pohl (13)," however, pointed out that when KH is formed 
in KBr the system KH-KBr is a true solution, since the lattice 
constant of potassium bromide is a hnear function of the 
potassium hydride content. 

When both hydrogen and alkaU metal vapour were in 
contact simultaneously with the crystal, which contained a 
constant amount of potassium, the amount of potassium 
hydride at the surface was fixed by the equations 

KHi^^K + iHa, 

K = ^^jif Q (Mass Action). 

(KM) 

It was established through the optical absorption of successive 
thin layers of the crystal that the potassium hydride gradient 
decreased inwards nearly linearly with distance, and it was 



112 GAS FLOWIN CRYSTALS AND GLASSES 

then found that two equations for the depth of penetration, x, 
of hych-ide could be derived ( 12 ) according as the hydrogen 
diffused as molecules (1) or as atoms (2): 

x^ 
1 



^^16i)^Vi>o7 



(K) 



(1) 



a;2 1 



(2) 



In the relation (1), k is the Henry's law solubility coefficient 
for molecular hydrogen in the crystal. Experiment showed (12) 
that equation (1) was correct, so that hydrogen is both dis- 
solved as molecules and diffuses as molecules. The derived 
value for D was 2-3 x lO-^cm.^sec.-i at 680° C. 




r- X. "'<7Vt* 

Fig. 28. Diagrammatic representation of penetration by hydrogen of KBr 
containing an initial uniform excess of K. 

If a crystal containing a given uniform concentration, N , 
of alkali metal is exposed to hydrogen, after the lapse of 
time t, an approximately linear hydrogen concentration 
gradient may be assumed to extend into the crystal (Fig. 28) 
a distance x. During the time interval dt the gradient will 
extend a further distance dx. If dnjdt denote the hydrogen 
stream (as molecules), 

dn'^Ndx^ -^ {B.^)^^f^ 
dt 2 dt X ' 



whence D 



Nx^ 1 



if h is defined by {)^2)x=q = ^'JPa- Thus 



p^t 4k' 
Nx^jt should be a linear function ofp^, a prediction which the 



DIFFUSION IN ALKALI HALIDE CRYSTALS 



113 



diagrams of Fig. 29 show is substantially correct. The diffusion 
constants computed from this relationship were 



D 



600OC 



= 5-5 X 10~*cm.^sec.~^, 



'520OC. — 3-5 X 10~*cm.2sec.~i 
The measurements on the movement of bromine and iodine 
were made along similar lines, the halogen diffusing into the 
salt containing thallium as indicator. The concentration of 




■/^ 









y 






A 


/ 


> 




/' 






f 






■Jb 

t 




i 


20' 



Concentration N 
• 4,0-10 6] 
X 1,6 -lO'' [K/cm? 
7 -10" ) 
n 5 .10'' Br/om? 



/(? 20 30 W 
H^-pressure in Attn. 
Concentration N 
• 1,5-lOiM 
X 4,4-10'^ [K/cm? 
O9,l.l0'M 
n 3,2-10" Br/cm? 



25 


■fo" 














. 










X / 


20 








/ 








/ 










f 


15 






















1/ 






10 
5 








/ 


X 


/o 


/ 




/. 










y.^ 




600' 




1 / 




1 



20 30 UO 



Concentratiou N 
• 4.4-lO'M 
X 14,-10'« [K/cm? 
O2,2-l0'8J 
D 3,2-10" Br/cm? 



Fig. 29. The linear relation between 'Nx^ft and p^. 



the thalhum within the crystal was large compared with the 
concentration of absorbed halogen (Fig, 30). If n is the 
quantity of halogen (expressed in atoms) diffusing through 
unit cross -section in time t, one may write 



dn dx 
—- = — xN 
at at 






since an approximately linear halogen gradient was estabUshed. 
Thus 2 



2t 



N = D{C),^„ 



114 



GAS FLOW IN CRYSTALS AND GLASSES 



and so according as diffusion occurs as atoms (or ions), or as 



Boundary 
of Crystal 



Halide 



CKcess Ihallium 



X 

molecules, one would have -^.N a linear function of ^2^ or p 

respectively. When the experiments were made it was shown 

that the diffusion occurred mainly as molecules (H). Table 18 

then gives the results for D for 

several temperatures and at several 

halogen pressures. In Table 18 are 

included some diflPusion constants 

for potassium vapour, determined 

by an electrical method (H). It is 

possible that some diffusion of 

bromine occurs as a diffusion of ions 

or atoms, but parallel measurements 

of Z)gjr by the electrical conductivity 

method showed that at high pressures 

(above 1 atm.) transport of bromine Pig. 30. Schematic representa- 
as Br~ was much less important tion of the permeation of 

thallium- containing halide 
by halogen. 



Halogen 
Vapour 




than its transport as Brg. Below 

1 atm. a greater percentage of 

dissociation occurred, with consequent enhancement of the 

percentage of transfer as ions. 

Since the diffusion of hydrogen and of halogen is mainly as 
molecules, and since they are dissolved within the haUde in the 
molecular state, one may include them in the same family of 
diffusion systems as gas-silica (p. 117) and gas-rubber systems, 
i.e. non-specific activated diffusions. They differ from the 
latter systems in one respect — the solution processes in the 
hahdes are highly endothermic (p. Ill) compared with the 
very sHght endothermicity of hydrogen solubihty in siUca 
(p. 140) and the sKght exothermicity of hydrogen solubihty in 
rubbers (p. 418, Chap. IX). The temperature coefficients of 
bromine diffusion in potassium bromide are small (Table 18), 
and correspond to energies from 3k.cal./g.mol. at 1 atm. 
pressure to 8k.cal./g.mol. at 16 atm, pressure. For hydrogen 
the energy is about 8 k.cal./g.mol. in potassium bromide, while 
for potassium it is about 16 k.cal./g. atom in the same crystal. 



DIFFUSIOIJ IN ALKALI HALIDE CRYSTALS 115 

In addition to these studies of solution and diffusion in 
halide crystals, so important when one considers photochemical 
and conductivity phenomena in crystals, there has been a 
number of investigations upon the sorption of dipole gases by 
alkali halide crystals. That water may penetrate a rock salt 
crystal has been shown from the infra-red absorption spectrum 
of the crystal after dipping it in water. In the interior of the 

Table 18. Diffusion constants of Brg and of K in KBr 



Temp. ° C. 


500 


600 


700 ! 


Big pressure (atm.) 


1 


4 


16 


1 


4 


16 


1 


4 


16 


in cm.^ sec.~i 
( X 10-*) 


2-33 


1-33 


0-9 


2-64 


2-28 


1-65 


3-33 


30 


2-7 


D^ from an electrical 
conductivity 
method ( x IQ-*) 


0-8 


3-0 


8-0 

1 



crystal the infra-red spectrum of water may be observed. 
No study has been made of the velocity of sorption of water 
by alkali hahde crystals, but a number of workers have 
observed and measured slow sorption processes for the 
following systems: 

SO2 inNaCl(i3a), 

HCl in KCl(i4), 

NH3 inNaCl(i3&,i3ff), 

In the case of the two latter systems the data obtained showed 

that a slow activated diffusion occurred into the crystal 

dC d^C 

substance, which followed the Fick diffusion law -^ = D— — 

Ot OX" 

and gave as the temperature coefficient of D 



E 



HCl-KCl 



7000 cal./mol. 



^NH3-Naci~6300cal./mol. 

These experiments were performed upon polycrystalline 
masses, and it is probable that diffusion occurred down grain 
boundaries. No experiments upon single crystals designed to 



116 GAS FLOW IN CRYSTALS AND GLASSES 

test this point are, however, available. Little experimental 
work has as yet been carried out upon the diffusion of water 
in hydrated crystals. The progress of such a diffusion may be 
followed by using heavy water as diffusing hquid in a crystal 
with Ught water of crystaUisation. Prehminary measurements 
of this kind have been made by Kraft (i3d) upon the dijffusion 
of D2O into alum. Kraft obtained data at 75, 65 and 55° C. 
which conform to the expression 

X> = 0-56x10-'^ e-6ooo/-R2^. 

Diffusion of helium through single crystals 
of ionic type 

The passage of heUum gas through a perfect ionic crystal has 
not so far been detected with any degree of certainty. On 
account of the laminar structure of mica, and the crystallo- 
graphic perfection of the laminae, it would be of great interest 
to establish heUum diffusion across the laminae. Rayleigh 
attempted to measure the helium permeabiHty both at 

Table 19. The helium permeabilities of ionic crystals 



Substance 


Permeability at ° C. 

in c.c./kr./cm.^/mm./ 

thickness/atm. pressure 


Quartz (cut _L to optic axis) 

Mica (cleavage plate) 

Calcite (cleavage plate) 

Fluorite 

Rocksalt 

Selenite (cleavage plate) 

Beryl (cut J_ to optic axis) 

Beryl (cut !| to optic axis) 


<0-05xl0-8 
<0-06 X 10-9 
<005 X 10-8 
<0-2 xlO-8 
<0-2 xlO-" 
<0-7 xlO-9 
<0-l xlO-8 
<0-15 X 10-' 



20°C. (15) and at 415°C. (I6). At the latter temperature the 
permeabiUty was still below 7 x 10"^ c.c./day/cm.^/mm. thick- 
ness/atm. pressure, more than 10~^ times as small as the heHum 
permeabihty of silica glass at 20° C. The figure given was 
about the limit of sensitivity of the apparatus. 

No better success has attended the efforts of Urry (17), or of 
Rayleigh (15) to measure the helium permeabihty of quartz, 
the densest of the crystalline forms of silica, and of beryl. The 



DIFFUSION OF HELIUM 117 

latter substance is of special interest since channels run parallel 
to the optic axis, of a diameter slightly greater than that of 
a helium atom. Table 19 gives the results of Rayleigh's 
attempts to measure heHum permeabiHties of a number of 
crystals. The permeabilities are less than the sensitivity limits 
of the apparatus. 

The PERMEABILITY OF GLASSES TO GASES 

That silica glass is permeable at high temperatures has been 
known for a considerable time. Reference to this property was 
made by Watson (18), who used it for the interesting object of 
purifying hehum for an atomic weight determination by means 
of its selective diffusion through silica glass. Some earUer 
and some later references have been made by Villard(i9), 
Berthelot(20) and others (21,22,23). Among the earhest quanti- 
tative measurements were those of Wiistner(24), who studied 
both permeation rates and absorption coefficients up to 
800 atm. Alty(25) recognised the phenomena as examples 
of activated diffusion and apphed to them a treatment akin 
to that evolved by Ward (26) and Lennard- Jones (27) for the 
activated sorption of hydrogen by copper. The experimental 
and theoretical aspects were extended by Barrer(28) who 
succeeded in identifying the type of interaction involved in 
the migration process. 

Methods of measurement of gas flow through 
glasses and crystals 

It is usual in the study of gas flow through glasses to use an 
apparatus which is in essence a chamber containing the gas 
separated from an evacuated chamber by the membrane whose 
permeabihty is being studied. The pressure in the gas-filled 
chamber is sensibly constant, since the rate of passage of gases 
through a glass is small, and under ordinary conditions will 
be varied from one experiment to another from a few centi- 
metres to an atmosphere. The rate of growth of pressure on 
the high .vacuum side is followed by pressure-measuring 
devices such as the McLeod gauge, Pirani gauge or ionisation 



118 



GAS FLOW IN CRYSTALS AXD GLASSES 



gauge. The problem of mounting the membrane offers little 
difficulty where soda glass, pyrex, borosilicate glass, silica glass 
and other glasses are concerned, for these glasses may be blown 
into double-walled vessels of the types illustrated in Fig. 31. 
When the glass is available in the form of plates, the latter 
may be mounted only with difficulty (cf. Fig. 32). Urry(33) 
mounted flat disks of quartz, basalt, and other rocks upon 
the ground-glass flanges of a glass tube which led to a gas- 
analysis apparatus. The specimen was made to adhere to the 




TcMcLtoo ^ 
Caucc 



'foscD Silica 
— Sorr Glass 






/\ 



- PY/f£X 



[b] 



Fig. 31. Types of double-walled diffusion cell. 

ground-glass plate by means of a little tap grease, and was 
sealed off from leakage between it and the gromid-glass 
flanges by fiUing the apparatus with mercury up to the top 
edge of the specimen. 

The experiments of Wustner(24) were made at very high 
pressures and temperatures, so that special requirements had 
to be met. The diffusion cells were contained in a platinum 
oven which in turn was in the interior of a steel bomb. The 
pressure was transmitted to the bomb from the pump (with 
oil as transmitting fluid), through the separator which was a 
U-tube containing mercury, to water, which filled the bomb. 
The requisite temperature within the bomb was reached by 
the use of the platinum oven. The silica diffusion cells inside 



PERMEABILITY OF GLASSES TO GASES 



119 



the oven contained the hydrogen imprisoned by mercury seals. 
This hydrogen was compressed when the pressure was raised, 
its pressure being given by the 
gauge, and its temperature by 
thermo elements leading from 
the oven. The pressure of 
hydrogen on one side of the 
sihca was thus high (700-1000 
atm.); on the other side it was 
negUgible, and diffusion con- 
sequently occurred through the 
small thin-walled sihca bulb 
into which the hydrogen was 
compressed. The ingoing and 
outgoing sides of the sihca wall 
were both open to the pressure- 
transmitting hquid, so that 
there was no nett pressure 
gradient across the wall, and 
so no danger of mechanical 
rupture. The volume of hydro- 
gen which diffused through the 
silica was estimated from the 
movement of the mercury seal 
in the sihca diffusion cell. 

Rayleigh (16) succeeded in the 
technically difficult task of 
mounting a mica plate for dif- 
fusion experiments at tempera- 
tures up to 4 1 5° C . The principle 
of his method was to keep an 
outer annulus of the plate cold 
and thermally insulated as far 

as possible from the inner part of the plate which was kept 
hot (Fig. 32). The mica plate was held between steel disks, 
which had a hot inner section and an outside ring water-cooled. 
The outside ring was separated from the inner disk by narrow 




3^^y^ 



120 GAS FLOW IN CRYSTALS AND GLASSES 

steel ribs which gave rigidity to the whole, and reduced heat 
conduction from the hot central disk to the outer annulus. 
The mica disk was sealed with red wax to the steel annuli on 
either side of it. In its central part, through which diffusion 
was to occur, the mica bore sufficiently Ughtly upon the central 
steel disks for the hehum to pass easily between mica and steel 
and so to collect in the low-pressure chamber. The mounted 
membrane was fixed across a steel tube, the two halves so 
formed serving as chambers to supply the hehum at constant 
pressure, and to collect it. Not only the annuh but also the 
steel tube was water-cooled. The heating was carried out by 
means of nichrome wire bent back and forth and passing 
through siUca tubing, there being one heating coil on each side 
of the mica disk. The diffused gas was removed for analysis by 
a Toepler pump. 

The principal characteristics of the permeation process 

The main features observed when one studies the passage of 
gases through membranes are: 

dC 
(1) The Fick diffusion law P = D~ (P denotes the per- 
meability and D the diffusion constant) is true in the stationary 

state (33, 24, 32), 

V (2) Stationary flow is estabhshed in a period of minutes, the 

actual time depending on the temperatures, 

(3) The permeation rates are usually proportional to the 
pressure and inversely proportional to the thickness of the 
membrane (29), 

^ (4) The velocity of diffusion is only slightly altered by 
roughening the outgoing surface (29). 

(5) The passage of an electric discharge through either the 
glass wall or through the gas (hydrogen or helium) has no effect 
upon the diffusion rate(i7), 

(6) The process of permeation through these glass mem- 
branes is highly selective, and markedly temperature de- 



PERMEABILITY OE GLASSES TO GASES 121 

pendent, so that by proper choice of temperature zones one 
could in theory very effectively separate certain gases (Og and 
He; air and He; argon and He; Hg and argon, etc.). 

Dependence upon pressure of the rate of permeation 

A Knear pressure dependence of the permeation rate is indi- 
cated by the results of various observers (24, 30,28, 31,32). It was 
immaterial whether the gas was monatomic (He) or diatomic 
(Hg, Og, Ng), the same linear relation was obtained. There is 
one contrary result (33), data on the velocity of permeation of 
helium through iDyrex, soda, lead, and Jena 16"i glasses being 
reported as conforming to an equation dpjdt = ap"^. The 
values of n were given as follows : 

Pyrex glass n = 0-88, 

Lead glass n = 0-56, 

Soda glass n = 0-64, 

Jena l&^^ glass n = 0-66. 

It was also stated that the value of n was nearly proportional 
to the amount of SiOg + BgOg. These results need further 
testing, since hehum cannot be dissociated in diffusing (which 
would give a law dpjdt = a,p^), so that the only possibihties 
remaining are that helium entered the glass from an adsorbed 
phase which does not obey a distribution law x = hp (where x 
denotes the amount adsorbed), or that irreversible changes 
were occurring in the structure of the glass. At room tem- 
perature on glass helium more than all gases should be 
-adsorbed according to Henry's law {x = kp). 

The nature of the permeation process through silica glass 

The observation that the diffusion rate (save for the exceptional 
data of Urry) is proportional to the pressure is typical of a very 
wide class of diffusion membranes of sihca, glass, basalt, rubber, 
porcelain, cellulose, collodion, and various polymeric products 
from styrene, vinyl acetate and similar substances. It may be 
characteristic of activated diffusion (p. 125) or of various types 
of flow down tubes (Chap. II). On the other hand, there are 



/ 



122 



GAS FLOW IN CRYSTALS AND GLASSES 



systems where permeation velocities are proportional to ^^p 
{p denoting pressure), exemplij&ed by the diffusion of hydrogen 
through nickel, iron, copper, palladium, platinum and other 
metals. In this type of system diffusion appears to be a specific 
property of the system considered. No trace of helium will 
diffuse through palladium (34) or copper (35), though the helium 
atom is much smaller than the hydrogen m^olecule. In ex- 
planation of all the facts, one may regard specific activated 
diffusions as occurring when the membrane can dissociate the 
diffusing molecule into atoms or ions. An insight into the 
nature of the gas-solid interaction in the case of the non- 
specific type of activated diffusion is provided by the experi- 
ments of Barrer (28) on gas-silica glass systems. It was observed 
that the temperature dependence of the permeation rate in 
calories per mole, though much greater than the sublimation 
energies, follow in the same order (Table 20). 

Table 20. Heats of siiblimation of gases, and the temperature 
dependence {in caL/mol.) for flow through silica glass 







Temperature 




Sublimation 


coefficient 


Gas 


heat 


horn P =Poe-^l^'^' 
for flow 


He 


126* 


5,600 


Ne 


590 


9,500 


Ha 


,529* 


10,300 


0, 


2150 


31,200 


N2 


1860 


26,000 


^ 


2030 


32.100 



* Calculated(i9> from the force law E= -a/R^ +/]/R^~, when the constants 
a and fi are derived from the virial coefficients of the equation of state of the gas. 

Table 20 suggests that van der Waals and repulsive forces 
contribute to the temperature coefficient of flow (in cal./mol.) 
just as they do to the latent heat of sublimation; in the latter 
instance the attractive forces, Avhich are small, are pre- 
dominant, but in the former the much larger repulsive forces 
are predominant. This provides a basis for the classification 
of activated diffusion processes into specific types (Hg-Pd) and 
non-specific types (He-SiOg), the basis being the nature of the 



PERMEABILITY OF GLASSES TO GASES 



123 



interaction between solid and gas. Table 21 attempts a 
classification of some diffusions having exponential tem- 
peratm-e coefficients into specific and non-specific activated 
diffusions. In addition to these "natural" diffusions there are 
numerous "forced" diffusions wherein ions may move under 
an impressed potential. These will be considered separately 
(Chap. VI). 

Table 21. Dijfusion processes 



Nature of 
system 


Nature of interaction 


Specific 


Non-specific 


Gas-solid 


Ha-Pd, Ni, Pt, Cu 
Oa-Ag, CugO, FeO 
Na-W, Mo, Fe 


H^-KBr 

Hg-SiOa, pyrex, rubber, 

cellulose 
Oa-SiOg 
Na-SiOg 
Ar, Ne, He-SiOg, B2O3, 

borates, silicates, rubber, 

cellulose 


Vapour-solid 


HgO-zeolites? 

NHg-zeolites? 
HCl, NHg-NaCl? 


NHj-pure rubber 
HgO-pure rubber, cellulose? 
COg-pure rubber 


Solid-solid 


S-FeS 

Ca++ K+, Na+-glass, 

zeolites, ultramarines, 

clays 
Metal-metal systems? 


' 


Liquid-liquid 


NH3-H2O? 
D2O-H2O? 


CeHsOH-CHgOH 
CeHsOH-CeHe 
s-C2H2Br4-s-C2H2Cl4 
D2O-H2O? 


Gas-liquid 


COa-NaOH aq. 


H2-H2O 

O2-H2O, NaOH aq. 



= At'' 



The influence of temperature on the permeation rate 

Three formulae have been employed to express the permeation 
rate as a function of temperature. Tfie formulae are 

dn 
~~dt 

dn 
~~dt 

dn 
~'dt 






(1) 
(2) 
(3) 



124 



GAS FLOW IN CRYSTALS AND GLASSES 



in which A, m; B, a; and C and E/R, are characteristic 
constants. The most satisfactory agreement over the high 



4-5 


\ 








'G 

i- 




X 








N 






C 
o 

o 

Q> 




> 


\ 










\J 


V 


S 








\ 


ff.ff 


, 




1 


X 



10 

Fig. 33 



7-4 

lOOO/T (Tin^K.) 

of helium through silica. 



78 





\ 








4-5 










"y. 








"Si 


\ 








u 




V 






o 




^^ 






w 




dv 










^^^ 






c 




\^ 






o 














^v 






o 




©^ 






Q> 










E 




\^ 
















.2: 






V 




bo 






\ 




—J 






\ 




5-5 






\ 








\ 












o 



7-0 7-5 

lOOOJT (T in "K.) 

Fig. 34. Passage of hydrogen through silica. 

temperature range is undoubtedly given by equation (3) 
(Figs. 33, 34), which has also the advantage of having a 
theoretical basis. Thus when an energy of activation is required 



PERMEABILITY OF GLASSES TO GASES 125 

to make the molecule enter the pore and to move it along the 
energetically periodic pore length, one would by analogy with 
the Arrhenius theory of chemical reactions expect the number 
entering the pore to be given by 

where N-^ denotes the number of molecules available to enter 
the pore and A denotes the chance that a molecule having a 
sufficient energy will actually enter the pore. While a kinetic 
theory of permeation of a more elaborate nature can be 
derived, the appHcabihty and significance of equation (3) 
will now be assumed, and from the experimental data the 
temperature coefficients in cal./mol. for the permeation process 
are collected in Table 22. These coefficients include the tem- 
perature variation of N-^^ (which may refer to adsorbed 
molecules, or even dissolved molecules). Since the heat either 
of sorption or of solution (p. 140) is small, these temperature 
coefficients may be approximately identified with activation 
energies for diffusion within the soHd. 

Irreversible effects 

Several interesting points arose from Barrer's(28) study of the 
influence of temperature upon the permeation rate. First, the 
permeabihty towards the heavier gases is affected by prolonged 
heating to high temperatures. This is shown by the permeation 
rates of air through a sihca tube, given as a function of tem- 
perature for various periods of heat treatment of the sihca 
(Fig. 35). The flowing curves are drawn through points 
determined in sequence, and the permeabihty decreases as the 
heating is prolonged. That the effect was solely a surface one 
was shown by treatment with hydrofluoric acid, which restored 
the permeation rate to its original value. This superficial 
change was probably connected with a visible clouding of the 
surface, and it was thought that it might be due to the 
formation of tiny crystals, possibly platy (3) or else of /5-cristo- 
bahte type. With an ensuing decrease in permeabihty there was 
an increase in the activation energy which is clearly indicated 











. 




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gfe 








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2 (=^ 


5S 

42 




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s a 


§s 








e 








"OD 
















O 









ft 






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tn 






bO 






3 












'm 






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C7 




T3-d 


tifi 




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erv 
erv 


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as .a 




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tion 
tion 


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C3 CO c6 


ft 

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<D <D (U 


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Hogness(30) 
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3 

o 



PERMEABILITY OF GLASSES TO GASES 



127 



in Table 22 . These effects were most marked for the heavy gases 
(Ng, Ar, and air), whereas for hydrogen and hehum the influence 
of heating upon the permeabiUty was slight. This fact, and the 
close agreement of the temperature coefficients of the per- 
meabihty when expressed in cal./mol.,* determined by various 



1-25 -_ 



o 100 



§ 0-75 

1=! 



I 



0Z5 



800 WOO 1100 

Temperature, °K. 

Fig. 35. The rate of flow of air through silica, as affected by- 
temperature and time of heating of the silica. 

workers on different samples of sihca led to the postulate (28) 
that there are two types of activated diffusion in silica 
glass: 

(1) A structure sensitive diffusion down faults, and cracks 
of molecular dimensions, predominant when argon, nitrogen 
and air diffuse. 

(2) A structure insensitive diffusion through the anionic 
network of the glass itself, and corresponding to a solution 
process. This diffusion is predominant when helium, neon and 
hydrogen pass through sihca glass. 

* -The permeabihty constant P, diffusion constant D and solubility h are 
related by P = Dh Ap/l, where Ap denotes the pressure difference across, and I the 
thickness of the specimen. When as for Hj and He (p. 140) the temperature 
coefficient of k is small, the temperature coefficient of P in cal./mol. is that for D, 
i.e. the activation energy for diffusion. 



128 



GAS FLOW IN CRYSTALS AND GLASSES 



Even for helium, neon and hydrogen, grain boundary 
diffusion must occur simultaneously with "lattice" diffusion, 
and the predominance of one or the other is conditioned by 
the temperature. As the temperature is lowered, the more 



-4-5 



\/ 



-5- 



fl _ 



P^ 



6-5 



-7-5 



-85 






10 



1-5 

T 



20 



Fig. 36. The structure sensitive and structure insensitive 
regions in the diffusion of helium in silica glass. 

temperature sensitive "lattice" diffusion is increasingly sub- 
merged in the grain boundary diffusion. So one should find a 
flattening of the curve log (permeation rate) against IjT 
( T in ° K.) for small values of T. Figs. 34 and 36 show this effect 
for hydrogen and hehura. 



PERMEABILITY OF GLASSES TO GASES 



129 



On the other hand, a specimen of sihca glass which had 
undergone prolonged heating gave a log (permeation rate) 
against \[T curve Knear even at room temperature, suggesting 
that the heating had diminished (by the surface change noted 
(p. 125)) the structure sensitive part of the diffusion. 

Table 23 gives mean values of the temperature coefficient 
of the permeation rate for diffusion processes occurring in 
various zones of temperature. Table 23 also illustrates in what 

Table 23. Temperature coefficients of permeability in 
cal./mol. at low temperatures* 









Nature of 


Energy 




Gas 


Glass 


Worker 


predominating 


cal./ 


Temp. 

°C. 








diffusion 


mol. 


Hj 


Fused silica 


Barrer(28) 


Lattice 


10,800 


>400 




Fused silica 


Barrer(28) 


Grain boundary 


4,300 


193 


He 


Fused silica 


T'sai and Hog- 
ness(30) 


Lattice 


5,700 


>300 






Burton, Braaten 


Grain boundary 


4,190 


110 too 






and Wil- 




3,040 


Oto -41 






helm(3i)* 




2,310 


- 41 to - 78 




Pyrex 


van Voorhis(36) 


Lattice 


8,700 


>300 






Urry(]7) 


Grain boundary 


5,840 


283 to 172 








Grain boundary 


4,540 


172 to 81 




Jena 16"i 


Urryd?) 


Uncertain 


8,720 


283 to 134 








Grain boundary 


6,900 


134 to 22 



* In a later communication (29) two of l^be authors state that no further 
diminution in E could be found below - 20° C. Their new values of E were : 
from 180 to 562° C, 5390 cal.; from 180 to - 78° C, 4800 cal. 

way the sihca content affects the activation energy for the 
process. The smaller the sihca content, the larger is the 
activation energy and the smaller the permeation rate (cf. 
p. 137). The passage of heUum through a number of glasses of 
known composition (36) also revealed that acidic oxides such 
as BgOg or SiOg increased the permeabihty, while basic oxides 
such as KgO, NagO, or BaO decreased it approximately in 
proportion to their amount. Feebly basic or amphoteric oxides 
such as PbO or AlgOgCse) were stated to have httle effect 
upon the permeability. It is to be noted that AlgOg may 
replace SiOg in the anionic network of the sihca membrane. 

* See footnote, p. 127. 
BD .9 



130 GAS FLOW IN CRYSTALS AND GLASSES 

Roeser(40) studied the permeability of various samples of 
unglazed and glazed porcelain in air. He found many samples 
with channels so large that stream-line or Knudsen flow 
occurred in them; but in other more perfect specimens only 
a strongly temperature sensitive diffusion was observed. The 
experiments, which were made up to 1300° C, gave varied 
agreement with the law 

Permeability constant* — P^e"^/^^, 

the products of some manufacturers giving linear curves 
of log (P) versus IJT, while other manufacturers' products 
behaved more or less capriciously. The slope of the cui've 
log(P) versus l/T even at the highest temperatures varied 
from specimen to specimen. The slopes observed at high 
temperatures for one set of porcelain tubes varied from 29,000 
to 63,000 cal. All these results suggest that grain-boundary 
diffusion is a more usual process than lattice diffusion, in 
conformity with Barrer's(28) findings for the migration of the 
heavier gases through silica glass. In this connection it may 
be mentioned that Roeser's results on sihca glass and on glazed 
and unglazed porcelain give permeabihty constants of similar 
magnitude. Porcelain may be considered to consist of crystals 
of mulhte (3Al203.2Si02) embedded in a glass magna, and 
often with undissolved quartz or clay in the structure. It is 
clear that such a chemical will be far from homogeneous, and 
to this chemical inhomogeneity as well as to its physical 
inhomogeneity may be ascribed its capricious behaviour. 

Further studies on irreversible phenomena associated with 
helium diffusion through pyrex glass were made by Taylor and 
Rast(4i). The effects noted by them were not confined to the 
surface of the glass, as were those observed by Barrer (28). They 
found that the permeability constant rose by 10 % at about 
550° C, after anneahng at that temperature, and that there- 
after the new permeabihty-temperature curve lay above the 

* Even with a single energy of activation for each gas, this law could not be 
rigidly obeyed when air diffuses, since air is a mixture of Ng and O2 and the law 
should be P =P,e-^oJRT^p^^-EsjRT^ 



PERMEABILITY OF GLASSES *rO GASES 131 

old one. The effect was considered to be the result of strain 
removal in the original glass, the curve log (permeabihty) 
against IjT being now continuous into the region where the 
glass could be regarded as a viscous hquid. The diminution in 
slope at lower temperatures which have been interpreted (28) 
as increasingly important contributions of grain-boundary 
diffusion were also noted, the slopes giving values of 

-^350-440° c. ~ 6480 cal./atom, 
which may be compared with the values of E in Table 23. 
However, the authors regarded this diminution as due to a loss 
of rotational or vibrational freedom in the sihcate complex, 
occurring at a critical temperature. As a further change in the 
slope of the log (P) versus IjT curve had occurred at 225° C, 
one must on this theory suppose a second loss of rotational or 
vibrational freedom. 



On the relationships between some types 
of molecular flow 

It will be interesting to point out at this stage how several 
important types of mechanism for gas transference through 
sohds are related. In the normal stream -hne flow of fluids the 
diameter of the pore and the pressure are such that the number 
of coUisions on the pore wall in unit time per unit area is com- 
pletely outweighed by the number of coUisions in unit time 
and in unit volume of the gas phase. When the pressure and 
pore diameter are such that colhsions with the waU completely 
outweigh colhsions in the gas phase, one finds not stream - 
hne flow but molecular streaming. Also as one continually 
restricts the pore diameter, the gas molecules must spend 
a greater and greater fraction of time in the surface field 
of the sohd, that is adsorbed on the sohd. Ultimately, as a 
logical conclusion to this process of restricting the pore 
diameter, the diffusing molecules are always within the surface 
fields of the sohd. Owing to the periodic or crystalhne nature 

9-2 



■y 



132 



GAS FLOW IN CRYSTALS AND GLASSES 



of the solid, the energy distribution along the surface is also 
13eriodic, and a molecule diffusing along the surface finds itself 
moving to successive energy hollows, separated by energy 
barriers. 

When the periodic fields of the opposite sides of the pore 
overlap, energy will be needed to make the molecule enter the 
pore. In illustration of this, Fig. 37 shows the energy needed to 




; 1,2 
Distance , (A). 

Fig. 37. Energy needed to make an argon atom pass through 
a square of argon atoms. 

make an argon atom pass through a square of argon atoms, 
when the diameter of the square is varied to make the atomic 
force fields overlap to varying extents. 

Experiments upon the transition region from molecular 
streaming to activated diffusion are few. Rayleigh(i5) has 
recently made interesting experiments upon the passage of air 
and helium through very narrow artificial channels. When 
optically plane glass plates were placed in contact and heated 
to remove adsorbed gases, but not heated sufficiently to destroy 
the planeness of the glass, it was found that contact was so 
intimate that no flow of helium between the two plates could 
be measured, although the rate of flow of helium through silica 
glass can be measured at room temperature. In another 
experiment the plane glass plates were placed in contact, 



PERMEABILITY OF GLASSES TO GASES 



133 



but not heated. Their distance apart was of the order of 
10 A,, and the permeation rates for helium and for air were 
9-5 X 10-2 cu. mm. /day and 1-4 x 10"^ cu.mm. /day respectively. 
The ratio of these velocities is 



instead of the ratio 



which Knudsen's law of molecular streaming would require. 
That is, Rayleigh succeeded in passing beyond the Umits of 
Knudsen flow into what must be at least a transition region to 
activated diffusion. In this region the ratio of the permeation 
rates would be governed principally by a ratio of exponentials : 





He 

Air 


6-8 

1 




He 


2-7 


/ 


28-8 


Air 


1 


-J 


4 



He 

Air 



Q—EssIRT 

g-EAirlRT • 



Numerical values of the permeabilities 

The volumes in c.c. at n.t.p. of gas diffusing per sec/cm.^ 
through a glass wall 1 mm. thick, when a pressure difference of 
1 cm. is maintained across the wall, is given in Tables 24-34. 



Table 24. He-SiO, 



Temp. 

°C. 


Permeability constant x 10* 
(c.c. at N.T.P./sec./cm.2/mm./cm. Hg) 


A 


B 


C 


D 


150 
200 
300 
400 
500 
600 
700 
800 
900 
1000 


0-73 
1-39 
315 
6-15 
10-4 
16-4 
21-9 
28-5 
36-^ 
45-4 


0-78 
1-52 
4-13 
8-25 
13-8 
19-3 


0-48 
0-99 
1-72 
300 
4-25 
5-50 
■ 6-72 
8-42 


0-19 
0-46 
0-92 
2-06 
4-62 



Authors: A=T'sai and Hogness; B=Braaten and Clark; C=Barrer; 
D = Williams and Ferguson. 



134 



GAS FLOW IN CRYSTALS AND GLASSES 



Table 25. He-SiOa/row -200 to 150° C. 







Permeability constant x 10* 


1 


Temp. 


(c.c. 


at N.T.P./sec./cm.7mm./cm. Hg) | 


° C. 










A 


A * 


A 


B 


-200 





. 





0-0028 


-180 


— 


— 


— 


0-0035 


-160 


— 


— 


— 


0-0038 


-140 


— 


— 


- — 


0-0044 


-120 





— 


— 


0-0053 


-100 


— 


. — . ' 


— 


00066 


-80 


000070 


— 


— 


0-0084 


-70 


0-00176 


— 


— 


0-0101 


-60 


0-00315 


— 


— 


0-0121 


-50 


0-0052 


— 


— 


0-0145 


-40 


0-0077 


— 


— 


0-0179 


-30 


0-0109 


— . 


— 


0-0224 


-20 


0-0174 


— 


. — 


0-028 


-10 


0-022 


— 


— 


0-037 





0-035 


0-029 


0-028 


0-050 


10 


0-051 


0-046 


0-040 


0-073 


20 


0-070 


0-062 


0-055 


0-104 


30 


0-090 


0-080 


0-073 


— 


40 


0-114 


0-106 


0-095 


. — 


50 


0-135 


0-132 


0-119 


— 


70 


0-205 


0-198 


0-176 


— 


90 


0-304 


0-29 


0-264 


0-274 


110 


0-44 


0-42 


0-39 


0-45 


130 


0-62 


0-59 


0-52 


0-55 


150 


' 0-79 


— 


— 


— 



Authors: A = Braaten and Clark; B= Burton, Braaten and Wilhelm. 



Table 26. Ne-SiO., 



Temp. 
°C. 


Permeability 


constant x 10* 


(c.c./sec./cm.7mm./cm. Hg) 


500 


0-139 


600 


0-282 


700 


0-50 


800 


0-81 


900 


1-18 


1000 


1-63 



Authors: T'sai and Hogness. 



PERMEABILITY OF GLASSES TO GASES 



135 



Table 27. ArgonSiO^ 



Temp. 

°C. 


Permeability constant x 10^ 
(c.c. /sec/cm. ^/mm. /cm. Hg) 


A 


A* 


B 


850 

900 

950 

1000 


0-0161 
0-062, 0031 


0-00022 
0-00050 


0-58 



Authors: A = Barrer; B= Johnson and Burt. 
Decreased permeability due to long heating of silica glass specimen. 



Table 28. Hg-SiOs 



Temp. 
°G. 


Permeability constant x 10^ 
(c.c./sec./cm.7mm./cm. Hg) 


A 


B 


B 


B 


B 


C 


D 


200 
300 
400 
! 500 
600 
700 
800 
900 
1000 


0-022 
0-099 
0-366 
0-70 
1-43 
2-52 
4-25 
6-4 
10-0 


0-48 
0-92 
1-75 
3-1 

4-8 


0-44 

0-84 

1-54 

^•70 

4-4 

7-0 


2-45 

4-0 

5-9 


0-50 

1-06 

2-16 

3-9 

6-0 




2-00 
2-76 
4-5 


0-051 

0-275 

0-58 

0-81 

1-70 

2-53 

3-6 

5-1 



Authors: A = Barrer; B= Williams and Ferguson; C=Wustner; D= Johnson 
and Burt (mean of three samples). 



Table 29. Na-SiOg 



Temp. 
0. 


Permeability constant x 10^ 
(c.c./sec./cm.2/mm./cm. Hg) 


A 


B 


B* 


650 
700 
750 
800 
850 
900 
950 


0-065 

0-132 

0-268 

0-43 

0-80 

1-19 


0-066 

0146 

0-271 

0-39 

0-64 

0-95 

1-44 


0-161 
0-65 



Authors: A = Johnson and Burt; B=Barrer. 
* Decreased permeability due to long heating of silica glass specimen. 



136 



GAS PLOW IN CRYSTALS AND GLASSES 



Table 30. He-pyrex glass 



Temp. 


Permeability constant x 10" 


°C. 


(c.c./sec./cm.^/mm./cm. Hg) j 





A 


B 


00037 


— 


20 


0-0064 


— 


50 


0-0128 


— 


100 


00264 


— 


150 


0058 


— • 


200 


0-124 


0-069 1 


250 


0-229 


— 


300 


0-38 


0-243 


400 


— 


0-70 


500 


— 


1-57 



Authors: A = Urry; B=van Voorhis. 

Table 31. He- Jena 16«i 



Temp. 


Permeability constant x 10' 


°C. 


(c.c./sec./cm.Vmm./cm. Hg) 


20 


0-0000095 


50 


0-0000471 


100 


0-000071 


150 


0-000183 


200 


0-00077 


- 250 


0-00176 


300 


0-00362 



Author: Urry. 

Table 32. He-miscellaneous glasses 



Temp. 

°C. 


Glass 


Permeabihty constant x 10* 
(c.c./sec./cm.Vnmi./cm. Hg) 


283 
283 
610 


Lead 
Soda 
Pyrex 


0-0037* 
0-0098* 
l-9t 



Authors: *=:Urry; t= Williams and Ferguson. 

Table 33. He-Thuringian glass 



Temp. 


Permeability constant x 10* 


°C. 


(c.c./sec./cm.7nini./cm. Hg) 


100 


0-00000106 


200 


0-000117 


300 


0-00084 


400 


0-0045 


500 


00132 



Authors: Piutti and Boggiolera. 



PERMEABILITY OF GLASSES TO GA-SES 

Table 34. Air -porcelain 



137 



Temp. 


Permeability constant x 10* 
(c.c./sec./cm.-/mm./cm. Hg) 


Sample I 


Sample II 


25 

400 

600 

800 

1000 

1200 

1300 


0-00106 

0-00106 

0-00212 

0-0032 

0-0161 

0-077 

0-32 


0-0012 

0-022 

0-117 



Author: Roeser. 

In deriving these permeability data graphs of the experimental 
measurements were used to obtain figures at comparable 
temperatures. Several features of this group of tables are 
interesting. First one may consicier the selectivity of the 
permeabihty of siHca glass towards a number of gases, illus- 
trated by the following series: 



For He-SiOgCSO) at 900° C, 
H2-Si02(28) at 900° C, 
Ne-SiO2(30) at 900° C, 
Na-SiOaCSS) at 900° C, 
Ar-Si02(39) at 900° C, 



P = 36-2 X 10-^ 

p = a•4xlo-^ 

P= 1-18x10-9, 
P = 0-95x10-9, 
P = 0-58 X 10-9. 



It must, however, be remembered that permeabilities towards 
certain of the gases vary from specimen to specimen of glass; 
thus much smaller permeabihties both to argon and to hehum 
have been reported (28). In yet another series may be given 
the permeabihty of a number of glasses to hehum : 



For He-SiO2(30) at 300° C, 
He-pyrex(i7) at 300° C, 
He-soda glass (17) at 283° C, 
He-lead glass (17) at 283° C, 
He-Jena 16ni(i7) at 300° C, 
He-Thuringian glass at 300° C. 



P = 3-15x10-9, 
P = 0-38x10-9, 
P = 0-0098 X 10-9, 
P = 0-0037 X 10-9, 
P = 0-0036 X 10-9, 
P = 0-00084 X 10-9, 



138 



GAS FLOW IN CRYSTALS AND GLASSES 



and from the series one observes the high sensitivity of 
permeabiHty to chemical composition (cf. p. 129). Further 
data along these hnes are provided by Rayleigh's studies on 
the hehum permeabihty of membranes, including sihcate 
glasses, sihca glass, and boron trioxide melts, as Avell as a 
number of metaUic and organic membranes. The following 
Tables (35 and 36) give the hehum permeabihties of a number 
of glasses; and also allow a comparison of the permeabihties 

Table 35. The helium permeability of glasses at room 
temperature in c.c.lsec.jcm.'^lmm. thick/cm. Hg pressure 
difference 



Substance 


Permeability 


Substance 


Permeability 


Silica 


0-058 X 10-9 


Fused B2O3 


0-055x10-9 


Optical silica 


0-043 X 10-" 


Fused B2O3 


0-056 X 10-9 


Silica sheet 


0-025 X 10-9 


Fused borax 


<0-000103x 10-11 


Thin silica tube 


0-053 X 10-9 


glass 




Pyrex 


0-0040 X 10-9 






Corex glass (mainly 


< 0-000040 X 10-9 






Ca3(P04)2) 








Extra white sheet 


< 0-000025 X 10-1" 






glass 








Micro cover glass 


<0-000020xl0-" 






Flint s;lass 


< 0-000030 X 10-11 






Soda glass 


< 0-000043 X 10-" 







Table 36. The helium and air permeabilities of various kinds 
of membrane at room temperature hi c.c./sec./cm.^/inm. 
thick I cm. Hg pressure difference 





Helium 


Air 


Ratio of 


Substance 


permeability 


permeability 


permeabihties 


Cellophane 


0-023 x 10-9 


0-049 X 10-11 


48 


Sihca 


0-052 X 10-9 


< 0-052 X 10-" 


> 10,000 


Gelatin 


0-14 X 10-9 


0-077 X 10-11 


184 


Celluloid 


6-1 xlO-9 


0-305 X 10-9 


20 


Rubber 


12 X 10-9 


4-16 X 10-9 


2-9 



of certain organic membranes with the inorganic glasses. 
All the measurements reported were at room temperature. 



PERMEABILITY OF GLASSES TO GASES 139 

and the constants have been converted to c.c./sec./cm.^/mm. 
thickness/cm. of mercury. 

Examination of Table 35 shows an extreme variation in the 
hehum permeabihty of the siKca membranes of only 2-3-fold 
at room temperature, while Table 24 indicates an extreme 
variation of 6-4-fold at 500° C. The sensitivity of the permea- 
bihty to the composition of the glass is again apparent, and 
it may be that the variations in Table 24 are due to small 
amounts of the alkaU metal oxides in the less permeable of 
the sihca glasses as well as to a greater amount of grain- 
boundary diffusion in the more permeable glasses. It is 
interesting that fused BgOg compares with fused SiOg in 
permeabihty, and also that the organic membranes cellophane 
and gelatin have permeabihties of the same order as the 
inorganic oxides. The ratio of hehum to air permeabihty 
clearly bears no relationship to molecular masses, being 
mainly governed by the ratio of exponential terms of the 
type e-^l^^. 

The SOLUBILITY OF GASES IIST SILICA AND DIFFUSION 
CONSTANTS WITHIN IT 

When discussing the temperature coefficients of the per- 
meabihties, we considered these temperature coefficients to 
be approximately those for the activated diffusion process 
within the sihca. This viewpoint was justified because, as 
the data now to be given show, the solubihty of hydrogen 
and hehum in sihca varies only to a very minor extent with 
temperature, and the permeabihty constant (P), diffusion 
constant (D) and solubihty k are related by 

V 

where Ajpll is the pressure gradient across the membrane. If 
the values of P and of k are known one may compute the more 
fundamental quantity D. The values of A; (24, 42) are given in 
Table 37. It may be noted that the solubihties are quite 



140 



GAS FLOW IN CRYSTALS AND GLASSES 



comparable with the solubilities of gases in rubber mem- 
branes (p. 418), in liquids and in crystals (p. 111). Further, 
the differences in permeability which one encounters for gases 
in sihca and pyrex are, one is now led to beheve, governed 
mainly by the differences in D, which are in their turn (p. 125) 
governed partly by an exponential term.* 

Table 37. The solubilities of hydrogen and helium in silica 







Solubilities (average values in c.c. 


System 


Temp. 


at N.T.p./c.c./atm.) 


Wiistner(24) 


Williams and Ferguson (42) 


Ha-SiOa 


1000 


00103 







900 


00102 


— 




800 


0-0109 


— 




700 


0-0099 


— 




600 


0-0082 


— 




400 


0-0057 


0-0095 (extrapolated) 




300 


0-0055 


0-0099 


He-SiOa 


500 





0-0101 




450 


— 


0-0103 


He-pyrex 


500 


— 


0-0084 



When the permeability constant is expressed as c.c. /sec./ 
cm.^/mm. thick/atm. pressure, and the solubihty as c.c./c.c. of 
siUca/atm. pressure, the,]iise of the equation 



P = Dk 



Ap 

T 



leads to a value of D expressed as cm.^sec."^ These values of 
D are of interest for purposes of comparison with corresponding 
yalues of D obtained for liquid-liquid or gas-liquid systems ; 
for gas-metal, and gas-rubber systems; and for ion-ionic lattice 
diffusion systems. This comparison will be made elsewhere 
(p. 426). In Table 38 the values of D are computed from the 
solubilities of Table 37 and the permeabiUties of Tables 
24-34. 

* But where grain-boundary diffusion predominates the number of internal 
surfaces, which is determined by the history of the specimen, is also 
important. 



PERMEABILITY OF GLASSES TO GASES 



141 



Table 38. The diffusion constants inside silica' 
and glass in cm.^ secr^ 



Solid 


Gas 


Temp. 


Solubility 
taken 


D 


Authors 






°C. 


c.c./c.c. solid 


cm.2 sec.-^ 




SiOa 


He 


20 


0-01 


0-024-0-055 X 10-« 


Rayleigh 


SiOa 


He 


500 


0-01 
(WiUiams and 
Ferguson) 


0-017-0-14 X 10-« 


Authors of 
Table 24 


SiOa 


Ha 


500 


0-01 
(Williams and 
Ferguson) 


0-006-0-011x10-8 


Authors of 
Table 28 


SiOg 


H, 


500 


0-0055 

(Wiistner) 


0-012-0-021 X 10-6 


Authors of 
Table 28 


SiOa 


Ha 


200 


00055 


0-05-0-08 X 10-8 


Barrer 


Pyxex 


He 


20 


0-0084 


0-0045 X 10-8 


Rayleigh 


Pyrex 


He 


500 


0-0084 
(Williams and 
Ferguson) 


0-02 X 10-6 


van Voorhis 



The diffusion constants of Table 38 may be written as 

and then D takes the following values: 

= (7-9 - 3-5) 10-6 e-560P/i?T cm.^ sec.-i, 
= (5-2 - 0-64) 10-6 e-seoo/iJT cm.^ sec.-i, 
= (8-3- 14.5) 10-6e-i«i«o/^2^cm.2sec.-i, 
= (13-7-35) lO-^e-ioioo/iJTcjjj 2gec.-i, 
= l-3x 10-*e-870o/i?rcm.2sec.-i, 
= 5-5 X 10-6e-87oo/i?Tc,jjj_2gec.-i. 



n2ooc. 

-^He-SiOz 

n500°c. 

-^He-Si02 

n500°c. 

-^Ha-SiOz 

r)200°C. 
-^H2-Si02 

r)2oo-c. 

"^He-pyrex 

7)500° c. 
He -pyrex 



The result for the He-pyrex system at 20° C. studied by 
Rayleigh suggests that grain-boundary diffusion (see Table 23) 
has been taking place, with a lower energy of activation than 
the 8700 cal. assumed, which is the activation energy* for 
"lattice" diffusion. With this exception the values of i)^ are 
all of the same order of magnitude, and do not alter markedly 
with temperature, but do show random fluctuations which 
would conform well with the theory (p. 127) of mixed grain- 
boundary and lattice diffusion. 



* See p. 139. 



142 GAS FLOW IN CRYSTALS AND GLASSES 

Theories of the diffusion process 

Several authors (33, 27, 25) have attempted' theories of the 
diffusion process. Urry (33) who considered flow to be a process 
of molecular streaming, as observed when rarefied gases pass 
through capillaries, is obviously incorrect for sihcate glasses. 
The other theories (27, 25) are based upon more acceptable 
premises, but do not lead to any notable advance in the study 
of the subject. They need not therefore be discussed here. 

REFERENCES 

(1) Taylor, W. H. Proc. Roy. Soc. 145 A, 80 (1934). 

(2) Barrer, R. M. Proc. Roy. Soc. 167 A, 392 (1938). 

(3) Shishacow, N. A. Phil. Mag. 24, 687 (1937). 

(4) Nagelschmidt, G. Z. Kristallogr. 93, 481 (1936). 
(4a) Farkas, A. Private communication. 

(5) Tiselius, A. Z. phys. Chem. 169 A, 425 (1934). 

(6) Z. phys. Chem. 174 A, 401 (1935). 

(7) Hey, M. Miner. Mag. 24, 99 (1935). 

(8) • Phil. Mag. 22, 492 (1936). 

(9) MoUwo, E. Z. Phys. 85, 56 (1933). 

10) Rogener, H. Ann. Phys., Lpz., 29, 387 (1937). 

11) MoUwo, E. Ann. Phys., Lpz., 29, 394 (1937). 

12) HUsch, R. Ann. Phys., Lpz., 29, 407 (1937). 

13) E.g. see Faraday Society Discussion, "Chemical Reactions in- 

volving Solids", pp. 883 et sec?. (1938). 
13a) Dm-au, F. and Schratz, V. Z. phys. Chem. 159 A, 115 (1932). 
136) Herbert, J. Trans. Faraday Soc. 26, 118 (1930). 
13 c) Tompkins, F. C. Trans. Faraday Soc. 34, 1469 (1938). 
13 cZ) Kraft, H. Z. Phys. 110, 303 (1938). 

14) Bradley, R. Trans. Faraday Soc. 30, 587 (1934). 

15) Rayleigh, Lord. Proc. Roy. Soc. 156A, 350 (1936). 

16) - — Proc. Roy. Soc. 163 A, 377 (1937). 

17) Urry, W. J. Amer. chem. Soc. 55, 3242 (1933). 

18) Watson, W. J. chem. Soc. 97, 810 (1910). 

19) Villard, P. C.R. Acad. Sci., Paris, 130, 1752 (1900). 

20) Berthelot, M. C.R. Acad. Sci., Paris, 140, 821 (1905). 

21) Jaquerod, A. and Perrot, F. C.R. Acad. Sci., Paris, 139, 789 

(1904). 

22) Richardson, O. and Richardson, R. C. Phil. Mag. 22, 704 (1911). 

23) Bodenstein, M. and Kranendieck, F. Nernst Festschrift, p. 99 

(1912). 

24) Wustner, H. Ann. Phys., Lpz., 46, 1095 (1915). 

25) Alty, T. Phil. Mag. 15, 1035 (1933). 



REFERENCES 143 

(26) Ward, A. F. Proc. Roy. Soc. 133A, 506, 522 (1931). 

(27) Lennard-Jones, J. E. Trans. Faraday Soc. 2S, 333 (1932). 

(28) Barrer, R. M. J. chem. Soc. 378 (1934). 

(29) Braaten, E. O. and Clark, G. J. Anier. chem. Soc. 57, 2714 (1935). 

(30) T'sai, L. S. and Hogness, T. J. phys. Chem. 36, 2595 (1932). 

(31) Bxirton, E., Braaten, E. O. and Wilhelm, J. O. Canad. J. Res. 

21, 497 (1933). 

(32) Williams, G. A. and Ferguson, J. B. J. Ainer. chem. Soc. 44, 

2160 (1922). 

(33) Urry, W. J. Amer. chem. Soc. 54, 3887 (1932). 

(34) Paneth, F. and Peters, K. Z. phys. Chem. IB, 253 (1928). 

(35) Smithells, C. and Ransley, C. E. Proc. Roy. Soc. 150 A, 172 (1935). 

(36) van Voorliis, C. C. Phys. Rev. 23, 557 (1924). 

(37) Piutti, A. and Boggiolera, E. R.C. Accad. Lincei (5), 14 (1923). 

Also R.C. Accad. Sci. Napoli (3) 29, 111 (1923). 

(38) Mayer, E. P%s. -Rev. 6, 283 (1915). 

(39) Johnson, J. and Biu-t, R. J. opt. Soc. Amer. 6, 734 (1922). 

(40) Roeser, W. Bur. Stand. J. Res., Wash., 7, 485 (1931). 

(41) Taylor, N. W. and Rast, W. J. chem. Phys. 6, 612 (1938). 

(42) Williams, G. A. and Ferguson, J. B. J. Amer. chem. Soc. 46, 635 

(1924). 



CHAPTER IV 

GAS FLOW THROUGH METALS 

Introduction 

It is difficult to prepare a pure metal, for not only do metals 
contain traces of other metals, carbon, sulphur, or phosphorus, 
but they also contain combined or occluded gases — hydrogen, 
oxygen, nitrogen, and sulphur dioxide. These impurities exert 
in many instances a profound effect upon the properties of the 
metal. The present discussion concerns the behaviour of such 
gas-metal^ystems. Pioneer researches on hydrogen-palladium 
systems were made by T. Graham (i) in 1866; even earher 
observations on the system hydrogen-iron were made by 
Cailletet(2), who in 1864 found that some of the hydrogen 
evolved when an iron vessel was immersed in dilute sulphuric 
acid was absorbed in the iron. Deville and Troost(3) first 
showed that hydrogen diffused through platinum, and the 
interesting permeability of silver towards oxygen was observed 
by Troost(4) in 1884. These workers have been followed by 
others (5, 6, i, 8), amongst whom must be mentioned Richardson, 
Nicol and Parnell(9), who developed an equation for the flow 
of gas through a metal which is to-day the basis of interpre- 
tations of diffusion processes : 

where P denotes the permeability constant, k and b are 
constants, I denotes the thickness, and^ and T are respectively 
pressure and temperature. 

It has also been found (2, 5,7,10,11,12) that hydrogen gas in 
nascent form can penetrate metals such as palladium, iron, or 
nickel at room temperature, if the metal is made the cathode 
during electrolysis or if the hydrogen is generated by chemical 
reaction at the surface. Pickling a metal in hydrogen in this 



GAS FLOW THROUGH METALS 145 

way may alter its mechanical properties to a marked degree (i3), 
and the process of absorption is also very sensitive to traces 
of poisons (14). The earUest observations on this type of 
diffusion we owe to Bellati and Lussana(7), Cailletet(2), and 
Nernst and Lessing(i2), but the field is one which has not 
been extensively studied, although the results should give 
information concerning interface reactions, and diffusion 
within the metal. 

Before discussing the experimental data upon the per- 
meability of metals to gases it will be of advantage to consider 
the cognate subject of gas solubility in metals. The differences 
in behaviour met with there may be reflected in the per- 
meabihties (P), since the latter are defined by 

dx' 

where D is the diffusion constant and dC/dx is the con- 
centration gradient, defined in certain circumstances by the 
solubility of the gas in the metal. 

The solubility of gases in metals 

For the study of gas-metal systems one has all the apparatus 
and technique developed for measuring the sorption of gases 
by sohds (15), The powerful X-ray method then gives the crystal 
habit of the products (iG), or shows the effect the absorption has 
upon lattice constants (17), so that one may construct the phase 
diagrams for the system. Absorption equilibria have been 
studied for over a decade by Sie verts and his co-workers (18) 
and by many others, and as a result these often remarkable 
systems are being increasingly understood. Summaries of 
findings on gas-metal equilibria will be found in the books of 
McBahi(i5) and of Smithells(i9), and it is not intended to give 
more than a resume of the data here. 

The metals which absorb common gases are summarised in 
Table 39. Oxides are not mentioned in the table; but apart 
from oxygen, hydrogen interacts most freely with metals, not 
as a rule to give hydrides but rather alloy systems, or solid 



146 GAS FLOW THROUGH METALS 

solutions. Solution of gases such as hydrogen, oxygen, or 
nitrogen occurs with dissociation, as is shown by a pro- 
portionahty between the solubility (at small concentrations) 
and the square root of the pressure. Compound molecules, 
such as sulphur dioxide, ammonia, carbon dioxide, or carbon 

Table 39. Summary of the reactivity of metals 
towards gases 







Metals which 




Metals which do not 


Gas 


Group 


dissolve gas 


Group 


dissolve gas : 


Ha 


Ia 


Hydrogen gives salt- 


IB 


Au 1 






like hydrides 


II B 


Zn, Cd 




Ib 


Cu, Ag (slight) 


IIIb 


In, Tl 




II A 


Hydrogen gives salt- 
like hydrides 


IV B 


Ge, Sn, Pb (but give 
covalent hydrides) 




III A 


Al. Rare earths Ce, 
La, Nd, Pr 


Vb 


As, Sb, Bi (but give 
covalent hydrides) i 




IV A 


Ti, Zr, Hf, Th 


VI B 


Se, Te give covalent 1 




Va 


V, Nb, Ta 




hydrides 




VIA 


Cr, Mo, W 


VIII 


Rh 




VII a 


(Mn) 








^^II 


Fe, Co, Ni, Pt, Pd 






0, 


Ib 

IV A 

VIII 


Cu,Ag 
Zr 

Fe, Co, (Ni) 




" 


Na 


III A 


Al (molten) 


Ib 


Cu, Ag, Au 




IV A 


Zr 


II B 


Cd 




Va 


Ta (nitrides) 


IIIb 


Tl 




VIA 


Mo (nitrides only). 


IV B 


Sn, Pb 






W (nitride) 


Vb 


Sb, Bi 




VII A 


Mn (various nitrides) 


VIII 


Rh 




VIII 


Fe (various nitrides) 






CO 


VIII 


Ni,Fe (above 1000° C.) 


Ib 


Cu 


SOa 


Ib 


Cu (Hquid), Au (liquid) 


VIII 


Pt 


He 








Do not dissolve in any 


Ne 








metal so far studied 


Ar 








either when liquid 


Kr 








or solid 


Xe 










COa 


VIII 


Fe 


VIII 


Rh, Pt 



monoxide, must also dissociate before they can penetrate into 
the body of a metal. Temperature alters the solubility according 
to an exponential law, so that curves of log (solubility) versus 
l/T are often linear, save where two alloy phases co-exist 
(Hg-Pd, Hg-Th, Hg-Ti), Avhere a limiting composition is 
approached (Hg-Pd, -Th, -Ti, -Zr, -V), or where various 



SOLUBILITY OF GASES IN" METALS 



147 



allotropic forms of the metal exist, and show different 
capacities to dissolve the gas (Hg-Fe). 

The greatest diversity of interaction is shown by hydrogen, 
which reacts with metals to give three different types of 
products : 

{a) covalent hydrides: BgHg, SbHg, SiH4, AsHg, 
(6) salt-hke hydrides: [Na+] [H-], [Ca++][2H-], 
(c) alloys: PdHo.55. 
Those systems with which experiments on hydrogen per- 
meability have to do are alloy systems, at least over an 
appreciable range of compositions. The hydrogen exists in the 
metallic lattice most probably in an ionised or partially ionised 
condition, 

Metals 
A metallic crystal is to be regarded as a giant molecule, 
composed of positive nuclei and electrons, so that electrostatic 
forces ensure the cohesion of the whole. The electrons occupy 
definite energy levels, there being two electrons in each level 



5 7 9 
6 8 10 



13... 



Fig. 38. Representation of electron distribution in a metal at 
and 300° K. (after Emeleus and Anderson(20)). 

at 0° K. At room, temperature a few of the electrons by virtue 
of thermal energy are promoted to higher energy levels. In 
Fig. 38 (20) one sees the number of electrons per level plotted 
against the number of levels, the full curve representing the 



148 GAS FLOW THROUGH METALS 

distribution at 0° K., and the dotted curve that at 300° K. In 
a metal with the electron distribution obtaining at 0°K. no 
electrical conductivity can occur because, since all levels are 
filled, the Pauli exclusion principle indicates that no nett flow 
of electrons may take place under any external potential, for 
no level can receive further electrons. There can be con- 
ductivity with the 300° K. distribution because there are some 




Fig. 39. Graphical representation of electron distributions in metals 
and semi-conductors (after Emeleus and Anderson(20)). 



levels with only one electron, or no electrons. These levels may 
receive electrons and there may be a nett flow of electrons 
when a potential is applied. 

The distribution curve of Fig. 38 is highly idealised. In the 
actual lattice the positive nuclei cause a periodic variation in 
the potential encountered by an electron moving through the 
lattice. The solution of the wave equation for such a potential 
distribution leads to the result that the electrons cannot 
assume any energies from zero to a maximum, but that there 
are bands or zones of permitted energies alternating with 



SOLUBILITY OF GASES IN METALS 149 

bands of forbidden energies. If there are fewer electrons than 
levels in a given band, or Brillouin zone, the condition of 
partially filled levels and so of metallic conduction is fulfilled. 
Correspondingly, if two Brillouin zones overlap (Fig. 39 a) (20), 
and there are not enough electrons to fill both zones, metallic 
conduction is again observed. If the zones do not overlap 
but are adjacent, and one zone is full and the other empty 
(Fig. 39b), the substance is a semi-conductor, for clearly the 
input of a small activation energy will promote electrons 
from the first zone to the second, and so leave incompletely 
filled levels in both zones, with consequent electrical con- 
duction (Fig. 39c). In an insulator the completely occupied 
zone and the unoccupied zone are so far apart that with 
normal activation energies no electrons are promoted, and so 
electrical conductivity is absent. 

Hydrogen-metal systems 

The nature of alloy systems of hydrogen with metals is now 
easily understood. The alloying hydrogen atom may provide 
electrons which fill the empty levels in a band, while the metal 
loses its para-magnetic properties (20). An Hg-Pd alloy ceases 
to be para-magnetic at the composition HQ.ggPd and then, since 
no more electrons can be supplied, very little more hydrogen 
will dissolve. Hydrogen and deuterium give the well-known 
isobaric curves (2i) of Fig. 40, in which there is an apparent 
invariant region, and a hysteresis effect. This is due to the 
formation first of an a-alloy, and then of a /?-aUoy, the two 
alloys CO -existing along the vertical lines of Fig. 40. The 
occurrence of two phases may be explained by supposing that, 
as the concentration of hydrogen atoms in the palladium 
lattice rises, the atoms interact with each other as well as with 
the lattice. When a critical interaction energy is reached, some 
of the atoms gather together in closer association in the 
palladium lattice, and so a new phase appears which is in 
equihbrium with the original more dilute phase. As the 
concentration of hydrogen in the lattice increases, the dilute 
phase diminishes and the concentrated phase increases in 



150 



GAS FLOW THROUGH METALS 



amount, and one travels along the invariant part of the curve. 
When the initial dilute or a-phase is all consumed the system 
becomes once more univariant, and also the lattice is nearly 
saturated. The hmiting composition is not certain but is of 



7U0mm. Isobar 
per gm. Pd 




Fig. 40. The solubility of hydrogen and deuterium in 
palladium at atmospheric pressm-e. 

the order PdH^.gg to PdHo.59. Lacher(22) analysed the solu- 
bility-pressure-temperature data for the hydrogen-palladium 
system and concluded that the heat of absorption of hydrogen 
in palladium could be expressed as 



AH 



cal./mol. 



20407ipr -f- 



225171%^ 



where the second term expresses the interaction energy 
between dissolved atoms, or protons, n^ is the number of 



SOLUBILITY OF GASES IN METALS 



151 



gram -atoms of hydrogen dissolved and n^ is the number of 
gram-atoms of potential energy "holes" in the palladium*. 
Various other hydrogen-metal systems have properties ana- 
logous to the hydrogen-palladium system, except that the 
interaction energy between dissolved hydrogen atoms or 
protons is not usually sufficient to cause the separation of two 
phases, at some critical composition. In Fig. 41 are given 
absorption isobars for metals which dissolve hydrogen (23). 
It is evident from the slopes of such isobars tb.at the solution 
process is strongly exothermic for metals such as V, Th, Zr, 




400 600 800 
Temperature, °C. 



1200 



Fig. 4L Some isobars for metal hydrogen systems. 

Ti, Ta, but endothermic for Cu, Fe, Co, and Ni. From the 
slopes of log (solubility)- l/T curves (24) (Fig. 42) the heats of 
solution given in Table 40 have been calculated. In some of 
the metals, notably those in which hydrogen dissolves endo- 
thermically, there is no appreciable alteration in the lattice 
constants on solution of the gas in the metal. On the other 
hand, when hydrogen dissolves in palladium one may have 
at saturation 10 % expansion of the lattice. In the metals 
Ti, V, Zr, Th, and Ta, where great quantities of hydrogen are 

* The dissolved hydrogen is supposed to occupy interstitial positions in the 
palladium lattice. These positions of minimum energy are referred to as potential 
energy "holes" in the lattice. 



152 GAS FLOW THROUGH METALS 

absorbed, the systems approach a limiting composition and 
have a new lattice structure. The densities and limiting com- 
positions in the expanded states are illustrated by Table 41, 




8 SO 12 H 1t> 18 20 22 

10.000/T 

Fig. 42. Observed solubilities s of hydrogen in various metals subjected to one 
atmosphere pressure of Hj, shown by plotting log^s against lO*i'T. The 
solubility s is the volume of H, gas (reckoned in c.c. at n.t.p.) absorbed by 
100 g. of metal. 

The methods of statistical mechanics (24, 22) have provided 
an approach to the problem of gas-metal solubility. The 
metal can be regarded as containing a series of holes of low 



SOLUBILITY OF GASES IN METALS 



153 



potential energy distributed periodically according to the 
lattice structure of the metal. The hydrogen molecules in the 
gas phase dissociate and are absorbed into the lattice where as 
atoms or protons they vibrate in the holes, among which they 

Table 40. Heats of solution of hydrogen in metals 



Exothermic 


Endothermic 


Metal 


Heat 
(cal./mol. H2) 


Metal 


Heat 
(cal./mol. Hg) 


Ti 

Zr 

Th 

V 

Pd 


10,000 

17,500 

22,500 

7,700 

2,040 


Cu 
Co 
Fe 
M 
Al 
Pt 
Mo 
Ag 


14,100 
7,300 
7,000 
5,600 

45,500 

35,400 
3,500 

11,600 



Table 41. Densities and limiting compositions 
of some alloy systems 



Metal 


Density 


Limiting 
composition 


Density 


Density 
ratio 


Ti 
Zr 
Ta 
V 


4-523 
6-53 
16-62 
6-11 


TiHa 
ZrHa 
TaH 
VH 


3-91 

5-67 

15-10 

5-30 


0-864 
0-867 
0-906 
0-867 



are distributed at random. Diffusion occurs by jumps from 
one hole to another, when a sufficient activation energy has 
been acquired. The partition function is then constructed for 
the systems 

(i) H2 molecules in the gas phase, 

(ii) H atoms in the gas phase, 

(iii) H atoms (or protons) in the metal lattice, 

and so an expression for the equilibrium between gas molecules 
and absorbed atoms is obtained: 

{27rmkT)^ 

'pY h^ 

Jct) \'{27T{2m) kTf- SnHk'. 



v.^ 



w^ exp 



F 



h^ 



2/^2 






154 GAS FLOW THROUGH METALS 

where Vg = concentration of dissolved atoms, 

k = Boltzmann constant, 

h = Planck's constant, 

m = the mass of the hydrogen atom, 

/ = the moment of inertia of the hydrogen molecule, 

w?2 = the weight of the normal electronic state of the 
hydrogen molecule, 

Xs = the heat of solution of a hydrogen atom in the 
metal, 

Xd = the heat of dissociation of a hydrogen molecule. 
Smithells and Fowler (24) showed that if one defines s, the 
solubility, as the number of c.c. of molecular hydrogen at 
N.T.P., dissolved in 100 g. of metal, the formula above 
reduced to ^ 

S ^ 101-21 -^— e-^X.s+iXd)lkT 

where /o^ = the density of the metal. These formulae are 
apphcable to solutions of hydrogen in metals such as Fe, Co, 
Cu and Ni. Those metals such as Ta, V, Ti, Zr, or Th, where 
the amount of hydrogen absorbed reaches a Hmiting value 
and is thereafter constant, can also be treated. The same 
methods led to the equation 






Sq-s " {kTJ r {27ri2m)kTf87TUkT y 



e~(Xs+iXd)!kT 



where Sq denotes the saturation solubility. Inserting standard 
numerical values gives 



= l()-l-S2 i^ Q-(Xs+hXd)lkT 



Lacher(22) was able to use the statistical mechanical 
approach in the same way to explain the occurrence and form 
of the peculiar isothermals of hydrogen-palladium systems 
(Fig. 40). All that it was necessary to add to the previous 
treatment was the assumption that as the concentration of 



SOLUBILITY OF GASES IN METALS 



155 



hydrogen atoms increased they interacted with one another 
until at a critical concentration they formed clusters in the 
lattice of the palladium instead of being distributed uniformly. 

Oxygen-metal systems 

When definite chemical compounds are not formed it should 
be possible to apply statistical considerations similar to those 
above to other gas-metal systems. Often, however, as in the 
system Og-Ag, the nature of the solution process is not 




200 400 600 T~°C ^^ 

Fig. 43. The solubility of oxygen in silver. 

clear (25, 26, 27). The oxygen molecule is dissociated and must 
be associated with the silver in at least two ways. First, an 
unstable oxide AggO is formed, but this oxide decomposes as 
the temperature rises, until, above 400° C, it should under 
ordinary pressures disappear. However, an endothermic 
solubility has now set in, and so the isobaric solubility- 
temperature curves (Fig. 43) (26) first have a negative slope 
(dissociation of the oxide) and then a positive slope (endo- 
thermic solution). Other oxygen-metal systems exist where 
there is a solubihty of oxides in each other or in the metal, 
or of oxygen in oxides. Here one has really to consider 



156 



GAS FLOW THROUGH METALS 



compounds which do not conform exactly to the law of fixed 
proportions, and where there is a lattice excess of one or other 
component. This may occur when some lattice spaces of one 
component are vacant, or when the other component can 
occupy interstitial as well as lattice positions. Compounds 



1700 



7500 



mo 



noo 



$00 



700 



Liquid(fe,0) 
Liqutd(FeO,Fe) 



S+ Liquid Fe 



y+FeO 



Solid Solution 
OxLfgen in Iron 



oL + FeO 



CL+Fee04. . 



05 



7-0 




20 25 

Oxygen Atoms. % 



Fig. 44. Temperature-composition phase diagram for Og-Fe. 

having a variable composition are usually referred to as 
Berthollide compounds, and the phenomena is encountered 
with the following typical substances: 

Oxides of Fe, Co, Ni 1 ^^^^ ^T' ""^^^^^ ^""^ sulphides 
Sulphides of Fe, Co, Ni '''^^ ^rm solid solutions with 

j oxygen and sulphur). 
Tungsten bronzes) 
Spinels (variabiHty range considerable). 

Zinc oxide (can contain excess metal). 



SOLUBILITY OF GASES IN METALS 157 

Nickel oxide, for example, can vary in composition between 

NiOi.ooo and NiOi.005. 

The solubility of oxygen in metals is complicated by the 
formation of oxide phases, so that complex phase diagrams 
result. As an example, the system Fe-Og may be taken 
(Fig. 44) (28, 29). The temperature-composition diagram given 
shows that oxygen is probably more soluble in y- than in 
a- or ^-iron. The solution of oxygen in molten iron occurs 
endothermically, since the solubihty increases with rising 
- temperature (it is 147 c.c./lOOg. at the melting point and 
387c.c./100g. at 1734° C). The diagram illustrates the 
great variety of sohd solutions possible, and thus indicates 
how iron-oxygen systems may deviate from the law of fixed 
proportions. 

The oxygen-copper system (30) has received considerable 
attention, and also shows a complex temperature-composition 
phase diagram (31). The solution of oxygen in metallic copper 
occurs endothermically, as the solubility data of Rhines and 
Matthewson(30) show: 

Temp. ° C. 600 800 950 1050 

'Solubihty (c.c./lOOg. metal) 5-0 6-6 7-0 10-9 

Other systems which have been studied are 02-Co(32), 
02-Ni(33), and 02-Zr(34). One interesting feature of oxygen- 
metal systems is the capacity of hquid metals to dissolve 
large quantities of oxygen or oxides, which on cooling are 
frozen out, as oxide or as bubbles of oxygen. 

Nitrogen-metal systems 

In order for nitrogen to be absorbed by a metal it is necessary 
that the metal should be capable of forming a nitride. Thus, 
while nitrogen is not absorbed by Cu, Co, Ag, and Au, it is 
taken up by the metals Fe, Mo, W, Mn, Al, and Zr. A great 
variety of phases is observed in some of these systems. 
Nitrogen-molybdenum (35, 36) gives the following: 

a (Mo) : body- centred cubic. No sohd solution with nitrogen. 



158 GAS FLOW THROUGH METALS 

/? (M03N): face-centred tetragonal. Stable only above 
600° C. Contains 25 % atomic nitrogen. 

7 (M02N): face-centred cubic. Stable at all temperatures, 
and contains 33 % atomic nitrogen. 

S (MoN) : hexagonal. Contains 50 % atomic nitrogen. 

Because of the use of iron catalysts for ammonia synthesis, 
the nitrogen-iron phase diagram is especially interesting (37, 38) 
(Fig. 45). Other systems, such as nitrogen-manganese (39), 
nitrogen-aluminium (40), and nitrogen-zirconium (34), need not 



"C. 
800 



600 



400 

5 10 15 20 25 

Nitrog&n Atoms, % 

Fig. 45. Temperature-composition phase diagram of Nj-^e. 

a = solid solution N2 in Fe, 7 = solid solution Ng in Fe, 7' = Fe4N, 
e = Fe3N. 

be considered here, since the behaviour already described for 
nitrogen-molybdenum and nitrogen-iron is typical. It is 
interesting that nitrogen dissolves in molten aluminium to 
give an approximately Unear plot of log (solubility) against 
IjT. This is characteristic also of solutions of oxygen in 
silver at high temperatures, and of solutions of sulphur 
dioxide in liquid copper (4i, 42). In the latter case, as in the 
two former, the solubility is proportional to the square root of 
the pressure, indicating a dissociation of the sulphur dioxide 
molecule in solution. 



N. 














x> 


y-^e 


' 7^ 








h' 


{ 


<3Ci 


■y 




y^A 


\ 



The solubility of gases in alloys 
The study of gas-alloy systems has been confined principally 
to hydrogen. One might recognise two possible classes of alloy, 
one compounded of two hydrogen-dissolving metals, and the 



SOLUBILITY OF GASES IN ALLOYS ' 159 

other composed of one hydrogen-dissolving metal and a second 
metal inert to hydrogen. The behaviour in the second of these 
systems is somewhat unexpected, for as the summarising data 
below show, instead of the hydrogen solubihty diminishing 
steadily to a zero value as the percentage of the inert metal 
increases, one often finds a solubihty first rising to a maximum 
and then falhng towards zero as the percentage of the inert 
metal increases. When hydrogen dissolves in both metals A 
and B of the alloy, however, one usually finds a continuous 
change in solubihty from 100 % A to 100 % B. The system 
Hg-Mo-Fe appears to be an exception: 

(a) Alloys in which the gas solubihty changes continuously: 

Fe-V (H2)(4.3), 

Pt-Pd (H2) (44), 

Al-Cu(H2)(44), 

Ag-Au (02)(45). 
(6) Alloys in which the gas solubility exhibits a maximum : 

B-Pd(H2)(46), 

Au-Pd(H2)(47), 

Ag-Pd(H2)(47), 

Fe-Mo (H2) (48). 

In the following figures are given data illustrating these 
types of behaviour. The maxima are not always sharp as in 
the Hg-B-Pd system, nor are they always so well defined. The 
curve of log (solubility) against l/T in. Fe-Mo aUoys, as Fig. 42 
shows for many hydrogen-metal systems, is linear (49). 

It would be out of place to give further details of gas-metal 
systems here. The purpose of this summary has been to 
illustrate the very interesting types of equUibria observed, so 
that one may have some idea of the state of combination of 
the gas diffusing in the solid. We have seen that hydrogen will 
diffuse in the form of atoms or protons, that oxygen or nitrogen 
diffuses after dissociation, and that oxygen-metal and nitrogen- 



160 



GAS FLOW THROUGH METALS 



metal reactions lead much more frequently to the formation 
of new phases — oxides or nitrides — than do hydrogen-metal 
reactions. Thus we must think of the diffusion of oxygen or 
nitrogen as a handing on of the 
dissolved gas atom by successive 
decompositions of unstable oxides 
or nitrides. There is little evidence 
to show what is the condition of the 
diffusing particle in its transition 
state, but it would be anticipated 
that oxygen and nitrogen, being 
much more electro-negative than 
hydrogen, may move as negative 
ions while hydrogen may very well 
be considered to diffuse as protons. 
It has been seen that even the 
solution of sulphur dioxide follows 
dissociation of the molecule, al- 
though the nature of the fragments remains unknown. Gases 
such as carbon monoxide, carbon dioxide, or ammonia must 



" 10 



1138° 










.183\^ 






















^ 


^ 


=== 





oPd 10 20 30 ^%nso 
Fig. 46. The solubility of Hg in 
, Pt-Pd alloys. 




5 10 15 20 25 

fttomi °/a of Boron 
Fig. 47. Solubility of H^ in B-Pd alloys. 



in the same way undergo dissociation into their components 
before they can penetrate a metal. The specificity of some of 



SOLUBILITY OF GASES IN ALLOYS 161 

the systems is one of their most fascinating features, and a 
great deal of further work must be undertaken before the 
reactions occurring can be properly understood. 

The measurement op permeation velocities 
The usual high vacuum technique employed in studying 
sorption equilibria and kinetics may be used in obtaining the 
permeability of metals (is, 19). In addition a special problem 
arises, the mounting of metal membranes in a manner which 
will be vacuum-tight and which will permit of heating the 
specimen to high temperatures. 

Ham (50) in an early apparatus mounted a sheet of platinum 
between two heavy steel tubes with flanged ends. The flanges 
were ground flat, and the joint put under great pressure by 
means of bolts passing through heavy steel rings on either side 
of the flange. Unless the surfaces are very smooth, or the 
pressure so great as to cause flow of the metal, this arrange- 
ment is not entirely vacuum -tight. In later arrange«ients(5i) 
the membrane was welded across a tube by atomic hydrogen. 
This method is very satisfactory, but it may not always be 
possible to do the welding. Such systems have the advantage 
that effects of temperature variations along the furnace are 
obviated. 

In other- arrangements tubes of the metal may be heated 
in a furnace. One end of the tube is closed and the other is 
open to a manometer. The whole tube may be surrounded 
by the diffusing gas. This very simple and useful method was 
used by Borelius and Lindblom(52), 

It is best to have the tube in the form of a bulb of large 
area in the centre of the furnace, with a narrow-necked tube 
leading from it out of the furnace. The narrow tube may then 
be fitted to glass by a ground joint outside the furnace. 
The arrangement avoids the effects of temperature inhomo- 
geneities near the ends of the furnace. Another way of 
doing this requires a short tube of the permeable metal 
(e.g. palladium) sealed at one end by gold solder, and at the 
other end welded by gold solder to a less permeable metal 



162 



GAS FLOW THROUGH METALS 




such as platinum or nickel. If the metal is welded to platinum, 
the platinum may be sealed through soft glass, and the system 
taken up to 370° C. under vacuum for the diffusion measure- 
ments. Another variation of this arrangement is to solder 
(with gold or platinum solder) a plug of palladium across the 
mouth of a platinum tube (53), which is again sealed through 
soft glass. In this connection it should be remembered that 
copper may be brazed to soft glass, and tungsten sealed 
through pyrex, provided the arrange- 
ment is such that these seals need 
not withstand high temperatures. 
The pressure on the high-vacuum 
side may be given by mercury mano- 
meters, McLeod gauges, or Pirani 
gauges according to the rate at which 

the pressure rises on this side. At Fig. 48. Apparatus for mea- 

high pressures robust methods of '""'^f ^fl^^ **'™"^'' 

° ^_ metals at mgn pressures. 

mounting membranes across the tubes 

are necessary. A suitable arrangement is that in Fig. 48(54), 
used to measure permeabiHties up to 112 atm. The mem- 
brane^ was brazed between the faced ends of two stout-walled 
tubes B, the joint being heated by a small external furnace. 
The apparatus required to measure the diffusion of nascent 
hydrogen is simpler, since permeation occurs at low tem- 
peratures. It is usual to use a hollow tube of the metal being 
studied, as cathode, and by a ground joint, an inseal, or by 
welding to another metal which is then insealed, to comiect 
it with a manometer system. A typical apparatus of this 
kind is that of BoreKus and Lindblom(52). If the hydrogen is 
generated by chemical means, the apparatus of Edwards (55) 
may be employed (Fig. 49), where the hydrogen diffused is 
measured by displacement of mercury . It would also be possible 
to adapt Borelius and Lindblom's apparatus to this case. 



The influence of temperature upon permeability 
The velocity of diffusion through a metal increases very 
rapidly as the temperature rises, as Fig. 50 illustrates for 




Fig. 49, Apparatus for measuring the rate of passage of 
nascent hydrogen through iron. 




500 



800 



700 



1200 



800 900 1000 1100 

y in°K. 
Fig. 50. The effect of temperature upon the permeation velocity of 
hydrogen through platinum at various pressures. 

II-2 



164 GAS FLOW THROUGH METALS 

the passage of hydrogen through platinum. One notes the 
similarity of these curves to the exponential rise of vapour 
pressure with temperature, or of chemical reaction velocities 
with temperature. As is well known,- this is due to an 
exponential term e~^^l^^ , e-^l^T ^n the vapour-pressure 
equation, or expression for the velocity constant respectively; 
and a similar term arises in the permeability constant P: 

However, some of the earhest studies of permeation velocities 
expressed these velocities as 

Rate = ^T", 

Rate = Ae^T^ 

Rate = Ae-^lT. 
For example, Winkelmann (8) gave for the passage of hydrogen 
through iron n~5\ while Johnson and Larose(56) for the 
diffusion of oxygen through silver gave a value oi n = 14-6. 
The second of these equations has been applied to hydrogen- 
nickel (57) and hydrogen-palladium (58) (for the permeabihty) ; 
and to carbon-iron, and nitrogen-iron (for the diffusion 
constants) (59) with some success. There is no theoretical 
interpretation for these first two equations however, and the 
agreement is in general better when the expression P = Pq e~^/^^' 
is used, and log P plotted against IjT. This has been done by 
Smithells and Ransley(60) for most of the gas-metal systems 
whose permeability has been studied, and some of their 
diagrams are reproduced here (Figs. 51-53). They found 
the equation to be very satisfactory indeed, although ex- 
ceptions may occur where allotropic modifications of metal 
exist in the temperature range considered (e.g. iron (58)), or 
if the diffusing gas can form two or more types of alloy or 
phase with the metal (Hg-Pd, Ng-Fe). 

From the slopes of the curves logP against 1/T, one may 
calculate the temperature coefficient in cal./atom of gas 
transferred. The first attempt to give a meaning to this 
temperature coefficient is due to Richardson, Nicol and 
l*arnen(9), and no subsequent theories have improved greatly 























4 t 


/ 




















. 


1 ^ 

V 


/ 














■B 






/ 


el 

.3 E 

■§3 


f 














I" E 

4 






/ 


pi 


/ 








/ 










/ 
' 


i 

-I 


f 








/ 








/ 














A. 






/ .t 


-a 


/ 












/ 


"? 
'f 






<A 


/ 












/ 


/ 


/- 


\/' 


</ 


/ 














y 


/ 


/ 


yy 


/ 
















</ 

























d^OT 




166 



GAS FLOW THROUGH METALS 



upon their treatment. They solved Fick's law for the special 
case of a gas dissociating in the medium and diffusing as atoms. 
In the stationary state, when d^C/dx^ = 0, they found 

1 D ik\^ 
Rate of permeation = y — I — I ^3^, 







P- 


95-4 mm. 

v 














-p-4- 

N 


> 

57mm. 


^ 


>v. 












\ 


X 


N 






o 






0-Oemrr 


\ 


\, 


\o 


V 


^ 




\ 


\ 




°\ 




\ 


c 
o 




\, 




So 




Ny O 




0) 




\ 




\ 




\ 




fc 




\ 


I 




\^ Hydrogen | 






V 




oS. 






cr> 
C 

_J 






\ 







\„ 






Nitrogen \ 






V" 








P-13 


3mm. iJ 


















\ 
















\ 

























Fig. 53. Effect of temperature on the diffusion of hydrogen and 
nitrogen through molybdenum. 

where D = the diffusion constant, 

hi = the dissociation constant of hydrogen m the metal, 
Sq — the solubility of molecular hydrogen in the metal, 
p = pressure. 
From the experiments they considered that 



INFLUENCE OF TEMPERATURE 167 

where C = constant and CD/2sq = 8-59 x 10"'; and so for the 
hydrogen-platinum system 

Q = 6-60 X 10-9^^ e-36.5oo/2i?2^^ 

where Q denotes the mass of gas diifusmg per sec. /cm. ^ of 
platinum of thickness I, from a pressure p into a vacuum. We 
now know of course that the heat of dissociation of hydrogen 
in the metal is not 36,500, because the diffusion coefficient of 
hydrogen atoms in a metal is itself exponentially dependent 
on temperature:/) = DqE^^^^I^'^. Richardson's equation should 
then become 

Q = 6-60 X 10-9^^ Do e-^o/^2^e-(36,5oo-2£o)/2i2r^ 

and until Eq is known one cannot find the heat of dissociation 
of hydrogen in platinum. The term in T- can as a rule be 
neglected in a formula such as that above, and it is usual to 
write P = Pq^-^I^^, where P denotes the permeability constant 
(c.c. at N.T.p. diffusing/sec. /cm. 2/mm. thick/cm., or atmo- 
sphere, of mercury). The values of E and Pq for a number of 
systems are given in Tal)le 42. As Smithells and Ransley have 
pointed out, the variation in the term Pq (lO^-fold) is small 
considering the variety of systems hsted. Since permeability 
constants depend upon so many variables, this consistency 
is notable, and so far susceptible to no interpretation. In 
systems like Hg-Ni the temperature coefficient is nearly the 
same in all the specimens considered; but in Hg-Pd a remark- 
able variation in the temperature coefficient occurs, and this 
coefficient appears to depend upon the previous history of 
the specimen. In its active most permeable state, the tem- 
perature coefficient is low; but in impermeable palladium 
it is high. When hydrogen passes through composite metal 
sheets of Cu-Pd and Ni-Pd, the temperature coefficient 
approximates to the values for the least permeable metal 
(copper or nickel), while for Ni-Pt and Pt-Ni the temperature 
coefficient is that of the metal at the outgoing face. It should 



168 



GAS FLOW THROUGH METALS 



Table 42, Permeability data for gas-metal 
systems {P^Pq e-^l^^) 





P. 




i 


System 


(c.c./sec./cin.7 
mm. thick/ 
atm. press.) 


E 
(cal./g. atom) 


Author 


Hg— Ni 


— 


14,600, 13,100 

(below Curie 

point) 


Post and Ham(6i) 




— 


13,100, 12,040 


Post and Ham(6i) 






(above Curie 








point) 






1-3 xlO-2 


. 15,420 


Lombard (57) 




0-85 X 10-2 


13,860 


Deming and Hendricks (62) 




1-4 xlO-2 


13,800 


Borelius and Lindblomoa) 




1-05 X 10-2 


13,400 


Ham (50) , 




1-44 X 10-2 


13,260 


Smithells andRansley(60) ) 




— 


13,400 


Post and Ham(ai) 


H2— Pt-Ni 


— 


13,400 


Ham (50) 


Hj— Pt 


1-41 X 10-2 


19,600 


Richardson, Nicol and 
ParneUo) 




1-18x10-2 


18,000 


Ham (5-)) 




2-6 xlO-i 


19,800 


Jouan(63) 


H2— Ni-Pt 


— 


18,000 


Ham (50) 


H5,— Mo 


0-93x10-2 


20,200 


Smithells and Ransley(60) 


H2— Pd 





5,000 


MelviUe and Rideal(64) 




- 


17,800 


Melville and Rideal(64) 




2-3 xlO-i 


4,620 


Lombard, Eichner and 

Albert (53) ! 




3-0 xlO-2 


10,500 


Barrer(53) i 


Hg— Ni-Pd 


— 


14,300 


Melville and Rideal(64) j 


Ha— Cu 


2-3 xlO-» 


16,600 


Smithells and Ransley(60) 




1-5 xlO-3 


18,700 


Braaten and Clark (65) 


H2— Cu-Pd 





13,700 


MelviUe and Rideal(64) ! 




— 


11,400 . 


MelviUe and Rideal(64) 


Ha— Fe 


1-63 X 10-3 


9,600 


SmitheUs and Ransley(eo) 




1-60 x 10-3 


9,400 


BoreUus and Lindblom(52) 




2-40 X 10-3 


11,000 


Ryder (66) 




— 


8,700 
(below 900° C.) 


Post and HanKsi) 




— 


18,860 
(above 900° C.) 


Post and Ham(5i) 


Ha— Al 


3-3^-2 


30,800 


SmitheUs and Ransley(67) 


O2— Ag 


3-75 X 10-2 


22,600 


Spencer (68) 




2-06 X 10-2 


22,600 


Johnson and Larose(56) 


Nj— Mo 


8-3 xlO-2 


45,000 


SmitheUs and Ransley(60) 


Na— Fe 


4-5 xlO-2 


23,800 


Ryder (66) 


CO— Fe 


1-3 x10-» 


18,600 


Ryder (66) 



INFLUENCE OF PRESSURE 169 

be remembered that in every case the diffusing molecule 
dissociates ; when carbon monoxide passes through iron, for 
example, the carbon and oxygen diffuse separately. 



The influence of pressure on the permeability 
of metals to gases 

In this field some very thorough studies have been made 
for hydrogen-metal systems, especially by Lombard and 
Eichner(69), Smithells and Ransley(60) and Post and Ham(6i). 
Even for hydrogen-metal systems, however, the interpretation 
of the permeation rate-pressure isotherms is not clear. Studies 
of glass, crystal, and organic membrane permeability have 
shown that two possible permeation rate-pressure relationships 
emerge : 

(i) Activated diffusion without dissociation: 

dpldt= kpe-b'^. 

(Gas diffusing in SiOg, KBr, zeolites and organic polymers 
such as rubber, bakeUte, ebonite, cellulose esters.) 

(ii) Activated diffusion with dissociation: 

dp[dt = kpie-blT. 

{H2, O2, Ng, SO2, CO diffusing through metals.) 

It is with the second type of diffusion system we have now 
to do. The first type requires an open crystal structure with 
large interstices (e.g. zeolites); in metals the crystal form is 
never sufficiently open, and so one finds that inert gases 
cannot either dissolve in or pass through a metal. This is true 
of any gas which cannot in some way react specifically with 
the metal under consideration. One criterion of the specific 
interaction is the ^Jp law in the expression dpjdt = kp^ e'^/^. 

Richardson (9) deduced the first specific expression for the 
permeation velocity through a metal, and his expression 
(p. 167), dp/dt = Ap^T^ e~V^, can be taken as the basis of the 
present discussion of isotherms. However, not all experiments 
gave an exact relationship dpjdt = A-^_^Jp. Thus one finds, on 



170 GAS FLOW THROUGH METALS 

writing dpjdt = A-^jp^^, the following values of n given for 
hydrogen and palladium: « 

Schmidt (1904) (70) n = \, 

Holt (1915) (71) n=\, ' 

Winkelmann (1901)(8) r«, = 0-7, 

Lombard and Eichner( 1932) (72) n = 0-8, n = 0-62, 
Lombard and Eichner (1933) (69) (i) n = 0-58, 0-59, 

(ii) ?^ = 0-56 (mean of a 
number of results). 
Since Schmidt's and Holt's results are not very accurate, 
and could also be expressed approximately by a ^^'p law, one 
is justified in saying that the exponent is less than one, and 
indeed the later values show it to be very nearly one-half. 
Finally, Ham and Sauter (73, 74) working with very pure 
palladium obtained many values of the exponent n between 
0*535 and 0-50, They found a maximum deviation from the 
^p law at 248° C. when n — 0-585. One may summarise the 
position by saying that there seem to be small and somewhat 
variable deviations from an exact ^p law, but that this law 
is very nearly fulfilled. 

One explanation of a ^_p law is, as previously indicated, that 
diffusion occurs as atoms. One then has the following relations : 

^^) n r ~ ^1' ^^^ small concentrations (Nernst dis- 

Gh (gas) 

tribution law). 

C^ (gas) 
(ii) -.^ , ' , = ^2 (law of mass action). 
C'H,(gas) '^ ^ _ 

(iii) Rate of permeation, P = —D ^^ (Pick's law). 

ax 

From (i) and (ii) 

(iv) Ch (solid) = ki^h^{Cji, (gas)}. 

dC C — C * 
Then if one writes -~ = 9 ^ i ^ and substitutes from (iv) 

for Ch (soHd), one finds 

(V) P = Dk,^k,{XCnJ-^{iCH.)}, 

* oCh denotes the concentration of H-atoms just inside the ingoing surface 
(a; = 0) and jC'n the concentration at the outgoing surface (x — I). 



INFLUENCE OF PRESSURE 



171 



which gives the observed relationship if ■\|{lC^J<t^J{QC-^^), as 
is the case when one side of the metal is held at a near-vacuum. 
This preliminary treatment is, however, very much too simple. 
It was first noted by Borelius and Lindblom (52) that even when 
one side of the metal was held under a vacuum, the diffusion 
isotherms appeared to obey a relationship (Fig. 54) 

P = k{^p-^Pt). 

They therefore suggested that a threshold pressure must be 
reached before permeation commences — an explanation 
difficult to base upon theory. 



P 

30 

20 



70 











jorc. 


^ 








y 


y 








A 


/' 


532'C. 


— 


^ 


(^ 






360X, 



70 75 20 25 30 
■^, in mms. 
Fig. 54. Permeation rate-pressure isotherms of Borelius and Lindblom (52). 

Then Smithells and Ransley(60) noted that the diffusion 
isotherms bent round at low pressures and thus did pass 
through the origin. They suggested that at low pressures the 
rate of permeation was proportional to the fraction of the 
surface covered by an adsorbed layer, d, as well as to the square 
root of the pressure. Thus, at low pressures the permeabihty P 

i^gi^^^^y F = kd^p, 

and so steadily increases until d~l. Thereafter the relationship 
is P = k^Jp. Smithells and Ransley(60) plotted many of their 
own and other workers' permeability -pressure isotherms to 
illustrate their suggestion. Some of their curves are reproduced 
in Figs. 55 and 56. The figures show, however, as do the 
authors' calculations, that d approaches unity at lower 
pressures the greater the temperature. That is, at high tem- 



172 



GAS FLOW THROUGH METALS 



peratures the ^p law is obeyed at lower pressures. It would 
be necessary to suppose that adsorption was endothermic for 
this to be true, whereas it is well known that for these systems 



















/ 


7aO°K 






















/ 


/ 






















J 


/ 
























/ 
























/ 
























/ 


/ 
























/ 














■' 1 








7 














^ 


645°K 








/ 


^ 








^ 














^ 


/ 


-^ 


-^ 



















5 10 15 20 25 VP 

Fig. 55. Permeation rate-pressure isotherms for H2-Pd. 



3 
O 














































.y 


^ 


CD 

c 


















■s^ 


^ 


723°K. 


















\y 


«^ 








+3 
to 
a> 
E . 












y 


y' 












E 1 

Q- 








--^ 


y 






- 














^^ 


^ 


^ 

















' 2 4 6 8 10 VP 

Fig. 56. Permeation rate-pressure isotherms for Hg-Cu. 

adsorption occurs in atomic form with evolution of heat. The 
results of Borelius and Lindblom on the diffusion of hydrogen 
through iron at 702° C. (Fig. 54) and at 100° C. (Fig. 57) will 



INFLTJENCE OF PRESSURE 



173 



illustrate the seriousness of this objection. Fig. 54 shows that 
at 702° C. if the rate is expressed as P = H^jp — ^jPt): Pt ^^^ ^ 
value of about 4 mm., while Fig. 57 shows that at 100° C. 
Pf has a value of nearly 8 atm. These, according to Smithell's 
theory, must give approximately the pressure needed to 
saturate the surface — 1500 times as great a pressure at 100° 
as at 702° C, although adsorption occurs exothermally. 



-?!- 



o 

^ 



/ 



/ 



L 



1 2 3 * ,/ -5^ 

Vp(Atmr) 

Fig. 57. Hj-Fe, the permeation velocity as a function of pressure, at 100° C. 

In other ways Smithells and Ransley's theory explains the 
observed facts. For if 

(Langmuir isotherm for sorption with dissociation), one has 

P — k k^p at low pressures, 

P — k^Jp dtAi high pressures, 

expressions which cover the observed facts. 

Another viewpoint has been developed by Ham(6i) and his 
co-workers. These authors find that the slopes of permeation 
rate-pressure isotherms for hydrogen-nickel systems are not 
exactly 0-5, as Fig. 58 shows. The deviations from a ^Jp law 
are at a maximum near the Curie point, and are related to 
the purity of the specimen of nickel employed. The isotherms 
do not have a slope of 0-5 near the Curie point unless the 
nickel is thoroughly decarburised. Similarly, the isotherms 
for carbonyl iron(6i) at 500° C. remained with an exponent 
well above 0-5 until the carbon was removed. The authors 



174 



GAS PLOW THROUGH METALS 



considered that all deviations from the exponent of pressure 
of 0-5 could be attributed to volume effects — some of the 
diffusing substance was associated with the impurities (often 
carbon, and possibly nitrogen or oxygen) in atom pairs, or as 
ions H2+. They pointed out that SmitheUs and Ransley's 
materials had not been decarburised, and attributed the 
deviations from a ^Jp law to this. 



0-6 


























1 








































^ 


y 






\ 






















/ 




























y 


' 














\ 










n-K 


y^\ 
















^ 


\. 









500 



ACQ 500 

Temperature, "C. 



06 



<:, 



0-5 

















~ 






















rr<~^ 














i. 










/ 


y 




\ 




















/ 








\ 


















/\ 










d 












^ 


1 








_ 


\v 


A^ 







500 



400 300 

Temperature °C. 



Fig. 58. Exponents n in expression permeation rate = A;jp"(H2-Ni) 
for two different samples of Ni. 



However, it is likely that these authors were deahng with 
two different phenomena. A satisfactory explanation of 
Figs. 54-57 may be given in terms of phase-boundary pro- 
cesses both at the ingoing and outgoing surfaces (53). Adopting 
the nomenclature of p. 170, one may suppose that the trans- 
ference of an atom from both 

(i) adsorbed layer to solid at the ingoing surface, 
(ii) solid to adsorbed layer at the outgoing surface, 
is comparable in speed with diffusion in the solid. Then much 
of the material transferred across the ingoing interface is 
removed by diffusion, and 

Actual (oCjj)< Equilibrium (oCjj), 



INFLUENCE OP PRESSURE 175 

as defined by equation (i) p. 170. At the outgoing interface, 
the slow rate of transfer across the interface, but considerable 
diffusion velocity in the solid, leads to an accumulation just 
inside the outgoing surface: 

Actual (iC^) > Equilibrium dC^). 

Thus the actual concentration gradient is smaller than the 
concentration gradient assumed in equation (v), p. 170. The 
dependence of this gradient upon the pressure is complex, but 
the curve of the permeation rate plotted against the pressure p 
must pass through the origin, because when p = both qC^ 
and iC^ are zero, and therefore the permeation rate is zero. 
At intermediate pressures, the permeation rate depends upon 
a complex function of pressure (see the feet of the curves of 
Figs. 54-57), but as the pressure rises, or at high temperatures, 
experiment shows that 

oCn-iCji = k^p (Figs. 54.-57). 

The universality of this relation must have important im- 
plications for the actual phase -boundary processes. This point 
will be discussed in more detail on pp. 178 et seq. 

Permeation velocities at high pressures 

Wiistner's (75) studies upon the hydrogen permeability of sUica 
glass (Chap. Ill) showed that even up to 800 atm. the per- 
meation rate-pressure isotherm was linear: 

Rate = A;f e-^/2^. 

A few comparable studies have been made upon the per- 
meabilities of metals to oxygen and hydrogen. Smithells 
and Ransley(76) observed that at pressures of 112 atm. the 
permeation rate-pressure isotherm obeyed accurately the 
Richardson equation: 

nate = k^ Ti e-'^l'^ . 

V 

Their data are shown in Fig. 59. A similar result was obtained 
by Lombard and Eichner(69) at pressures of 26 kg. /cm. ^ for 



176 



GAS FLOW THROUGH METALS 



the Ha-Pd system, and by Borelius and Lindblom for H2-Fe 
at 28 atm. The system Og-Ni, on the other hand, gave a 
limiting permeation velocity at high pressures (76) (Fig. 60). 
In this case a visible film of oxide forms on the surface of the 
metal, and it is likely that in the presence of the solid oxide 



f ^2 

o 



S4 



afl> 



^ 5 iO 

v'P (Atmospheres) 
Fig. 59. Diffusion of hydrogen through nickel at 248° C. 



*^ 1 

CO *■ 



o 

a /^' 2 



10 



V'P (Mm.) 



Fig. 60. Diffusion of oxygen through nickel at 900° C. 



the concentration of dissolved oxide (or oxygen) in the nickel 
in contact with the oxide film had reached a saturation value, 
for this would lead to a limiting permeation rate, according 
to Fick's law 



Rate of permeation = —D 



I 



where C^ is the saturation oxygen concentration at a; = 0, and 
Ci its value at the outgoing surface x = I. The contrary must be 



PERMEATION VELOCITIES AT HIGH PRESSURES 177 

true of the systems Hg-SiOg, Hg-Ni, Hg-Pd. The fact that many 
interface processes may occur in the diffusion can, as is shown 
subsequently (p. 182), lead to the conclusion that in the 
stationary state of flow concentrations within the solid may 
be very much less than their values in equiUbrium systems. 
Among the possible phase-boundary processes one might 
include the following: An adsorbed gas molecule or atom is 
driven into the solid by molecular bombardment by an 
activated gas molecule. Smithells and Ransley tested this 
possibility by introducing argon at 100 atm. into a diffusion 
system Ha-Ni where the hydrogen pressure on the ingoing side 

(T)-(2) Molecule (TWT) ^lo^^^^ule 
/ • \ (S) Adsorbed atom 



(ii) 



Metal Metal 

(T)-(D Molecule 



1; Adsorbed atom 



/////////M //////7///(^// 

Metal Metal ^^issolved atom 

Fig. 61. Models of possible phase- boundary processes. 



was 4 atm. No difference in the permeation velocity was 
observed, so that at least certain types of penetration process 
by molecular bombardment do not occur. The experiment does 
not eliminate the two types of penetration by bombardment 
represented by the diagrams above (Fig. 61 ). The gas molecules 
are supposed to be hydrogen, which are adsorbed in atomic 
form. To distinguish the atoms they have been numbered. 

Other diffusion systems in which conditions might corre- 
spond to very high pressures indeed are met with in the 
passage of nascent hydrogen through metals. For example, 
Barrer(53) found that hydrogen supplied at a current density 
of 0-44 amp. /cm. 2, by electrolysing sulphuric acid at 20° C, 
passed through a hollow palladium cathode lO^-fold as rapidly 
as hydrogen diffused from molecular hydrogen gas through 



178 GAS FLOW THROUGH METALS 

a sample of palladium in a similar state of activity. If the 
phenomena encountered are essentially the same^ then 



P 



elect. / i^elect. 



-'^thermal N i^thermal 

and as i?thermai — ^ ^m., Select ~ 10* atm. While in some ways 
the analogy may be false, there is no doubt that the pressures 
or concentrations of hydrogen atoms at the surface of the 
metal were extremely great. Similarly, Borelius and Lind- 
blom (52) found that if hydrogen were generated electrolytically 
at the surface of an iron tube some of it passed through the 
tube and the rate of permeation P was given by 
P^kyi-^It^e-^l^T^ 

and was completely analogous to the expression for thermal 
diffusion P = Kyp-^p,-\e-^I^T, 

even the temperature coefficients E being the same. Assuming 
the complete analogy, Borelius and Lindblom. found from 
their data _ -. „ ^^^ j 

i'(atmos.) ~ ^ ''^^^^(amp./cm^j- 
Their own experiments were conducted at current densities 
of up to 0-043 amp. /cm. ^ By carrying out these experiments 
at large current densities, it should be possible to reach a 
current density at which the surface layers of metal were 
completely saturated. The permeation velocity should then 
no longer increase as the current density increases. This point 
has been tested (53), and has important implications in the 
discussion on mechanisms of diffusion in the next section. 
It was found that only in very active palladium tubes could 
a limiting permeation velocity be reached (see Fig. 62). 

Some mechanisms for the process of flow 

In 1935, Ham (77) proposed a specific equation for the rate 
of flow of gas through a metal based upon kinetic theory. 
Ham's equation approximates to 



t-^m-^-"i^^ 



DETAILED MECHANISMS OF PROCESS OF FLOW 179 

where A, y, a and b are constants, and ^~|. It will be seen 
that this equation is similar in form to the Richardson 
equation, but since it involves a number of assumptions it 
need not be considered further. 

Riemann(78) has suggested another variant of Richardson's 
equation based upon thermodynamic considerations and 
Tick's law. His equation took the form 

In this equation a = Cp/R {Cp denotes the atomic heat of 
dissolved gas atoms), Pi,P2 = pressures of gas at the ingoing 
and outgoing surfaces respectively of a plate of thickness I, 
and AHq is defined by AH = zl^o + (|- 2a) ET {AH denotes 
the heat of desorption of two atoms of gas from the metal to 
the gas, as a molecule). But since we are dealing with an 
activated diEfusion, D can be further written asD = D^e^^^^^', 
where E denotes the activation energy for diffusion. 

Riemann's equation, Richardson's equation (p. 167) and 
equation (v), p. 170 all suffer from one grave defect. It has 
been assumed that the concentrations just inside the sohd are 
equilibrium concentrations, and this is only true where there 
are no rate- controlling phase-boundary processes. There do 
in fact seem to be no slow phase-boundary processes for 
diffusion through rubbers (79), but this is not so for diffusion 
through metals. Melville and Ridea] (64) made the first attempt 
to include possible phase-boundary processes in the equation 
of flow. Other possible phase-boundary processes were given 
by Smithells and Ransley(76), Wang(80), and Barrer(8i). They 
regarded the following as possible: 

(i) An adsorbed atom passes into the metal: 
Rate = y^i^il-^) 

{kj^ is the velocity constant of the reaction, 6^ the fraction of 
the surface covered by adsorbed atoms, C^ is the concentration 



180 GAS FLOW THROUGH METALS 

of hydrogen gas just inside the metal, and Cg is the concen- 
tration of hydrogen in the metal when saturated with the gas). 

(ii) A dissolved atom re-enters the surface: 

Rate = k^C^{l-d^). 

(iii) A molecule strikes the surface and is adsorbed as atoms : 

Rate = ^'3^H.(l-'^l)^• 
(iv) Two adsorbed atoms evaporate as a molecule: 
Rate = k^d\. 

(v) A molecule strikes the surface, one atom being absorbed 
and the other adsorbed: 



Rate = A:5i>H.(l-^i)(l-^) 



(vi) An absorbed atom combines with an adsorbed atom, 
and evaporates as a molecule: 

Rate = ^6 ^'h 'bl- 
ether processes may be conceived, but they are not probable 
ones, and in any case the six listed above are adequate for 
this discussion. Smithells and Ransley, considering the first 
four of the preceding rate equations, deduced as a general 
expression for the velocity of permeation 

^ = ^{V[i+^/M-i}, 

where A and K are terms involving 6, and considered that a 
simple form of this equation 

^ = .4{V(1 + A'^i>)-1} 

agreed with their experimental findings that at low pressures 
djpjdt approximates to Bp, where 5 is a constant; and at high 
pressures to C^^, where C is also a constant. 

Wang (80) and Barrer(8i) gave equations defining the per- 
meation velocity which included all six of the processes above. 



DETAILED MECHANISMS OF PROCESS OF FLOW 181 

Wang's treatment led him to conclude that the permeation 
velocity would increase indefinitely with pressure, according 
to a ^J'p law when p is large. This would explain Smithells 
and Ransley's observation that even at 100 atm. the rate 
of permeation of hydrogen through nickel was still propor- 
tional to the square root of the pressure. At these pressures 
it had previously been supposed that the surface layer would 
be saturated with adsorbed gas, and so the processes (i) to (iv) 
would give a Kmiting value for the permeabiUty. Wang 
considered that the inclusion of (v) and (vi), however, made 
this no longer necessary, and so his equations indicated 
that some such processes as (v) and (vi) must have occurred. 
However, Barrer(8i) pointed out that Wang reached these 
conclusions because the rate equations given by him and by 
Smithells and Ransley took no account of the approach 
towards saturation in the metal itself. The rate equations (i) 
and (v) were in fact written by them as 

(i) Rate = k^d^. 

(v) Rate = k^]}{\-d-i). 

The correct equations for the permeability showed that it 
always reached a hmiting value at infinite pressure. Barrer (82) 
also showed that the phase-boundary processes could result 
in very great concentration discontinuities at the surface of 
a metal, and that under these conditions, even at high pres- 
sures, the concentration just within the metal could be a 
fraction only of the equilibrium value found in the absence of 
phase-boundary processes. He advanced the view that in the 
expression 

Permeation rate = D—^ — ^, 

the concentration C^ just inside the ingoing surface was even 
at 100 atm., for the Hg-Ni system, nowhere near its saturation 
value, so that Wang's deduction of a ^^'p law is valid at these 
pressures. The manner in which concentration discontinuities 
arise is indicated on pp. 174-5. 



182 



GAS FLOW THROUGH METALS 



. Later (53), it was shown that for certain Hg-Pd systems 
(Cj — C2) was indeed much less than its equihbrium value. This 
was indicated by measuring both the permeability constant and 
the dififusion constant (Chap. V). When gas flow occurred from 
a finite pressure through palladium into a vacuum the values 
of (C*i — C2) were those in Table 43, for palladium samples of 
low permeability. On the other hand, when a palladium 
sample was alternately oxidised and reduced the permeabihty 
was high, and for some measurements of Lombard and his co- 
workers (69) the values of (C^ — C2) approached the equilibrium 
ones (Table 43). Here the phase-boundary processes have 

Table 43 



T°C. 


D 

cm.^sec.-^ 
xlO-'^ 


Pi 

(Barrer) 

c.c./sec./cm.^/ 

mm./cm.Hg 

xlO-* 


(C1-C2) 
(Barrer) 
at 1 atm. 
pressure 
c.c./c.c.Pd 


PjCestrap.) 

(Lombard) 

c.c./sec.,'cm.-/ 

mm./cm.Hg 

xlO- = 


(C1-C2) 

(Lombard) 

c.c./c.c.Pd/ 

atm. 


(equilibrium) 

c.c./c.c.Pd/ 

atm. 


' 350 
334 
310 
272 


6-8 
5-4 
3-7 
2-0 


0-65 
0-52 
0-366 
0193 


0-73 
0-73 
0-75 
0-74 


0-84 
0-78 
0-70 


11-8 
12-6 
16-5 


12-0 
12-5 
13-6 



accelerated so much that diffusion and equilibrium solubility 
at the interfaces control the permeation velocity. Finally, 
in support of this view Barrer (53) found that the permeation 
rate through a very active palladium sample was independent 
of the current density, but that as the activity decreased it 
became dependent upon current density, approaching more 
and more nearly to the relation 

Rate = k^I 

found for inactive samples (Fig. 62). 

Earlier attempts were made to fix the nature of the rate- 
controlhng process by noting the type of kinetic expression 
followed(83, 84, 85). Wagner (83) claimed to have found in an 
Hg-Pd system that various laws were valid under different 
conditions : 



(i) 



dC 

dt 



D-^ (difFi;sion). 



DETAILED MECHANISMS OF PEOCESS OF FLOW 183 



dC 
(ii) -^ = k' ^JJp^^ — k-^G (jDenetration of atoms into the 

metal). 

dC 
(iii) -^ = ^"jjg^ — A^aC'^ (sorption of hydrogen as molecules). 

Mechanisms based upon kinetic expressions are to be 
accepted with some reserve. 



6-0 



a 4-0 



2-0 







O- 



® Series I 



Series II , 




1-395 



0-465 0-93 

^/(/ = current in amps.) 

Fig. 62. Influence of current density upon diffusion of Hg through Pd in varying 
stages of activity. Series I, most active; Series II, less active; Series III, 
still less active. 



The BEHAVIOUR OF HYDROGEN ISOTOPES IN DIFFUSION 
AND SOLUTION IN METALS 

There are several reasons why hydrogen and deuterium should 
react at different velocities: 

(1) They have different masses. Kinetic theory leads one 
to expect that on this account the reaction velocity constants 

k 
should be in the ratio t^' = J'2. 
kr^. 



184 GAS FLOW THROUGH METALS 

(2) They have different zero point energies. Hydrogen has 
the greater zero-point energy, and so should not need so large 
an additional activation energy before passing over a given 
energy barrier as would deuterium. The effect of these 
differences may be expressed by an exponential term q-'^eirt^ 
where zlE' is a small energy increment. 

(3) According to quantum theory there exists a finite 
probability that the atoms H or D may pass through an energy 
barrier without having sufficient energy to surmount it. The 
mass appears in a negative exponential term, and so may 
result in very great velocity differences, hydrogen always 
reacting more rapidly. It may here be said that hitherto no 
velocity difference has been great enough to suggest that this 
quantum mechanical leakage occurs (86). 

In activated diffusions as in chemical reactions a component 
of the system passes over an energy barrier, or succession of 
energy barriers, in passing from its initial to its final state. 
Thus the same phenomena which govern the one process 
govern the other also. No difference in the diffusion velocities 
of hydrogen and deuterium is large enough for it to be necessary 
to assume a quantum mechanical leakage. The processes 
involving (1) and (2) remain to be considered. In their earliest 
publication on the subject, A. Farkas and L. Farkas(87) 
reported a permeation velocity ratio for hydrogen and 
deuterium which could be expressed as 

P 

Subsequently, however, Jost and Widmann(88) measured the 
diffusion velocity of hydrogen and deuterium inside the 
palladium lattice, and found 

^302-5°C ^ 1-35 
2)302.5°C- 1 ' . 

which was nearly the value ^J2 indicated by simple kinetic 
theory. Further, Jouan(89) stated that the permeation rate 
ratio of hydrogen and deuterium through platinum was 



BEHAVIOUR OF HYDROGEN ISOTOPES 185 

constant at approximately .^2 : 1 = Pg, : P^, from 550 to 
950° C. However, a re -examination of these authors' results 
shows that this is by no means so, the mean ratio at a series 
of temperatures being : 



T°C. 550 650 750 


850 


950 


Ph^/Pd^ 1-55 1-50 1-35 


1-36 


1-27 


These ratios conform to the expression 






P 

^ Ho _ ^680/i2r 







with some accuracy. A. rarkas(90) reinvestigated the per- 
meabiUty of iDalladium to hydrogen isotopes, using the half- 
life of diffusion as a measure of the permeation velocity. He 
also used the half -life of the process of conversion at the 
surface of para-hydrogen to equilibrium hydrogen to measure 
the velocity of sorption or desorption of hydrogen at the 
surface. It must be remembered, however, that the solution 
of the diffusion equation is in the form of an infinite series 
of exponentials, and thus the "half -life period" does not 
accurately measure the velocity of diffusion* and is, moreover, 
a function of the thickness. Also, the conversion to the 
equilibrium mixture of para-hydrogen need not occur by 
sorption and desorption of an atomic layer of hydrogen, but 
may result from an exchange reaction between a molecule 
and an atom in the hydride layer. However, the ratios of 
the half -life periods for diffusion of hydrogen and deuterium 
depend on temperature, and are often too large for ex- 
planations involving only a factor ^^2. These velocity ratios 
depend at low temperatures in a somewhat capricious manner 
upon the history of the palladium specimen. 

A convincing proof that an exponential factor e~^^'^^ 
governs the permeation rate ratio is provided by the measure- 

* As it does for a first order reaction, where, since C = C^e'^^, one has 

C 
In — " = kt, and kh=]n2 

C - ' 

if t^ denotes the half-life period of the reaction. 



186 



GAS FLOW THROUGH METALS 



ments of Melville and Rideal (64) on the diffusion of the isotopes 
through palladium, using tubes and disks of palladium. Their 
data are presented in the following tables: 

Table 44. Area Pd tube 1-51 cm.^, thickness 0-1 mm. 
Volume of system, 41 c.c. 



T°C. 


Gas 


Quarter- 
. life of 
permeation 
(min.) 


Half-life 
of per- 
meation 
(min.) 


Three- 
quarter- 
life of 
permeation 
(min.) 


Ratio 
H/D 


AE 
(kg.eal.) 


228 
281 
322 
362 


H 
D 

H 
D 

H 
D 

H 
D 


16-7 
30 

2-55 
4-40 

1-40 

2-52 

0-65 
0-80 


42-2 
79 

8-0 
12-7 

40 

6-6 

1-76 
2-30 


18-4 
29-0 

8-95 
15-1 

3-95 
4-95 


1-80 
1-58 
1-73 
1-26 


0-76 
0-68 
0-86 
0-4 



Table 45. Area Pd disk 0-78 cm.'^, thickness 0-075 mm. 
Volume of system 44- 1 c.c. 



T°C. 


Gas 


Half-life of per- 
meation process 
(min.) 


Ratio 
H/D 


AE 
(kg.eal.) 


1.54 
167 
189 


H 
D 

H 
D 

H 
D 


3-30 

8-0 

2-31 
4-41 

215 

3-75 


2-42 
1-90 
1-75 


104 

0-90 
0-83 



Farkas (90), in another set of data, gave the ratios as 
T°C. 186 131 106 20 

Pjj/Pd 1-24 1-36 1-40 1-84 

which conform satisfactorily to the relationship 
Ph/Pd = 0-6e66o//?T_ 

Data have also been obtained (04) for composite membranes 
in which copper or nickel were deposited electrolytically upon 
palladium. The deposited films were very thin, about 10"^ cm. 



BEHAVIOUR OF HYDROGEIST ISOTOPES 187 

thick, and the possibility of there being holes in the copper 
membrane was checked by measuring the apparent activation 
energy, which is different for copper and for palladium. As 
a measure of the permeation velocity the time required for a 
given pressure drop was used — a procedure which may give 
only an approximate measure of the actual permeation rate 
process. The deposition of copper films reduced the per- 
meation rate by a large factor; the mean AE for the process 
was, for a series of measurements involving membranes 
Cu-Pd-Cu and Pd-Cu-Pd, 770 cal. Analogous measurements 
for Ni-Pd membranes gave a zl^ of 600 cal. It is interesting 
that the deposition of a copper layer 8-4 x 10^^ cm. thick, on 
one side of the palladium, gave a velocity twice that observed 
when a copper film 8-4 x 10~^ cm. thick was also deposited on 
the other side of the disk. Therefore the palladium did not 
appreciably affect the characteristics of the copper membranes', 
which alone governed the -velocity of gas permeation. 

Since the permeability constant depends on the diffusion 
constant, upon phase boundary processes and upon the solu- 
bility I P = — D — I , it is not possible to interpret these 

results in any exact way. The difference in cal./mol. of the 
temperature coefficients for hydrogen and deuterium may 
depend on the following factors: 

(i) The activation energy for adsorption and for desorption. 

(ii) The activation energies for penetration from the ad- 
sorbed layer into the sohd, and the converse process. 

(iii) The activation energy for diffusion in the metal lattice. 

(iv) The heat of solution of the gas in the lattice. 

Even when all phase-boundary processes occur much more 
rapidly than any other processes, the temperature coefficients 
of the permeability constants are governed by the terms (iii) 
and (iv) above. But one notes the general correspondence 
between the differences AE in the temperature coefficients of 
permeability of metals to hydrogen and deuterium, and 



188 GAS FLOW THROUGH METALS 

between differences in the temperature coefficients of a variety 
of heterogeneous reaction velocity constants (Table 46). 

There is also an agreement in the order of magnitude of 
the A E observed in diffusion and in heterogeneous reactions, 
and the AE calculated by Sherman (95) as the difference in 
zero-point energy of a large number of oscillators such as 

Table 46. Some differences in temperature coefficients for 
heterogeneous processes involving H and D 



Reaction 


-E?(apparent) ^ _ -/^EIRT 

cal./mol. kj) 


Author 


Ha + 2C(aolid)->2CH(surface) 
Da + 2C(eoIid>^'2CD(aurface) 

Tungsten 
2NH3 ^N2 + 3H2 

filament 

Tungsten 
2ND3 ^Na + SD, 

filament 

Tungsten 
2PH3 ^2P(soiid) +3Ha 

filament 

Tungsten 
2PD3 >2P(aoiid) +3H2 

filament 

HaO + Al4C3^4Al(OH)3 + 3CH4 
D2O + Al4G3^4Al(0D)3 + 3CD4 

CH^ + 3C(solid)->4CH(aurtaoe) 
CD4->3C(solid) + 4CD(8urtace) 

Ha + CuO^Cu+HaO 
Da+CuO-^Cu+DaO 

Hydrogenation of styrol 

(Pd-BaSOi catalyst) 
Ha, Da + iOo(Ni)->HaO, DgO 
Ha, Da + Na6(Ni)->HaO, DaO + Ng 
Ha, Da + C2H4(Ni)^C2H6, CaH^Da 


^ 15,700 
42,400 

32,200 

14,200 
^26,700 


Zl£ = 700cal. 

800, 790, 
890, 900 

510, 550 

750 

780 
400 

540 

750 
720 
700 


BarrerOD 
Barrer(92) 

Barrel (92) 

Barrer(S6) 

BarrerOi) 

Alelville and 
Rideal(64) 

Cremer and 
Polanyi(93) 
Melville (94) 
Melville (94) 
Melville (94) 



Ni-H,Ni-D {AE = 0-7 k.cal.)orPt-H, Pt-D {AE = 0-5k.cal.). 
The calculations are only approximate; they assume that 
the metal atom behaves as though it were independent 
of all other lattice atoms, save only the hydrogen atom; 
and that the same is true of its associated hydrogen atom. 
They also assume that the oscillation is a simple harmonic 
motion. 



BEHAVIOUR OF HYDROGEN ISOTOPES 189 

The solubility of the isotopes in palladium has been made 
the subject of an experimental study by Sieverts and his co- 
workers (21), and shows that in an equilibrium system there 
are considerable solubihty differences, Sieverts and Danz's (2i-) 
results are shown in Fig. 40. The higher temperature results, 
where Nernst's distribution law, S = fc-^/p, may be expected 
to hold {S denotes the solubility and A; is a constant), lead to 
the following solubihty ratios at one atmosphere pressure : 



T°C. 


200 


220 


240 


260 


280 


300 


320 


340 


350 


400 


^d/'S'h 


0-60 


0-63 


0-64 


0-67 


0-685 


0-68 


0-675 


0-71 


0-71 


0-74 



The heat of solution per g.mol. of hydrogen dissolved is, in 
the range 220-350° C, 2200 cal., while the heat of solution of 
deuterium is 1760 cal. /mol., using the ratio S-^IS-^ = q'HOIBt 
calculated from the above data. Melville and Rideal in an 
analogous temperature range obtained 

AHj^^ = 2500cal./mol., 

Zl^H= - ^^D, = ^^40 cal./mol. 

It 'is then clear that interpretation of relative permeation 
velocities may have to take account of this large difference in 
solubility. In another study (37) the solubihties of hydrogen 
and deuterium were measured in iron, giving results of which 
the following are typical: 



T°C. 


600 


700 


800 


950 


1000 


1200 


1350 


1450 


Sn, 


1-8 


2-4 


(3-2) 


5-9 


6-6 


9-0 


(10-8) 


12-6 


(c.c./lOO g. Fe) 


















(C.c./lOO g. Fe) 


1-5 


2-2 


2-9 


5-5 


6-1 


8-5 


10-0 


11-7 


















SjlJSr,, 


1-20 


1-09 


MO 


1-07 


1-08 


1-06 


1-08 


1-08 



A theoretical study of the solubility of the isotopes in 
palladium (22) led to the expression 

where AH denotes the heat of absorption of gaseous hydrogen 
in palladium, Xo is the heat of solution of a hydrogen atom 
in the lattice, Xd ^^e heat of dissociation of a hydrogen 



190 GAS PLOW THROUGH METALS 

molecule into atoms, and V{T) is the j)artition function of a 
dissolved atom. The subscripts H and D refer to hydrogen and 
deuterium respectively. The difference in the zero-point 
energies of the isotopes in solution may be calculated if the 
vibration frequency v is known. Assuming 
j^H = 8-75x10-12 sec.-i, 

and that — — Hn^ , one finds 



Mj, 



and 



%?-A:? = -470cal. 
ET^^log^^^ = +260cal. {T - 1200°K.). 



An additional assumption in the latter calculation is that 

V{T) = {1-6-^"/^^)-^. 
Finally one knows, from an argument analogous to that for 
computing (;\'?-%?), that Ux^-Xd) = 900 cal., and so 
AH^-AHj)^ -470 + 900 + 260 = 690 cal. 

Lacher found he could express Sieverts and Zapf's(2i) data on 
the solubility of the isotopes in palladium at high tempera- 
tures as 1 1 KA 

\ogpl, = log ^H - -^jT + log 129, 

410 
\ogp\y = log ^D - -^ + log 104 

{p in atmospheres; 6 = fraction of saturation hydrogen 
content). There is then a good agreement between the experi- 
mental and calculated heat difference. Although fitting 
equations of the type given above to experimental data is 
prone to small errors, these are greater in the temperature - 
independent than in the temperature-dependent term. 

One can thus see that many factors control the relative 
behaviour of the isotopes in solution, diffusion, and permeation 
in metals. The solubility ratio combines the effects indicated 
above ; the diffusion constant ratio involves differences in the 
zero -point energies of dissolved hydrogen and deuterium 



BEHAVIOUR OF HYDROGEN ISOTOPES 191 

atoms in initial and transition states respectively; and the 
permeability constant ratio combines the effects of differing 
diffusion constants, and differing solubilities, or, if phase- 
boundary processes are also important, differences in the 
relative rates of reaction such as (i) to (vi) on pp. 179 and 180. 
One may express these different possibilities in a single 
potential-energy distance diagram, in which a molecule is 
supposed to approach a membrane, be adsorbed, penetrate 
into the solid, diffuse through it, emerge into the adsorption 
layer on the other side, and be desorbed into the gas phase 
from this layer. The heights of the various energy barriers 
may vary greatly from case to case. 

The estfluekce op phase changes fpon permeability 

It has been found that in the region of concentrations where 
two hydrogen-palladium alloys coexist (the perpendicular 
sections of the curves of Fig. 40) it is not possible to trace out 
the same isobaric curve on sorption and desorption. There is 
a hysteresis effect illustrated by Fig. 40. It is interesting to 
find in the study by Lombard, Eichner and Albert (96) that the 
permeability -temperature curve follows m an inverse manner 
the absorption isobar, there being a great increase in the 
permeability in the region 180-200° C. .It may be that this 
rapid alteration in permeability marks the change from 
^-phase to a-phase alloy. 

The same type of hysteresis loop which was noted in the 
absorption of hydrogen in palladium, where a- and y^-phases 
co-exist, is observed also in the absorption of hydrogen by 
iron (31), where now the two phases are produced by allotropy 
in the metal. The permeability-temperature curve shows a 
break at this point (Fig. 63) (5i). Indeed, wherever a phase 
change occurs one may look for a variation in the permeability, 
so that the property of permeability may be used to determine 
transition points. The change in permeability of nickel towards 
hydrogen has similarly been used to characterise the Curie 
point in nickel (62). 



192 



GAS FLOW THROUGH METALS 



When the phase changes are brought about by alloying two 
metals, one would anticipate that in the same way the per- 
meability would vary with the composition of the alloy, and 
discontinuously so wherever a new phase is formed. Evidence 
upon these points is scanty, but the data of Baukloh and 
Kayser(97) indicate that discontinuities do exist in the 
permeability-composition curve of alloys of nickel with copper 
and with iron (Fig. 64). 

xJO'- 



W ■ 




o; 









/ 






600'C. 


/ 






1 






> 


/ 






/ 


650°C. 


^ 






y" 





20 40 60 
Per cent /Nickel 



60 



Temperature (in mi ///volts) 

Fig. 63, Fig. 64. 

Fig. 63. The permeability curve for iron. 

Fig. 64. Effect of composition upon rate of Hg-iDermeation through Ni-Cu alloy. 



The influence of pre-treatment upon permeability 

A metallic crystal, like a salt crystal, normally consists of a 
mosaic of small crystallites — dendritic, block -like, or columnar. 
Impurities exist in part at surfaces of separation of the 
crystallite components, and in part in true solution. Per- 
meability is, like diffusion, a structure sensitive property and 
one is accordingly likely to find permeation anisotropy 
in aleotropic single crystals ; diminution in permeability with 
growth of crystallites following annealing, if grain boundary 
diffusion occurs; effects due to impurities in the metal; and 
sejisitivity to the state of the metal surface, its roughness, 
degree of oxidation, or the extent of foreign metallic films. 



INFLUENCE OF PRB-TRE ATMENT 



193 



The experimental difficulty of preparing and mounting 
metallic single crystals has so far prevented the discovery 
of any diffusion anisotropy in metallic systems, with few 
exceptions such as bismuth (Chap. VI). Effects which may be 
due to any of the other possibilities are more frequently 
encountered, and in a few instances have been systematically 
studied. Some of this e^^idence may be discussed. 




10 IS 

Time in hours 



ZO 



Fig. 65. Hg-permeability of nickel as a function of time of heating. 



Baukloh and Kayser(97) showed that while the hydrogen 
permeabiMty of nickel was not affected by prolonged heating 
at temperatures of 600° C. or even higher, at temperatures of 
950-1050° C. a decrease in permeabihty followed. The influence 
of heating upon the metal is not the same, at a given tem- 
perature, for all samples, as their figures show. One such 
diagram is given in Fig. 65 and illustrates a rather extreme 
case, where a diminution can be observed even at 680° C. These 
effects could be attributed either to movements of impurities, 
or to recrystaUisation with a diminution in the amount of 
grain boundary diffusion. Ham and Sauter (73, 74) also made the 
observation that the permeability of both palladium and iron 
was very much affected by the heating given the metal. 



13 



194 GAS FLOW THROUGH METALS 

One of the most complete investigations upon the influence 
of heating upon permeability has been carried out by Lombard 
and his co-workers (90). Their observations upon the per- 
meabiUty of hydrogen to palladium led them to the conclusion 
that it was possible to reduce the permeabihty of palladium 
100-fold or more by heating the metal. This loss in the per- 
meability of pure palladium towards hydrogen, which occurred 
jDrogressively and the more rapidly the higher the temperature, 
was irreversible, and could amount almost to a total loss. By 
heating palladium to 500-520° C. for specified periods and 
then cooling it to temperatures below 450-500° C, any steady 
state of permeabihty could be attained. Heating the metal 
in air at 500° C, and then cooling it in air and subsequently 
reducing it in hydrogen at 150° C. partly if not whoUy restored 
the permeability; but oxidising the metal at 500° C. and 
reducing it again at this temperature failed to increase its 
permeability . The authors found that a given sheet of palladium 
could be regenerated a number of times by oxidation and 
reduction, and pointed out that this behaviour recalls the 
preparation of metaUic catalysts for hydrogenation, and 
focuses attention upon the state of the surface of the mem- 
brane. The loss of permeability was ascribed to a process of 
agglomeration of fine particles at the palladium surface, with 
a consequent reduction of the surface, and decrease in ease of 
access of hydrogen to the lattice of the palladium. This 
temperature of agglomeration was considered to be in the 
vicinity of 500° C. for pure palladium, and to be retarded by 
some impurities and accelerated by others. One membrane 
lost its diffusing power at 315° C. Experiments by Barrer(53) 
also revealed a great diminution in the permeability of 
palladium due to long heating at temperatures between 
270 and 360° C. , the palladium at these temperatures tending to 
approach a final steady state. Thus two states of permeability 
may be possible for palladium, one when the metal surface is 
activated by suitable oxidation and reduction, and one when 
the metal has undergone the maximum crystallisation at the 
surface. In the former state surface processes have their 



INFLUENCE OF PRE-TRE ATMENT 



195 



minimum effect on the diffusion velocity; in the latter they 
have their maximum. 

In Fig. 66 is illustrated the effect of prolonged heating upon 
the permeability of palladium for a number of heating and 
cooling cycles, with regeneration of the palladium between 
the cycles. In the first cycle the metal was maintained at a 
constant temperature, and its permeability fell steadily; in 
other cycles the temperature was raised or lowered and the 



50 



^40 



30 



20 



JOG 200 300 400 500 600 700 
Temperature in °C. 

Fig. 66. Effects of Heat-treatment upon the permeability of palladium (96). 



Jill 




1 J 1 1 


1 1 1 1 


1 M 1 


MIL 


: 






/ 
/ 
■A 




/ 1 
- / 1 
■■" 1 


ii. - 


: 




/ 


/ 

/ 




1 
, 1 


// z. 


; 


/ 


/ 


' 


^ 




f 


: 


/ 
/ 
/ .■ 


X 






' 


/ : 


1 KM--^ 


/ / 
/.-••■ 

•| 1 M 


1 1 1 1 


MM 




i 1 1 


1 1 . r 



permeabilities were measured simultaneously. The different 
permeabihties which palladium may possess have usually 
different temperature coefficients. This may be illustrated by 
Table 47. 

Smithells and Ra,nsley(67) made a study of the effects of 
oxidation, reduction, polishing, and etching upon the per- 
meability of nickel and iron. PoUshed nickel membranes were 
less permeable than oxidised and reduced nickel. Etching 
increased the permeability of iron more than oxidation and 
reduction, but oxidation without adequate reduction of the 
iron poisoned it, and rendered it impermeable to hydrogen. 
Table 47a gives the data obtained. Another metal, the per- 
meability of which is sensitive to surface treatments, is 
aluminium, also studied by Smithells and Ransley. The 
apparatus was so arranged that the aluminium could be 

13-2 



196 



GAS FLOW THROUGH METALS 



scratched with a steel brush without exposing it to air. Results 
of a number of the experiments are shown in Fig. 67. The 
effects are not simple; oxidation reduced the permeability; 
scratching the surface appeared to reduce the permeability 
when only the outside was abraded; but increased the per- 
meability when the inside was abraded as well. The most 

Table' 47 . The per7neabilities for a number of 
palladium membranes 



Sample 


Permeability 
c.c./cm.2/hr./atm. 


Condition of sample 


1 

Average for a 
number of samples 

3 

4 

1 


8-3 X 102 e-«2o/i?T| 
18T-i e-366o/J?r | 
10-7 X 102 e-'5oo/iJ2' 

9-5 X 102 e-i»5oo/Er 


High permeability (69) 

High permeability (96) 

Inactive (by heating) (96) 
Inactive (by heating) (53) 



Table 47a. Effect of surface treatment upon permeability 











Permeation rate 


Metal 


Treatment 


Temp. 
°K. 


Pressure 
mm. 


at this pressure 

c.c./sec./cm.2/mm. 

thick 


Ni 


Polished 


1023 


0-042 


1-39 X 10-6 




Oxidised and reduced 


1023 


0-042 


2-70 X 10-6 


Ni 


PoUshed 


1023 


0-091 


2-91 X 10-6 




Oxidised and reduced 


1023 


0-091 


4-23 X 10-6 


Fe 


Polished 


673 


0-77 


0-47 X 10-' 




Etched 


673 


0-77 


4-4 xlO-' 


Fe 


Pohshed 


863 


0-073 


1-28 X 10-' 




Oxidised and reduced 


863 


0-073 


0-76 X 10-' - 




at 600° C. 










Oxidised and reduced 


863 


0-073 


1-54x10-' 




at 800° C. 









important effect was the steady diminution in permeability 
with the time of heating, whatever the surface condition. This 
phenomenon may compare with the surface changes in 
palladium (96), or the effect of heat treatment of nickel (97, 98, 99). 
Sometimes the chemical treatment of a metal with a gas may 
increase its permeability to a second gas; for example, Ham 



INFLTJElSrCE OF PRE-TRE ATMENT 



197 



and Sauter noted tkat heating iron in nitrogen increased the 
velocity of permeation of hydrogen by 10 to 15-fold, although 
baking out reduced the permeabihty to its former value. 




3 

Hours 

Fig. 67. Effect of surface treatment on the permeability of 

aluminium to Hg at 580° C. 

Run 1. Al exposed to air. 

Run 2. Outer surface scratched. 

Run 3. Outer surface scratched again. 

Run 4. Outer surface scratched again. 

Run 5. Inner and outer surfaces scratched. 

Run 6. Outer surface anodically oxidised. 



GrAIN-BOUI^DARY and lattice PERMEATIOlSr 

Photo -micrographs may illustrate both the mosaic structure 
of metals, and a concentration of impurity between crystal 
blocks. If diffusion occurs mainly down these boundaries, the 
permeability of a metal would be governed by their number 
and nature. The mere fact that oxide is fouiid concentrated 
in these boundaries does not, however, necessarily imply that 
diffusion of oxygen occurs along them, since the oxide may 
be formed within the lattice and then be thrown out of solution 
in the zones between crystallites. That grain-boundary 
diffusion plays an important part in diffusion processes such 



198 



GAS FLOW THROUGH METALS 



as that of thorium in tungsten, of ions in microcrystaUine 
sodium chloride, or of gases through sihca glass is undoubted, 
and these and other examples have been considered elsewhere 
( Chaps. VII andlll ) . It is not easy, however, to establish similar 
cases when gases diffuse through metals. It is true that great 
variations in permeability may be encountered with different 
samples of metal (e.g. Hg-Pd systems), but these differences 
may also be attributed to the variable influence of phase- 
boundary processes. The evidence available casts doubt upon 
the special importance of grain boundary diffusion for 
hydrogen- nickel (97) and hydrogen-iron systems (lOO) at high 
temperatures. In both cases it was shown that crystal size had 
no marked effect upon the permeability. Edwards (55) found 
the same permeability towards hydrogen in a single crystal 
plate of iron before heating it, and after heating it to refine 
the grain. Smithells and Ransljey(i.c.) observed that a single 
crystal iron tube had the same permeability as a similar tube 
with 100 grains/mm. 2 (Table 48). Grain boundary diffusion is 

Table 48. Permeability to hydrogen of iron of 
different grain sizes 



Temp. 

°C. 


Pressure 
mm. 


Permeation rate 
c.c./sec./cm.7mm. 


Fine grain '. 
xlO-« 


Single crystal 
xlO-« 


245 
413 
621 

779 


140 
140 
140 
140 


2-4 

17-6 

92-8 

203-0 


1-2 

171 

89-5 

205-0 



more easily observed the lower the temperature, and thus 
experiments of this type should be conducted at the lowest 
possible temperature. 

The permeability of hydrogen-iron systems has received 
considerable attention at low as well as at high tempera- 
tures. Ham and Rast(ioo) studied the permeabihty from 55 to 
920° C. In addition to discontinuities in permeability at every 



grahst-boujstdary and lattice permeation 199 

thermal critical point, they observed a hysteresis loop in the 
permeabihty -temperature curve illustrated in Fig. 68. At 
room temperatures, when the diffusing atoms were supphed by 
electrolysis, Barrer(53) found that a bright steel tube was at 
first impermeable, but quickly became more and more per- 
meable to hydrogen in successive diffusion experiments. 
Poulter and Uffelman(ioi) found that hydrogen diffused 



iO 



i\ 


Ni 








\^4 
1 


1 


\_ 












\ 




tse'c 








\ 








o D 


■3 

onn 




M 












\1 


\mp 












V 



12 1-C 16 fa so B-2 S-i 



Fig. 68. The low- temperature permeability of hydrogen to iron(ioi), 
showing a typical hysteresis loop. 

through iron at room temperature, but not until a pressure of 
4000 atm. was applied. When the diffusion had thus been 
started the steel was permeable in a second run at only 100 atm. 
Some if not all of these observations point to an opening up of 
grain boundaries as diffusion proceeds. The grain structure 
can develop due to desorption between grains, of hydrogen 
which has diffused in the lattice. Thus the system may 
commence as one where lattice diffusion predominates (p. 198 
and Table 48), and end as one where grain boundary diffusion 
predominates. 



200 GAS FLOW THROUGH METALS 

Not only may diffusion occur through the lattice, and 
probably in certain conditions down grain boundaries, but 
also in each crystallite preferred directions of diffusion may 
be encountered. Smith and Derge( 102,103) prepared palladium 
specimens of various grain sizes and showed that grain size 
did not affect the sorptive capacity. They considered, from 
crystallographic evidence, and from the marked influence of 
deformation upon the velocity of uptake, that shp planes in 
each crystal grain are of special importance in absorption. 
Etching rolled foils previously charged with hydrogen revealed 
that fissures had developed inclined at 45° to the direction of 
roUing. It was thought that the fissures were caused by 
preferential penetration of hydrogen along slip planes which 
were dilated as hydrogen diffused into the lattice. 

Much more evidence of this kind is necessary before any 
generahsation concerning the relative importance of grain- 
boundary, slip-plane, or isotropic lattice diffusions may 
be made. 



Flow of nascent hydrogen through metals 

Atoms of hydrogen may be generated by electrolysis or by 
chemical reaction at the metal surface. Some of the atoms of 
hydrogen thus liberated may leave the surface as molecules 
and others may enter the metal and diffuse through it. The 
possible sequence of processes is 

■"-2 (gas) 

Except for the processes which liberate the atomic hydrogen, 
the reactions are the same as those occurring during thermal 
diffusion. Morris (i04) showed in agreement with the scheme 
above that some hydrogen gas, generated by the action of 
citric acid on iron, diffused through it, and some was evolved 
(Fig. 69). The fraction diffusing was greater the more rapid 
the rate of corrosion of the steel (the acid strength being 
constant). 



NASCENT HYDROGEN THROUGH METALS 



201 



The possible variables in this type of diffusion include 
current density, temperature, the dimensions of the mem- 
branes, concentration of acid, time, and the concentration of 
added salts. Bodenstein(ii) considered that his results on the 
diffusion of hydrogen through iron obeyed the relation 

Permeation velocity = k ^JI. 





10 20 30 40 
Hours 
Fig. 69. 



Of 0-2 

V7 1 / in amps per cm.') 
Fig. 70. 



Fig. 69. Volumes of hydrogen [simultaneously evolved and diffused by the 

action of citric acid on fast corroding steel. 
Fig. 70. The electrolytic diffusion of hydrogen through iron as a function of 

current density. Fe, 0'18%C. x = untreated. 0= annealed. 



Borehus and Lindblom(52), however, showed that over a- 
considerable range of current densities the relationship was 

Permeation velocity = k{^JI — aJI(), 

where ^JIl is a constant, called the threshold current density 
(Fig. 70). The relation is analogous to their equation for the 
thermal permeation rate (p. 171): 

Permeation velocity — k^i^jp — ^jpi)- 
The explanation that the rate of permeation can be expressed 
as P = kd ^JI and that the surface is not fully saturated at 
low- current densities is not more tenable than Smithells and 
Ransley's earlier explanation of the same facts for thermal 
diffusion (p. 171). The most satisfactory explanation is that 
advanced by Barrer (p. 174), 



202 



GAS FLOW THROUGH METALS 



The influence of temperature upon the permeation velocity 
of hydrogen through iron and through palladium (53) shows 
that the rate of flow rises exponentially with the temperature. 
The slope of the curves log (permeability) against 1/ T (T = ° K.) 
for the Hg-Fe system (Fig. 71) is the same at all current 
densities from 0-0075 to 0-045 amp./cm,2, and is very nearly that 
found for the thermal permeability (^eieet. = 9400 cal./atom, 




Fig. 71. Electrolytic diffusion of hydrogen through iron as a function of tem- 
perature. Curves for current densities: 0-0075 amp./cm.^, 0-015 amp. /cm.", 
0-030 amp./cm.^ and 0-045 amp./cm.". 



compared with ^thermal = 9100 cal./atom). This relationship 
and the similar dependence of the permeability upon current 
density and gas pressure (j). 178) led Borelius and Lindblom (52) 
to assume that all rate-controUing processes were the same for 
thermal diffusion and diffusion of nascent hydi'ogen. 

In Edwards' (55) experiments, the gas was generated by the 
action of hydrochloric and sulphuric acids upon the iron 
membrane. In the steady state the permeation velocities 



NASCENT HYDROGEN THROTJGH METALS 



203 



gave a linear logP—l/T curve whose slope gives E = 7300 
cal./atom for the data using iV^H2S04. Barrer's(53) data for 
the Hg-Pd system also gave linear logP—l/T curves, and 
the slopes corresponded to E = 8500 and 9200 cal./atom. 

As Edwards' and Barrer's data show, the steady state of 
flow through the membrane is established slowly (Fig. 72), and 
the interval required depends strongly upon the temperature. 
During this interval the metal is absorbing a quantity of gas 
and setting up its steady state concentration gradient. It is 




Time in minutes 

Fig. 72. Time lag in establishing steady state of flow of Hj through 
a steel cathode (53). 



connected therefore with the velocity of diffusion of the gas 
within the bulk metal, and the slower this diffusion the more 
slowly is the steady state approached. 

Other studies of the diffusion of nascent hydrogen have given 
empirical relations between the rate of passage through the 
metal and the concentration of the acid generating the gas at 
the surface of the metal (55); and for the influence of various 
capillary active substances (i05) upon the velocity of per- 
meation. For example, the addition of mercuric chloride 
caused an acceleration in the rate of permeation. This accelera- 



204 GAS FLOW THROUGH METALS 

tion in the passage of hydrogen results in a more rapid decrease 
in the mechanical strength of the iron(i3) which is a normal 
consequence of "pickling" the metal in acid. 



REFE]RENCES 

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(15) See McBaiQ, J. W. Sorption of Gases by Solids, Routledge ( 1932)„ 

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(18) A full Kst of references to work of Sie verts and his co-workers 

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(20) Eineleus, H. and Anderson, J. Modern Aspects of Inorganic 

Chemistry, Chap. 13, Routledge (1938). 

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(22) Lacher, J. Proc. Roy. Soc. 161 A, 525 (1937). 

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(25) Sieverts, A. and Hagenacker, J. Z.phys. Chem. 68, 115 (1910). 

(26) Steacie, E. and Johnson, F. Proc. Roy. Soc. 112 A, 542 (1926). 

(27) Simons, J. H. J. phys. Chem. 36, 652 (1933). 



REFERENCES 205 

(28) Mathewson, C, Spire, E. and Milligan, W. J. A?mr. Steel Treat. 

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(30) Rhines, F. and Mathewson, C. Trans. Amer. Inst. min. (metall.) 

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(31) Smithells, C. Gases and Metals, Fig. 123. 

(32) Seybold, A. and Mathewson, C. Tech. Publ. Amer. Inst. Min. 
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(33) Merica, P. and Waltenbiu-g, R. Bur. Stand. Sci. Pap. 281 (1925). 

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(1936). 

(35) Sieverts, A. and Briining, K. Arch. Eisenhuttenw. 7, 641 (1933). 

(36) Hagg, G. Z. ^%s. C/iem. 7 B, 339 (1930). 

(37) Sieverts, A., Zapf, G. and Moritz, H. Z. phys. Chem. 183 A, 33 

(1938). 
Earlier references are given in Smithells, C, Gases and Metals, 
p. 171. 

(38) Smithells, C. Gases and Metals, Fig. 119. 

(39) Hagg, G. Z. phys. Chem. 4B, 346 (1929). 

(40) Rontgen, P. and Braxm, H. Metallwirtschaft, 11, 459 (1932), 

(41) Sieverts, A. and Krumbharr, W. Z. phys. Chem. 74, 295 (1910). 

(42) Smithells, C. Gases and Metals, Fig, 107. 

(43) Kirsehfeld, L. and Sieverts, A. Z. Elektrochem. 36, 123 (1930). 

(44) Sieverts, A. Z. Metallkunde, 21, 37 (1929). 

(45) Toole, F. and Johnson, F. J. phys. Chem. 37, 331 (1933). 

(46) Sieverts, A. and Bi-iining, K. Z. phys. Chem. 168, 411 (1934). 

(47) Sieverts, A. and Hagen, H. Z. phys. Chem. 174, 247 (1935). 

(48) Cf. ref. (35). See also Takei, T. and Miirakami, T.^Sci. Rep. 

Tohoku Univ. 18, 135 (1929). 

(49) Smithells, C. Gases and Metals, p, 186, Fig. 132. 

(50) Ham, W. J. chem. Phys. 1, 476 (1933). 

(51) Post, C. and Ham, W. J. chem. Phys. 5, 915 (1937). 

(52) Borelius, G. and Lindblom, S. Ann. Phys., Lpz., 82, 201 (1927). 

(53) Barrer, R. Trans. Faraday Soc. 36, 1235 (1940). 

(54) Smithells, C. Gases and Metals, p. 84 (1937). 

(55) Edwards, C. J. Iron and Steel Inst. 110, 9 (1924). 

(56) Johnson, F. and Larose, P. J. Amer. Chem. Soc. 46, 1377 (1924). 

(57) Lombard, V. C.R. Acad. Sci., Paris, 111, 116 (1923). 

(58) Lombard, V., Eichner, C. and Albert, M. Bull. Soc. chim. Paris, 

4, 1276 (1937). 

(59) Bramley, A. Carnegie Schol. Mem., Iron and Steel Inst., 15, 155 

(1926). 

(60) Smithells, C. and Ransley, C. E. Proc.Roy.Soc. 150 A, 172 (1935). 

(61) Post, C, and Ham, W. J. chem. Phys. 6, 599 (1938). 

(62) Deming, H. and Hendricks, B. J. Am,er. chem. Soc. 45, 2857 

(1923). 

(63) Jouan, R. J. Phys. Radium, 1, 101 (1936). ' 



206 GAS FLOW THROUGH METALS 

(64) Melville, H. and Rideal, E. Proc. Roy. Soc. 153A, 89 (1936). 

(65) Braaten, E. and Clark, G. Proc. Roy. Soc. 153 A, 504 (1936). 

(66) Ryder. Elect. (CI.) J. 17, 161 (1920) 

(67) SmitheUs, C. and Ransley, C. E. Proc. Roy. Soc. 152A (1935). 

(68) Spencer, L. J. chem. Soc. 123, 2124 (1923). 

(69) Lombard, V. and Eiehner, C. Bull. Soc. chim. Fr. 53, 1176 

(1933). 

(70) Schmidt, G. Ann. Phys., Lpz., 13, 747 (1904). 

(71) Holt, A. Proc. Roy. Soc. 91 A, 148 (1915). 

(72) Lombard, V. and Eiehner, C. Bull. Soc. chim. Fr. 51, 1462 

(1932). 

(73) Ham, W. and Sauter, J. D. Phys. Rev. 47, 337 (1935). 
(74) Phys. Rev. 47, 645 (1935). 

(75) Wiistner, H. Ann. Phys., Lpz., 46, 1095 (1915). 

(76) Smithells,C.andRansley,C.E. Proc. i?02/.<Soc. 157 A, 292(1936). 

(77) Ham, W. Phys. Rev. 47, 645 (1935). 

(78) SmitheUs, C. Gases and Metals, p. 90 (1937). 

(79) Barrer, R. M. Trans. Faraday Soc. 35, 628, 644 (1939). 

(80) Wang, J. S. Proc. Camh. phil. Soc. 32, 657 (1936). 

(81) Barrer, R. M. PM. Magr. 28, 353 (1939). 

(82) Phil. Mag. 28, 148 (1939). 

(83) Wagner, C. Z. phys. Chem. 159 A, 459 (1932). 

(84) Engelhardt, G. and Wagner, C. Z. phys. Chem. 18 B, 369 (1932). 

(85) Doehlemann, E. Z. Elektrochem. 42, 561 (1936). 

(86) Barrer, R. M. Trans. Paroc^aj/ aSoc. 32, 486 (1936). 

(87) Farkas, A. and Farkas, L. Proc. Roy. Soc. 144 A, 467 (1934). 

(88) Jost, W. and Widmann, A. Z. phys. Chem. 29B, 247 (1935). 

(89) Jouan, R. J. Phys. Radium, 7, 101 (1936). See Fig. 4. 

(90) Farkas, A. Trans. Faraday Soc. 32, 1667 (1936). 

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(92) — Trans. Faraday Soc. 32, 490 (1936). 

(93) Cremer, E. and Polanyi, M. Z. phys. Chem. 19 B, 443 (1932). 

(94) Melville, H. J. chem. Soc. 1243 (1934). 

(95) Eyring, H. and Sherman, J. J. chem. Phys. 1, 345 (1933). 

(96) Lombard, V., Eiehner, C. and Albert, M. Bull. Soc. chim. Fr. 

3, 2203 (1936). 

(97) Baukloh, W. and Kayser, H. Z. Metallk. 26, 157 (1934); 27, 281 

(1935). 

(98) Lewkonj a, G. and Baukloh, W. Z. Meto^Z^^ 25, 309 (1933). 

(99) Baukloh, W. and Guthmann, H. Z. Metallk. 28, 34 (1936). 

(100) Ham, W. and Rast, W. Trans. Amer. Soc. Metals, 26, 885 (1938). 

(101) Povdter, T. and Uflfehnan, L. Physics, 3, 147 (1932). 

(102) Smith, D. and Derge, G. Trans. Amer. electrochem. Soc. 66, 253 

(1934). 
(103) J. Amer. chem. Soc. 56, 2513 (1934). 

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(105) Aten, A. and Zieren, M. Rec. Trav. chim. Pays-Bas,49, 641 ( 1930). 



CHAPTER V 

DIFFUSION OF GASES AND NON-METALS 
IN METALS 

Introduction 

In the previous chapter were considered the main phenomena 
of gas flow through metals. It was shown there that the gas 
flow (P) is governed by the equation 

ax 

where D is the diffusion constant, but so far we have had Httle 
to say concerning the important constant D. It has, however, 
been stated (Chap. IV) that the diffusion constant obeys an 
exponential law t) ^ j^ ^-e/rt 

analogous to the expression 

P = P^e-^iiRT 

for the permeability constant. It is not difficult to see why 
an energy of activation should be involved in diffusion pro- 
cesses in solids. The solid lattice contains atoms distributed 
in a periodic manner, to give the regular crystalline array. 
Just as when the solubility of gases in metals was discussed 
(Chap. IV, p. 153), one may regard the interstitial positions in 
the lattice as positions of minimum potential energy — potential 
"holes" — separated by energy barriers. The diffusing atom 
requires an activation energy before it may pass from one 
minimum of energy to another, exactly as does an atom under- 
going a chemical reaction. In this way an exponential term 
Q-E/RT jg introduced. While the value of ^ in a diffusion process 
is a relatively easily interpreted quantity, the Ej^ in the 
expression for the permeability is a complex quantity, which 
may be made up of E for diffusion processes, and phase- 
boundary processes, and the heat of solution (AH) of the gas 
in the metal. Thus the energy of activation for diffusion can 



208 DIFFUSION OF GASES AND NON-METALS 

be related to the crystal structure and force fields, and the 
dimensions of the diffusing particle, but the temperature 
coefficient for the permeability is not easily related to these 
quantities. 

When one examines the literature one finds that relatively 
few measurements have been made of the diffusion constants 
of gases in solids. One has to be sure that phase-boundary 
processes do not control the velocities of absorption, but this 
is not always easy to establish. Again, the mathematical 
analysis of the data often presents difficulties, and very 
frequently a number of phases co-exist which render inter- 
pretation even more uncertain. One has sometimes to deal 
with the case when the diffusion constant is a function of the 
concentration, necessitating the use of Fick's law in the form 

dt dx\ dx 

However, the mathematical handling of the data in this instance 
is now possible (Chap. I), and the formal treatments of Tick's 
law for the great proportion of the possibilities which arise 
have been completed. 

The measurement of diffusion constants in metals 

To measure the concentration gradient in a metal, one may 
conveniently employ the method of Bramley (i, 2, 3, 4, 5) and his 
co-workers. They heated iron in a suitable gas atmosphere of 
which the following are examples: 

CO and CO2 (diffusion of oxygen or carbon), 

CO and CH3CN (diffusion of nitrogen or carbon), 

CO and <^ ^N (diffusion of nitrogen or carbon), 

CO and <! ^CHg] 

> (diffusion of carbon), 
CH3 
CO and paraffins (diffusion of carbon), 




MEASUREMENT OF DIFFUSION CONSTANTS 209 

CO and NHg (diffusion of carbon or nitrogen), 

Ng and NHg (diffusion of nitrogen), 

Hg and PHg (diffusion of phosphorus), 

CS2 and ■{ /CIH3 (diffusion of carbon and sulphur). 

After the metal in the form of bars had been heated to a 
definite temperature for a suitable period it was removed, and 
successive thin layers taken off in the lathe and analysed. 
In this way the concentration gradient was determined and 
the solution of Fick's law in the form 



-<'-iSl 



'xl2V(Dt) \ 



was used to evaluate D. Here Cq is the concentration at a; = 0, 
supposed constant and conditioned by the gas atmosphere. 
Bramley and Lord (4) showed that bars of the same steel heated 
for the same time and at the same temperature but in very 
different atmospheres gave the same distribution of carbon 
in all cases. 

The concentration-distance curves into the metal are, 
however, not always of a shape to which a simple solution of 
the Fick law is applicable. Figs. 73 and 74 give different types 
of concentration-distance curve, for which the solution given 
above is manifestly not correct. In the former case a saturated 
carbon layer has established itself at the surface, and in the 
latter the surface has been to some extent decarburised by 
hydrogen present in the carburising atmosphere. An attempt 
has been made to treat diffusion problems of the type illus- 
trated by Fig. 73(6), but a complete treatment of Fig. 74 is 
not available. However, Bramley and his co-workers neglected 
the initial parts of these unusual forms of concentration 
distance curve, and applied the simple solution of Fick's law 
only to the tail of the curves. The error involved in evaluating 
D is therefore smaller. 

In addition to this analytical method of determining the 
concentration gradient it should in some cases be possible to 

BD 14 



210 



DIFFUSION OF GASES AND NON-METALS 



use an X-ray method. This would reveal the presence of 
embedded crystals of carbide, nitride, or other compounds 
of the diffusing element and the metal, and the intensity of the 
typical pattern may be used as a measure of concentration. 
The method might be used where the lattice is expanded by 
its content of dissolved gas, for the expansion is often a 



1-0 



5t 



a. 



DEPTH IN MM. 

Fig. 73. Types of concentration gradient. 

function of the charge of gas. The X-ray method has been 
employed for following the inter-diffusion of metals; it has 
also been used in studies(7, 8,9, lO) of the hydrogen-palladium 
system in which the lattice constants both for the a- and 
/?-phases increase with the concentration of hydrogen in the 
lattice (Fig. 75). However, while the method is frequently 
available in investigating equilibrium (ii, 12, 13) systems, it is 
not so often employed in measuring concentration gradients. 
When one does not require the concentration gradient, but 
simply the total quantity absorbed or desorbed, one may use 



MEASUREMENT OF DIFFUSION CONSTANTS 211 



1-2 
l-l 
1-0 
0-9 
0-8 
^0-7 


^ 


V 





















k 


V 




















\ 
















u 






\ 
















q:0-6 

Q. 






\ 




















-A 
















io-s 

m 
q: 
< 
O0-4 






^\ 




















^ 


^^ 


<. 












0-3 








A^ 


X 


V 










0-2 








\ 




\. 


















\ 


X 


s,^ 








0-1 










\. 




X 


<^ 














^ 




1 „ 




■-^ 


:.-^ 



1-5 2-0 2-5 30 3-5 4-0 4-5 5-0 

DEPTH M M. 



Fig. 74. T3rpes of concentration gradient. 






3-97 



3-37 



Fig. 75. Lattice constants in Ha-Pd alloys as a function of ^. 



14-2 



212 DIFFUSION OF GASES AND NON-METALS 

the metal in the form of a wire, and measure its electrical 
resistance. This method has been used for carbon-oxygen 
(carbon at very high temperatures is believed to dissolve 
oxygen(i8)), and for hydrogen-palladium (14,15). The assumption 
is usually made that the change of resistance is a linear function 
of the charge of hydrogen. Similarly Coehn and Jurgens(i6) 
followed the absorption of hydrogen by a large number of 
palladium-silver alloys by measuring their electrical resistance . 
It is probable that the absorption of gases by any other metal 
or oxide which reacts with or dissolves the gas appreciably 
could be followed by electrical resistance changes. As suitable 
systems one may give 

Ha-Ta* Oa-CuaOcH) 

H2-V Oa-NiO (18) 

Ha-Ti 02-Zr(i9) 

Ha-Ce, La, and certain rare earths Ng-ZrciD) 

Ha-Th 

Na-Fe 

N2-M0 

Na-Mn 

The E.M.F, produced by a palladium wire charged with 
hydrogen and immersed in a solution containing hydrogen 
ions is different from the e.m.f. given by uncharged palladium, 
and Coehn and Specht(2i) measured the diffusion of hydrogen 
along a palladium wire whose centre was electrolytically 
charged with hydrogen, by measuring the potential at other 
points along the wire as a function of time. The speed of 
movement of a ]Doint of constant potential was related to the 
time by an expression 

4:Dt = x^, 

and from curves such as those in Fig. 76 the authors were 
able to measure D. The numbers of the curves relate to the 
point on the wire at which the potential was measured. 

* Sie verts and Bruning(20) have shown that the resistivity rises in direct 
proportion to the quantity of hydrogen absorbed, being 30 % greater when the 
metal is saturated. 



MEASUREMEISTT OF DIFFUSIOlSr CONSTANTS 213 

Another method for following the diffusion of hydrogen 
along filaments was devised by Coehn and Sperling (22). The 
centre of a palladium wire was charged with hydrogen gas, 
and the wire clamped under a photographic plate. The 
hydrogen liberated at the surface of the wire blackened the 
plate. As before, the movement of the black strip, by means 



-" -t-oz - 




80 



280 



320 



160 20O 2¥0 

Time in hours 
Fig. 76. Flow of hydrogen in a palladium wire followed by measuring 
the change of potential with time. 

of the relationship x^ = 4:Dt {x denotes a point of a given 
darkness on the plate), was used to calculate D. 

The thermo-electric properties of metals are also altered 
when they absorb gases. If a thermocouple is made between 
a pure metal and a hydrogen-saturated metal, the pure metal 
is electronegative, and the thermal e.m.f. per 1° C. is (23) 



Fe 


0-705x10-' v., 


Pt 


0-223x10-'^ v.. 


Ni 


1-08 X 10-7 v.. 


Pd 


174-5 X 10-"^ V. 



As the capacity of the metal for dissolving gas increases, so 
does the thermal e.m.f. The method ought therefore to be 



214 DIFFUSION OF GASES AND NON-METALS 

applicable to the solution of hydrogen in tantalum, titanium, 
cerium, thorium and other metals which dissolve hydrogen in 
large quantities. 

Euringer (24), who measured the evolution of hydrogen from 
nickel wires, gave a method for measuring both the solubihty 
and the diffusion constant. The method, which should be of 
general apphcabiUty, was based upon the following solution of 
Tick's law (Chap. I): 

where a^ is the wth root of the Bessel function of zero order, 
and Cq is the initial uniform concentration in a wire of radius 
Tq. The rate of evolution, P, of gas from the wire is 



since ^iM = _^^(.,^,). 

The rate of evolution of gas therefore depends upon a constant 
^CqDIt and an infinite series of exponentials. The shape of the 
curve of log P against log t does not therefore change through 
alterations in Cq or D, save to undergo bodily displacement. 
Thus an experimental curve can be compared, for a given Cq 
and D with a calculated one, upon a log-log scale of axes. 
When the curves are correctly ascertained, the theoretical 
curve may be made to coincide with the experimental one, 
by suitable horizontal and vertical displacements. D and Cq 
are then observed as follows: 

Use the suffix "e" to denote experimental, and "c" to 
denote values assumed in calculation. Take a point {t^, P^) on 
the log-log curve which after displacement coincides with a 
point {t^,Pe). For this point the products Dt must be equal, 

00 

and so the series 2 e~^""^' are equal. Thus D^t^ = D^t^, and 



MEASUREMENT OF DIFFUSION CONSTANTS 215 



X^ is calculated. Similarly, if the exponential series are equal, 
one must have 



•^^Oe _ ^c^Oc 



CoA 



c,a: 



or C. =^^^ 
^ D Pr 



and so one may calculate Cq^, the experimental value of Cq. 
One might also find the slope and intercept of the asymptotic 
curve 

2CqD DoClt 



logP = log- 



2-303' 




Fig. 77. 



vahd for large values of t, but Euringer pointed out that then 
changes in P may not be easy to evaluate, since they have 
become small. He therefore advocated the method already 
indicated. 

Van Liempt(25) outlined a method of obtaining D approxi- 
mately from the evolution of gases from wires and plates. The 
Fick law for diffusion into a semi-infinite sheet is 



C_ 



= 1 



2 rxl2V(Dt) 



V^ 



Wo 



dw. 



If olie plots C/Cq against a;/2 ^J{Dt), the typical curve of Fig. 77 
is obtained. The curve encloses the same area CQ|^J7^ as the 



216 



DIFFUSION OF GASES AND NON-METALS 



triangle of which- the hypotenuse cuts the a;/2 Aj{Dt) axis at 
the point 2/^7r = 1-13, where C/Cq = 0-11. This point of inter- 
section was called the "apparent penetration depth", a, or 
"apparent outgassing depth" if one is considering desorption. 
When the real curve is replaced by the straight line in this way, 
one can find simple formulae for the diffusion constant D in 
terms of plate thickness, d, and breadth B{B^d) (Fig. 78). 
If ^ = gas evolved/cm. length of the plate being degassed. 



\ 



Fig. 78. 



one has so long as a^^d or QIQq^O-50: Q = 2[|aCo]. The 
original quantity absorbed was 



dCnB, 



so that 



Vo — ""-^0 



Q^_a_4 jDt 

Qq d dsj TT ' 



Other cases were given which are listed below: 

(i) Degassing a plate, where a'^^d oi QIQq^O-50: 



V q) VU)i6- 



MEASUREMENT OF DIFFUSIOISr CONSTANTS 217 

(ii) Degassing a wire of radius Vq, where Q/Qq < 0-67: 
^ ^ o^,2f 3-2Wo-V(9-12Wo) l 

(iii) Degassing a wire of radius Tq, where Q/Qq > 0-67: 

n 7rr§ 

144^(1 -Wo)'" 

It must be remembered that the values of D calculated by 
these formulae were not exact, although all the values of D 
were of the correct magnitudes. Table 49 illustrates the data 
obtained by outgassing a nickel sheet 3 cm. wide and 0-15 mm. 
thick. The composition of the evolved gases was not given. 

Table 49. Diffusion constants by van Liempt's method 



Temp. 


Time 


Q/Qo 


D 


Mean value of D 


°C. 


mm. 


cm.2 sec.~^ 


cm. 2 sec.-^ 


700 


2 


0-33 


4 X 10-8 






5 


0-43 


1-8 X 10-8 


2-5 X 10-8 




10 


0-50 


1-8 X 10-8 




800 


1 


0-41 


12 X 10-8 






2 


0-52 


9-9 X 10-8 


10 X 10-8 




10 


0-70 


9-4 X 10-8 




20 


0-85 


8-5 X 10-8 




900 


2 
4 


0-67 

0-80 


2-1 X 10-^ 
2-9 X 10-^ 


2-5 X 10-' 



Another method, which depends upon the time-lag in setting 
up the stationary state of flow through a membrane, is 
applicable to gas flow in metals. The first treatment was given 
by Daynes(26), and later extended by Barrer(27,28,29). The 
Initial concentration in the plate is Cq, and at the ingoing and 
outgoing surfaces, O^ and Cg respectively. Then if one plots 
the pressure on the high-vacuum side against the time, the 
curve takes the course shown in Fig. 79, and tends to the steady 
state asymptotically. In the simplest case the asymptote cuts 
the time-axis at the point 



d^ 1 rc, c, c,i 

DCi-CaLe 3 2j' 



218 DIFFUSION OF GASES AND NON-METALS 

where d is the membrane thickness. If Cg — 0, Cq = 0, 

^ d^ 

and D may readily be calculated. This formula is appUcable 
only if interface reactions are fast compared with the diffusion 

process (28. 29). When the rates of the processes 




25 50 

Time (min.) 

Fig. 79. Time-lag in setting up the steady state of flow of hydrogen 
through a palladium tube (29). 

(i) passage of adsorbed atom or molecule into the metal 
(velocity constant k-^), 

(ii) emergence of dissolved atom or molecule into the surface 
(velocity constant k^) 

are much smaller than D, one can see that the values of L 
are no longer governed by D. The value of X is a complex 
function of k^ and k2 where the process of permeation is 
controlled solely by these two phase-boundary reactions. 



MEASUREMENT OF DIFFUSION CONSTANTS 219 

When k^, k^ are much greater than D, the expression for L 
reduces to 

Thus the expression L = d^jQD gives a value of D which is 
more nearly equal to the real value the more the phase- 
boundary processes are accelerated. The method can be 
applied equally to the diffusion of gases through hot metals, 
or to the diffusion of nascent gases hberated at room tem- 
peratures by electrolysis or by chemical reaction at the metal 
surface. 

This review of the possible ways of measuring diffusion 
constants in metals indicates the lines of approach which have 
been developed. Future work should aim at the establishment 
of the relative importance of phase-boundary reactions and 
volume diffusion; the estimation of the influence of concen- 
tration upon the diffusion "constant"; the understanding 
of the role of impurities and mechanical treatment; and 
measurements of che influence of temperature and other 
variables upon the diffusion constants. In this way the in- 
vestigations will ultimately contribute to the problems of 
mobiHty and reaction in solids. Unfortunately there is so 
far a great scarcity of data. 



A COMPARISON OP METHODS OF MEASURING 
DIFFUSION CONSTANTS 

It is perhaps unfortunate that the only system for which the 
diffusion constant has been measured by a great many different 
methods is the hydrogen-palladium system, for palladium 
membranes have shown very capricious permeabilities 
(Chap IV, p. 194), and this variability seems to extend also to 
diffusion constants. The permeability to hydrogen of nickel 
membranes, on the other hand, has shown itself to be more 
constant, as the data of Table 42, p. 168, indicate, so that 
hydrogen-nickel systems might prove a better testing ground. 



220 DiFFUSioisr of gases and non-metals 

The published data for the diffusion constants, in the form 
of log Z) against 1/T curves, are shown in Fig. 80, These data 
are by no means concordant, and their divergences from one 
another may be explained in part by the difficulty of the 
technique employed. In part, however, differences in the 
properties of the palladium may be responsible, for, as Tam- 
mann and Schneider showed, the rate of absorption of hydrogen 
depends on the state of the metal, e.g. whether it is soft or 
tempered. Also one may anticipate effects due to impurities, 
and strongly developed grain boundaries. Duhm's(32) result 
*(9-5 X 10~^ cm.^/sec. at room temperature) is so far away from 
all other published data that it has not been given in Fig. 80. 
The equations of the different curves in Fig. 80 are given in 
Table 50. The most unlikely data are in square brackets.* 
By drawing a straight line through the various points in 
Fig. 80 one may obtain a mean diffusion equation for all the 
different palladium samples and methods. 

It is known that when a potential difference is maintained 
between the ends of a palladium wire charged with hydrogen, 
the hydrogen moves towards the negative end(2i,32). It 
therefore carries a positive charge, and may move as protons. 
These protons are probably well screened by electrons, for 
their effective charge has been found to be q^q to 2^^ of its 
value for a free proton. Coehn and his co-workers, and also 
Duhm, measured the mobility of the hydrogen under an applied 
potential, by measuring the e.m.f. or resistance of sections of 
the wire after intervals of passing a current. The temperature 
coefficient of the mobility, U, obeyed the expression 

where E was 6-3 and 4-9 k.cal. respectively by the two methods. 

* These figures are considered doubtful not because of the value of E but of 
Dg. The value of Dq is such that a different mechanism of diffusion would have to 
be postulated in the transition state, involving a large disturbance in the 
surrounding palladium lattice, and thus a large entropy of activation, or large 
Dp. If the diffusing particle is a proton this disturbance is most unlikely. 



MEASUREMENT OF DIFFUSION CONSTANTS 221 




d -5'0 

a 

o 

w -6-0 



-7-0 



^V 



V, A 




0-0021 



0-0026 
l/T(Tin°K.) 



0-0031 



0-0036 



Fig. 80. Summary of data on diffusion constants of hydrogen in palladium. 

© =Time lag (Barrer). 

X =Rate of sorption (Jost and Widmann). 

Q] =Variations in e.m.f. (Coehn and Specht). 

j\^ =Action on photographic plate (Coehn and Sperling). 

* = Rate of sorption (Tammann and Schneider). 

X =Rate of sorption (Tammann and Schneider). 



Table 50. The diffusion constants, D, in palladium, defined 
by D^D^e-^lRT 



Method of 
measurement 


Diffusion constant 
cm.^/sec. 


Author 


Time lag 

Rate of absorption 

Rate of absorption 
[Rate of absorption 

E.M.F. of wire 
[Effect on photo- 
graphic plate 


2-5 X 10-1 e-io""/^^ 
5-4 X 10-» e-^-"t<^lRT 
2-6 X 10-2 e-'wo/-^^^* 
1-07 X 103 e-i29oo/2?T| 
7-4 X 10-1 e-'2oo/iJr 
2-5 X 108 e-20'»o/Br 


Barrer (29) 

Jost and Widmann (30) 
Tammann and Schneider (3i) 
Tammann and Schneider od] 
Coehn and Specht (2i) 
Coehn and Sperling (22)] 


Mean curve from Fig. 80 


1-5 X lo-2e-68oo/«r 





* Soft. 



f Tempered. 



222 

The diffusion constants of various 
elements in metals 

Euringer(24) employed the method described on p. 214 to 
interpret his results on the desorption of hydrogen from 
nickel. He obtained the values for D = D^e^^i^^ given in 
Table 51. There are no data available with which to compare 
Euringer's measurements, since van Liempt's data (Table 49) 
on the outgassing of commercial nickel sheets and wires refer 
to a mixture of gases (Hg, CO, COg). 

Table 51. Dijfusion constants D = D^e^^^^^ of 
hydrogen in nickel 



Temp. 

°C. 


D 




Solubility of hydrogen 
in nickel in c.c. at 
N.T.p./c.c. metal at 
760 mm. pressure 


165 
125 

85 


10-5 xlO-8 
3-4 xlO-8 
1-16 x 10-» 


2-04 X lO-Se-s'oo/'B^ 


0192 
0194 
0-202 



Edwards' (33) measurements on the diffusion of nascent 
hydrogen may be interpreted by the time-lag method (p. 217) 
to give Umiting values to the diffusion constants of hydrogen 
in iron. The curves of Fig. 81 show a considerable time lag in 
setting up the steady state permeation velocity, and by writing 
D — l^/QL, where L is the intercept on the time axis and I the 
thickness of the sheet, the data of Table 52 were obtained. 

Hydrogen gas may diffuse in a metallic lattice as protons, 
which will not greatly disturb the lattice in the act of dijEFusion, 
since the dimensions of the proton are small. When oxygen, 
nitrogen, phosphorus, sulphur, or carbon diffuse in solids, there 
will be a tendency for the diffusing particle to be negatively 
charged or to diffuse as an atom ; its size will not therefore be 
less than that of the free atom, and considerable distortion 
may occur in solution in the lattice and in the act of diffusing.* 
There is therefore a tendency for compounds of these sub- 



* In metals such as iron, however, the ratio 



radius of diffusing atom 



is small 



radius of iron atom 

enough for the atoms of nitrogen or carbon to exist interstitially in the iron 
lattice. 



DIFFUSION CONSTANTS 



223 



stances with the metal to separate out of the lattice as crystals 
embedded in the solid metal, or aggregated at grain boun- 
daries (34). When elements such as sulphur diffuse in iron, the 
process may be regarded as the passing on of the diffusing 
atom by alternate dissociation and formation of sulphides, 
7'5r 




2-5-, 



100 
Time (min.) 



150 



200 



Fig. 81. Diffusion of nascent hydrogen through iron sheet (thickness 0-005 in.). 

rather than as the inter-penetration of alloying metals, by 
place change, or as the zeohtic type of diffusion occurring 
"when hydrogen passes into a metal as atoms or protons. 



Table 52. Diffusion constants of hydrogen through iron 
(Hg liberated by HCl) 



Temp. 

°C. 



Intercept L 
min. 



D^iyQL 
cm.^ sec.~^ 



D=Doe-^IRT 



10 
20 
40 
50 

75 
100 



270 

173 
67 
39-4 
15-6 
3-6 



1-66 X 10-9 
2-59 X 10-9 
6-68 X 10-9 
1-14 X 10-8 
2-87 X 10-* 
1-24 X 10-^ 



1-65 X lo-2e-920o/i?T 



224 



DIFFUSION OF GASES AND NON-METALS 



Table 53. Diffusion constants of carbon in iron 









Values of D 


Temp. 


Carburising 


DxlOs 


(cm.2 sec.-i) at 1000° C, 


°C. 


mixture 


cm.^ sec."^ 


as obtained by 
various authors 


950 


CO + xylene 


7-0 


19-3x10-' (6) 




CO + toluene 


9-6 


4-2, 11-2 xlO-'(35) 




CO + benzene 


10-5 






CO + petrol 


12-0 


10-4, 12-4x10-' (36) 


1000 


CO + xylene 


120 






CO + toluene 


15-0 






CO + petrol 


21-5 




1050 


CO + toluene 


24-0 






CO + petrol 


40-0 




1100 


CO + toluene 


42-0 






CO + petrol 


48-0 





The values of the diffusion constants for a number of the 
electro-negative elements (C, 0, N, P, S) in iron are given in 
Tables 53 and 54. Many authors (6, 35, 36) have given figures for 
the diEFusion constant of carbon in iron, and the diffusion has 
been followed by heating the metal in a variety of gaseous 
atmospheres (1,2, 3, 4, 5) (p. 208, and Table 53). The diffusion 
constants are similar also if nitrogenous hydrocarbons replace 
the hydrocarbons of Table 53. In Table 54 are given the 

Table 54. Diffusion of various elements into steel 
(D in cm? sec~'^) 



Temp. °C. 


800 


850 


900 


950 


1000 


1050 


1100 


1150 


Diffusion system 


















D X 108 for C-Fe(i) 


— 


1-7 


3-8 


8-7 


20-0 


— 


— 





DxlO^forC-Fed) 


1-5 


3-8 


7-5 


11-7 


20-0 


28 


45 





Z)xl08forN-Fe(i) 


1-2 


3-0 


6-0 


10-8 


13-5 


25-0 


40 





Z)xl0i»forS-Fe(5) 


— 


— 


— 


30 


5-5 


7-0 


10 


13 


i)xlO»forP-Fe(5) 


— 


— 


— 


0-72 


1-31 


2-5* 







DxlO"forO-Fe(5) 


— 


— 


— 


— 


1-0 


— 


— 


— 



* At 1040° C. 



diffusion constants obtained by Bramley and his co-workers 
for different elements in steel at various temperatures. In 
considering these data it must be remembered that the diffusion 
constants are often very different in the presence of other 



DIFFUSION CONSTANTS 



225 



impurities in the lattice; for example, the diffusion of sulphur 
and phosphorus is retarded by carbon (p. 236), while the 
diffusion of nitrogen is accelerated by oxygen but retarded (36a) 
by carbon (Table 54a). 

Table 54a. Dijfusion of Ng in a-Fe at 550° G., as a 
function of the carbon content 



% C in Fe 


0-01 


0-06 


0-54 


0-82 


1-40 


D X 108 cm.2 sec.-i 


214 


1-16 


0-33 


0-11 0-05 



The data in Table 54 may be presented in the form 

D = D^e-^l^^, 

and the following formulae are applicable: 

C-Fe :i) = 5-5 x io-2e-32>2oo/i2T (i). ^^d 3-5-4-7e-3i40/iJr (i)^ 

N-Fe:i) = 1-07 x io-ie-34,ooo/j?T (i)^ 

S-Fe : D = 4-8 X io-6e-23.4oo/i?r (5)^ 

P-Fe :i> = 4-5 x io-2e-43,30o/iJT (5) 

The expressions D = D^e'^l^'^ are derived from data con- 
siderably less accurate for S-Fe and P-Fe than for C-Fe, 
N-Fe systems. 

A very recent series of measurements by Wells and Mehl(36fe) 
applied Matano's method of analysis (Chap. I, p. 47) to 
determine D for carbon in iron over a range of carbon 
concentrations of 0-1 to 1-0 % C. by weight. They found 
that the diffusion, which was unaffected by the grain size 
of the samples of iron used, followed the expression 

( — 32 000\ 
— ' I cm.2 sec.-i. 

Fig. 81a summarises all data on the. diffusion of C in steel. 
It may be noted that the most satisfactory analysis of the 
experimental data is that of Wells and Mehl. 



15 



226 DIFFUSION OF GASES AND NON-METALS 



Degassing of metals 

The evolution of gases from metals is a process of some 
technical importance. The commonest gases to be evolved are 
hydrogen, oxides of carbon, and nitrogen (Table 55). When 

-5-0 



-6-0 



'7-0 



■8-0 




-1-0wt.%C 

0-lwt.%C 







l/rxlO*(y in°K.) 

Fig. 81a, Summary of data on diflfusion of carbon in iron and steel. 

-□- Tammann and Schonert(35) 

Wells and Mehloej) 

A Runge (6) 

- - - - Paschke and Hauttmannoec) 

-X- Bramley and Allen (36rf) 

the metalloid, graphite, for example, is degassed (37) the com- 
position of the evolved gas alters with the period of heating, 
and the evolution of gas is more rapid the smaller and less 
perfect the component crystallites, so that charcoals more 
readily evolve gases than graphite. It is very difficult indeed 
to remove the last traces of gas from carbon, the general 
behaviour being that at a given temperature a state is reached 
where no appreciable gas evolution occurs, but as soon as the 
temperature is raised again a fresh burst of gas is obtained. 



Table 55. Gases evolved in heating metals 



Metal 


Temp. 

°C. 


Gases 
evolved 


Remarks 


Mo (38) 


>1000 


Mainly H, 


Amo.unts of gas slight by 




1000 


CO, CO,, and H, 


1760° C. 




>1200 


Oxides of carbon 
and Ng 




W(38) 


^2430 


H2,CO,C02,N2 


No further gas evolved after 
outgassing at 2430° C. 
Amounts of gas relatively 
smaU by 1760° C. 


C(38) 


1300 


H, (52%) 


Graphite (tungar anode) 






CO (44%) 


could be freed of gas com- 






CO, (4%) 


pletely by prolonged heat- 




1600 


H, (30%) 
CO (48%) 
C0,(6o/o) 
N, (16%) 


ing to 2150° C. 




1900 


H, (11%) 
CO (13%) 
CO, (5%) 
■N, (71%) 






2110 


CO (90/0) 

CO, (4%) 
N, (87%) 




Fe(39) 




H,, oxides of 


GuUlet and Roux(39) found 






carbon, nitro- 


30 c.c./lOOg. metal 






gen 




Zn(40) 




Almost pure H, 
from electrode- 
deposited Zn 




M(38,41) 




CO, H,, and CO, 


As much as 100 c.c./lOO g. 
Mainly CO for Mond nickel, 
and mainly H, for electrode 
deposited nickel 




750 


CO, (12%) 


Total 0-67 c.c. in 20 min. 






, CO (54%) , 


(from 100 g. metal) 






H, (34%) 






850 


CO, (7-7%) 
CO (84-6%) 
H, (7-7%) 


Total 0-39 c.c. in 20 min. 




950 


CO, (5%) 
CO (90%) 
H, (5%) 


Total 0-60 c.c. in 20 min. 




1050 


CO, (9-5%) 
CO (84-8%) 
H, (5-7%) 


Total 0-53 c.c. in 20 min. 




1150 


CO, (11%) 
CO (48<%,) 
H, (41%) 


Total 0-27 c.c. in 20 min. 


Al(42,43) 




CO, H„ CH^ 


Quantities from 2 to 30 c.c./ 
100 g. reported. SolubUity 
of gases in aluminium small, 
and therefore gases evolved 
and retained in pinholes. 
Hydrogen occluded as a 
result of the action of water 
upon aluminium 


Cu(43) 




SO, (60%) 


Total gas extracted 2 c.c./ 






CO (20%) 
H, (14%) 


100 g. 



15-2 



228 DIFFUSION OF GASES AND NON-METALS 

These characteristics of graphite filaments are common to all 
metal filaments, and it is also usual to find that if the filament 
is degassed at a high temperature and cooled in vacuum to 
room temperature it remains free of gas at this temperature 
for a long period. This is the result of the high temperature 
coefficient of the diffusion constant. The data in Table 55 give 
some typical results on the degassing of metals. The com- 
position of the evolved gases, besides dej)ending upon the time 
of heating and the temperature, depends also upon the nature 
of the metallurgical process by which the metal has been 
obtained. Therefore the data in this table are by no means 
comprehensive, and are intended only to give some typical 
experiments on degassing. In addition to considering the 
effects of time of heating and temperature upon the gas 
evolution, one may also have to consider the place of origin 
of the gas. Eltzin and Jewlew (37) considered the gases evolved 
from graphite to be different in composition according as they 
were evolved from the "surface" of the filament, or from the 
"interior" of the filament,* their analysis giving: 

Surface COa 34% Interior CO 68% 

CO 58% Na 32% 

N^ 8 % 

It is interesting to find nitrogen among the products evolved 
from graphite, for its presence as a stable compound in the 
lattice may mean that it replaces carbon atoms in the edges 
of graphite laminae, or is firmly attached to peripheral carbon 
atoms in the laminae. 

Kinetic studies of the evolution of gases from metals have 
been made which indicate that the processes involved are 
often true diffusions. The evolution of carbon monoxide from 
nickel wires (4i) shows that the curve log (gas evolved) against 
time has the typical shape associated with a difiusion process 
(Chap. I, Fig. 7), and from this curve the diffusion constants 

* Their concept of "surface" and "interior" is however rather vague. They 
considered the first gases evolved to come from the surface, and subsequent 
gases to come from the interior. 



DEGASSING OF METALS 



229 



may be calculated. The method which van Liempt(25) applied 
(p. 215) to measure the mean diffusion constant of gases from 
commercial nickel should give comparable constants, since the 
main gas evolved is carbon monoxide. Some of these diffusion 
constants are given in Table 56, from which it is evident 
that the two methods agree in order of magnitude only. 
Van Liempt also calculated "diffusion" constants for the 
outgassing of molybdenum wire, obtaining a mean value 
of 7-6 X 10-9 cm.2 sec.-i at 900° C. 

Table 56. Diffusion constants of CO in nickel 



Author 


Form of 
nickel 


Diffusion 

constant 

cm.2 sec.-i 


Temp, 
°C. 


Smithells and Ransley(4i) 
van Liempt (25) 

(Using Smithells and 
Ransley's data) 


Wire 
Thin sheet 

Wire 


4-0 X 10-8 
14-0 X 10-8 

2-5 X 10-8 

9-9 X 10-8 

2-1 X 10-' 

Final 14 x 10-' 

Initial 91 x 10"' 

Mean 6 x 10-8 


950 
1050 

700 

800 

900 

1050 

950 



Filaments glowed in gas atmospheres will frequently absorb 
the gas, by reactions which are in part chemical and in part 
processes of diffusion. Tantalum (44) absorbs nitrogen slowly 
at 1300° C. and rapidly at 1800° C; while oxygen is taken up 
at 730° C. and rapidly at 1500° C. Tantalum will also decom- 
pose hydrocarbon vapours at high temperatures, the carbon 
diffusing into the filament (from 1700 to 2500° C.) with the 
formation of carbides, while evacuation at 2200° C. causes 
the carbon to evaporate again from the metal until only pure 
tantalum remains. Similarly, nitrogen (38) can be absorbed 
by molybdenum wire, setting up an equilibrium whose tem- 
perature variation impKes a heat of 38,500 cal./mol., while the 
rate of establishment of the equilibrium involves an apparent 
activation energy of 26,600 cal. The analogous tungsten 
nitride could also be formed. Another type of filament-gas 
reaction may also be found in which the metal filament 



230 



DIFFUSION OF GASES AND NON-METALS 



evaporates and the condensing metal combines with or adsorbs 
an otherwise inert gas. In this way metal filaments may be 
used in the clean-up of residual gas (45). 



Diffusion and absorption of gases in 
finely divided metals 

When a gas is allowed to come into contact with a thoroughly 
outgassed finely divided metal, there may take place an 
instantaneous adsorption and one or more slow processes. 

Table 57. Some data on the sorption of gases by 
finely divided metals 



System 


Behaviour 


Hg, D2-Cu(46) 

H2-CU(47) 

H2-CU(48) 


- 78° C. rapid initial and slow subsequent sorption. 

0° C. amount taken up by slow sorption increased 
Initial rapid sorption and slow subsequent sorption 

at all temperatures studied 
Slow process observed 


Hg, D2-Ni(49) 

H2-Ni(50) 
Hg-NioD 


Two processes, fast and slow. Hg more rapidly sorbed 

than Dj 
One and possibly two slow processes observed 
Slow process observed 


H2-re(52) 
H2-Fe(53) 


One rapid sorption process and two slow ones 

suggested 
- 190° C, van der Waals' sorption. Above 0° C. 

two slow processes suggested 


N2-Fe, Al203(64) 

Na-Fe, Fe-AlgO,, 
K2O 

02-Ag(55) 

1 


Slow uptake of nitrogen by iron, as well as initial 
rapid sorption 


Slow uptake of oxygen by silver, as well as initial 
rapid sorption 



The nature of the slow uptake of gas has been the cause of 
considerable discussion. The general similarity of the pheno- 
mena observed for a number of sorption systems is illustrated 
in the summarising data of Table 57. In all the cases given, at 
least two processes have been established. The initial rapid 
process is correctly associated with the van der Waals' ad- 
sorption on the surface, and it might be anticipated that 
the slow process was the solution of the gas in the metallic 
lattice, either forming an alloy (Hg) or a compound (Ng) 
with the metal. However, the behaviour is not necessarily 



DIFFUSION OF GASES IN METALS 



231 



so easily explained, for not only do some authors describe 
two slow processes, but in a large number of studies of 
sorption very similar slow processes have been observed 




200° 400° 

Fig. 82. Sorption of hydrogen by Mo-Si catalysts. 

where solution is not probable. Among these systems are 
the following: 

Ha, D.^-Cr203{57,58), 

Ha-CrgOg, ZnO(56), 

Oa-CuCraO^, ZnCr204, CoCr204, MCr204, BeCr204(59), 

H2-M0O3, SiO2(60), 

H2-C (Charcoal (61), Graphite (62), and Diamond (63)), 

CH4-C(61), 
C2H4-M(64). 



232 DIFFUSION OF GASES AND NON-METALS 

All these systems give the characteristic high-temperature 
isobar illustrated in Fig. 82, in which at first the amount 
sorbed increases with rising temperature, and then decreases. 
It is considered that the rising part of the isobar denotes a 
non-equilibrium condition, but that the falling part is rever- 
sible for such systems as Hg-C, Ha-ZnO . CraOg. In other cases, 
however, no part of the curve denotes a reversible equiUbrium 
(CH4-C; C2H4-M). It is also a characteristic of these reversible 
and irreversible chemical sorptions that the velocity of 
sorption increases exponentially with temperature, and from 
this increase in sorption velocity an apparent energy of 
activation may be calculated, whose magnitude is that of 
ordinary chemical reactions or of activated diffusions. The 
apparent energy of activation increases with the amount of gas 
sorbed, and so the sorption velocity decreases strongly as the 
charge of gas is increased. This means that at low temperatures 
one will not get saturation of the available surfaces in any 
finite time. 

Now one may compare these properties with those noted in 
the processes of slow sorption of gases by finely divided metals. 
Once more one finds high temperature isobars in which the 
amount sorbed at first increases with rising temperature and 
then decreases (Fig. 83). The decreasing part of the isobar is 
reversible, and the increasing part is irreversible. The velocity 
of sorption increases strongly as the temperature rises, and an 
apparent energy of activation is observed. Once again the 
irreversible part of the isobar may be interpreted as due to 
variable apparent activation energies which increase with gas 
charge, so that at low temperatures the velocity of sorption 
is so diminished that the available sorption volume is not 
saturated in any finite time. 

It seems therefore that activated diffusion into the bulk of 
a metal and reversible chemical adsorption (called by H. S. 
Taylor (65) activated adsorption) may be similar in their 
observable properties. Indeed there has been considerable 
argument as to whether reversible chemical adsorptions may 
not be activated diffusions, and vice versa. The situation is 



DIFFUSION OF GASES IN METALS 



233 



clarified by the proof that activated diffusion processes into 
soHds (e.g. H2-Pd(30)) can occur to give, eventually, homo- 
geneous solutions; but that reversible chemical adsorptions 
which do not involve inter- or intra-lattice diffusion also occur 
(e.g. H2-C(6i,62, 63)). There is no need to strain either the 
hypothesis of activated diffusion or of activated adsorption to 
include all the features observed in considering gas -solid 




•200 







50 



100 



150 -100 -50 

Temperature, °C. 
Fig. 83. Sorption isobars of hydrogen on nickel, showing low and high tem- 
perature sorption. Curve 1, 2-5 cm. pressure; curve 2, 20 cm. pressure; 
curve 3, 60 cm. pressure. 

systems. In those cases where two slow processes of sorption 
of hydrogen by metals have been postulated, it may be that 
a slow chemical adsorption process occurs at the surface, 
followed by a slow absorption process by the metal. The 
adsorption process can be either on the metal surface itself, or 
as seems likely under some experimental conditions, it may be a 
chemical adsorption of the hydrogen by an oxide monolayer. 
One of the few quantitative analyses of sorption kinetics in 
gas metal systems was made by Ward (47), who after heating 
and evacuating finely divided copper in hydrogen a number of 
times, concluded that he was measuring an activated diffusion 



234 DIFFUSION OF GASES AND NON-METALS 

process, and analysed the data on the rate of sorption by means 
of Tick's law. In accordance with this law, for small values of 
the time, t, Ward found that Q, the quantity of hydrogen 
absorbed, was proportional to ^^t. If Uq denotes the concen- 
tration in the adsorbed film of gas from which solution in the 
metal occurs, one should have, from Lennard- Jones' (66) 
analysis of this problem, 

Q 1 

and so by plotting j against -^ one may measure the 

activation energy for the solution process. The figure obtained 

was 

E = 14,100 cal./g. atom, 

which may be compared with the temperature coefficients for 
the permeability of copper to hydrogen of 1 8,700 and 1 6,600 cal. 
(Chap. IV, p. 168). 



The influence qf impurity upon diffusion constants 

It has been observed that the presence of carbon in nickel 
decreases the permeability of nickel towards hydrogen below 
700° C, but above this temperature increases the perme- 
ability (67, 68). Also in steel it is very important to know the 
effects which various possible impurities (0, N, S, P, C, Si) 
have upon the permeability and diffusion velocity of other 
elements in the steel. While few experiments have been made 
on the permeability, the diffusion velocity within the material 
has been very thoroughly studied by Bramley and his co- 
workers (1,2,3,4, 5). Ham and Sauter(69) showed that nitriding 
the surface of steel increased its hydrogen permeability 
10-15 times, and that the original permeability was restored 
after out-gassing the metal. Bramley and his co-workers 
found that the nitriding of a steel rod was accelerated by 
the presence of small amounts of oxygen in the metal, as the 
following data indicate (Table 58). The nitriding process occurs 



INFLUENCE OF IMPURITY UPON DIFFUSION 235 

on heating in an atmosphere of ammonia, and the processes 
taking place may be visualised as 

4re + 2NH3 -> 2re2N + SHg, 

2Fe2N -> 2re2N (dissolved), 

FcaN ^ 2re + N" (dissolved). 

The role of the oxygen or oxide may be to fix the nitrogen 
atoms as oxides of nitrogen, which prevents the formation and 
subsequent evolution of molecular nitrogen. These two 

Table 58. Nitriding after various treatments, involving 
solution or removal of oxygen 





Diffusion 


Condition of Fe before nitriding 


constant 




for N2 X 108 


Swedish Fe, heated 200 hr. at 1050° C. in dry Hg 


1-8 


Swedish Fe, in original state 


2-1 


Swedish Fe, in original state oxidised in CO-CO2 (75- 


2-5 


25%) forSOhr. 




Axmco Fe, in original state 


2-6 


Swedish Fe, oxidised for 100 hr. in maUeabUising 


2-9 


furnace at 1000° C. 




Swedish Fe, oxidised for 200 hr. in malleabiUsing 


3-5 


furnace at 1000° C. 





systems (Hg diffusing in nitrided Fe; and N2 diffusing in 
oxidised Fe) are the only ones where acceleration occurs. More 
usually one finds a retardation, as for instance when sulphur 
diffuses through steel of increasing carbon content. Fig. 84 
shows how the diffusion constant decreases with the percentage 
of carbon and is nearly inhibited by a large quantity of this 
element. Analogous experiments were made by heating a 
mixture of toluene and carbon bi-sulphide in contact with 
steel. The shape of the curves was attributed to a de-sulphuri- 
sation brought about by the hydrogen liberated from the 
toluene. However, the analysis of the deepest part of the 
curves led to the conclusion that the diffusion constant of the 
carbon was decreased by increasing the sulphur content, while 
the diffusion constant for sulphur was decreased by increasing 
the carbon content. 



236 



DIFFUSION OF GASES AND NON-METALS 



The last of this series of experiments was a study of the 
phosphorisation of steels. Phosphorisation occurred on heating 
the steel sample in a hydrogen-phosphine mixture, and con- 
centration gradients of phosphorus into the metal were 
established. It was found on analysing the concentration- 
distance curves that carbon retarded the diffusion of phos- 
phorus very strongly, and that phosphorus entering the metal 
swept the carbon in front of it. 



6 -t 

- \ 

a \ 



o 0'£ (y* o-e OS 1-0 

CARBON PERCENT. 

Fig. 84. The decrease in the dififusion constant of sulphur 
in steels as the carbon content increases. 



REFERENCES 

( 1 ) Bramley, A. and Jinkings, A. Carnegie Schol. Mem., Iron and Steel 

Institute, 15, 17 (1926). 
Bramley, A. and Beeby, G. Carnegie Schol. Mem., Iron and 

Steel Institute, 15, 71 (1926). 
Bramley, A. and Jinkings, A. Carnegie Schol. Mem., Iron and Steel 

Institute, 15, 127 (1926). 
Bramley, A. Carnegie Schol. Mem., Iron and Steel Institute, 15, 

155 (1926). 

(2) Bramley, A. and Lawton, G. Carnegie Schol. Mem., Iron and 

Steel Institute, 15, 35 (1927). 

(3) Bramley, A. and Turner, G. Carnegie Schol. Mem., Iron and 

Steel Institute, 17, 23 (1928). 

(4) Bramley, A. and Lord, H. Carnegie Schol. Mem., Iron and Steel 

Institute, 18, 1 (1929). 

(5) Bramley, A., Heywood, F., Cooper, A. and Watts, J. Trans. 

Faraday Soc. 31, 707 (1935). 

(6) Rimge, B. Z. anorg. Chetn. 115, 293 (1921). 



REFERENCES 237 

(7) Hanawalt, J. D. Phys. Rev. 33, 444 (1929). 

(8) Linde, J. and Borelius, G. Ann. Phys., Lpz., 84, 747 (1927). 

(9) Kruger, F. and Gehm, G. Ann. Phys., Lpz., 16, 174 (1933). 

(10) Owen, E. A. and Jones, J. Proc. phys. Soc. 49, 587 (1937). 

(11) Hagg, G. Z. phys. Chem. 7B, 339 (1930). 

(12) Z. phys. Chem. 4B, 346 (1929). 

(13) Meyer, L. Z. phys. Chem. 17B, 385 (1932). 

(14) Wagner, C. Z. phys. Chem. 159 A, 459 (1932). 

(15) Sieverts, A. and Hagen, H. Z. phys. Chem. 174 A, 247 (1935). 

(16) Coehn, A. and Jiirgens, H. Z. Phys. 71, 179 (1931). 

(17) Dunwald, H. and Wagner, C. Z. phys. Chem. IIB, 212 (1933). 

(18) V. Baumbach, H. and Wagner, C. Z. phys. Chem. 24B, 59 (1934). 

(19) de Boer, J. H. and Fast, J. Rec. Trav. chim. Pays-Bas, 55, 459 

(1936). 

(20) Sieverts, A. and Bruning, K. Z. phys. Chem. 174 A, 365 (1935). 

(21) Coehn, A. and Specht, W. Z. Phys. 62, 1 (1930). 

(22) Coehn, A. and Sperling, K. Z. Phys. 83, 291 (1933). 

(23) Franzini, T. R.C. 1st. Lomhardo, [2] 66, 105 (1933). 

(24) Euringer, G. Z. Phys. 96, 37 (1935). 

(25) van Liempt, J. Rec. Trav. chim. Pays-Bas, 57, 871 (1938). 

(26) Daynes, H. Proc. Roy. Soc. 97 A, 286 (1920). 

(27) Barrer, R. M. Trans. Faraday Soc. 35, 628 (1939). 

(28) Phil. Mag. 28, 148 (1939). 

(29) To be published. 

(30) Jost, W. and Widmann, A. Z. phys. Chem. 29B, 247 (1935). 

(31) Tammann, G. and Schneider, J. Z. anorg. Chem. 172, 43 (1928). 

(32) Duhm, B. Z. Phys. 94, 34 (1935). 

(33) Edwards, C. A. J. Iron and Steel Inst. 60, 9 (1924). 

(34) Smithells, C. Oases and Metals, p. 173 (1937), Figs. 120, 121. 

(35) Tammann, G. and Schonert, K. Stahl u. Eisen, Diisseldorf, 42, 

654 (1922). 

(36) Calculated by Runge(s) from observations of Giolotti and 

co-workers. 
(36a) Eilender, W. and Meyer, O. Arch. Eisenhiittenw. 4, 343 (1931). 
(366) Wells, C. and Mehl, R. Metals Technol. A.I.M.E. 1940, Tech. 

Publ. No. 1180. 
(36c) Paschke,M.andHauttmann,A. Arch.Eisenhiitt.9,^05(l9Z5-<o). 
(36d) Bramley, A. and Allen, K. Engineering, 11 March 1932. 

(37) E.g. Eltzin, I. and Jewlew, A. Phys. Z. Sowjet. 5, 687 (1934). 

(38) Norton, A. and Marshall, F. Trans. Amer. Inst. min. (metall.) 

Engrs, Feb. 1932. 

(39) Guillet, L. and Roux, A. Rev. Metall. 26, 1 (1929). 

(40) Rontgen, P. and Moller, H. Metallwirtschaft, 11, 685 (1932). 
Burmeister,W. andSchloetter,M. MetallwiHschaft,13, 115 {1934). 

(41) SmitheHs, C. and Ransley, C. J. Proc. Roy. Soc. 155 A, 195 (1936). 

(42) VUlachon, A. and Chaudron, G. C.R. Acad. Sci., Paris, 189, 324 

(1929). 

(43) Hessenbruch, V/. Z. Metallk. 21, 4:6 (1929). 



238 DIFFUSION OF GASES AND NON-METALS 

(44) Andrews, M. J. Amer. chem. Soc. 54, 1845 (1932). 

(45) E.g. Bryce, G. J. chem. Soc. p. 1513 (1936). 

(46) Beebe, R., Low, G., Wildner, E. and Goldwasser, S. J. Amer. 

chem. Soc. 57, 2527 (1935). 

(47) Ward, A. F. Proc. Roy. Soc. 133 A, 506, 522 (1931). 

(48) Leypiinsky, O. Acta phys.-chim. U.R.S.S. 2, 737 (1935). 

(49) Magnus, A. and Sartori, G. Z. phys. Chem. 175 A, 329 (1936). 

(50) lijima, S. Sci. Pap. Inst. phys. chem. Res., Tokyo, 23, 164 (1934). 

(51) Benton, A. and White, T. J. Amer. chem. Soc. 52, 2325 (1930). 

(52) Harkness, R. and Enunett, P. J. ^mer. c/iem. ^oc. 56, 490 (1934). 

(53) Morosov, N. M. Trans. Faraday Soc. 31, 659 (1935). 

(54) Hammett, P. and Brunauer, S. J. Amer. chem. Soc. 56, 35 (1934). 

(55) E.g. Benton, A. and Elgin, J. J. Amer. chem. Soc^l, 7 (1929). 

(56) Pace, J. and Taylor, H. S. J. chem. Phys. 2, 573 (1934). 

(57) Taylor, H. S. and Diamond, H. J. Amer. chem.Soc.56, 1821(1934). 

(58) KohlscMtter, H. Z. phys. Chem. 170A, 300 (1934). 

(59) Frazer, J. and Heard, L. J. phys. Chem. 42, 855 (1938). 

(60) Griffith, R. and HUI, S. Proc. Roy. Soc. USA, 195 (1935). 
HoUings, H., Griffith, R. and Bruce, R. Proc. Roy. Soc. USA, 

186 (1935). 

(61) Barrer, R. Proc. Roy. Soc. 149 A, 231 (1935). 

(62) Trans. Faraday Soc. 32, 481 (1936). 

(63) J. chem. Soc. p. 1256 (1936). 

(64) Steacie, E. and Stovel, H. J. chem. Phys. 2, 581 (1934). 

(65) Taylor, H. S. J. Amer. chem. Soc. 53, 578 (1931). 
Trans. Faraday Soc. 28, 131 (1932). 

(66) Lennard-Jones, J. Trans. Faraday Soc. 28, 333 (1932). 

(67) Lewkonja, G. and Baukloh, W. Z. Metallk. 25, 309 (1933). 

(68) Baukloh, W. and Guthmann, H. Z. Metallk. 28, 34 (1936). 

(69) Ham, W. and Sauter, J. Phys. Rev. 47, 337 (1935). 



CHAPTER VI 

DIFFUSION OF IONS IN IONIC CRYSTALS 
AND THE INTERDIFFUSION OF METALS 

iNTRODUCTIOlSr AND EXPERIMENTAL 

The study of the interdiffusion of soUds probably begins with 
the empirical facts of carburisation of steel, an art many 
centuries old. That solid metals will interdiffuse was early 
observed (1); but the velocity of this interdiffusion was not 
realised untU the quantitative measurements of Roberts - 
Austen (2) revealed that at 300° C. gold would diffuse through 
lead faster than sodium chloride would diffuse through water 
at 18° C. The first alloys were prepared by Faraday and 
Stodart(3) in 1820, by heating together mixtures of metal 
powders. It is interesting to find that this original method is 
employed today (4,5) for the preparation of special alloys (6, 7). 
One may also trace, from early studies of carburisation of 
steel (8, 9), and with increasing research on intermetallic 
diffusion, the development of nitriding, chromizing, calorizing, 
sherardizing, and siliconizing, and the formation of bi-metal 
strip and veneer metals (lo, ii). The processes of homogenisation 
of segregated alloys, rates of transformation in metals, and 
of precipitation of crystals in solids (e.g. Fe3N in Fe), are all 
closely connected with processes of diffusion in solids. One 
may see therefore the technical importance of a knowledge of 
the laws governing intermetalKc diffusion. The subject has 
not yet reached a completeness in itself or in relation to cognate 
topics such as the diffusion of gases in metals (Chaps. IV and V) 
•or of ions in ionic lattices. Since the diffusion usually occurs in 
the lattices of the solids, it is likely to yield much information 
on the physics of crystals, and phenomena such as annealing, 
age-hardening, plasticity, recrystallisation, and order-disorder 
transformations in alloys. 



240 INTERDIFFUSION OF SOLIDS 

The problem of diffusion in ionic lattices has been studied 
principally by the indirect method of measuring the con- 
ductivity of the crystal. Some of the earliest measurements 
of conductivity in sohds are due to Faraday (12). The study of 
conductivity in electrolytes has been developed along three 
main lines: 

(1) The movement of ions in ionic lattices. 

(2) The movement of ions in molten ionic liquids (e.g. 
molten NaCl). 

(3) The movements of ions in solution. 

While we are not concerned with (2) and (3), it is interesting 
to note that the conductivity of silver haUdes shows no dis- 
continuous jump on passing from the solid to the liquid 
state (13, 14), although it is more usual to find discontinuities 
whenever a phase change occurs (KI(i4), HgCu2l4(i5), 
Hgl2(i6)). Early measurements upon the conductivity of solid 
oxides (17, 18, 19) showed that the current carriers were ions, and 
also led to the development of the Nernst lamp, using a filament 
of zirconia, with thoria and rare earths. One of the earhest 
observations upon the increase of conductivity of micro - 
crystalline salts under pressure was made by Graetz(20), and 
upon photoconductivity by Arrhenius(2i), who noted that the 
conductivity of silver chloride and bromide was altered by 
light. The modern developments of the subject we owe 
especially to von Hevesy, Seith, Jost, Wagner, and Tubandt. 

Apphcation of the mobility of hydrogen or sodium ions in 
glass is made in the glass electrode, or the preparation of pure 
sodium by electrolysing sodium ions from molten sodium 
nitrate through glass. Processes of base exchange in zeohtes 
(as in water softening by "permutit"), or even in clays, must 
occur in part by diffusion of ions down concentration gradients 
in the individual crystallites composing the mass. Zeolites 
contain large interstitial channels down which such a diffusion 
is possible. 

The determination of diffusion constants in metals may be 
made by a number of rather special experimental techniques, 



INTRODUCTION AND EXPERIMENTAL 241 

only some of which permit the whole concentration gradient 
to be measured. Dunn (22) followed the diffusion of zinc from 
a-brass by vaporising the zinc in vacuo, and measuring the 
loss in weight of the sample. This method involves the 
assumption that the concentration of zinc at the outgoing 
surface is zero and, like all methods of averaging, does not 
easily show whether the diffusion "constant" depends on the 
concentration. Another method of averaging has been em- 
ployed to find the rate of diffusion of carbon and nitrogen in 
iron (23). The diffusing element is removed as a gas (CO, or Ng) 
as soon as it reaches the surface A similar method (24) was used 
to measure the diffusion rate of oxygen in y-iron, the oxygen 
being removed at the surface by hydrogen. If the metal is in 
the form of wires, the electrical conductivity may be used to 
give the average composition of the wire (25, 26, 27) The method 
has proved successful for hydrogen-palladium, carbon- 
tantalum, and other systems. The lattice parameter, in a few 
cases (28), undergoes a steady change as the concentration 
increases, and this change may be employed to find the mean 
concentration. 

When it is desired to find the actual concentration gradient, 
one may use several methods. The first, used by Bramley and 
his co-workers (23) to measure diffusion coefficients in iron, is 
to heat the metal with the diffusing substance, and to remove 
and analyse chemically thin layers at that interface from 
which diffusion proceeded. Other methods of analysis are 
available besides chemical ones. The shavings may be 
examined by means of an X-ray camera, and variations in 
lattice parameter, or occurrence of known alloy phases noted, 
and used to establish concentration gradients. Similarly, 
spectroscopic analysis of the shavings by giving the position 
and intensity of spectral lines will allow the concentration 
gradients to be measured. This method was used to follow the 
diffusion of a number of metals in sLlver(29). A micrographic 
method of estabhshing concentration gradients in Cu-Al, 
Mg-Al systems has also proved successful (30). 

A very interesting method (3i), which could be used either as 

BD l6 



242 



INTERDIFFTJSION OF SOLIDS 



r< 






tq 



^ 



o 









I I 



ci 



+ I 



I cr 



+IIII II++II+ +I+IIIIIII 



I + I I I I I 



I 2 



^ O 
c4 



ffl 



a 

00 '^ 



iffl ^ ^ 



«;^ a; 



1 -3'd 
) o o 

(M "* i2 



>? a a a g 

rH 55 (» 2i2 , 









a .a 



I s a 



5 a 



T3 a 



CO CO t> » 

^-co5 a:5^ja rS a a jj3 j" 

?'Q0C^^Q0^O<£>Ja^-*00-^OTJ< 
Q0rH^C0C0ClC<|(M'-HOl>^Mt>T)* 






a s 



O «0 CO <>■ I> fc 00 



o a 
a ® 

o a 






IlSrTRODUCTION AND EXPERIMENTAL 



243 



I I I I 

I I I T I 

I I I T T ^ T 



I I ^ 



« pjij ^ , 



3 CO 



-I o "5 



I CO C<1 






-=• 1 



2ri 



r^ 00 



T-l ^ r-H (M rt fO HH lO 00 r-H <N 






' a" fl 

1 -^ lO CO 



'T3 



rt (N O rt CO ^ 

CO ■— ( ■— I --( I— I O 



o ® 

g ^5 as 05 CO o 
2 ° 03 05 o o 

(N '— I ""I <M s^ 

«05 ^ „ „ „ 

S<l 1— I I> 00 lO Tfl 03 

OS 03 OS o o o 
. rt rt F-4 cq M S>1 



O 3t3 bp 



G0O3 .— i<miocOI>-05i>*<MCOtJHiOCOI:^ *" 



l6-2 



244 INTERDIFFUSION OF SOLIDS 

an averaging method, or to determine the actual concentration 
gradients, from the radioactive emission of successive thin 
layers, is the radioactive isotope technique. The method was 
originally employed to determine self-diffusion constants in 
lead, using radium D or thorium B as indicator. Thorium B 
may be condensed on the metal foil or single crystal. The path 
of a-particjes in lead is only 50 /i, so that when some of the 
thorium B atoms have penetrated more deeply than this their 
radioactivity can no longer be detected, and the radioactivity 
of the lead sheet falls off. Instead of following diffusion by 
measuring the ionisation produced by a-rays, the recoil rays 
accompanying emission of a-particles were measured in some 
cases (32), the improvement effected by this method being that 
recoil particles in lead have a range of only 0-5 x 10~^ cm., so 
that diffusion coefficients as small as lO^^^cm.^day"^ can be 
evaluated. The a-ray and recoil-atom methods have been used 
also to follow the self-diffusion of bismuth using thorium C as 
indicator. The results obtained by the two methods are in 
satisfactory agreement. 

The radioactive indicator method has been used in just the 
same way to follow the diffusion of lead ions in lead chloride 
and lead iodide (32). With the discovery of artificial radio- 
activity the method seems capable of very wide appHcation 
indeed. Gold, for example, has been rendered radioactive by 
neutron bombardment, and then used (33) to measiu:e the self- 
diffusion constant of gold in gold; and radioactive zinc has 
been used to measure self-diffusion in copper (33a). Some of the 
substances which may be rendered radioactive by bombard- 
ment with neutrons, deuterons, protons, or y-rays, and whose 
half-life period seems adequately long for the duration of 
possible diffusion experiments, have been collected in Table 59. 

In addition to the normal radioactivity of radium, thorium, 
polonium, and uranium, radioactivity may be induced in these 
elements. The cyclotron has made it possible to obtain high- 
energy particles in considerable concentrations, and so it may 
be .anticipated that artificial radio elements will become 
increasingly accessible to research workers. It is for this reason 



INTRODFCTION AND EXPERIMENTAL 245 

that the radioactive isotope method is regarded as extremely 
important. 

The diffusion of elements from the interior to the surface of 
a metal will usually change its thermionic emission (35), photo- 
electric emission (36), or contact potential (37). These methods 
have been used especially in following the grain-boundary 
diffusion of thorium in tungsten, and the surface migration of 
barium, caesium, sodium, and potassium over tungsten (Chap. 
VIII ) . Cichocki (38 ) demonstrated the diffusion of metals from 
salts through copper, silver, and gold foil by making use of 
the positive-ion emission. The salt was enclosed in the foil, and 
heated, and after an interval positive ions escaped from the 
outer surface. 

Diffusion constants in ionic lattices are in many instances 
calculated from the conductivities. The actual transfer in the 
ionic lattices may be demonstrated by the method of 
Tubandt(39) who employed cells such as 

Pt/AgCl/Ag,S/Ag,S/Ag,S/Ag, - 

and passed a current through the ceU, so that silver dissolved 
at the silver electrode, and was precipitated at the platinum 
electrode, the quantity transferred being found by weighing. 
If sticks of two salts are pressed together and heated they 
diffuse into one another, and the composition could be found 
by dividing the material into sections, and determining their 
density or chemical composition. 

Types of difftjsion gradient 

The diffusion gradients established on the interdiffusion of two 
solids may take a variety of forms. When two slabs of metal, 
e.g. Cu and Ni, are heated together, the simplest type of 
diffusion gradient which may be established is that illustrated 
in Fig. 85(40). The figure shows that the presence of a third 
substance may alter the shape of the concentration gradient. 
The form of the concentration gradient on either side of an 
interface is not always of this, smooth form. Bramley and his 
co-workers (23) heated steel bars in various atmospheres from 



246 INTERDIFFUSION OF SOLIDS 

which the elements C, N, O, S, and P diffused into the steel. 
The variety of shapes of concentration gradient which they ob- 
tained are illustrated in Chap. V, Figs. 73 and 74. Under certain 
conditions they obtained gradients with a maximum, as well 




0€mm 



0-? 03mm 



0-? ^ 0-1 

Distance from Boundary 
Fig. 85. Diffusion of copper into nickel (Grube and Jedele(40)). 

Pure M. 

Ni containing Mn. 

Curve after heating at 1025° C. for 120 houi-s. 



60 

o 40 

o 

to 


K 
























N 


\ 
























\ 


I 
























M 


Sv. 
























( 


H 


^ 




- 






















r^ 


r-c^ 


^^ 


























hc^ 






4 


6 


u 


' K 


- ^ 


? e* 


/ 2i 


» A 


» A 


5 ■<« 


J \ 


^ 40 



Distance in inches x 10'^ 
Fig. 86. Diffusion of chromium into iron (Hicks(4i)). 
Gradient after 96 hours at 1200° C. 

as S-shaped. Obviously the analysis of such curves to give 
diffusion constants will be difficult. Another complicating 
factor arises when two phases co-exist. In this instance con- 
centration discontinuities occur (Fig. 86) (4i). Systems in which 
concentration discontinuities exist may often be analysed 



TYPES OF DIFFUSION GRADIENT 247 

readily to give the diffusion constant, D.^ The discontinuity 
usually marks points of fixed concentration, and the rate at 
which this discontinuity progresses into the solid is therefore 
likely to be governed (for linear diffusion) by the law 

where x denotes the distance of the discontinuity from the 
origin, and t is the time. 



The structure of real crystals 

The ideal crystal consists of a perfectly ordered array of atoms, 
ions, or molecules in three dimensions. This ideal is difficult 
to attain experimentally, although in one instance at least 
(NaCl)(42) it appears to have been closely approached. There 
cannot be any ionic conductivity or atomic diffusion in a 
perfect lattice which conserves its ideal order in all circum- 
stances, and the observation that conductivity and diffusion 
do occur is only one of a number of lines of evidence which 
lead to the conclusion that a real crystal is not perfectly 
regular under all conditions. 

As a result of a large number of researches it has been 
established that two types of fault system may exist in crystals 
which may be called reversible and irreversible fault systems. 
The former have reproducible properties in many respects, 
but properties of the latter depend upon the previous history 
of the specimen. In particular, irreversible fault systems 
give rise to the "structure-sensitive" diffusion and con- 
ductivity data described in the following chapter, while 
reversible fault systems give rise to reproducible conductivity 
and diffusion phenomena. Some of the reproducible and 
equilibrium types of fault systems may be briefly discussed 
before the conductivity and diffusion data are given. 

Besides defects which have been artificially introduced in 
crystals, other types of imperfection exist of the greatest sig- 
nificance for understanding ionic mobility in crystal lattices. 
Lattice imperfections of this kind are: 



248 INTERDIFFUSION OF SOLIDS 

(1) Ions existing interstitially within the normal lattice. 
This leaves holes in the lattice, and electrolytic conduction or 
dijBFusion may jiroceed by jmnps of the ions from one inter- 
stitial position to another; or by diffusion of the holes. 

( 2 ) Some positions in the normal lattice are vacant, although 
there are no interstitial ions. Again the holes may diffuse, by 
jumping of ions from an adjacent lattice place into the hole, 
leaving a second hole. 

The equihbrium between holes, interstitial ions, and the 
normal lattice is maintained at temperatures upwards of about 
100-200° below the melting-point (43, 44, 45); but at low tem- 
peratures the amount of disorder in the lattice depends upon 
the history of the specimen. This non-equilibrium distribution 
of points of disorder in the lattice is referred to as " irreversible 
gitterauflockerung"* by von Hevesy(46). It may have the 
most remarkable influence upon the ionic conductivity, and, 
with the subject of grain-boundary diffusion, is discussed later 
(Chap. VII). The equilibriutn types of disorder existing in a few 
lattices are especially interesting when considering models for 
the diffusion process at high temperatures, such, for example, 
as that given by Frenkel(47). 

Some equilibrium types of disorder in crystals 

In silver chloride the current -carrying ion is the cation, and 
therefore only the cations are in disorder (Fig. 87) (48). In 
Fig. 87 the arrows indicate the possible processes of ionic 

Ag+ CI- Ag+ CI- D CI- Ag+ ■ CI- - 

Ag+— > / 

CI- Ag+ CI- Ag+/ CI- Ag+ CI- Ag+ 

Ag+ CI- Ag+ CI- Ag+ CI- Ag+ Cl- 

Ag+ 
CI- n Cl- Ag+ CI- Ag+ CI- Ag+ 

Fig. 87. Disorder in the silver chloride lattice according to Frenkel. 

mobility. These are of two kinds corresponding to migrations 
of holes and of interstitial ions respectively.There is an energy 

* Gitterauflockerung = lattice looseniag. 



STRUCTURE OF REAL CRYSTALS 249 

of activation for the types of jump illustrated in Fig. 87, as well 
as an endothermic heat offormationofthe interstitial positions. 
a-silver iodide (49), silver mercury iodide (oO), and similar com- 
pounds give the extreme example of the disorder illustrated 
in Fig. 87, for they show an almost perfect anionic lattice, 
but a nearly random distribution of cations within it. This 
accounts for the observation (p. 240) that the ionic con- 
ductivity of solid and molten silver iodide are nearly the same 
(~ 1 ohm^^cm."^). Diffusion in potassium chloride occurs, 
according to Schottky, because of a disorder in both anion and 
cation lattices (Fig. 88) (45) which does not produce inter- 
stitial ions.- The similar size of K+ and Cl~ ions renders the 

• K+ CI- n Cl- K+ Cl- 

CI- K+/ CI- K+ D'^ k:+ 

K+ • CI- K+ CI- K+ Cl- 
Fig. 88. Disorder in potassium chloride according to Schottky. 

interstitial spaces too small to accommodate K+ or Cl~ ions. 
Instead, there are equal numbers of cation and anion vacant 
spaces formed by movements of ions into the crystal surface 
where they tend to build new lattice layers. 

-It has been found that a number of compounds exist which 
do not rigidly obey the law of fixed proportions. In the spinel 
group of compounds, of which MgAlgO^ is the type, certain 
constituents of the lattice may be lost but the lattice still 
retains its structure. Spinel itself may undergo the continuous 
transition MgAl204 -> Mg2Al8/304 (y-alumina). The excess of 
one component is due either to the existence of gaps in the 
lattice where that component should be, or to an interstitial 
excess of another component. The behaviour is shown by 
certain oxides, sulphides and halides. 

As an example of an oxide with metal excess one may take 
zinc oxide{5i). At 600° C. some oxygen has been lost by dis- 
sociation, and a solid solution of zinc in zinc oxide remains in 
which the metal is dissociated into cations and electrons. The 



250 INTERDIFFUSION OF SOLIDS 

higher the oxygen pressure the less the zinc ion excess, and so 
the smaller the conductivity. Cadmium oxide (5i) behaves in 
a similar manner. 

Other oxides give systems with an oxygen excess {oxides of 
iron, cuprous oxide, and nickel oxide(52)). Cuprous oxide, for 
examjDle, takes up an excess of oxygen as 0" ions, as a result 
of which vacant cation sites appear in the lattice. The electrons 
for this process are liberated by the reaction Cu+ -^ Cu++ + e, 
and electronic conductivity occurs because electrons may be 
supplied by the electron transfer of the above reaction. The 
higher the oxygen pressure the more excess oxygen is dissolved 
in the lattice (e.g. -0-1% at 1000° C. and at SOmm.Hg 
pressure), and so the higher the conductivity. The sizes of the 
ions(0" = l-32A.;Cu+ = 0-96 A.) do not allow one to arrange 
the excess oxygen interstitially, and so the type of disorder 
given in Fig. 89(48) is postulated. X-ray studies (53) of ferrous 
oxide, sulphide, and selenide showed that empty cation posi- 
tions similar to those in Fig. 89 may indeed exist. 

Cu++ Cu+ n Cu+ Cu+ Cu+ 

0" 0" 0" 0" 0" 

Cu+ Cu++ Cu+ Cu+ D Cu+ 

Fig. 89. Disorder in cuprous oxide with excess oxygen. 

Cuprous iodide (54) and bromide (39) show in part an electronic 
conductivity due to the solution of an excess of iodine or 
bromine as ions, the electrons for the formation of the ions 
being supplied by the reaction 

Cu+->Cu++-fe, 

Potassium iodide if pure has a small conductivity ionic in 
type. The lattice may dissolve excess either of iodine or of 
metal, and the conductivity is then altered by the presence 
of iodine ions, or potassium ions: 

The alkali halides in general (55) can behave as solvents for small 
amounts of halogens, alkali metals, and even hj^drogen. The 
solute is not always in atomic or ionic form however ( Chap . Ill ) . 



251 



The influence op gas pressure upon conductivity 

Most important from the viewpoint of lattice disorder are the 
observations of Wagner and his co-workers upon changes of 
conductivity brought about by surrounding the crystal by a 
gas atmosphere of its electronegative component (oxygen for 
oxides, or halogen for halides). These measurements were the 
basis of the classification of lattices, as in the previous section, 
into those with anion excess, cation excess, or stoichiometric 
cation anion ratio. 

As an example of the influence of pressure we may consider 
Oa-CugO systems which contain excess of the electronegative 
component oxygen. It may be supposed that Cu+ ions 
release electrons, giving Cu++ ions, and an equivalent number 
of Cu+ ions diffuse from their lattice sites. These ions and 
electrons react at the surface of the crystal with gaseous 
oxygen, producing more crystalline CU2O. One denotes Cu++ 
by the symbol (e);, or electron defect site; similarly (Cu+); is 
a vacant lattice site. The corresponding occupied sites are 
denoted by (e)^^, (Cu+)^. The symbols "Z" and "gf" mean 
respectively "leerstelle " or "vacant place ", and "gitterplatz " 
or "lattice site". The reaction is then 

02(gas) + 4(e), + 4(Cu+), ^ 2(Cu20), + 4(e), + 4(Cu+),. 

Since (CuaO)^ is simply an array of (0 — ),-i-2(Cu+),, the net 
reaction is 

02(gas) + 4(e), ^ 2(0—), + 4(e),+ 4(Cu+),. 

Application of the law of mass action then gives 

(Cu+)f(e)f ^^ 

i'oa ' 

because (e), and (0"~), are nearly constant. But (Cu+); = (e),, 
and so / \ nz -m 

The conductivity, which in cuprous oxide is mainly electronic, 
is thus proportional to the eighth root of the pressure -of 
oxygen. While the theory predicts an eighth root, experiment 



252 



INTERDIFFUSION OF SOLIDS 



shows an approximate seventh root dependence of conduc- 
tivity on pressure (Fig. 90), a satisfactory agreement. 

Table 60, after Wagner (43), gives in a simple form all the 
available information concerning lattice disorder and the 
conduction process in certain oxides and halides. The use of 
the table may be illustrated with reference to silver chloride: 

Column 2. There are almost equal numbers of interstitial 
cations and vacant cation sites in the silver chloride lattice. 
The number of quasi free electrons or electron-defect sites is 
negligible. 




log p m mm. 

Fig. 90. The dependence of the conductivity of CugO on the 
oxygen pressure. (Dunwald and Wagneros).) 



Column 4. Neither silver nor chlorine is in any appreciable 
excess over their stoichiometric ratio. 

Columns 5, 6 and 7. Increased partial pressure of chlorine 
has no influence upon conduction due to cations, but may 
alter the electronic conductivity. Electronic conductivity 
(columns 2 and 9) occurs however only to a neghgible extent. 

Columns 8 and 9. Silver chloride is almost exclusively a 
cationic conductor. 



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254 interdiffusion of solids 

The energy of disorder in crystals 

The results of the previous sections lead us to consider im- 
portant types of equilibrium ionic disorder to be: 

Vacant cation sites, with an equal number of interstitial ions ; 

Vacant anion sites, with an equal number of interstitial ions; 

Equal numbers of anions and cations in interstitial positions ; 

Equal numbers of vacant anion and cation sites. 

The first three examples are known as Frenkel disorder, 
and the last example as Schottky disorder. Calculations of 
the energies of disorder (56) are likely to be of importance if 
they give the absolute values of the energy of disorder from 
known force laws and crystal parameters; and if they show 
which type of disorder is most likely to be met mth in 
dififerent salts. 

Let E denote the lattice energy, that is, the energy needed 
to dissociate 1 Mol. of the lattice into gaseous ions. Then EjN^ 
is this energy referred to one ion pair of the lattice; it is con- 
nected with the energy E^ needed to produce a Mol. of Schottky 
lattice defects, and the corresponding polarisation energy, 
£'poi, released around the vacant sites, by E = Eg + E^^^. In 
a preliminary calculation of E one may use the force law 

F ------- 

dr r^ r^ ' 

in which the first term gives the Coulombic attractive force, 
and the second the repulsive force.* Born's expression for 
the lattice energy is then applied, and the lattice energy per 
ion pair in sodium chloride for example is 



B 



E 

No 



= 1-74 



a \ nj 



where a denotes the lattice parameter, and e is the electronic 
charge. In addition, the polarisation energy involvedis approxi- 
mately equal to that when an ion is transferred from a medium 

* The repulsive exponent n is usually given a value of 9-13. 



THE ENERGY OF DISORDER IN CRYSTALS 255 

of dielectric constant e to a medium of dielectric constant 
unity, given for a pair of ions by 



iVo a 



(-1) 



Accordingly, the net energy of disorder per ion pair becomes 



/q ct \ nj a \ ej 



which is very much smaller than the lattice energy. However, 
it must be remembered that many approximations are involved 
in so simple a calculation as that above. The nature of these 
approximations is now indicated: 

(1) The force law used in calculating the lattice energy is 
inadequate since it neglects contributions of van der Waals' 
interactions, which may become appreciable in lattices where 
the mass of the ions is large, or where one type of ion exists 
largely in interstitial positions. Thus in a-AgI, or a-AgaS, where 
the cations are in almost complete disorder in an anion lattice, 
van der Waals' forces are considerable. 

(2) Interactions between induced quadrupoles also con- 
tribute to the polarisation energy. 

(3) When a vacant site is formed, surrounding ions will tend 
to rearrange themselves slightly, and the corresponding energy 
term must be allowed for. 

(4) A better method of expressing the repulsive potential 
replaces the term 

F -^ 

-^rep. — ^n 

(r+ + r_-r) ^ 

where r+ and r_ are the radii of a positive and a negative ion, 
distant r from each other, and b and/? are constants. Even this 
equation is inaccurate at very small separations. 



256 INTERDIFFUSION OF SOLIDS 

(5) The crystal cannot be regarded as a continuum, as was 
assumed in calculating the polarisation energy, ^poi., for one 
is dealing with phenomena on a molecular scale. Thus one is 
uncertain what value to give to e and what the effects of 
inhomogeneity of the medium upon the polarisation energy 
may be. 

Other refinements may be suggested (56), but enough has 
been said to indicate the nature of the calculation which must 
be made, and its difficulties. The energy of disorder may be 
determined experimentally, however, from conductivity data. 
Koch and Wagner (57) measured the conductivity of solutions 
such as PbClg, CdClg, in AgCl, and of pure AgCl. The addition 
of cadmium chloride will alter the number of vacant lattice 
sites in a silver chloride lattice with which it forms a homo- 
geneous solution. For in cadmium chloride the cation to anion 
ratio is 1 : 2, instead of 2 : 2 for silver chloride, and there is thus 
one cation too few for every Cd++ ion in the lattice, giving one 
vacant site for each cadmium ion incorporated. The authors 
have established that the electrical conductivity of solutions 
of cadmium chloride in silver chloride is in the main determined 
by the product of concentration and mobility of the vacant 
sites, the concentration of interstitial ions being much smaller. 
But the concentration of vacant sites is equal approximately 
to the concentration of the cadmium chloride, and so the exact 
analysis of the conductivity data permits one to find the 
mobility of vacant sites, and by extrapolating to zero con- 
centration of cadmium chloride to find the much smaller 
concentration of vacant sites in pure silver chloride, as well as 
their mobility. The results of this investigation appear in 
Table 61 (48), and from the measurements at various tempera- 
tures, by application of the van't Hoff isochore -„ = ^nm^ 
the following energies of disorder were found : 
AgCl : E ~ 25,000 cal./ion 
AgBr : E ~ 20,200 cal./ion. 
These energies show that the mathematical treatment of 



ENERGY OF DISORDER IN CRYSTALS 



257 



energies of disorder is correct in outline, since, as the theory 
indicates, the energy of disorder is much less than the lattice 
energy. 

The question as to which of the two main types of disorder 
(that of Frenkel or that of Schottky) will prevail in a given 
system can be treated analogously by the mathematical theory. 
Simple geometrical considerations are, however, sufficient to 
show the conditions which decide the actual types of disorder. 

Table 61. Concentrations and mobilities of interstitial 
ions and vacant 'positions in silver halides^ 



Compound 


Temp. 
°C. 


MobOity of 

vacant sites 

in Ag lattice 

cm./sec./v./cm. 


Fraction of ions 

in interstitial 

positions — fraction 

of vacant spaces 


AgCl 
Agl 


350 

300 
250 
210 

300 
250 
210 


6-6 X 10-* 
4-2 X 10-* 
2-3 X 10-* 
1-5x10-* 

7-6 X 10-* 
3-4 X 10-* 
2-0 X 10-* 


1-5 X 10-2 
5-5 X 10-* 
2-2 X 10-* 
8-1 X 10-5 

4-0 X 10-3 
1-8 X 10-' 
7-6 X 10-* 



* The concentrations are calculated assuming that the mobility of holes and 
interstitial ions is the same. This assumption may lead to errors in the data 
of Table 61. 

If the ions displaced from their regular lattice sites are small 
compared with the interstices, they may more easily exist in 
interstitial positions, and Frenkel's disorder is possible. If the 
anions and cations are of comparable radii, the lattice must be 
highly distorted for ions to exist interstitial ly, and Schottky's 
disorder becomes probable. 



The influence of temperature on diffusion in metals 

AND conductivity IN SALTS 

The earliest attempts to express conductivity data for salts 
made use of power series in the temperature ( T). Foussereau (58) 
represented a number of specific resistances by the expression 

logR = a-bT + cT^, 

BD 17 



258 INTERDIFFUSION OF SOLIDS 

where a, b, and c are constants. Rasch and Hinrichsen(59) and 
Konigsberger (60) independently employed the equation 

logK = a + blT, 

where K denotes the conductivity. Phipps, Lansing and 
Cooke (61 ) followed Konigsberger 's suggestion that the equation 
was really of the form 

logK = -E/RT+C, 

where E is an energy term concerned as we now know with 
the movement of the current-carrying ion or diffusing metal 
atom from one position of minimum potential energy in the 
lattice to another. All results are now expressed in terms of 
Konigsberger's formula, although sometimes two or more 
exponential terms are necessary when two or more ions 
participate in the transport of electricity. The general and 
very satisfactory applicability of the exponential law, both 
for diffusion in metals and conductivity in salts, is illustrated 
in Figs. 91-93. It can be seen from these figures and from 
Figs, 94 and 95 that one may classify diffusion and conductivity 
systems into three groups: 

(A) Substances which conduct by transport of one ion, or 
where the diffusion constant, D, obeys a simple exponential 
law D = D^e^^l^^. This is the normal and by far the most 
numerous group (Figs. 91 and 92). 

(B) Substances in which more than one exponential term 
is important, according to the temperature. Thus the diffusion 
constants of indium and of cadmium in silver (29), and the con- 
ductivity of lead iodide (32) involve two exponentials. One may 
write for lead iodide: 

because both ions carry the current. At low temperatures, 
one exponential predominates, and the behaviour is that of 
class (A). Another salt which may follow a similar law is 
silver chloride, for which Smekal(02) gives 



Temperature in °C. 
650 700 




Fig. 91. The diffusion constants of various metals in silver 
(Seith and Peretti(29)). 



Temperature in °C. >■ 

WO 200 



300 



^00 SOO 600 700 S00900 




Fig. 92. The conductivities of a number of salts (Seith(39)). 



17-2 



Temperature in °C. 

200 250 

T 



300 




Fig. 93. The diffusion and conductivity data for lead iodide (Seith(39)). 

-K^Pbia = conductivity of Pblj. 

Ky = conductivity of I' ions. 

irpb++ = conductivity of Pb" ions, 

Z)pb++ = self- diffusion constant of Pb" ions. 





















1 1 1 













J.'. 








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200' 



300° 



'tOO" 



500'C. 



Fig. 94. The conductivity of cuprous bromide as a function of 
temperature (Tubandt(39)). 



INFLUENCE OF TEMPERATURE 261 

Where one has a pair of mutually soluble salts the behaviour 
tends to be that of group (B). For example, in Table 62 (39) are 
given the constants for the conductivity of CuBr-AgBr mixed 
crystals. The conductivity may be represented by 



Agl 



x70'* 



£<■ 



ZO 



18 



■hi 



U 



-3- 





f 


■«-/iji 




- 


- 
























^ 








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1 


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if 



32 



30 



28 



26 



I xlO^ 



■ AgiHgh 



Fig. 95. Conductivity of silver iodide and silver mercury 
iodide (Seith(39)). 

Table 62. Constants in the equation 

K = ^ca+ e-^^/^^ + ^Ag+ e-^^/^^ (ohm-i cm.-i) 

for the conductivity of CuBr-AgBr mixtures 



Composition 
mol. % AgBt 


Aas+ 


EAg+ 

cal./ion 


Acu+ 


Ecu+ 
cal./ion 


100 
90 
80 
65 


1-5 X 10« 
1245 

850 
235 


20,600 

10,040 

8,460 

6,540 


66 
7-3 
17-3 


7540 
3720 
3960 



Similar data for CuI-AgI mixed crystals are given in 
Table 63(39). ' 

The tables illustrate the loosening of the lattice by the 



262 



INTERDIFFUSION OF SOLIDS 



addition of the component having the least tightly bound 
conducting ion. For example, the energy term required for 
the migration of silver ions in CuBr-AgBr mixtures falls from 
20,600 to 6540 cal. when 35 mol. % of CuBr have been added. 



PbCL* 0.005 KCL 




lO^T (Tin "/<.) 

Fig. 96. The influence of potassium chloride on the conductivity 
of lead chloride (Gyulai (63)). 

Table 63. Constants in the equation 

K = Aq^+ e-^il^^ + Aj^g+ e--^2/^3" (ohm-i cm.-^) 

for the conductivity of CuI-AgI mixtures 



Composition 


-4Ag+ 


-^Ag+ 


-4cu+ 


Ecu+ 


mol. % Agl 


cal./ion 


cal./ion 


100 


24-9 


4600 








95 


36 


5400 


3-7 


5400 


90 


26 


4960 


5-7 


4960 


70 


10 


3720 


8 


3720 


50 


6-3 


3270 


10-7 


3270 


30 


2-8 


2400 


11-2 


2400 





— 


— 


5-5 


1190 



The same loosening has been efifected for lead chloride by 
adding small amounts of potassium chloride, as Fig. 96(63) 
shows. It can be seen that the potassium chloride has reduced 



INFLUENCE OF TEMPERATURE 263 

the energy needed for rendering mobile the current- carrying 
ion, for the slope of the log K-l[T curve is greater for PbClg 
than for PbCl2 + 0-005 KCL 

For KCl-NaCl solutions Smekal(62) proposed an equation 
with three exponential terms, since all three ions may act as 
current carriers: 

-^%aCl-KCl = -^Na+ + -^Cl- + ^K+ 

(C) Some salts do not obey any simple conductivity-tem- 
perature law, as Fig. 94 illustrates in the case of cuprous 
bromide, and Fig. 95 for silver iodide and silver mercury 
iodide (Ag2Hgl4). Each transition point shows a sharp break 
in the conductivity-temperature curve, and the conductivity 
of a-CuBr and a-AgI can actually be greater in the sohd state 
than in the molten state. In the case of cuprous bromide one 
notes (Fig, 94) that in addition to the basic curve III some 
irreversible curves are also shown (I, Ila, 116). This peculiarity 
is due to the extreme sensitivity of the conductivity of cuprous 
bromide to excess halogen, and to the difficulty of excluding 
traces of such impurities, which are sufficient to raise the 
conductivity by several powers of ten. 

The extremely high mobility of silver ions in a-AgI and 
Ag2Hgl4 has been the subject of a number of researches. The 
solution of their unique behaviour came from the studies of 
Strock (49) and Ketelaar (50) on silver iodide and silver mercury 
iodide respectively. Strock showed that while the anion lattice 
was in perfect order the cation lattice was almost completely 
disordered. The work of rendering the silver ions mobile is thus 
very small, and their conductivity high. Ketelaar found that 
the same property explained the conductivity-temperature 
curve of AggHgl^. 

a-AggS, a-AggSe and a-AggTe have also extremely high 
conductivities, and it was some time before the explanation 
of the conductivity values was forthcoming (64). Ultimately, 
Wagner (65) pointed out that the conductivity of a-AggS was 
sensitive to the sulphur-vapour pressure in the surrounding 



264 



INTERDIFFUSION OF SOLIDS 



gas phase. Therefore, just as for CugO and shnilar oxides 
(p. 251), the conductivity is in part due to electrons. The elec- 
trons are suppUed by reactions in the lattice such as 

S"->8 + 2e, Se"^Se + 2e, Te"^Te + 2e. 

The values for the conductivity K, and the constants A 
and E in the formula K = Ae'^l^^ for electrolytic conductors 
in which the conductivity is predominantly by one ion, are 

Table 64. Conductivity, and constants in the conductivity 
formula K = Ae-^i^^ {Seith(39)) 



Salt 


Melting- 
point 

°C. 


K (at the 
melting- 
point) 
ohm-i cm.~^ 


A 
ohm-i cm.~i 


E 
cal./ion 


E 
e.V. 


LiF 
LiCl 


842 
606 


0-6 X 10-2 
1-5x10-2 


4 xlO^ 

5 xlO' 


51,000 L. 
38,000 L. 


2-20 
1-65 


NaF 
NaCl 
NaBr 
Nal 


992 
800 
735 
661 


1-7 X 10-» 
1-3 X 10-3 
1-3 X 10-=* 
4-0 X 10-3 


1-5 X 10« 
1 xlO« 
1 xl06 
1-5 X 10« 


52,000 L. 
44,000 L. 
41,200 L. 
33,000 L. 


2-25 
1-90 
1-78 
1-42 


KF 
KCl 
KBr 
KI 


846 
768 

728 
680 


8-0 X 10-4 
2-0 X 10-4 
2-0 X 10-4 
1-5 X 10-4 


3 xlO' 

2 xl06 
1-5 x 10« 

3 xlO* 


54,400 L. 
47,800 L. 
45,600 L. 
41,000 L. 


2-35 
2-06 
1-97 
1-77 


RbCl 
RbBr 


717 

681 


5-0 X 10-5 
3-5 X 10-5 


3 xlO« 
1-8 X 10« 


49,200 L. 
47,000 L. 


212 
2-03 


TlCl 
TlBr 


427 
457 


5-0 X 10-* 
5-0 X 10-» 


2-5 X 10» 
1-7 X 10» 


18,320 L. 
18,560 L. 


0-79 
0-80 


AgCl 
AgBr 
a-AgI 
AgaHgl^ 


455 
422 

522 


1-0x10-1 

6-0 X 10-1 

25 


3 xlO« 

3 xlO« 
5-5 

4 xl02 


22,200 T. 
20,600 T. 

1,186 T. 

8,600 K. 


0-96 
0-89 
0-05 
0-37 


PbClg 
Pbia 


501 
402 


5 X 10-» 
3 X 10-5 


6-6 

1-2 X 105 

(Pb++) 

9-8 X 10-4 

(I-) 


10,960 S. 
30,000 S. 

9,360 S. 


0-47 
1-30 

0-40 



L. Lehfeldt. 



T. Tubandt. 



K. Ketelaar. 



S. Seith. 



summarised in Table 64. Lehfeldt (66) showed that the energy E 
in the conductivity equation K = Ae-^'^'^ depends upon the 
radius of the halogen ions in the lattice to a remarkable extent. 
His representation of this effect is given in Fig. 97. As the 



INFLUENCE OF TEMPERATURE 



265 



anion increases in size, the energy needed to render the 
current-carrying ion mobile diminishes considerably. The 
effect is to be traced partly to the increasing polarisability of 
the anions with increasing radius. On the other hand, the 
polarising capacity of the cations increases in the order 
Rb<K<Na<Li. The polarisation energy liberated around 
vacant lattice sites tends to offset the work of forming the 
vacant site. 




f-6 t-8 

Radius of the anion in A. 



Fig. 97. Dependence of the energy E in the conductivity equation 
K =Ae~^l^'^ upon the anion radius (Lehfeldt(66)). 



The identity of the current-carrying ions 

The method of the radioactive indicator affords a simple 
means of determining whether the anion or cation is the 
conductor in a given salt. One has only to compare the 
expressions 

K = Ae~^l^^ , for the conductivity K, 

and D = D^e^^/^'^, for the diffusion constant Z). 



266 INTERDIFFUSION OF SOLIDS 

If the values of E are equal, the current carrier is the ion used 
as the indicator. An example of this method has been given 
elsewhere for lead iodide and lead chloride (p. 271). 

The second method is the measurement of the transport 
number, for which the experimental technique was de- 
veloped by Tubandt and his co-workers (39). It was early- 
discovered (67, 68) that Faraday's laws were valid for the salts 
barium chloride and silver chloride, and thus that the current 
carriers are ions. Tubandt and his school carried these in- 
vestigations much further by pressing salt cylinders together 
between metal electrodes and electrolysing the system. By 
weighing the cylinders and electrodes before and after electro- 
lysis the amounts of material transported were estimated 
directly. However, it was shown that in many instances 
threads of metal formed stretching from anode to cathode, so 
that conduction soon became metallic. a-AgI did not behave 
in this manner, and it was sufficient to coat the electrodes with 
a protective layer of this salt to suppress the formation of 
metal threads. With a cell arranged as below: 

Pt / a-AgI / Ag,S / Ag,S / Ag^S / Ag, 

it was possible to measure the total current, the weight of silver 
deposited at the cathode, the weight of silver dissolved from 
the anode, and the loss or gain of weight of any intermediate 
silver sulphide cylinder. In this way it was found that the 
transport number of silver was unity. Nevertheless the result 
does not mean that AggS is a pure cationic conductor, but 
only that any electrons in the silver sulphide lattice do not 
enter the silver iodide lattice. At the AggS/AgI boundary 
both Ag+ ions and electrons are liberated and removed, with 
formation of excess sulphur : 

Ag2S = S + 2e-h2Ag+. 

The Ag+ ions move to the cathode, while the electrons move 
to the anode. Therefore in the Agl phase all the current is 
carried by Ag+ ions, and the transport number measured by 
deposition of silver at the cathode is unity. The excess sulphur 
liberated at the phase boundary AggS/AgI eventually reacts 



IDENTITY OF CURRENT-CARRYING IONS 



267 



with silver from the anode. Other examples of Tubandt's 
method do, however, speak miequivocally for cationic or 
anionic conductivity, especiaUy if used in conjunction with 
a supplementary method such as that described in the next 
paragraph. The example of a-AggS serves to indicate the 
type of difficulty encountered. 

Finally, should the conductivity be sensitive to the pressure 
in a surrounding gas atmosphere of a component of the crystal, 
the investigations of Wagner and his school (51,52,43) have 
suggested that part of the conduction may be electronic 
(p. 251). 

By the application of these methods the current-carry- 
ing ions have been identified in the instances given below. 
Inspection of these examples shows that in salts with ions of 



Cationic 
conductors 


Anionic 
conductors 


Cationic 
and anionic 
conductors 


Ionic and 
electronic 
conductors 


Elec- 
tronic 


AgCl 


PbFj 


Pbig ■ 


a- and /J-AggS 


Metals 


AgBr 


PbClg 


Alkali halides 


a- and yJ-AggSe 


FegO, 


a-AgI 


PbBra 


near their 


a- and /J-AggTe 


PbS 


AgN03 


BaFa 


melting- 


a-CuI 




a-Ag^Hgl^ (Hg++ 

and Ag+) 
Alkali haljdes 


BaClg 
BaBrg 


points 


a-ZnO 
a-CujO 
a-NiO 




below 500° C. 






a-FeO 

a-FeS 
etc. 





different valency it is as a rule the ion of smallest valency which 
migrates under the applied e.m.f. The extent to which 
electronic and ionic conduction occur in the case of salts 
conducting by the two mechanisms may depend very much 
upon the temperature. Fig. 98(54) shows that in a-CuI the 
conductivity becomes 100 % ionic at high temperatures, and 
100 % electronic at low temperatures. 

In the studies of the conductivity of metaUic alloys (Hg-Pd, 
Au-Pb) it has been found that the alloying constituents may 
move in the lattice under the impressed e.m.f., so that the 
conductivity is not really completely electronic. The move- 
ments of dissolved hydrogen in palladium under an impressed 



268 



INTERDIFFUSION" OF SOLIDS 



E.M.F. have been used to measure the charge on the hydrogen, 
and its mobihty and diffusion constant (69) (Chai?. V). The 
transference numbers of hydrogen are of course small. Seith (70) 
showed that if carbon is dissolved in iron it will move in an 
electric field towards the cathode, so that it is positively 
charged. The transport number for the solution Fe + 1 % C 



80 
60 

20 










r 








/ 










/ 








/ 








^ 


^ 







200 



230 
Temperature In "C. 



360 



Fig. 98. The percentage of ionic conduction in a-CuI at various temperatures. 



at 1000° C. was ~ 10-^. The measurements were extended 
to alloys of gold in lead(7i), and of gold in palladium or 
copper (72, 73), transport numbers being respectively lO^-^'' at 
200° C. and 10-" at 900° C. However, although ionic mobility 
may be demonstrated, it is of very slight importance compared 
with electronic conduction. 



The relation between conductivity and 
diffusion constants 

Nernst(74), and later Einstein (75), found a relation between 
the diffusion constant in electrolytic solutions and the con- 
ductivity. The relation is 

^= AT*' 



CONDUCTIVITY AND DIFFUSION CONSTANTS 269 

where B denotes the steady velocity, or mobihty , of the solute 
under unit force. Von Hevesy and his co-workers (76, 32), 
Braune(77), Tubandt, Reinhold and Jost{78) made use of an 
analogous expression in dealing with the conductivity of ionic 
crystals. The results justified the use of the Nernst equation 
at least qualitatively. 

Wagner (79) made a quantitative calculation of the relation- 
ships involved, using thermodynamic and kinetic properties 
of the crystals. His treatment applied to mixed crystals of the 
type CugS + AggS, PbClg + PbBrg, AgCl + AgBr. If one denotes 
by Y^ the equivalent fraction of the species * of valence Z^ , then 



i:z,n/ 



where n^ gives the number of gram ions of the substance i in 
the solution. If the species "t" is cationic, the summation 
HZ^n^ refers only to the cations. If it is anionic, the summa- 
tion refers only to anions. The concentration of the species 
i in equivalents/c.c. is 

if V is the volume of the system. The mean velocity u^ of an 
ion of transport number U^ is given by 

UiK 

when u^ is the velocity under a gradient of 1 V./cm., F is the 
Faraday, and K the specific conductivity. Since a field of 
1 V./cm. is equivalent to a force of 3^ {Zi€) dynes on the ion 
(e denotes the electronic charge), one obtains as an expression 
for the mobility B^, defined as the stationary velocity attained 
by the ion under a force of one dyne: 

300w, 



Z,e • 
If in a mixture of CugS 4- AggS one denotes silver, copper, 



270 INTERDIFFUSION OF SOLIDS 

and sulphur ions as species 1, 2, and 3 respectively, Wagner 
showed by considering both forced and natural diffusion that 






^2 -S2(-^l ^1 + ^3 ^z) ^2 ^(/^Ag^s) 



In these expressions, /«Ag,s denotes the chemical potential of 
AggS, as defined by Gibbs. When the solution of AggS in CugS 

is dilute, ^jf'^ = „ „ , and the expressions for D^ and D^ are 

'2 -'2 

Z^B.iZ^B^ + Z.B^) RT 1 



372 -^2^2' 






Y^Z^B^ + Y^Z^B^ + Z^B^ N, Z^ 
^^B^iZ^B^-^- Z^B^) RT 1 



When the mobility of the anion is small compared with the 
mobility of the cations*, the terms containing ^3 may be 
omitted, and the equations reduce to 

n _n Z^B^B^ RT ^ RT 

^1-^2 Z.Y.B. + Z^Y^B^N.-^'N, 

for dilute solutions of one salt in the other. 

The mobility ^3 may be obtained from the transport number 
of the cations in the solid solution by means of the relationship 

300w, 300 U.K. 



5o = 



Z^e F {Z2e)a, 



The relationship gives, in the simplest case, the connection 

between D and K, 

RT 300 UK 



D 



Nq F {Ze)C' 



* In a binary salt mixture, with a common non-diffusing anion, elementary 
considerations of electrical neutrality throughout the crystal will also show that, 

for natural diffusion, -—^=-^, and JJi-:r^= - Dz -5-* , if the cations have the 

dx ox ax ox 

same valency. Thus Dy = Di. 



CONDUCTIVITY AND DIFFUSION CONSTANTS 271 

In this connection mention should be made of Frenkel's 
relation between K and D (Table 76). This is 

In this expression N-^ denotes the number of ions per unit volume, 
and the factor of proportionahty turns out to be nearly unity. 
That the factor is actually small is illustrated by the expressions 
for the conductivity of lead iodide (Fig. 93), and for the self- 
diffusion constant of lead ions in lead iodide as measured by 
the method of the radioactive indicator. Thus 



K 



Pb+ + 



30.000 

= 1-15 X lO^e RT ohm-i cm.-i, 

30,000 

3-43x10^6 ^r cm.2day-i. 



For lead chloride, on the other hand, only anionic conduction 
occurs, and the equations are 



10,960 



Kq^_ = 6-55 e RT ohm-i cm.-i, 



38,800 

6-65 X 105 e RT cm.May-^ 



Table 65. 8 elf -diffusion of Ag+ in Agl from conductivity 
and diffusion measurements 



Temp. ° C. 


454 


500 


551 


594 


651 


701 


744 


i)(ca.ic.) cm.2 day-i 
D(obs.) cm.2 day-i 
a 


214 
1-53 
0-71 


2-68 
2-00 
0-75 


3-32 

2-48 
0-75 


3-86 

2-88 
0-75 


4-60 
3-46 
0-75 


5-25 ' 5-85 
3-98 4-19 
0-76 0-72 



Wagner's (79) treatment shows that the relationship 

^ = f^ 

.is only approximate. Tubandt, Reinhold and Jost (78) used the 
relation „„ 

where a is a constant. They calculated the diffusion constant 
of silver in silver iodide from the conductivity, using Nernst's 



272 INTERDIFFUSION OF SOLIDS 

equation, and compared the value so obtained with the self- 
diffusion constant (see below). Table 65(78) shows the value 
of a to be about 0-74. 

Diffusion constants in metals and ionic lattices 

It has been indicated how the diffusion constant in a salt may 
be calculated from the ionic mobility of the current carrier 
(p. 268), or from the use of a radioactive isotope as an in- 
dicator (p. 244). Von Hevesy(3i) considers that it will be 
possible to use the latter method to follow the self-diffusion 
of numerous metals (Table 59). The method can be extended 
in a few instances by using as indicators small quantities of 
certain salts in solid solution in a closely related salt (77, 78). 
For example, (78) small amounts of CuCl or NaCl were dissolved 
in AgCl. These mixtures are all cationic conductors in the tem- 
perature and concentration range investigated. The diffusion 
constants Dj^g^+ or -Dcu+ were measured, and also the conduc- 
tivity, and transport numbers of each ion. These latter data 
served to calculate the ionic conductivities. The self -diffusion 
constant, D^^+, was found for pure AgCl by assuming the 
correctness of the relation : 

Specific Ionic Mobility* of Ag+ in pure AgCl 
Specific Ionic MobiUty of Na+ in mixed crystal 

Self-diffusion Constant of Ag+ in pure AgCl 
Diffusion Constant of Na+ in same mixed crystal * 

It was established that, at 238° C, 

D^^+ = 3-5 X 10-6 cm.2day-i, 

where the specific mobility ratio = 25, and 

i)c„+ = 2-1 X 10-2cm.2day-i, 

where the specific mobility ratio = O-Ol. 

Then the above relation gives for the self-diffusion constant 
9 X 10"? and 21 x 10"^ cm.'^day^^ respectively. The agreement 
is satisfactory. 

U K 
* Defined by K^' — .A , where Yi and U^ are the mol fraction and transport 

number of species 1 in a crystal of conductivity K. For pure AgCl, A''Ag+ = K. 



DIFFUSION CONSTANTS IN METALS AND LATTICES 273 

The results of a great many investigations are summarised 
in the following tables (Tables 66, 67), It will be seen that the 
diffusion constants in salts may roughly be divided into two 
groups, in one of which the energy needed to render the ions 
mobile is greater than 10 k. cal., and in the other it is less. 
In the second group of electrolytes faU those substances such 
as a-AgI, Ag2Hgl4, CuBr, whose conductivity is unusually 
high. The Langmuir-Dushman expression (p. 298) for the 
diffusion constant is often used as a means of correlating the 
diffusion data. This equation gives for D 

Noh 

where d is the lattice constant, E the activation energy for 
diffusion, and h is Planck's constant. The empirical nature of 
this equation is stressed elsewhere (p. 299), but its applicability 
in many cases is outstanding. In Tables 66 and 67 in the last 
column are given the values of E calculated from D and d. 
The point to be noted here is that for the salts of group II of 
Table 66 the equation does not hold (Smekal(62)), but that as 
a rule when the activation energy is large the equation is 
satisfactory. 

Those diffusion systems which have been marked with an 
asterisk in Table 67 represent metal pairs which form a con- 
tinuous series of mixed crystals. As a rule in such systems 
the value of Dq is small, an important exception being self- 
diffusion processes (Pb in Pb :i>Q = S-l cm.^sec."!; Au in 
Au-.Dq — 1-26 X lO^cm.^sec."^). The normal range of values 
of Dq hes between 10~^ and 10"^ cm. ^ sec. ~^. There are ex- 
ceptions, however; for example, the diffusions of sUicon and 
tin into copper give Dq = 1-0 x 10* and 6-7 x 10^ cm. ^ sec. "^. 
Dq for the self-diffusion of bismuth, in a direction perpen- 
dicular to the c-axis, reaches a value of (1-33 — 16-3) x 10^. It 
is difficult to assess the reproducibility of some of the data, 
since some of the diffusion processes listed are structure 
sensitive, but those for bismuth appear reasonably con- 
sistent (see Fig. 100). It is to be noted, that the higher the 

BD l8 



274 



INTERDIFFUSION OF SOLIDS 



activation energy the larger the factor Dq, although there is 
no simple relationship between them, and exceptions also 
occur. Some of these exceptions are, however, for structure- 
sensitive diffusions (Th-W, Mo-W) where diffusion does not 
occur solely through the lattice. Here the number of channels 
available for diffusion may be restricted to grain boundaries, 

Table 66. Diffusion constants of ions in salts according 
to the equation D = Dq q-^i^t 









E (cal./ion) 








calcu- 




Do 


E 


lated by 


System 


cm.^ sec.-^ 


cal./ion 


Langmuir- 
Dushman 
formula 


Group I. Salts with E > 10,000 cal./ion 


Ag+ in AgCl(46) 




23,000 


19,300 


Ag+ in AgBr(46) 


— 


19,000 


21,000 


Na+ in NaCl(46) 


— ■ 


11,800 


35,000 


CI- in NaCl(46) 


— 


47,200 


38,600 ! 


Pb++ inPbCl2(46,80,32) 


7-7 


36,800 


34,500 


Pb + + in Pbl2(46,80,32) 


4-9, 10-6 


30,000 


29,000 


CI- in PbCl2(46,32) 


— 


11,000 


18,000 


Se" in a-Ag2S(62,8i) 


67 X 10-5 


20,040 


— 


Ag+ in a-Cu2Te(80,78) 


2-4 


20,860 


— 


Group II. Si 


ilts with E < 10,00 


cal./ion 


i 


I' in Pbl2(46,32) 





9,300 


4,300 1 


Ag+ in a-CuI (62) 


4-5 xlO-» 


6,760 


— 


Li+ in a-AgI(80,62,78) 


58-3 xlO-* 


4,570 


— 


Cu+ in a-AgI(80,62,78) 


16-3 xlO-5 


2,260 


— 


Cu+ in a-Ag2S(62) 


46 X 10-5 


3,180 


— 


Cu+ in a-Ag2Se(63) 


15-5 xlO-5 


2,940 


— 


Cu+ in a-Ag2Te(62) 


3-85 x 10-5 


2,660 


— 


Ag+ in a-Cu2S(62.8i) 


32-7 xlO-5 


4,570 


— 


Ag+ in a-AgI (46) 


— 


2.260 


— 



and so the factor Dq becomes correspondingly small. On the 
basis of Eyring's(99) theory of diffusion (p. 302) a large value 
of Dq implies a big entropy increase on passing into the 
activated state, or a large disturbance in the lattice in this 
state. The larger disturbances are usually found in self- 
diffusion processes, where the chemical similarity between the 
diffusing atom and the solvent is a maximum. One has in the 
large and small values of Dq an analogy with the "fast'' 



Table 67. Diffiision constants in metals according 
to the equation D = DQe~^l^^ 









E (cal./ 






E 


atom) from 


System 


Do 


Langmuir- 




cm.2 sec.-i 


cal./atom 


Dushman 
equation 


*Pb in Pb (32, 46, 80) 


5-1 


27,900 


24,400 


^-Tl in Pb (46, 80, 82) 


3-7 xlO-2 


21,000 


22,400 


;^-Sn in Pb (46, 80, 82) 


3-4 xlO-i 


24,000 


23,200 


Au in Pb (46, 80, 71, 83) 


4-9 xlO-i 


13,000 


13,300 


Ag in Pb(8o,83) 


7-5 xlO-2 


15,200 


— 


Bi in Pb(80,82) 


7-7 xlO-» 


18,600 


21,900 


Hg in Pb(80,84) 


3-6 xlO-i 


19,000 


— 


Cd in Pb(8o,84) 


~2-l xlO-2 


-18,000 


20,000 


Zn (9-58%) in Cu (80,22) 


3-2 xlO-2 


42,000 


41,000 


Zn (29-08 o/o) in Cu (80,22) 


5-8 xlO-* 


42,000 


38,000 


Sn (10%) inCu(85,86) 


— 


40,200 


40,000 


*Au in Au(33) 


1-26 x 102 


51,000 


— 


*Pd in Au(80,87) 


Mix 10-3 


37,400 


— 


*Cu in Au(80,87) 


5-8 xlO-* 


27,400 


— 


*Pt in Au(80,87) 


1-24 x 10-3 


39,000 


— 


fTh in W (small grains) (85, 8S) 


7-5 xlO-i 


94,000 


96,700 


fTh in W (large grains) (85,88) 


4-1 xlO-3 


94,400 


118,200 


Th in W (volume) (35) 


1-0 


120,000 


— 


fU ia W(85,89) 


1-0 


100,000 


100,500 


tYt in W(83,89) 


0-46 


68,000 


70,100 


fCe in W(S5,89) 


10 


83,000 


82,700 


fZr in W(85,89) 


10 


78,000 


77,400 


*tMo in W (polycrystal)(46,80,90) 


5 X 10-3 


80,500 


— 


*Mo in W (single crystal) (46, so, 90) 


6-3 xlO-* 


80,500 


— 


N2 in Fe(85,23) 


1-07 X 10-1 


34,000 


38,100 - 


C in Fe (80, 23, 91) 


4-9 xlO-i 


36,600 


36,700 


tC in W2C(62) 


— 


108,000 


— 


Znin (Cu + 4%Zn)(92) 


1-57 X 10-3 


34,100 


— 


Alin (Cu + 4% Al)(92) 


6-34 X 10-2 


40,400 


— 


Si in (Cu + 4 0/0 Si) (92) 


1-0 xlO* 


64,200 


— • 


Snin(Cu + 4o/oSn)(92) 


6-7 xl02 


54,000 


— ■ 


*Cu in Ni(80,93) 


104 X 10-3 


35,500 


— 


Cu in Ag(29,80) 


5-9 xlO-5 


24,800 


— 


Sb in Ag(29,80) 


5-3 xlO-s 


21,700 


— 


Sn in Ag(29,80) 


7-9 xlO-5 


21,400 


— 


In in Ag(29,80) 


7-3 xlO-5 


24,400 


— • 


Cd in Ag(29,80) 


4-9 xlO-5 


22,350 


— 


*Au in Ag(80,94,95) 


M xlO-* 


26,600 


— 




5-3 xlO-i 


29,800 


— 


*Pd in Ag(8o,87) 


4-16 X 10-« 


20,200 


— 


Au in Ag-Au 02) 


5-2 xlO-* 


29,800 


, 


Ag-Pd (20% Pd) in Ag-Au 02) 


7-0 xlO-« 


20,200 


— . 


Au-Pd in Ag-Au (92) 


8-3 xlO-* 


37,400 


— . 


Cu in Ag-Au (92) 


106x10-3 


27,400 


— 


Au-Pt in Ag-Au(92) 


1-28x10-3 


39,000 


— 


fBi in Bi (_Lc-axis)(96,80) 


(1-33-16-3) xlO« 


137,000 


— 


fBi in Bi (||c-axis)(96,80) 


(2-2-6-5) x 10-* 
to 6x10-* 


30,000 


— ■ 


Cu in Al(97) 


2-3 


34,900 


31,400 


Mg in Al(97) 


1-5 xl02 


38,500 


29,000 


Cu in Cu(33a) 


1-1 xlOi 


57,200 


— 


Al in Cu(98) 


1-2 xlO-2 


37,500 


— 


Zn in Cu(98) 


8 X 10-1 


38,000 


44,000 


Sn in Cu(98) 


10 


45,000 


— 


Si in Cu(98) 


5-2 xlO-2 


39,950 


— 


Be in Cu(98) 


4-5 xlO-5 


27,900 


— . 


Cd in Cu(98) 


3-5 xlO-3 


8,200 


— 


ZninCu4-20% Zn(98) 


— 


31,000 


38,500 


AlinCu-hl60/o Al(98) 


— 


54,000 


39,000 



* Interdiffusing metals form a continuous series of solid solutions. 
j Diffusion may show structure sensitivity (see next Chapter). 



276 INTERDIFFUSION OF SOLIDS 

and " slow " reactions of chemical kinetics. In the former case 
the kinetic theory suggests accumulation of energy through 
many degrees of freedom, a viewpoint corresponding with 
a large disturbance of the lattice. A tentative explanation 
of the influence of chemical and physical similarity of 
solute and solvent upon the diffusion is advanced on 
p. 285. 

It is also noteworthy that on the whole the activation energy 
for diffusion and for conductivity (Table 64) in ionic lattices 
is smaller than the activation energy for diffusion in metaUic 
lattices. It is probable that this difference may be traced to 
the greater influence of polarisation forces in the case of salts, 
and to the greater density and so closer atomic packing in 
the case of many metals. 

Diffusion anisotropy 

When diffusion occurs with different velocities along different 
crystallographic axes one has the phenomenon of diffusion 
anisotropy. It is of interest to see what examples can be found 
of diffusion anisotropy in salts and metals. This property has 
already been encountered in the sorption of gases by zeoUtes, 
heulandite(ioo), for example (Chap. Ill, p. 102), giving a very 
great anisotropy indeed, and being in one direction (perpen- 
dicular to the anionic laminae of this layer crystal) quite 
impermeable, and in two other directions easily but different^ 
permeable. It should be noted that anisotropy in diffusion 
has never yet been observed in cubic crystals. One finds that 
diffusion of zinc into copper (loi), of oxygen into copper-silicon 
alloys (102), and of carbon (i03) or nitrogen (io3) into iron, occurs 
with equal velocities in all directions. 

Seith(82) showed that the conductivity of lead iodide could 
be expressed by the relation 

K = 9-78 X "10-4 e-93oo/i?r+ 1.15 X 105 e-3o.ooo//fr^ 

in which expression the first term refers to iodine ion transport 
and the second to the lead ion transport. The equation applied 



DIFFUSION ANISOTROPY 



277 



to a compressed pellet of iodide for conductivity in the direction 
of the pressure, which caused the crystallites to orient them- 
selves so that their c-axes coincided with the direction of 
pressure. The conductivity as shown by the following figure 
depends on the direction of diffusion (96) in the' crystal to a very 
marked extent. Fig. 99 shows log X(Z' = conductivity) plotted 
against l/T {T = temperature in ° K.), and the curves 2a and 
26 are logK-ljT curves of different samples of lead iodide 
perpendicular to the c-axis, while 1 shows this conductivity 
curve parallel to the c-axis. The former curves have slopes of 
about 9000 cal. and the latter has a slope of 30,000 cal. One 



275° 300° 325° 350° 375° WO' 




J9 18 77 

t/Tx 10* (T!n''K.}—*- 

Fig. 99. Diffusion anisotropy in lead iodide. 

concludes that the iodide ion carries most of the current normal 
to the c-axis, but the lead ion carries most of the current 
parallel to it. The anisotropy is several powers of ten at low 
temperatures, but is naturally temperature dependent, and 
seems to vary somewhat for different specimens. The curves 
3a and 36 are plotted from the two terms of Seith's equation (32) 
given in this paragraph For the self-diffusion of lead ions 
in lead iodide the following diffusion constants were obtained 
(Table 68). 

The table shows that diffusion anisotropy for lead ions may 
exist, but is slight compared with the anisotropy of the iodide 
ion. Like heulandite, lead iodide is a layer lattice, and it is 
the layer structure which is responsible for its behaviour in 
diffusion. 

Both Warburg and Tegetmeier(i04) and Joffe(i05) observed 



278 



INTEKDIFFUSION OF SOLIDS 



Table 68. Self-dijfusion of lead ions in lead iodide 
{cm.^ day~^) {using ThB as radioactive indicator) 



Number 



Temp. ° C. 



D (cm.^ day~^) 



Remarks 



323 
316 
306 

278 
262 

317 
314 
300 

268 



6-14 X 10-« 
5-02 X 10-« 
2-05 X 10-« 
5-94 X 10-^ 
3-31 X 10-' 

2-50 X 10-6 
2-57 X 10-« 
3-26 X 10-6 
2-67 X 10-' 



_L to c-axis 

_L 






anisotropy in the conductivity of quartz. Perpendicular to 
the crystal axis the conductivity 
is 10*~^-fold greater than parallel 
to the axis, although the tem- 
perature coefficient was the same 
in both directions. A third ex- 
ample of diffusion anisotropy is 
provided by bismuth (96). Self- 
diffusion parallel and perpendi- 
cular to the c-axis, using ThC as 
radioactive indicator, gave an 
anisotropy of up to 10^-fold 
(Fig. 100). The figure shows 
that parallel to the c-axis the 
activation energy for diffusion is 
30 k.cal., but that perpendicular 
to it the activation energy is 
137,000 cal. 

Another study of the same 
type (106) was made of the dif- 
fusion of mercury into single 
crystals of cadmium or zinc. In 
both cases the diffusion was a 
maximum parallel to the basal 
planes and a minimum perpendicular to them. On the basal 
planes mercury drops gave circular diffusion, while all other 



2 1 



i/tx 10t(Tin''K.) > 

Fig. 100. Self-diflFusion in Bi. 
(1) Parallel to c-axis, (2) per- 
pendicular to c-axis (Seith(96)). 



DIFFUSION ANISOTROPY 279 

planes gave ellipses with the major axis parallel to the basal 
plane. The ellipticity diminished as the temperature was raised. 
Similarly, Spiers (107) observed, during the spreading of mercury 
on tin, that the mercury diffused into ellipses. 



The influence of concentration upon the diffusion 
constants in alloys 

When silver ions move in a silver halide lattice the concen- 
tration of ions within the lattice is fixed by the number of 
lattice points available to silver, per unit volume. When, 
however, gold diffuses into lead, or lead into gold, concen- 
tration gradients of one metal into the other are established, 
and it is then necessary to find whether the diffusion constant 
D varies with the concentration, in order to interpret the data. 

dC d^C 

This means that one must use not the equation ^^ = D^^-j 

ot ox^ 

but the equation -^ = ^ I Z) ^ I and the method of 

Matano(i08) (Chap. I). The concentration gradients may be 
measured by any of the methods outlined for the normal Fick 
law, the most usual being by spectroscopic, X-ray, or chemical 
analysis of thin layers adjacent to the interface. 

The application of the equation -^ = ^ I Z) -^ I to the 

copper-nickel system then gives for the variation of D with 
the percentage of copper the curve of Fig. 101 (108). The value' 
of D does not alter very markedly until the copper constitutes 
80 % of the alloy. Then, however, a rapid increase occurs. 
Fig. 102 shows similar effects of composition upon the diffusion 
constants in Au-Ni, Au-Pd and Au-Pt alloys. As the per- 
centage of alloying metal increases the constants at first remain 
fairly steady with decreasing gold content, but after a certain 
threshold value a much more rapid increase is observed. 
Similarly, Table 69 shows the diffusion constants of a number 
of metals in copper at 750° C, along with other data which 
will be discussed later. It can be seen again that the diffusion 



280 



INTERDIFFUSION OF SOLIDS 



constants alter with changing concentration of the diffusing 
metal. 

The diffusion of Al, Be, Cd, Si, Sn and Zn in copper was 
subjected to the Matano method of analysis by Rhines and 
Mehl(98), with the result that in every instance the diffusion 
constant was found to vary with concentration. In all cases 
the diffusion in nearlypure copper was less rapid than diffusion 








r » 

d 




. 


/OOO'i 




3 




J 








• J 


V- 


* 






rir 




Tt" ' • 





%Cu 




CONC. IN ATOMIC PEHCENT AU 



Fig. 101. Fig. 102. 

Fig. 101. The diffusion constant as a function of composition for a nickel-copper 

aUoy (Matano (108)). 
Fig. 102. The diffusion constant at 900° C. for the inter-penetration of Au and 

Ni, Pd and Pt as a function of composition of the Au-Ni,^ Au-Pd, and Aii-Pt 

aUoys (Mehl(92) after Matano (los)). 

in the alloy (Fig. 103) and rose at first slowly and then rapidly 
with increasing amounts of the solute. It is made apparent 
from these data that the diffusion constant is generally con- 
centration-dependent, and that the application of the normal 
Fick equation to metal-metal systems can yield only an average 
value of D, and may therefore lead to errors in evaluating Dq 
and EinD = D^e-^l^'^ . Many more diffusion studies, using the 
Matano method, are essential before the relationships between 
Dq and E can be put upon a quantitative basis. 

From measurements of D which have been derived by the 



Matano treatment of the law 



dt 



dx \ dxj ' 



trends in the 



INFLTJENCE OF CONCENTR ATION 



281 



S5h 
O 



o 

=0 



C3 

CD 

Hi 

pq 
<! 





is 


CO 


^1 


fs 


o 

(M O , 


Co 




eS iM a> 


Xt 


fs 


M 1 ^t- K^M 




H ^<M 




•^ -1 


CO 1 1 ! 


(D >3 


00 


00 1 1 


CO O 






cSin 






rfl t3 




O 


Is 


^^ 


=pS 1 

CO 


^2 




o 1 1 ^ 1 S 




•^ ' p^ 1 ^ 






00 


S t*^ 


f-H 


o 1 1 






o 




5r< 


-^ 1 


2°" 


'-* o 


±100 




^ 1 


^^ 








^ 


1 
1 




o^fl 


^11 i 


g >> 


ON 


w M 1 


c3 O 




1 


-^^ 







i4 




«>5 1 




<n 


fl 6>- 






tSJ(N 


oN 




Q ^ ■* 






■^ 1 1 


* >> 


ON 


CO 1 1 


cS O 






^;=! 







i4 


$5fl 


>-o2 r 


^N 


- ^" ' ! 


d CD 




CO ^ ' 1 


l-CO 




^ 1 ftg? CO 2 


N'-' 


§^ 




-l^ j 






. <i 






^ --^ rt 






-S 


1 

o 




1 l^f? 
_ Eel t; ® a -5 i 

>; § :S ts 2 ° 


& 




i 




f r's^.ScS^ 


u 




• C ^ -►^ as ^ — '+^ 


1 




= jn <^ S '^ 


<o 




m 




rH Tl ^ ^-H (Lj ^ 1 

1 -g ,-H S c; M 

2 > ^ ^ &a -S 







60 



^20 
\ 







REVISED D AT 800 °C 






/ 


/ 












BERYLLIUM AND COPPER — . 1 
CADMIUM AND COPPER — (Extrapolated} 
SILICON AND COPPER——-' | 


/ 


/ 








TIN AND COPPER—^- 


— 




/ 






/ 






1 , 






/ 

/ 

/ 
/ 




1 
1 

1 






J' 


y 










/ 
/ 


f 


/ 






■^ 


/ 










^ 


-»rr: 




'^^ 


^ 


"^ 














y 







8 10 12 14 16 18 

CONCENTRATION IN ATOM PER CENT 



Fig. 103, Selected average diffusion coefficients interpolated to 800° C. for 
Cu-Al, Cu-Be, Cu-Cd, Cu-Si, Cu-Sn and Cu-Zn systems (Rhines and 
Mehl(98)). 



/ 



y 





^\ 



T 



ALUMINUM AND COPPER 
BERYLLIUM AND COPPER 
CADMIUM AND COPPER 
SILICON AND COPPER 
TIN AND COPPER 
ZINC AND COPPER 



2 4 6 a 10 12 14 16 

CONCENTRATION IN ATOM PER CENT 

Fig. 104. Relationship between E and the concentration for the systems 
Cu-Al, Cu-Be, Cu-Si, Cu-Sn, Cu-Zn (Rhines and Mehl m). 



INFLUENCE OF CONCENTRATION 283 

activation energy for diffusion may also be obtained. These 
trends (Fig. 104) in E with concentration are usually initially 
slight, but may become large and rapid at higher concen- 
trations. The data of Mehl(92) for the system Al-Cu follow an 
approximately linear law E oc [cone, of Al]^. The values of the 
energy of activation for the diffusion process can be seen in 
Fig. 104(98) both to increase and to decrease with increasing 
concentration of the alloying material. In some instances, 
therefore, the solute is more easily loosened in the lattice by 
the alloying process, in others the converse is true. 

Summary of factors influencing diffusion constants 

Some factors which influence the diffusion velocity and con- 
ductivity have already been indicated: 

{a) It has been seen that, when certain types of disorder 
exist in a crystal lattice, an increasing partial pressure of the 
electronegative component may alter the conductivity (O2 in 
FeO, CuaO, MO, ZnO, CdO; S in FeS; Brg in CuBr; I in Cul). 

(6) The constants D, Dq and E in the equation 

have been shown to depend very much upon the concentration 
of solute in the solvent metal. 

(c) The importance of polarisability has been observed. The 
data of Table 66 showed how a high polarisability of ions in a 
lattice may reduce the term E in the conductivity equation 
K = Ae-^l^T. In polarisable salts (Agl, AgaS, CuBr) the 
activation energy E is small ( < 10,000). 

(d) Lehfeldt's diagram (Fig. 97), showing the connection 
between E for a series of alkali halides and the radius of the 
halogen ions, indicated that with increasing radius of the halide 
ion the energy E diminished; and with increasing radius of 
cation E increased. This relation may also be due in part to 
polarisation. 

It is now interesting to see what other factors can alter 
diffusion constants. Tables 70, 71 and 72 give the diffusion 



284 



INTERDIFFUSION OF SOLIDS 






o 



o 
< 















'A s 
«.2 

^ 00 


7 o . 








S ® o o 




(M 




xg^F^;^ 


•-< 






P-i;g 


l> 




























CO 


1-6x10- 
24,000 

Tetrag 
29 


00 


IN 

CO 


























L<=> ■ 








1— < 


O o O 


i-H 






H 


31x1 
21,0 
F.c. 

79 


p-H 


O 
CO 
























m 


4-4 X 10- 
18,600 
Rhombi 
35 


00 




























2x10 
18,00 
C.p.h 
17 




(M 

CO 






(35 


^ 


o 

CO 






» 








< 


2 o o lo 


Tt* 


(M 




^^-'iS 


-* 


o 




coSfe^ 


'"' 








■* 












-ri 


1 






^ 


o 


o 






O a>3 


^ 


-2 








45 


=3 






S >.g 1 


ce 


o 














1 oj -^ 9 




O 

-»^ 1 




^ 


cm. 2 sec. 
ctivation 
ystal hab 
aximum t 
atomic % 
tomic rad 


-1 

a. 9 






Cl«50^ 


^<J 


u.g 



•I 



H 



O ti o 



(M 1—1 



SSI 



Oo o o 
'-' CO o O 

CO 



^ o -*^ 
^.-2;:-, 



J-, O SiD 

2o g, 



CO (ME-I 



3> . O 

X ,-H O 
tH (M -^ 



CO 



^ ^ o 

c« Qi O 3 

I OJ -rH 9 



' o • _ 

"^ >« -^3 -, M 

■— I CO ^ (^ S; 

^ (N O '^ 

05 






CO 



O 



Ci S cS 

« O J3 

^. OS cs g 



a<^ 

o ^^ 



o 

CO 



o 



3 C 



i.s 






a-s 



O ^ O (U 



FACTORS INFLTJENCING DIFFUSION CONSTANTS 285 

constants of metals in lead, in silver, and in noble metals (92). 
Table 69 (p. 281) gives the same data for copper (92). Examina- 
tion of these and other data indicates the following properties 
of diffusion systems : 

- (i) The melting-point and atomic radius of the solute show 
no direct connection with the diffusion constant in lead and 
sUver (Tables 70 and 71), since the trend is in opposite directions 
for lead and for silver. 

(ii) The diffusion constant is smaller the greater the melting- 
point of the solvent. 

The influence of the melting-point of the solvent upon the 
diffusion constant is large, and there exists in a number of 
instances a well-defined relationship between the energy E in 
the equation D = DqC^^'^^ and the melting temperature: 



System 


Cu in Au 


CuinNi 


Moin W 


Melting-point of solvent temp. ° K. 


1356 
20-2 


1728 
20-6 


3743 
21-3 



The influence of the melting-point upon the diffusion constant 
was the basis of attempts by Braune (94) and by van Liempt (109) 
to incorporate the empirical relationship of Table 76 in 
equations for the diffusion constant D = DqC^^/^^ by writing 
E = b^{T^/T), where 6 is a constant. These formulae will be 
referred to later (p. 301). 

(hi) The diffusion constant depends inversely upon the solid 
solubility, being least for metals which form a continuous 
series of mixed crystals, or for self-diffusion. 

The influence of solubility may be explained in the following 
way (cf. p. 274), When the atoms of solute and solvent are 
identical, the solute occupies a lattice site in the crystal without 
distorting the lattice. When the atoms of solute and solvent 
are dissimilar, the solute atom distorts the solvent lattice until 
in the extreme case it is thrown out of solution. The difference 
in degree of disorder between the normal and transition states of 
the solute atom before and in the act of diffusing respectively 



286 



INTERDIFFTJSION OF SOLIDS 



O 



o 




(© 




Ci 




,s£ 




•c<. 




o 




^ 




r>. 




S 


' ." 




O 


■ai 


o 




CO 


„ 


O 


< 


1 — 1 


(N 


g 


^ 

O 


o 

?5. 


1— I 


iJ^ 


II 


S 




•«<» 


5- 


(*S 




w 


bC 


S 


^ 


^ 




on 


a 


CO 


•iS> 


TtH 






1-1 
w 














i-( 











1 




^3 1=3 





1 


1 1 













iH i-< 






'S' 


^^^^ 


w 





p:^ 


X ^ X '^ 


00 

CO 





a 


6 6 


""^ 






e . 








-;< 






S^' 




1 




<j3 =8 




1 















rH 








1 fH 






^— V 


2 .2 .2 






P 


(M 


lO 


T3<| 


*t^ 1 1 'tH 1 


t> 


lO 


P-I 


x^ X - 1 

6 "^ 


CO 


»o 













^ 








1 '■' "-, 






'? 


2 2 


tH 


(M 


.9 


6 «= 


^ 






01 






3 


1 









X 1 


1 


1 


<1 *o 









'"' 


10 






T3 


OS 
1 _^ 






Ag-Pd 
alloy 
0%P 


3 

-7* (N 


1 


1 


M 








^ 








^'^ 
^ =* 


X 
«9 (M 






be 


OS 
1 






Au 
diffusin 
in Ag) 




X "^ 

CO oT 


00 

CO 


CO 




' — ' 


■-I 










:s2 










^jT 


c ^ 


1 


1c§ 


G 


.2 a 

03 03 


i^ % 


'0 


3 -is 


a 05 -S 




:2 ic 




Q =* ^ 


«^- 


»go 



FACTORS INFLUENCING DIFFUSION CONSTANTS 287 

is therefore much less when solute and solvent are dissimilar 
than when they are similar. The additional energy needed to 
loosen the lattice sufficiently for diffusion to occur is accord- 
ingly less for dissimilar than for similar atoms. At the same 
time the entropy of activation is greater for the similar than 
for the dissimilar atoms. This entropy of activation is defined 
by (p. 303) 



D 



B^e-^iRT^ 






3-5 



<fO 




Ag(Au) 



Fig. 105. Diffusion constants of metals in silver at 800^ C. 
(Z) in cm.^ day~^.) 

where A denotes the mean free path (identified normally with 
the lattice parameter), h is Planck's constant, and k is Boltz- 
mann's constant. These two effects act in opposite directions, 
but the net result is a diminution in D with increased solid 
solubility. 

Data giving the difi'usion constants of a number of metals 
in lead are given in Fig. 106(46,92). Here also one sees the 
diminishing diffusion rates with increasing soUd solubihty. An 
interesting method of representing the diffusion data in lead 
and in silver has been adopted by Seith and Peretti(29), and 
the data given (Fig. 105(56)) illustrate in another way this 
dependence of the diffusion rate upon the mutual solubility. 
These figures also show definite trends in D with the position 



288 



INTERDIFFUSION OF SOLIDS 



of the solute and solvent in the periodic table. This position 
governs in part the mutual solubility of the diffusing elements. 
It may be considered as surprising that so far no relationship 
between diffusion velocities and the atomic radii of solute and 
solvent has emerged. That such relationships may exist has, 
however, been shown by Sen (iio), 

- — =-— /v) pnn pv) inn 

who demonstrated the following 
rule. In a pair of sohds M and 
N the direction of most rapid 
diffusion is from M to N when 
the minimum distance of ap- 
proach of atoms in N is greater 
than the same distance in M. The 
rule is illustrated by Table 73. It 
is thus to be anticipated that 
small atomic size of solvent will 
usually favour slow diffusion, 
unless the solute diffusing is of 
even smaller atomic radius. 

Valence and polarisation ef- 
fects are observed in metals as 

in ionic lattices, especially where zinc, cadmium, mercury, and 
thallium act as solvent metals. Similarly with copper, silver, 



• 




^j^-^ 


^ 


.^^ 




^ 




- 






Y" 




s^fp^ 




^ 


1 







^i ^^4 ej ^^ ^l po i9 is it 

1/7x10* 

Fig. 106. Diffusion constants of 
various metals in lead (D in 
cm.^ sec.~^). 



Table 73. Ejfect of atomic radius upon direction 
of diffusion 





Minimum distance of 


Direction of 


System 


approach in cm. 


diffusion 


Cu-Pt 


Cu : 2-54 X IQ-* 


Pt : 2-78 X 10-8 


Copper into platinum 


Cu-Zn 


Cu : 2-54 X lO"* 


Zn:2-67 x 10-8 
2-92 xlO-8 


Copper into zinc 


Fe-Ag 


Fe : 2-54 x 10-^ 


Ag : 2-876 x lO-* 


Iron into silver 


Au-Pb 


Au: 2-88x10-8 


Pb:3-48 xlO-8 


Gold into lead 


Fe-C 


Fe : 2-54 x 10-^ 


C : 1-50 X 10-8 


Carbon into iron 



gold, and some B subgroup metals with atomic radii not very 
favourable for solid solution, and with which valence effects 
occur also to a certain extent, one obtains numerous alloy 
phases. The metals of group 8, on the other hand, have atomic 



FACTORS INFLUENCING DIFFUSION CONSTANTS 289 

radii favouring solid solution, so that electrovalence effects are 
subordinated and intermediate phases not so numerous. 
Although in metals ionisation and deformation are hard to 
measure, it has already been indicated (p. 267) that Au-Fe, 
Au-Pd, and C-Fe systems may be electrolysed so that there 
is definitely polarisation of the constituents. Frenkel's 
theory (47,44) (p. 293) of ionic conductivity illustrates the great 
importance of polarisation forces in decreasing the energy 
needed to render an ion mobile in an ionic lattice. The 
polarisation is a maximum when the electron affinity of the 
cation is large, and of the anion small. In Fig. 107 we may, 



I.SJe 





^hg^Sb 



Pig. 107. The relative mobilities of silver ions in various silver compounds, 
arranged around a parabola. On the right is the relative electronic con- 
tribution to conductivity. 



after von Hevesy (46), arrange the mobilities of silver in a series 
of different lattices (ionic and metallic) around a parabola. 
It is seen that the ionic mobihty rises strongly from fluoride 
to telluride as the electron affinity of the anion diminishes, 
or its polarisability increases. On the right-hand side of the 
parabola it is shown that the small self-diffusion rate of silver 
in silver is raised by the addition of tin or antimony, while the 
electronic conduction is decreased. Similar considerations 
apply when the anion is the same and is combined with various 
cations. 

The increased mobility of silver in a silver-antimony alloy 
as compared with a pure silver lattice may be considered as 
due to the loosening of the silver lattice by distortion due to 
introducing antimony. Von Hevesy (46) suggested that as a 

BD 19 



290 INTERDIFFUSION OF SOLIDS 

qualitative measure of lattice loosening in ionic compounds 
one might employ the ratio 

Conductivity above the melting-point 
Conductivity below the melting-point * 

The ratio is great in many instances, but in a few cases 
(a-AgI, a-CuI) it is less than unity (Table 74). As another 
method of comparing the relative loosening of two metallic 
lattices one may use the ratio 

Self-diffusion constant for first metal 
Self-diffusion constant for second metal " 

This ratio for lead and gold at 326° C. is 26,000, and one sees 
that the lead lattice is loosened but the gold lattice is not at 
this temperature. 



Table 74. The ratio of conductivities above and below 
the fnelting -point 



System 


Ag+in 
AgCl 


Ag+in 
A'gBr 


Ag+ in 
Agl 


Li+in 
LiCl 


Na+in 
NaCl 


Na+in i 
NaNOa ' 


Conductivity ratio 
above and below 
melting-point 


16 


2-5 


0-5 


1-0 
xlO* 


1-5 

Xl05 


10 
xl05 j 

1 



One of the most interesting phenomena alhed to the problem 
of diffusion in metals is the order-disorder transformation in 
alloys. The terms order and disorder are being used in a 
different sense from that previously used when discussing 
vacant sites and interstitial ions in a lattice, as will be seen 
when the transformation is described. When a binary alloy 
with constituents in the ratio 1 : 1 or 1 : 3 is cooled, the melt 
sohdifies to give a crystal. At these high temperatures all the 
atoms, though regularly arranged in sjDace, are interchanged 
at random through the lattice. A given lattice point is as 
likely to be occupied by an atom of one kind as the other. As 
the lattice is cooled a reorganisation occurs which sets in at 
a fairly 'definite temperature, in \Vliich certain of the lattice 
sites tend always to be occupied by one type of atom, and 



FACTORS INFLUENCING DIFFUSION CONSTANTS 291 

other definite sites by the second type of atom. In the alloy 
CugAu, for example, at high temperatm-es Au and Cu atoms 
are distributed at random in the face-centred cubic lattice, 
while at low temperatures gold occupies cube corners and 
copper face centres. The transitions may be followed by 
resistance or X-ray measurements, and as a result of many 
studies (110 a) it appears that the process of ordering may not 
reach its equilibrium state over a long interval of time at 
temperatures below the critical temperature of the trans- 
formation. This is because small zones of order are set up 
through the crystal which are out of phase with each other. 
The incorporation of one zone in another to give a homo- 
geneous ordered phase is then very slow. 

In any one zone undergoing the transformation disorder to 
order the time, t, required for a given fraction of the trans- 
formation should be 

where ^ is a constant and E is the energy barrier which must 
be surmounted before the reorganisation occurs. This energy 
barrier must be very similar to that which occurs during 
diffusion, for in both cases a momentary mobility must be 
imparted to the atom. Owing, however, to the numerous 
antiphase nuclei which form through the crystal, the simple 
expression T = ^e^'^^ is not vahd(iio&), for the slow coalescence 
of antiphase nuclei is occurring simultaneously, and one cannot 
measure the change in the separate nuclei independently. 

Models for conductivity and diffusion 

PROCESSES in crystals 

From a number of different viewpoints expressions have been 
derived for conductivity or diffusion constants in crystals. 
A number of types of conduction may be recognised as 
theoretically possible, and examples of some of these are well 
estabhshed. It is now convenient to review some of the 
models for diffusion and also equations for the diffusion 
constant which have been derived by a number of workers. 

19-2 



292 INTERDIFFUSION OF SOLIDS 

An important type of ionic diffusion is that which 
occurs in zeolitic structures. In zeohtes one has an anionic 
framework with large interstices in which exist the cations 
necessary for electrical neutrality. The anionic framework is 
sufficiently open to allow cations to pass through it, along 
cation channels, without any appreciable loosening of the 
anionic network. Thus one may often exchange one cation for 
another in these interstitial compounds. It is probable that 
the diffusion of hydrogen as atoms or ions in the metaUic 
lattices of palladium and similar metals may also be referred 
to this type. This is also the mechanism by which gases flow 
into zeolites, or through sihcate glasses and organic mem- 
branes. Von Hevesy(46) regarded the very rapid diffusion of 
gold into lead as zeohtic, and indeed considered that most 
examples of rapid diffusion in pure metals occurred by zeohtic 
diffusion. The process of amalgamation may perhaps be 
regarded as a zeohtic diffusion, accompanied simultaneously 
with a disintegration or reorganisation of the solvent lattice 
into amalgams. In order, however, for diffusion in metals to 
be truly zeohtic the solute atoms should be small enough to 
fit interstitially between the solvent atoms.* It is conceivable 
that a metaUic lattice could be formed in which one kind of 
atom was freely mobile, while the other kind of atom formed 
a more rigid lattice framework. Such a system would compare 
with the anionic disorder in lattices of Agl, or AggHgl^ (p. 263). 

A picture which is often given of the process of inter- 
diffusion of metal pairs supposes place exchange by a thermal 
loosening of the lattice sufficient for the molecules to pass 
round each other, as distinct from Frenkel's mechanism (p. 293) . 
It has been illustrated in Fig. 108. This model has not passed 
uncriticised. Bernal(iii), for example, considered that the 
activation energy needed for the loosening would be too 
great but that diffusion might occur by spontaneous small 
gliding processes along different planes. As with the Frenkel 
mechanism, however, the polarisation energy may consider- 

* The atomic radii of gold and lead are respectively 2-88 A. and 3-48 A., 
probably not sufficiently different for interstitial solution. 



CONDUCTIVITY AND DIFFUSION PROCESSES 293 

ably diminish the energy needed to create the momentarily 
disorganised lattice. 

On the quantitative side one may indicate the theory of 
volume diffusion and conductivity in ionic crystals due to 
rrenkel(47) and extended by Jost(44,56). The theory is based 
upon the concept of equilibrium disorder introduced in a lattice 



ooooo 
ooooo 

00(2)00 

oo®oo 
ooooo 
ooooo 

/ 

Undisturbed lattice 



OOOOO 

oo o oo 

OO @ X oo 

oo ^ ® oo 

oo o oo 

ooooo 

// 

Lattice momentarily loosened to 
permit place exchange of ' 
molecules A and B 



Fig. 108. A place exchange process as pictured in a crystal lattice. 



• ••••• 

Fig. 109. Frenkel disorder in a two-dimensional lattice. 

by the thermal energy (p. 248). Thus in Fig. 109 it may be 
supposed that the atom or ion at the position (a) has gained 
sufficient energy for it to migrate to the position (6), while at 
any temperature the average number of vacant sites and inter- 
stitial atoms or ions is constant and in equilibrium with the 
normal lattice. Diffusioncanensueby two mechanisms: an atom 
may move over a potential energy barrier E^^ into the vacant 



294 INTERDIFFUSIOlSr OF SOLIDS 

site, giving a diffusion of the vacant sites; or an' interstitial 
ion may move over an energy barrier to another interstitial 
position. When 

n — the number of vacant sites/unit volume, 
N — the number of lattice sites /unit volume, 
Eq = the energy needed to create a hole, 

Frenkel(47), by kinetic theory, and Jost(44,56), by statistical 
mechanics*, showed that ?^ ~ jSfe-^ol^R^. Only a fraction of these 
holes, or corresponding interstitial atoms, will diffuse how- 
ever, because an activation energy is necessary for diffusion. 
Thus the number moving is proportional to e~^^o+2^i)/2^2^. 
An approximate value of D, the diffusion coefficient, can be 
deduced if it is assumed that there are six ions around each 
hole, distant d from its centre, and capable of moving with 
the mean thermal velocity in any one of six directions. Each 
of the six particles may move in a single direction (to the 
hole), so the six of them are equivalent to a single particle 
free to move in all directions. Therefore the diffusion constant* 
for a hole is 

n^ = i _ Q-EyiRT 
.'^62 

and the diffusion constant for all the ions is 



n = 1 _ p-EJRT 

6 2 



/6n\ 



since GnjN is the fraction of ions around the holes. Then 

In this equation ^dv = 65 cm.^/day"^, when d = 3 A., and 
V = 5x 10* cm. /sec. The case of forced diffusion (electrical 
conductivity) may be obtained by using the Einstein equation; 

D = BkT {B denotes the mobility) 

and also K = N{ZefB, 

* For Schottky disorder the analogous relationship is n = a.Ne~^ol^'^\ The 
exponential does not contain the factor J, and a may have values as high as 10*. 



CONDUCTIVITY AND DIFFUSION PROCESSES 295 

where K denotes the conductivity, and N the number of ions 
per unit volume of valency Z, and charge Ze. Then 

K = N{Ze)^^e-^^o+^i)l^T^ 

The factor before the exponential has a magnitude similar to 
that for D, and normally one may write 

iC= (10-100) e-(^o+2Si)/2iJT^ . 

We have now to consider how accurately this theoretical 
expression predicts the experimental findings. The value of 
{Eq + 2^j) might, at first sight, be thought to be comparable 
with the lattice energy ( 1 00-200 k.cal.); whereas the experi- 
mental values are instead from 1 to 50 k.cal. The suggestion 
by Smekal(ii2) that the mobility occurs only along internal 
surfaces and outer surfaces leads to impossible values of the 
constant A in the equation 

K = ^e-(-E:o+2Si)/2Br^ 

Reasonable values of the quantity {Eq-^2E-^) may, however, 
be computed for lattice ion movement if the polarisation 
properties are also considered, for it has been found that 
polarisability and polarising properties and conductivity are 
connected (cf. pp. 254 and 265) (iw). One may proceed to 
calculate the energy, Eq , involved in the formation of a hole, 
and removal of an ion to an interstitial position, in a 
manner analogous to tha,t outhned on p. 254. To ^Eq must 
then be added E^, the energy required before the hole, or the 
interstitial ion, can migrate. Calculations of this kind have been 
made by Jost(44), He determined Eq for NaCl by evaluating 
the following energy terms: 

(i) Coulombic energies in normal and displaced positions. 
(ii) Repulsive energy in the normal and displaced position. 
(iii) Polarisation energies around the vacant site and the 
displaced ion. 

The calculation is subject to the limitations outhned on 
p. 255. The next step is to find E-^^, the energy of activation 



296 INTERDIFFTTSION OF SOLIDS 

for diffusion of either a vacant site or an interstitial ion. The 
term ^Eq + E^ might then be compared with the experimental 
figm-e for \Eq + E-^ obtained from conductivity data (44k.cals.), 
The theory indicates a quahtative agreement with experi- 
ment. 

Certain salts (e.g. a-Ag2S ; D = 10 x e-^^^^'^ cm.^ day-^) have 
a very high electrolytic conductivity; |^(^o + 2^i) is only 
3220 cal. For this salt the polarisation properties reduce 
^{Eq + 2E-^) to a very low value. Calculation gives: 

Eq = energy of "hole" formation = -2/couiomb + ^rep. + ^poi. 

= (2-31 -0-11 -2-15) e2/a 
= ~ 5-7 k. cal. 

A similar approximate computation of Ej^, the potential 
energy needed for migration from a displaced position, 
— h~h~i>'^^^ displaced position + 1, +^, + 1 gives 

^1 ~0-le2/a~10'7k.cal. 
and thus i(^o + 2^i)~ 13-1 k.cal., 

which may be compared with the experimental value 3220 cal. 
Qualitatively, if not quantitatively, the Frenkel theory can 
thus account for the exponential term. 
The constant A in the equation 

to which the theory of Frenkel gives a value 2 x 10^ to 2 x 10^, 
actually takes experimental values of 10 > ^ > 10^. To explain 
this discrepancy three possibilities arise : 

(i) By analogy with the theory of thermionic emission it 
may be assumed that there is a temperature eoefficient to the 
quantity 1{Eq + 2E^) ,w;hich has been neglected in the Frenkel 
treatment outUned. 

(ii) A molecule having once acquired the energy ^{Eq + 2E-^) 
may retain it while describing a mean free path far greater 
than corresponds to a single displacement. 

(iii) The activation energy may be stored through many 
degrees of freedom (p. 300) . 



CONDUCTIVITY AND DIFFUSION PROCESSES 297 

Braunbek(ii5) developed an analogous expression for the 
conductivity of an ionic solid. Using sodium chloride as his 
model lattice, he assumed a simple linear vibration of sodium 
ions, while the chlorine ions were supposed to remain fixed. 
Each sodium ion was situated in the mid-point of an octahedron 
of chlorine ions, and the sodium ion on acquiring sufficient 
energy could move from the centre of its octahedron through 
the mid-point of one of its faces, and enter a new vacant 
octahedron whose central sodium has diffused away by a 
similar process. The probability of occurrence of such a process 
during a vibration was calculated, and from this the self- 
diffusion constant, and the conductivity K: 

where e = electronic charge, 

d= 5-63 X 10-8 cm., 

T = the vibration period, ~2-l x IQ-^^gec. 

The quantity E was regarded as comparable with the energy 

of melting of the lattice. This equation should however also 

- . n No. of vacant Na+ ion sites , . . 

contain the ratio -= _ , , ,. „^-r , • -, ^ ? which 

N Total No. of Na+ ion sites 

gives the probability that the second octahedron wUl be 
empty to receive the migrating Na+ ion. The agreement 
with V. Seelen's'^i^) data is thus probably fortuitous. 

Cichocki(ii7) derived an expression for the self-diffusion 
constant in ionic or metallic lattices. He used a body-centred 
cubic lattice as his model, and then endeavoured to calculate 
the probability that a given atom would have the requisite 
energy and direction of vibration to move to a new position, 
at a time at which the atoms surrounding that position have 
the energy and direction of vibration to make an adequate 
interstice for it. His theory is interesting as an attempt to 
allow for correct timing in the diffusion. An analogous treat- 
ment by Dorn and Harder (ii8), in which the crystal is regarded 
as a periodic system of energy wells, is less satisfactory. It is 



298 INTERDIFFUSION OF SOLIDS 

not possible to assume constant potential energy walls about 

the hole in which the solute atom resides. Sometimes it may 

be almost impossible for the atom to escape in a given direction, 

and sometimes it may readily be able to do so. The height of 

the surrounding energy barriers fluctuates with time according 

to the differences in phase of vibrations in surrounding atoms 

in the lattice. Cichocki's formula, which allows for this, is 

nevertheless open to the same objection as Braunbek's (p. 297). 

Langmuir and Dushman(ii9) proposed a semi-empirical 

equation for diffusion in cubic lattices, which has proved 

a useful guide to the behaviour of diffusion j^rocesses in " 

ionic and metaUic lattices (Tables 66 and 67). It was derived 

by considering the lattice as composed of layers of atoms in 

planes a distance d apart, where d denotes the interionic or 

interatomic distance. It was assumed that d was also the mean 

free path of a diffusing ion or atom. The number of atoms per 

unit area is then d^, and the chance that an atom will leave this 

area in unit time is kd'^. By analogy with an early expression 

for the reaction velocity constant (120), Langmuir and Dushman 

W 
wrote k = pe-^vikT ^^y^^ j) _ _^ dz^-EiRT ^yj^ere v is the 

vibration frequency of the solid, N^hv = E, and h denotes 
Planck's constant, and the other terms have their usual 
significance. The Langmuir-Dushman equation is to be re- 
garded as empirical, the frequency v being fictitious, as the 
following correct derivation of the diffusion constant shows. 
Suppose two salts which form soHd solutions of cubic symmetry 
are interdiff using, salt A passing in the +x direction. Draw 
planes at x and x + d (Fig. 110), where d is the mean free path of 
an activated molecule, and also at x — ^d and x + ^d, normal 
to the x-co-ordinate. All salt A ions in the region x — ^d to 
x-\-\d can if they acquire the necessary activation energy with 
suitable direction ( + x) pass the intermediate plane x -f- \d. The 
chance that an activated ion will move in the -f x direction is 
one-sixth the chance it Avill move in any direction and this 
latter chance is assumed equal to the probability of activation, 
j^o e-^l^^ , where Vq = the vibration frequency in the lattice = the 



CONDFCTIVITY AND DIFFUSION PROCESSES 299 



number of vibrational collisions/second. The total number of 
ions of the salt A moving/sec. /unit area in the +x direction 
across the plane x + ^dis thus \dCvQe~^l^^; while the number 

moving in the — x direction is \d{ C — d-^ I VqC'^i^^. Thus the 

nett flow across the plane x-\-\d per second per unit area in 

the + X direction is \d^ ^=— Vr, e~^'^'^, which also equals D -tt- . 

ox ox 



Thus 



D = ^d\e-^iP^^ 



Fig. 110. 



(kinetic theory deduction), 



^^h 



(Langmuir-Dushman empirical ex- 
pression), 

so -that if there were any correspondence EJNqIi = \Vq; but 
since v — E/NqA, and as E can be even greater than 
90,000 cal./moL, Vq must if derived from the Langmuir- 
Dushman expression correspond m such a case to an energy 
of 540,000 cal. Clearly no physical significance can then be 
attributed to the term EJNqJi in the Langmuir-Dushman 
expression. When not one but two degrees of freedom are in- 
cluded for the storing of the activation energy, the rate R at 
which the ions are activated becomes 

E 
E 



1/72 ^ ,. p-EIRT 
® HT ' 



with D 

which is Bradley's expression ( 121 ). When^i degrees of freedom 



300 INTERDIFFUSION OF SOLIDS 

are included the appropriate expressions are those of 
Wheeler (122)* 

^"'^"U^/ {n-l)l 
( E \"-i 1 



and I> = ¥\-^) (^^^I^^oe-'-. 

Langmuir and Dushman's empirical relationship 

has been used fairly extensively as a guide to the behaviour 
of diffusion systems, perhaps unfortunately, since there has 
been a tendency to read a physical significance into the values 
of d thus computed. These happen in many cases to be of the 
order of magnitude of the inter- crystalline distances. In 
general a better agreement is found from the Langmuir- 
Dushman equation, between i^caic. ^^^ -^obs.' using for d the 
crystal parameters, when the systems have a high activation 
energy for diffusion, and this may perhaps be regarded as 
evidence that activation energies are then stored in only one or 
two degrees of freedom. However this may be, many systems 
of diffusing metal pairs obey Langmuir and Dushman's 
formula with some precision, as Table 67 shows. Agreement 
is much poorer for those ionic solids for which low activation 
energies are observed. The self-diffusion of lead may be cited as 

E 

an example of diffusion obeying the equation D = j^^-r dH~^l^'^ 

with precision. Fig. lll(4i>) shows experimental and calcu- 
lated curves of log D versus 1/T, in which the experimental 
curve (1) gives an E value of 27,900 cal. When d is taken 
as the shortest distance between two neighbours, 4-94 A., 
the whole logD-l/T curve is fixed by a single exj^erimental 
point and in the figure is given by curve (2). When d is 8*5 A. 

* These three formulae apply in their present form to zeolitic diffusion in 

dilute solution, or to place exchange diffusion. In concentrated zeolitic solution 

,, ,. No. of vacant interstices ^ , • , , j -r. j-rc • i -r. i ■ 

the ratio _, . . -t j^^— must be nicluded. For diffusion by Frenkel 

Total No. of mterstices 

or Schottky mechanisms the ratio -^ (p. 303, footnote) must be included. 



CONDUCTIVITY AND DIFFUSION PROCESSES 301 



(the unit cube diagonal) the agreement is a httle better and 
the logD-l/T curve is given by (3) in the figure. For this 
system the Langmuir-Dushman empirical equation is re- 
markably satisfactory, and a single diffusion constant would 
suffice, using crystallographic data, to map out the whole 
course of the log Z)-l/T curve. 

-4- 
-5 
-6 
7 



'9 . 



-70 



— . ^ff '*■ 



26 25 /4 23 22 27 20 79 



78 72' 



(1) 



1/r X 10*. 

Experimentally found curve, ^ = 27,900 cal./atom. 



(2) Calculated using c^ = shortest distance between Pb^atoms, 

and £ = 25,500 cal./atom. 
(3) Calculated using (Z = -y/S x shortest distance between'Pb 

atoms, and -K = 26,700 cal./atom. 



Fig. 111. Self- diffusion in Pb calculated from D 



_E_ 



^2g-EiRT calculated from 



a single experimental diffusion constant and the distance between neigh- 
bouring atoms {d). 

The formulae of Braune (94) and of van Liempt (109) recognise 
the dependence of the velocity of diffusion upon the melting- 
point by introducing in the exponential term the equality 



E 
RT 



36' 



T 

T 



or 



I = 362T^, 



wherein T^ is the melting-point. In these equations the 
constant b depends upon the atomic size, and polarisation 
properties of the lattice. The usual value of 6^ is about 2, and 
for lattices of very different melting-points such as lead, silver, 



302 



INTERDIFFUSION" OF SOLIDS 



and tungsten respectively (Table 75) the values of b^ are as 
1:2:1, while the absolute melting-points are as 1:3:10 (see 
also p. 285). 

Eyring(99) in 1936 showed how the transition state theory 
of reaction velocity could be apphed to viscosity, plasticity, 
and diffusion. This treatment has been apphed especially to 
liquids, and to organic polymers, but it should be still more 
apphcable to the problem of diffusion in crystals. The ion or 
atom in a body-centred cubic lattice, for examj)le, may diffuse 
through the centre of any one of six faces, over a potential 
energy barrier, to a neighboiu-ing vacant site or interstitial 
position. The velocity constant for passing over a potential 
energy barrier is 

k' 



F„ h 



In this expression 

a = the probabihty that a system having once crossed the 
top of the energy barrier will not recross it in the reverse 
direction before losmg its activation energy. 



Table 75. The dependence of the self-diffusion constant 
upon the melting -point 



Metal 


Melting- 
point ° C. 


Dis' c. in 1 
cm. 2 day-i 


Pb 

Ag 
W 


327 

961 

3400 


2-2 X 10-15 
9-6 X 10-20 
4-3 X 10-59 



F^ = thepartitionfunctionforthenormalstateof thesystem. 

Eq = the energy of activation for the transition from the 
initial to the final state. 

F% — the partition function for the activated state, ex- 
cluding the partition function for the co-ordinate in which 
the transition occurs. This latter partition function gives 
the frequency term kTjh {k — the Boltzmann constant; 
h — Planck's constant). 

If it is now supposed that there is a concentration gradient 
dC/dx in the -h x direction, and that the distance between two 



, CONDUCTIVITY AND DIFFUSION PROCESSES 303 

successive minima of potential energy in this direction is d, one 
finds the concentration C at one minimum and C + d{dGldx) 
at the next. The number of ions or atoms passing in the + x 
direction is thus NQdk'G, and in the reverse direction it is 

Ngdk'i G + d-^\ . The excess flow in the —x direction is thus 

where Nq is Avogadro's number. Finally*, 

B = dW = d^a^^e-^o/RT^ 

In the simplest cases F^/F^^ ~ (1 — e-^^l^^), and tends to unity 
for large v {v is the vibration frequency of the atom or ion in 
the lattice). If the equation is to apply to diffusion processes 
in solids for which the Langmuir-Dushman equation also 
holds, one may write E = ET + Fq and so 

E kT F* F* E I 

For the self-diffusion of lead at 230° C, this relationship 
becomes rr* 

There is so far no analysis of factors determining the ratio -^ 

for diffusion by place exchange or other mechanisms. In 
some instances, the considerations of the footnote below may 
have to be allowed for. 

Eyring and Wynne- Jones (99) have extended Eyring's equa- 
tion for the diffusion constant by introducing the entropy 
of activation A 8^, and the heat of activation AH^. The 
diffusion constant is now given by 

* For Frenkel or Schottky disorder a term 

n _ No of interstitial ions, or vacant sites 

N Total No. of potentially diffusible ions 

should be included (p. 294). In dilute zeolitic solution, or in place exchange 

•diffusion (Fig. 108), no such term arises. The latter may be the mechanism of 

interdiffusion of many metal pairs. (See p. 297, and footnote, p. 300.) 



304 INTERDIFFUSION OF SOLIDS 

It is then seen that the temperature independent factor Dq is 
related to the entropy of activation by the expression 



^""^ U/ 2-7: 



2-72 

Therefore a large Dq implies a large entropy of activation, or 
a considerable loss of order in the lattice on passing from the 
initial to the transition state. This is particularly noticeable 
for self-diffusion constants. 

These formulae based upon different conceptions of the 
diffusion process are collected in Table 76. It can here be 
seen what physical quantities have been correlated with the 
diffusion or conduction process. The process of diffusion is the 
necessary preliminary to many metallurgical and chemical 
processes involving sohds. In addition, mechanisms of self- 
diffusion and diffusion are of absorbing interest as a study in 
themselves. Yet the very variety of the attempts which have 
been made to obtain satisfactory models for diffusion, and the 
limited application of so many of them, demonstrate how httle 
progress has been made towards a comprehensive treatment. 
This is rather surprising when one considers the present 
knowledge of the crystalline state. Three main viewpoints 
have been advanced: the theory of equihbrium disorder 
in ionic and metallic lattices; the carrying over of kinetic 
theory, and of gaseous reaction kinetics to solid phases; and 
the application of statistical mechanics to diffusing systems. 

On the experimental side much more numerous and more 
accurate data are required. It will be necessary to know with 
exactitude how D, Dq and E in the expression D = DQe~^l^^ 
vary with composition in alloy systems. Many more self- 
diffusion coefficients obtained by the radioactive indicator 
method are required. The connections between polarisation, 
atomic radius and density, position in the periodic table, alloy 
formation, melting-point, and degree of lattice loosening must 
be placed upon a more quantitative basis than the present data 
permit. When these properties have been correlated among 
themselves, and with existing X-ray data" on crystal structure, 
it should be possible to understand more clearly phenomena of 
diffusion in metallic and non-metallic lattices. 



Table 76. Expressions for the diffusion constant and the 
ionic conductivity in crystals 



System 



Formula 



Author 



Ion-ionic lattice 
metal atom, 
metallic 
lattice 



Ion-ionic lattice 
(sodium . 
chloride) 

Ionic or metallic 
lattice (self- 
diffusion) 



Ionic or metallic 
lattice 



Ionic or metallic 
lattice 



Ionic or metallic 
lattice 



Ionic or metallic 
lattice 



d =mean free path 

£o = eDiergy of hole formation 

^1 = energy for diffusion of ion or atom 

into hole 
V =mean thermal velocity of diffusing 

atom or ion 
N{Ze) 



K^ 



kT ^ 
K = conductivity 
Z = valence _ • 

e = electronic charge 
iV^= number of ions per unit volume 



K^ 



3t(8-3) dE 

E = activation energy for ion transfer 

T = vibration period, ~ 2 x 10~^^ sec. 



Z)=2-43xl0'» 



^1 /T^ 

V V M^ 



g-{E^+E)lRT 



iM';i= atomic or molecular weight 
V = atomic or molecular volume 
T^ = melting-point 

^1 = energy needed to form interstitial hole 
E = energy needed to bring atom to inter- 
stitial position 

v= vibration frequency of lattice 
6 is a constant 

"^-qKrt) (/-I)! 

/=the number of degrees ,of freedom 

involved in the diffusion process 
/=2 in Bradley's equation 

DJ^(a-^\d^e-^I^T 

Fa* I En = ratio of partition functions tran- 
sition and normal states respectively 
(excluding for former the partition 
function for co-ordinate of diffusion 
process) 

a = transmission coefficient, i.e. proba- 
bihty that system having reached 
transition state will pass over the 
energy barrier to a new state 

«" \h ) 2-72 

AS^, AH^ denote respectively entropy 
and heat of activation 



Frenkel(47), 

Jost (44) 



Frenkel(47), 

Jost (44) 



Braunbekdio) 



Cichockidi?) 



Langmuir and 
Dushman(ii9) 

vanLiempt(io9) 



Bradley (121) 
Wheeler (122) 



Eyring(99) 



Eyring and 
Wynne- 
Jones (99) 



306 

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CHAPTER VII 
STRUCTURE-SENSITIVE DIFFUSION 

Types of irreversible fault in real crystals 

It is usual to classify diffusion processes into volume, grain- 
boundary, and surface diffusions. While the preceding and 
following chapters show that the problems of volume and 
surface diffusion have been attacked in a fairly adequate 
manner, the data with which the present chapter has to deal 
are much more fragmentary. Jost, Wagner, Frenkel, and 
Schottky(i) have developed the theory of disorder in equili- 
brium with order in crystals. It transpires from theory and 
experiment that the transitions 

Order ^ Disorder 

occur in both directions with an energy of activation, and that 
the change from left to right is an endothermic process. As 
with chemical reactions which proceed with an energy of 
activation, by suddenly chilling the system the high tem- 
perature equilibrium is frozen, and a non-reversible low 
temperature system results. The degree of disorder in a crystal 
increases with increasing temperature, so that the disordered 
crystal chilled suddenly should have more interstitial ions and 
vacant, lattice sites than it would possess in its equilibrium 
state, and should show an increased ionic conductivity. 
Anneahng, provided cracks and grain boundaries did not 
occur, should diminish the conductivity to a basic value. 

A second kind of structure-sensitive diffusion is also possible. 
The crystal, whether metallic or ionic, consists of a mosaic of 
small blocks,* separated by grain boundaries, or submicro- 
scopic flaws. It is well known that diffusion on an external 
surface (Chap. VIII) proceeds more easily than volume dif- 
fusion. By an analogy, which will later be supported by 
experimental data, one may suppose that diffusion along 

* The evidence for this point of view is presented in the next section. 



312 STRtrCTURE-SENSITIVE DIFFUSION 

these "internal surfaces " will occur more readily than volume 
diffusion. The velocity of diffusion is largely governed by the 
exponential term in the equation D = D^e-^l^^ , so that if 

-^surface "^ -^ grain boundary ^ -^lattices 

the velocity of grain-boundary diffusion may exceed that of 
lattice diffusion even though the number of paths available 
for lattice diffusion is much greater than the number of paths 
available for grain-boundary diffusion. 

A third tj^e of structure-sensitive diffusion process is to 
be attributed to im.purities in the lattice. These may be incor- 
porated during the growth of the crystal due to accidental 
impurity of the mother liquor, or of the vapour. They may also 
be introduced intentionally, as when alkali-halide lattices are 
heated in alkaU-metal vapours, halogen vapours, or mixed 
gas atmospheres of hydrogen and alkaU metal. Wagner (2) and 
his co-workers introduced excess of one component by heating 
the crystal in an atmosphere of its electronegative component 
(O2 for CdO, ZnO, NiO, CuaO; S for FeS; Brg for CuBr; Ig 
for Cul). It is not difficult to recognise the irreversible low- 
temperature diffusion processes resulting from this type of 
disorder because of the sensitivity of the high-temperature 
conductivity to gas pressure. In the high-temperature region 
there exists an equilibrium 

Gaseous component ^ Component in excess in the lattice, 

which Wagner (1) used to study the different kinds of equili- 
brium disorder occurring at high temperatures in crystals. 
When a crystal with an equilibrium excess of one component 
is chilled to low temperatures, irreversible conductivity and 
diffusion properties result (cf. Fig. 94, Chap. VI). 

When the conductivity cannot be attributed to excess of 
one component, i.e. when the conductivity is not sensitive to 
changes in partial pressure of a surrounding gas atmosphere 
of that component, and yet irreversible conductivity or 
diffusion phenomena are observed, the behaviour may be 
attributed either to grain-boundary diffusion, or to lattice 
disorder, frozen by rapid cooling of the high-temperature 



IRREVERSIBLE FAULT IN REAL CRYSTALS 



313 



equilibrium state of disorder. To determine to which type of 
fault the structure sensitivity is due requires that one should 
know a great deal concerning the properties of the crystal. 
The ways in which these properties have led one to envisage 
different lattice imperfections may now be briefly reviewed. 

Non-equilibrium disorder in crystals 

Some properties of crystals depend upon the previous history 
of the specimen, while others are insensitive to all treatments. 
On this basis a classification (3) of crystal properties may be 
made (Table 77). The structure sensitivity of some of these 

Table 77. Classification of crystal properties 



Property 


Insensitive 


Semi-sensitive 


Sensitive 


Character 
Examples 


Additive, contribu- 
tion of anomalous 
parts ' of smaller 
order of magnitude 
than that of nor- 
mal parts 

Specific gravity, 
specific heat, re- 
fractive index, X- 
ray interference, 
elastic properties 


Additive, but con- 
tribution of ano- 
malous parts of 
same order as that 
of normal parts 

Electrical conduc- 
tivity in ionic 
crystals, diffusion. 
X-ray extinction, 
vibration damping 


Selective 

Tensile strength, 
plasticity, dielec- 
tric strength, mag- 
netisation curve of 
ferromagnetic sub- 
stances 



properties has been related to various types of imperfection, 
distributed over the surface and in the interior in a random 
manner. 

The most obvious imperfections in crystals are cracks on 
and within the crystal surface. It is sometimes possible to 
see the cracks with the eye, or under a microscope, especially 
after suitable etching. -Examination reveals that the surface 
consists of blocks, or grains (Fig. 112) (4), whose interfaces 
extend into the solid as cracks. Cracks of this size result from 
thermal atid mechanical treatments of the crystal, arid also 
according to Smekal(5) as a result of strains due to primary 
flaws in the crystal. The primary flaws are considered by 
Smekal to be due to occluded impurity, or to too rapid rates 



314 STRUCTURE-SENSITIVE DIFFUSION 

of growth of the crystal which leave gaps or local variations 
in orientation. It is also possible that they result from ex- 
tension of the very shallow surface cracks which Lennard- 
Jones and Dent (6) consider to occur as a result of unbalanced 
forces at the crystal surface, which cause a lateral contraction 
of the surface. The reality of contraction due to surface tension 
is demonstrated by the experiments on the electrical con- 




Fig. 112. Appearance of bright platinum surfaces ( x 100), 
showing crystallites, furrows and grain boundaries. 

ductivity of thin films mentioned in the following chapter (7), 
Due to the break up of evaporated films on surfaces, under 
surface tension forces, the resistance rises after a time interval. 
The presence of cracks and grain boundaries is revealed by 
a number of additional experiments. Poulter and Wilson (8) 
found that water, ether, and alcohol would penetrate into glass 
or quartz for considerable distances when a pressure of 
15,000 atm. was maintained for a quarter of an hour. If the 
pressure was released quickly, the glass was shattered, due to 
expansion of the diffused liquid in the grain boundaries. The 
measurements suggested also an upper limit to the thickness 



NON-EQUILIBRITJM DISORDER IIST CRYSTALS 315 

of the crack, since larger molecules such as oil, or glycerine,* 
caused no analogous shattering. Similarly, the diffusion of 
hydrogen through iron which can be made to occur at pressures 
of 9000-4000 (9, 10) atm., at room temperature, while in part 
a true lattice diffusion, also opens up grain boundaries, so 
that it is possible to force mercury and oil through the metal 
after diffusing hydrogen through it. These cracks in the iron 
were not visible to the naked eye. 




Fig. 113. Particles on SiOg glass, under strong grazing illumination ( x850). 

The presence of cracks, especially at the surface, governs the 
tensile strength of crystals (ii). If a specimen of rock-salt is 
dipped in hot water the tensile strength rises by 20-fold. 
Etching glass or silica fibres in hydrofluoric acid increases the 
tensile strength 5-fold. Some vapours sorbed on silica fibres 
diminish their tensile strength (water, 3-fold; alcohol, 3-fold; 
benzene, 2-fold). These effects may be explained as penetration 
of cracks by the sorbed liquid, increasing the bonding between 
grains or blocks in the solid rock-salt when water is sorbed, 
but diminishing it in glass, or silica, possibly due to hydration 
and swelling of silica powder down the grain boundaries and 
cracks, so thrusting the grains apart. Evidence of surface 

* It is to be noted, however, that these larger molecules are not spherical. 



316 STRUCTURE-SENSITIVE DIFFUSION 

cracks was adduced from experiments on the condensation of 
metallic atoms on diamond and silica surfaces (12). The metal 
atoms aggregated in lines along the surface (Fig. 113), and 
these lines were thought to trace out the course of surface 
cracks. Also the tensile strength of glass fibres is often found 
to increase as their diameter is diminished. This may be 
explained by supposing that cracks exist in the fibre, but that 
the probability of finding a sound fibre is greater the less its 
diameter, because the surface or volume of the fibre, in which 
the cracks exist, is in this way also diminished. 

The continuation of cracks throughout the crystal results 
in a mosaic structure of the crystals. The existence of these 
secondary flaws through crystals of rock-salt may be demon- 
strated by heating sodium chloride in sodium vapour (13). 
Some vapour is absorbed and the crystal becomes coloured. 
Aggregation of the dissolved sodium occurs, under suitable 
conditions, along faults in the crystal, and the course of these 
faults may be traced by examination using the ultramicro- 
scope. Aggregations of sodium of four kinds have been found 
(Fig. 114) (14), corresponding to tree-like, spheroidal, striated, 
and nearly homogeneous distributions of the metal. 

When a crystal is placed on an X-ray goniometer and 
rotated in the X-ray beam, the intensity of reflection rises to 
a maximum and then falls away. The more ideal the crystal 
the sharper is this "sweep curve". Calculations have been 
made of the angular breadth of the "sweep curve" for an ideal 
crystal, which show that only a few seconds of arc will cover 
the whole breadth of the curve. Analogous calculations for a 
mosaic crystal indicated a much greater angular breadth of 
the "sweep curve", which may amount in extreme cases to 
several degrees. In Table 78 (15) are illustrated the observed 
breadths of sweep curves at their points of half intensity ; and 
also values of the "integrated reflection" defined as 

(Total reflected energy for uniform velocity of rotation) 

_ X (velocity of rotation) 

Intensity of incident radiation 



NON-EQUILIBRIUM DISORDER IN CRYSTALS 317 

It is seen that Reiininger's(i5) artificial rock-salt crystal re- 
presents as near an approacK to the ideal crystal as can be 
obtained, as far as the mosaic structure type of fault is 
concerned. 

There has been discussion as to the size and distribution of 
the Smekal blocks in a mosaic crystal. Zwicky(i8) suggested 
that a lattice is subdivided into a periodic block structure with 
definite spacings, basing his theory in part upon the appearance 




Fig. 114. Types of crystal fault in rock-salt, revealed by aggregation of 
absorbed sodium, and examination ia TyndaU liglit(i4). 



of regular triangular etch pits on bismuth (19), and on the 
persistence of structure in liquids near their melting-points. 
He made calculations which purported to show that a crystal 
with such a super-lattice would be more stable than a crystal 
without the super-lattice. Since the calculations are not 
correct (20), and the evidence from the etch pits on bismuth 
does not necessarily imply a super-lattice of Zwicky type, his 
view may be discarded in favour of that of Smekal. The latter 
considered that any real crystal tends to become an a-periodic 
mosaic of blocks. The boundaries of the blocks grow from 



318 



STRUCTURE-SENSITIVE DIFFUSION 



primary flaws, or from mechanical and thermal treatment, and 
penetrate through the mass. Since dififusion processes dovm 
these systems of faults obey a law 

where E is greater than the corresponding energy term for a 
surface diffusion (p. 312), it may be concluded that the Smekal 
cracks can be of molecular dimensions, so that the crystal 
force fields on either side of the crack overlap. 

Table 78 



Crystal 
(NaCl) 


Half-breadth 
of sweep 

curve 

(200 face) 

sec. 


Integrated 

reflection 

xl06 




200 


400 


600 


Natural crystal, 

polished cleavage 
Natural crystal, 

untouched cleavage 
Artificial untouched 

cleavage, 

Renningerdo) 

Calculated for ideal 
crystal : 
Darwin (16) 
Prins(i7) 


900 

40 to 50 

7-1 

4-2 
4-9 


270 
102-5 

47-8 

45-0 
410 


45 

26-3 

10-5 

121 
9-9 


16 
9-8 
4-6 

6-9 
5-1 


- 
For CuKa 
radiation 
and NaCl 



It must be remembered that in addition to the subdivision 
of the crystal into Smekal blocks, each block may have its 
own glide planes and that gUde planes may act as regions for 
further break-up of the crystal block, or for preferential 
diffusion into the block. Palladium, after sorption of hydrogen, 
shows not only a block structure, but also a herring-bone 
pattern on the surface of each grain or block (2i, 22) wliich has 
been attributed to preferential penetration of hydrogen down 
slip planes. 

The manner in which a mosaic crystal may grow from a melt 
is shown by Buerger's (23) studies on dendritic crystals. A 
needle-Hke crystal first forms, the crystal branches into other 
needles, these yet again into more needles, and so all the space 
is quickly occupied by the dendritic mosaic crystal. Typical 



NOlSr-EQUILIBRITJM DISORDEE, IN" CRYSTALS 319 

dendritic mosaics of bismuth are illustrated in Fig. 115. 
Dendritic mosaics can be grown from solution, where once 
again the peculiar lineage structure may be recognised. Also 
galena and quartz may show tree-like markings and boundaries, 
while haematite crystals may be grown which are a mosaic of 
plates, laid down in a direction approximately normal to the 
direction of growth. Buerger considers that a complete range 
of structures is possible from dendritic mosaics, parallel crystal 
intergrowths, crystals with multiple terminations, or block 
structure, to crystals which, like certain specimens of gypsum 




Fig. 115. A dendritic surface of rapidly cooled bismuth ( x 2). 

and calcite, show no imperfections under optical or X-ray 
examination. This lineage theory of crystalgrowth allows one to 
visualise readily how mosaics of various kinds may be formed. 
The non-equilibrium disorder which is due to interstitial 
ions and vacant lattice sites, obtained by heating the crystal 
to high temperatures and then chilling it quickly, can be 
revealed by the absorption spectrum, if the concentration of 
the disordered points is large enough. It is generally necessary 
to incorporate impurities, however, to obtain an adequate 
concentration of interstitial ions or vacant sites for spectro- 
scopic examination. For example, the blue crystal obtained 
by heating an alkali haUde in alkaU metal vapour gives a new 
absorption band(i3) (Fig. 116). The breadth of the band, H, at 



320 STRUCTURE-SENSITIVE DIFFUSION 

its point of half -intensity allows the concentration of colour 
centres, or i'^-centres, as they are called, to be estimated 
(cf. Chap. Ill) by the equation 



N = 1-31 X 1017 



(/^2+2)2-^°^^^ 



H, 



where -fi^max denotes the absorption coefficient at the peak of 
the absorption curve, and ju is the refractive index for light 
having the wave-length of the absorption maximum. The 
number of colour centres/c.c. (N) is lO^^-lO^' with the sodium 
vapour pressures normally employed. The i^-centres are con- 
sidered to be due to electrons occupying vacant chlorine-ion 
sites in the lattice (24). They may be electrolysed out of the 



¥00 




600800 100 

TT 



Wave-length (jn\i.,) 
600800 VOO 600800 WO 



Naai 



Ka 



60080 wo 600 

"I II I' l l 



Rba 




3 2 3 2 3 2 3 2 

Energy (eV.) 

Fig. 116. Absorption bands of colour centres in solution in 
alkali halides at 10° C. (PolildS)). 

crystal, and move towards the anode giving a sharp colour 
boundary in the crystal (i3). Irradiation with blue light causes 
the nature of the absorption band to change, and a new type 
of colour centre appears, the ^'-centre, regarded by Mott(25) 
as consisting of two electrons in a vacant lattice site. Irradia- 
tion with infra-red Ught, or heating, causes the reverse change 
into i^-centres to occur, so that a photo -equilibrium can 

result: , , t u^. 

blue ught 

i^-centres , "^ i^'-centres 

infra-red radiation or heat 

Heating a crystal of alkali halide containing dissolved 
alkali metal in hydrogen discharges the colour of the crystal 
and causes the absorption spectrum to change (13). A new peak 
occurs in the ultra-violet. These centres, which are usually 
designated as Z7-centres, are in a reversible equilibrium with 



NON-EQUILIBRIUM DISORDER IN CRYSTALS 321 

^-centres, into which they may be transformed by heat or by 
ultra-violet light. f/-centres consist of alkali-hydride (13) that 
has been formed in the lattice, and dissociates on heating or 
irradiation as follows: 

KH^K+ + H + e. 

The electron occupies a vacant chlorine-ion site and con- 
stitutes an i'^-centre. 

These centres of disorder may contribute to the conduc- 
tivity, under the influence of light and of heat. 

Structure-sensitive conductivity processes 

Conductivity data may be employed to demonstrate the 
properties of structure-sensitive diffusion in crystals. It is 
often found that the conductivity -temperature curve of ionic 
crystals divides itself into two sections, a reversible high- 
temperature curve, obeying a law 

K = Ae-^IRT^ ^ 
and families of low-temperature curves obeying analogous laws 

The positions of the low-temperature curves depend upon the 
treatment accorded the specimen and are therefore structure 
sensitive. The high -temperature curve is obtained for all 
specimens of a given crystal, and is structure insensitive (27) 
(Fig. 117). When crystals of sodium chloride were heated for 
periods of 10 hr. at a series of temperatures, the conductivity 
rose as the temperature of heating rose, corresponding to an 
increased number of faults and flaws resulting from the pre- 
heating (Fig. 118). Smekal (26) expressed the results of Figs. 117 
and 118 for the conductivity (in ohm~^ cm.~^) by the expression 
K = J[je-iO'3oo/2^ + 1-4 X 10^ q-23,oooit + 3-6 x 10^ e-25,70o/r 
Structure sensitive Anion conductivity 

• — — y 

Cation conductivity 

showing that the slopes of the structure-sensitive conductivity 
curves were all nearly the same, but the temperature inde- 
pendent factor altered very markedly. 



322 



STRUCTURE-SENSITIVE DIFFUSION 



Single crystals of sodium nitrate (27), because of the smaller 
internal surface, show a lower conductivity than the poly- 
crystalline mass solidified from the melt. If a powdered salt 
is put under pressure, the coherent block, composed of a'mosaic 




~''o90 1-00 Tid HO 1-30 1-VO t-50 1'SO 1-70 7-80 ISO 
7/Ty10K(Tin''K.) 

Fig. 117. The conductivity of various NaCl crystals. (The lowest 
curve is for a suigle crystal.) 

of small crystallites, shows a higher conductivity than a single 
crystal (28). Quartz sand when mixed with the crystals of 
sodium nitrate (62 % SiOg), by increasing the number of grains 
and diminishing their size, was observed to double the con- 
ductivity (29). Rock-salt crystals, prepared by crj^staUisation 
from aqueous solution, possessed at 90° C. a conductivity 



COISTDTJCTIVITY PROCESSES 



323 



100-fold smaller than that of a rock-salt polycrystalline mosaic 
prepared from a melt(30). These observations lead to the 
conclusion that internal surfaces are of great importance in 
conductivity measurements. 




ISO 



2-00 



2-50 



3-00 



3-50 



llTxlO^(Tin''K.) 



Fig. 118. Kock-salt crystals, heated for 10 hr. at a series of temperatures, show 
an increasing conductivity. After 10 hr. at 160° C. (the lowest curve), rising 
to 200, 300, 400, 500, 600, 700, 780 (the highest curve). 

Another kind of experiment which has thrown light upon 
the nature of structure-sensitive conductivity requires the 
addition of small quantities of impurity to the crystal lattice. 
As early as 1897 (3i) it was noted that the addition of sodium 
chloride to lead chloride caused an increase in the conductivity 
of the latter. One of the most remarkable examples of this 
phenomenon was given by Ketzer(32), who by adding 0-001 % 
of rock-salt to lead chloride raised the conductivity of the lead 
chloride 50-fold. Gyulai (33) repeated these experiments, adding 



324 STRUCTURE-SENSITIVE DIFFUSION 

small amounts of potassium chloride, and showing that in the 
equation ^ ^ Ae-^l^^, 

both A and E altered (Table 79). Lehfeldt(34) reversed the 
procedure of Gyulai, and added small amounts of copper and 
lead salts to potassium chloride, obtaining as usual an increase 
in the conductivity of the solvent salt. Since the slope of the 
curves log (conductivity) against l/T {T in °K.) are almost 

Table 79. The effect of adding KCl to PbClg (Gyulai) 
on the constants of the equation K = Ae-^'^^ 



PbClg melted in Clj gas 


PbCl2 + 0-005% KCl 
melted in Clg gas 


A 


E 


A 


E 


1-41 
1-65 
1-29 
1-08 


10,620 
10,860 
11,000 

10,840 


4-36 
6-88 
4-04 
902 


8680 
9000 

8720 
9360 


PbClg mel 


ted in Ng gas 




PbCl2 + 0-005% KCl 
melted in Nj gas 


A 


E 


A 


E 


512 

6-55 
6-29 
1-43 


10,920 

11,0701 Q, , 

10,860f ^^^ 

11,700 Cry 

frc 


4-78 

limed 1 

staUised j- ^®* 
m solution | 


4400 

th (35) 



the same, the change in the conductivity is due to an in- 
crease in the factor A in the equation K = Ae~^'^^. In 
general, however, the effects encountered are in part due to 
a loosening of the lattice (see Table 79, KCl in PbClg), when 
the impurity is added, and in part to a decrease in the size 
of constituent crystal grains. A loosening of the lattice sug- 
gests an increase in the number of interstitial ions and vacant 
sites, and this viewpoint is supported by Lehfeldt's(34) obser- 
vation that long-continued electrolysis will free some crystals 
of impurity (e.g. of PbCl2 in KCl (34)). On the other hand, 
Tubandt and Reinhold(36) found no redistribution of solute 
and solvent by electrolysis for NaCl or KCl in PbCla, so 



CONDUCTIVITY PROCESSES " 325 

that this experiment supports the viewpoint that the solute 
here enhances the conductivity solely by increasing the 
number of crystal grains in a given mass of crystal. 

When a current is passed through a crystal, the resistance 
in the low-temperature structure-sensitive conductivity region 
rises rapidly, as a counter-electromotive force is set up. This 




Fig. 119. The two conductivities for rock-salt. (1) True conductivity, 
(2) conductivity after space-charge redistribution (Beran and Quittnero?)). 



counter-electromotive force is not always due, however, to 
polarisation at the electrodes, but to a redistribution of charges 
in the body of the crystal (28, 37). The experiments therefore 
suggest that interstitial ions exist in the crystal which are not 
in reversible equilibrium with the lattice. Both the initial, or 
true, conductivity of the crystal, and the conductivity after 
the redistribution of space charge in the crystal, conform to 
the well-known exponential equation X = Ae~^/^^ (Fig. 119), 
the values of E being respectively 7510 and 9600 cal./ion. 

Mechanical deformation will create centres of disarray in 
a crystal (38) which result in a momentary increase in con- 



326 STRUCTURE-SENSITIVE DIFFUSION 

ductivity. When a rock-salt crystal was put under a series of 
pressures rising by steps from 20 to 700 kg. /cm. 2, each 
successive step caused a momentary increase in the conduc- 
tivity, while releasing the pressure resulted in no new effect. 
When the pressure was again applied there was no further 
conductivity jump until the previous maximum pressure was 
exceeded, when a momentary increase in conductivity ap- 
peared once more. The jump in the conductivity was shown 
by Stepanow(39) to be an increase in the true conductivity 
rather than a decrease in the counter-electromotive force due 
to polarisation. It was later found that a crystal, if put under 
pressure and then annealed, would give a conductivity jump 
when it was subjected to a second compression, even when 
this compression did not exceed the initial load. The suggestion 
by Jofife (40) that these phenomena result not from an increase 
in the number of centres of disorder, but from a displacement 
of the charge in the crystal was contradicted by Gyulai(4i). 
The experiments reviewed show the multiplicity of effects 
which can influence the non-reversible disorder of crystals and 
so the conductivity or diffusion. The conductivity depends 
upon the few mobile ions in the crystalline mass which are 
perhaps 10~* or less of the total number of ions (42). The energy 
for loosening these ions is considered by Smekal to be only 0*4 
of the energy for loosening a lattice ion. This estimate may be 
compared with the values given in Chap. VIII, p. 363, for the 
ratio of the activation energy for volume and surface diffusion, 
which may vary from 0-2 to 0-5. The latter is the ratio for 
the thorium-tungsten system, the former for caesium on 
tungsten. 



327 



Structure-sensitive diffusion processes 

Structure-sensitive diffusion in metallic systems is of fairly- 
common occurrence. The self-cliffusion of bismuth, while 
strongly anisotropic, is to a certain extent dependent upon the 
bismuth crystal (43) employed: 

D{\\ to 111 plane) = (1-33-16-3) x lO^^Q-isifioo/RT qj^2 gee -i, 
D{^ to 111 plane) = (2-22- 6-5) x iQ-^g-^^'^^'^/^^ cm.^ sec -^. 

The variation in the above equations for D is in the temperature 
independent factor, Dq, but only a small range of values of Dq 
is found. Bugahow and RybaIko(44) made a study of the 
diffusion of zinc and copper in brass in which they found that 
the diffusion constants increased when one passed from single 
crystals to polycrystalline masses because the diffusion con- 
stant depended on grain size. On passing from a single crystal 
to a polycrystalline mass, both E and Dq (in the equation 
D = D^e-^l^'^) changed; but in all polycrystalline samples the 
E values were the same and only the values of Dq altered. 
It has proved possible to measure changes in the state of 
tungsten surfaces very readily by following the thermionic 
emission, which is extremely- sensitive to adsorbed films. 
Molybdenum or tungsten filaments are used containing a 
certain amount of thorium {in the intergranular boundaries).* 
By flashing the filaments at very high temperatures the 
surface may be momentarily freed of thorium; and if the 
filaments are then kept at some lower temperature (2050° K. 
is usual for tungsten), the thorium diffuses slowly from inner to 
outer surfaces and the process may be followed by measuring 
the thermionic emission. It was found that the rate at 
which this diffusion outwards occurred depended on the 
size of the crystallites which comprised the filaments (45, 46) 
(Fig. 120), the extreme variation in D being in the instance 
cited 300 : 1 . It is to be noted, however, that the slopes of the 
three lines of Fig. 120 are all the same — the activation energy 

* The solubility of thorium in a tungsten lattice is negligible. 



J 




^ 








V 


..^^^^ \ 


<5 
















"^ 












t2 


, 




















/ 






^^^ 












""^^ 















■^ 



^■3 



if-S 



> fx7(P*- 



t^P 



if9 



328 STRUCTURE-SENSITIVE DIFFUSION 

does not depend on grain size. By mechanical treatment of a 
tungsten single crystal (46) it was possible to increase the 
diffusion rate without appreciably changing other properties. 
This phenomenon can be contrasted with the behaviour of 
the malleable metal lead in which mechanical working caused 
no change in the self-diffusion co- 
efficient (47). 

Gehrts (48 ) showed that the therm- 
ionic activation of tungsten and 
molybdenum filaments, by diffu- 
sion of thorium from inside to the 
surface, obeyed the law 

e =\- Ce-(i?/r2)«i2/ _ 1 _ Q^-U^ 

where a^ = 2-406 (the first root of 
the Bessel function 
of zero order), 

d — the fraction of the 
surface covered by 
thorium after flash- 
ing, 

T> = the diffusion constant, 

r = the radius of the crystallites composing the 
filament, 

(7 = a constant. 

From the data of Fonda, Young and Walker (4ti) he calculated 
the intergranular diffusion constant at 2050° K. to be 

P = 0-5 X 10-i0cm.2sec.-^ 
Langmuir's(49) data gave 

D = M X 10-10 cm.2 sec.-i at 2055° K. 
Mehl(50) gave for the volume diffusion of thorium in tungsten 

the equation 

D = i.oOe-i2o,ooo//?3' cm. 2 sec.-i, 

and for grain-boundary diffusion 

D = o-74e-34.ooo/R7' ^^2 gec.-i. 



Fig. 120. Diffusion of thorium 
in tungsten crystallites of 
various sizes (46). 

A. Particle diameter 5-3/t; 

B. Particle diameter l-Sfi; 

C. Particle diameter 3000/1. 



DIFFUSION PROCESSES 



329 



Some numerical values of Dq and E, showing the influence of 
grain size, are given below: 

Table 80. Constants in the equation D =DQe-'^l^^ 
for a grain-boundary diffusion 



System 


Particle 
radius in /i 


E 

cal./atom 


^0 

cm. 2 sec.-i 


Th in W 


3000 
7-3 
5-3 


94,400 
95,600 
93,600 
94,600 


3-0 X 10-» 
4-8 X 10-1 
7-9 X 10-1 
8-4 X 10-1 



The diffusion of a number of elements through tungsten has 
been followed by the thermionic emission method (5i ). None of 
the films formed at the tungsten surface is as stable as a 
thorium film, but the results are analogous. They demonstrate 
that a large energy of activation is necessary for diffusion, and 
that the velocity of diffusion depends upon the grain size of 
the tungsten. The diffusion data for these metals are collected 
in Table 81 . Since the values for D and Dq are dependent upon 

Table 81. The constants D, Dq and E in the equation 
D = DQe^^/^^ for diffusion in particular samples of tungsten 



Diffusing 
metal 


D X 1011 
cm. 2 sec.-i 
at 2000° K. 


^0 

cm.2 sec.-i 


E 

cal./atom 


Atomic 

weight 


U 
Th 
Ce 
Zr 

Yt 
C in V/2C(56) 
C in single W crystal 

at 2460° K. (55, 56) 


1-3 

5-9 
95 
324 

1820 

5x10* 


10 

0-75 

1-0 

10 

0-46 


100,000 
94,000 
83,000 
78,000 
62,000 

-108,000 
72,000 


238-5 
232 
140-3 
91 

89 

12 
12 



the grain size, they are not to be taken as more than a 
measure of these constants for particular specimens of tung- 
sten. The diffusion of carbon in tungsten (52) was followed 
by measuring the conductivity of the wire whose surface was 
maintained saturated with carbon. The conductivity fell 
linearly with its carbon content until at the composition W2C it 



330 STRUCTURE-SENSITIVE DIFFUSION 

was only 7 % of that of pure tungsten. Further diffusion in the 
carbide WgC resulted in the formation of WC. The process was 
reversible, when the surface carbon was removed by evapora- 
tion, or with oxygen as carbon monoxide. The diffusion 
constants in tungsten were those in Table 82, when a constant 
concentration at the surface of 0-002 % of carbon was 

Table 82. Diffusion of carbon in tungsten 

(a) for 7 mil. pure W 



D X 10' cm.^ sec.-i 
T'C. 



5 10 

2185 2355 



(6) for 4 mil. W, 0-5% ThOg 



D X 10' cm.2 sec.-i 1-6 4-8 
T°Q. 2070 I 2188 



7-8 
2300 



18 
2400 



assumed at all temperatures. Zwikker's(53) data on the same 
system emph&,sise the influence of grain boundaries, since he 
found values of the diffusion constant varying in the ratio 
30 : 1 at 1970°K. for different tungsten specimens. 

Van Liempt (54) made a study of the diffusion of molybdenum 
in tungsten single crystals * and poly crystals, and found once 
again a dependence upon the size of the individual crystalHtes. 
His data for the two cases may be expressed by 

{a) "Single crystal": i) = 1-6 x iQ-s e-so-ooo'^^', 
(6) PolycrystaUine mass: D = 2x io--2e-80'Ooo/i?z'^ 

The energy of activation is the same in the "single crystal" 
and the polycrystalline mass, but the temperature indepen- 
dent factor is different. 

Preferential penetration down grain boundaries may some- 
times be shown by taking microphotographs of the crystal in 
which diffusion has occurred (50). Fig. 121 gives a cross-section 
of a bi-crystal of brass, stained so that the preferential loss of 
zinc from the grain boundary is very clearly indicated. 

* Since E was the same for the supposed single crystal and for the poly- 
crystal, and as difiFusion occurred down grain boundaries for the latter, it may 
be inferred that grain boundary diflEusion occurred also in the former, which did 
not therefore remain a single crystal. 



DIFFUSION PROCESSES 



331 



It might be thought that all metals showing well-defined 
grain boundaries would show preferential penetration down 
those boundaries, but the photographic evidence is often very 
decisively against such an hypothesis. The photographs show 
that carburising and nitriding of iron or the penetration of 
zinc into a copper bi-crystal do not occur preferentially 
down the grain boundaries. It is therefore rather remarkable 
that evaporation of zinc from brass can occur preferentially 




Fig. 121. Bi-crystal of brass keld for 1 hour at 790° C. in vacuum. 
The loss of Zn has occurred around the grain boundary. 



down a grain boundary (Fig. 121), and that Bugahow and 
Rybalko(44) found that both zinc and copper diffuse in brass 
more rapidly when grain boundaries are present (p. 327). It 
is also noteworthy that structure-sensitive diffusion processes 
in metals occur most often when the metals are hard, and 
have a high melting-point. The diffusion of metals in lead 
for example (melting-point 327° C.) is always a true lattice 
diffusion, while in tungsten one finds predominantly a grain- 
boundary diffusion. The malleability of lead makes it capable 
of being deformed without actually breaking the crystals into 



332 STRFCTTJRE-SENSITIVE DIFFUSION 

small crystallites, while the effect of mechanical working upon 
any hard single crystal is to cause it to change into a poly- 
crystaUine mass. 

It has been observed in a number of studies of the oxidation 
rate of metals (55) that, after sintering the oxide film, further 
oxidation of the metal obeys a law 

x^ = kt+C, 

where x denotes the thickness of the oxide film, and k and C 
are constants. The form of the above equation suggests that 




800" 900* 1000° 1100° 



Fig. 122. The velocity constant for oxidation of some metals as a 
function of temperature (Dunn)(65). 

oxygen attacks the underlying metal by diffusion of oxygen 
or of metal ions and electrons through the intervening oxide 
layer. The constant k obeys the usual exponential formula 
k = k^e-^/^^ (Fig. 122), and on the hjqDothesis of diffusion as 
a rate-controlling factor the slopes of these logA;-l/T curves 
give the activation energies for diffusion. It is noted that 
a break occurs in the oxidation velocity of copper at about 
660° C, although the corresponding curves for the samples 
of brass are linear down to 580° C. Wilkins and Rideal(5ii) 
suggested that the break in the curve for copper was due to 



DIFFUSION PROCESSES 



333 






£ to t. 



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m g ® S^ 



S cS !h O o 









fe 2 ^ ce S 



f^1 



P3 



: ^ -^ 



m Q 



o 



p? 



rO O O O 






+ 






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II II 

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.9 ft 
^ c€ 



ft o 



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s fe 



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d =? 

03 

ft" 



CD 



iS.2 " 






(D d b 
S © 3 



id ® 






05 2 .5 Jg d 
d t5 M tS d 
c «tt r< ® o 



^ ^'fi-^fi 



.2 d 
oD :a 

s !=l 



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9. « £ § § ^ 

dW f=^2 ® M^- 





d 








•+2 




>H 




<B 




N 


% 


H 


A-'^ 


03 


HO 


O 



,d 

2 § 

"d- -S 

% 8 



CO 



* o 
tn O 

d -^ 
ft_d 
o '3 
r i 

^ O P d 

d fc, d d 
OQOO 



Pm 



d.2 



C5 



d-" 

-H^.2 

-t3 03 += 

03 g ^ 



334 STRUCTURE-SENSITIVE DIFEUSIOIS" 

a low-temperature grain-boundary diffusion through cuprous 
oxide merging into a high-temperature lattice diffusion. 

When gases (He, Hg, Ar, Ng, Og) diffuse through silica glass, 
one sometimes gets a continual change in slope of log (per- 
meation rate)-l/y curves at low temperatures ((57,58,59); see 
also Table 23 and Chap. Ill), thought to be due to grain- 
boundary diffusion. 

Diffusion through metals may pass from being predominantly 
lattice diffusion to grain-boundary diffusion by the action of 
the diffusing gas upon the metal. Steel becomes brittle when 
exposed to the continued action of hydrogen. Similar obser- 
vations upon brittleness created in metals by diffusion have 
been collected by McBain(60). Copper became brittle and 
fissured after diffusion experiments (6i), and the diffusion rate 
increased rapidly even at constant temperatures and pressure, 
suggesting that grain boundaries have developed, and even 
become macroscopic channels. Similarly palladium in hydro- 
gen becomes disintegrated to a considerable extent, developing 
a thready structure, with longitudinal fissures (62). 

One may conclude this chapter by giving in Table 83 a list 
of those systems in which grain-boundary diffusion can play 
an important part. There are undoubtedly many others which 
have not been studied, or have been inadequately studied, and 
a number of properties of structure-sensitive diffusion not yet 
revealed. 



REFERENCES 

(1) Chapter VI, pp. 247 et seg., also 292 et seq. 

Wagner, C. and Schottky, W. Z. phys. Chem. 11 B, 163 (1930). 

Frenkel, J. Z. Phys. 35, 652- (1926). 

Jost, W. J. chem. Phys. I, 466 (1933); Z. phys. Chem. 169 A, 129 

(1934). 
Wagner, C. Z. phys. Chem, 22 B, 181 (1933). 

(2) Baiunbach, H. and Wagner, C. Z. phys. Chem. IIB, 199 (1933). 
Dunwald, H.'and Wagner, C. Z. phys. Chem. 22B. 212 (1933). 

(3) Orowan, E. /n<. Con/. i^%s. 2, 81 (1934). 

(4) McBain, J. W. Sorption of Gases by Solids, p. 279. Routledge 

(1932). 

(5) Smekal, A. Int. Conf. Phys. 2, 93 {lQ3i). 



REFERElSrCES 335 

Lennard-Jones, J. E. and Dent, B. Proc. Roy. Soc. 121 A, 247 

(1928). 
Lovell, A. Proc. Boy. Soc. 166 A, 270 (1938). 
Appleyard, E. Proc. Phys. Soc. 49, 118 (1937) (extra part). 
Poulter, T. and Wilson, R. Phys. Eev. 40, 877 (1932). 
Bridgman, P. Rec. Trav. chim. Pays-Bas, 42, 568 (1923); Proc. 

Amer. Acad. Arts Sci. 59, 173 (1924). 
Poulter, T. and Uffelman, L. Physics, 3, 147 (1932). 
Joffe, A. Int. Conf. Phys. 2, 77 (1934). 
Andrade, E. Int. Conf. Phys. 2, 112 (1934). 
Andrade, E. and Martindale, J. Philos. Trans. 235, 69 (1935). 
Hilsch, R. and Pohl, R. Trans. Faraday Soc. 34, 883 (1938), 

where numerous other references may be found. Also Pohl, R. 

Proc. Phys. Soc. 49, 1 (1937) (extra part). 
Smekal, A. Handbuch d. Phys. 24/2, 835, Berlin: Jiilius 

Springer (1933), 
Renninger, M. Z. Kristallogr. 89, 344 (1934). 
Darwin, C. G. Phil. Mag. 27, 315, 675 (1914). 
Prins, I. Z. Phys. 63, 477 (1930). 
Zwicky, F. Rev. Mod. Phys. 6, 193 (1934). 
Goetz, A. Int. Conf. Phys. 2, 62 (1934); Z. Kristallogr. Sonder- 

heft, 1934. 
Orowan, E. Z. Phys. 79, 573 (1932); 89, 774 (1934). 
Smith, D. and Derge, G. Trans. Amer. Electrochem. Soc. 66, 253 

(1934); J. A?ner. chem. Soc. 56, 2513 (1934). 
Barrer, R. M. To be published. 
Buerger, M. J. Z. Kristallogr. 89, 195 (1934). 
de Boer, J. H. Rec. Trav. chim. Pays-Bas, 56, 301 (1937). 
Mott, N. F. Trans. Faraday Soc. 34, 822 (1938). 
Smekal, A. Handbuch d. Phys. 24/2, 883 (1933). 
V. Hevesy, G. Z. phys. Chem. 101, 337 (1922). 
Tammann, G. and Veszi, G. Zeit. anorg. Chem. 150, 355 (1926). 
V. Seelen, D. Z. Phijs. 29, 125 (1924). 
Goethals, C. Rec. Trav. chim. Pays-Bas, 49, 357 (1930). 
Smekal, A. (with Quittner, F.). Z. Phys. 55, 298 (1929). 
Fritseh, C. Ann. Phys., Lpz., 60, 300 (1897). 
Ketzer, R. Z. Elektrochetn. 26, 77 (1920). 
Le Blanc, M. Z. Elektrochem. 18, 549 (1912). 
Gyulai,^Z. Z. Phys. 67, 812 (1931). 
Lehfeldt, W. Z. Phys. 85, 717 (1933). 
Seith, W. Z. P%5. 56, 802 (1929). 

Tubandt, C. and Reinhold, H. Z. Elektrochem. 29, 313 (1923). 
Reran, O. and Quittner, F. Z. Phys. 64, 760 (1930). 
Wenderowitsch, A. and Drisina, R. Z. Phys. 98, 108 (1936). 
Gyulai, Z. and Hartley, D. Z. Phys. 51, 378 (1928). 
Stepanow, A. Z. Phijs. 81, 560 (1933). 
Joffe, A. Z. Phys. 62, 730 (1930). 



336 STRUCTURE-SENSITIVE DIFFUSION 

(41) Gyulai, Z. Z. Phys. 78, 630 (1932). 

(42) Smekal, A. Z. Techn. Phys. 8, 561 (1927). 

(43) Seith, W. Z. Elektrochem. 39, 538 (1933). 

(44) Bugahow, W. and Rybalko, F. Tech. Phys. U.S.S.R. 2, 617 

(1935). 

(45) V. Hevesy, G. Z. Elektrochem. 39, 490 (1933). 

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(47) V. Hevesy, G., Seith, W. and Keil, A. Z. Phys. 79, 197 (1932). 
Seith, W. and KeQ, A. Z. Metallk. 25, 104 (1933). 

(48) Gehrts, A. Z. Techn. Phys. 15, 456 (1934). 

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(50) Mehl, R. Trans. Amer. Inst. min. (metall.) Engrs, 122, 11 (1936); 

J. Appl. Phys. 8, 174 (1937). 

(51) Dushman, S., Dennison, D. and Reynolds^ N. Phys. Rev. 29, 903 

(1927). 

(52) Andrews, M. J. phys. Chem. 27, 270 (1923). 

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(1932). 



CHAPTER VIII 

MIGRATION IN THE SURFACE LAYER 
OF SOLIDS 

Introduction 

Since molecules, ions and atoms can move in solid lattices (as 
when ammonia is sorbed by natrolite, or two metals or salts 
inter diffuse), it is not difficult to visualise a similar migration 
of particles along external surfaces. Grain-boundary diffusion 
occurs more readily than lattice diffusion (for example, the 
activation energies are 90 and 120k.cal. respectively for 
thorium diffusing in tungsten (i)), so that surface migration 
might be expected to occur more readily still. The main lines 
of experiment which have led to the present knowledge of 
surface migration are: 

(1) The study of the growth and dissolution of single 
crystals. 

(2) Phenomena of condensation and aggregation of con- 
tinuous films on solid surfaces. 

( 3 ) Examination of stable monolayer or multi-layer systems 
by photoelectric and thermionic methods. 

Much of the early evidence of the reahty of surface migration 
came from the first source, and this more or less classical 
evidence may now be reviewed. 

Evidence op mobility from growth and 
dissolution op crystals 

Volmer and Estermann(2), when studying the rate of growth 
of mercury crystals at — 63° C. from mercury vapour at 
— 10°C., noted a remarkably rapid rate of growth of thin 
hexagonal crystals in the directions of the plane of the hexagon. 
The linear growth of a hexagon was 3 x 10^^ cm./min., which 
was 1000-fold greater than could be explained by the kinetic 



338 MIGRATION IN SURFACE LAYER OF SOLIDS 

theory. This growth must occur either by surface migration, 
or because impacting molecules entered the lattice and ex- 
panded it laterally. Volmer and Ad1iikari(3) then pointed out 
that surface mobility was implied in certain phenomena of 
crystal growth from melts. Crystal needles often project above 
the surface of the melt, and the needle can only form in this 
way if lateral diffusion of ions on its surface takes place. 

Even more definite evidence was forthcoming in a number 
of studies with benzophenone (3, 4) and with phthalic anhydride, 
coumarin, salol and diphenylamine(5). In the earliest experi- 
ments (3) with benzophenone a succession of mercury drops 
was allowed to brush a long crystal of the organic sohd which 
was slowly worn away. Not only was benzophenone removed 
at the point of contact of the mercury, but also for some 
distance away. In a later series of experiments (4) a stream of 
mercury brushed the edge of a glass plate on which was 
benzophenone 0- 1-1 mm. away from this edge. The benzo- 
phenone was removed by the mercury although it was un- 
touched by it. Moll's (5) experiments were made by depositing 
a very thin film on glass, the edge of which was rinsed by 
dropping mercury. The films were shown to diminish in 
thickness by noting changes in the interference colours 
observed with transmitted light. Moll could find no evidence 
of mobihty with paraffin and cetyl alcohol. 

Richter and Volmer (6) attempted to measure the diffusion 
rates of benzophenone over a mica surface as a function of 
temperature. The quantity of benzophenone in units of 
10~^g. moving per hour over 1 cm.^ of mica increased as the 
temperature rose, although the sorption decreased. Thus the 
diffusion has a large temperature coefficient which suggests 
that it is an activated process. 

An interesting new method for studying the growth of 
crystals from solution, which suggests that lateral migration 
occurs, has been developed by Berg (Qa). Two plane glass plates 
form a wedge, so that transmitted light will give interference 
colours. The crystallising solution is placed in the wedge, and 
lateral growth of a sodium chlorate crystal takes place. 



MOBILITY FROM GROWTH AND DISSOLUTION 339 

Concentration gradients caused distortion of the interference 
bands, which Berg succeeded in interpreting in terms of the 
concentration gradients estabhshed. The concentration dis- 
tribution was not uniform, being greatest at the edges of a 
square plate of sodium chlorate. Thus the highest rate of flow 
occurs at the middle of each edge, and to explain the continued 
production of plane faces it was argued that a mobile surface 
film must have formed. 

Evidence of mobility from the condensation and 
aggregation of metal films 

The structure of condensed films 

There can be two types of film on solids, those which are stable 
in monolayers and those which tend to aggregate into three- 
dimensional structures. The conditions for stability of mono- 
layers or aggregates are similar to those governing the 
stabihty of films at Hquid surface, which either give stable 
monolayers (e.g. fatty acids with long chains on water) or 
gather into lenses (paraffin, or ethylene dibromide on water). 
The monolayer is stable if the spreading occasions a nett 
decrease in free energy, when the various interfacial free 
energies are considered. Another necessary condition for the 
reorganisation of a film is surface or bulk mobility. 

There are a great number of metal films which are thermo- 
dynamically unstable in this sense. Such films can only be 
maintained as glassy deposits if the temperature is so low 
that no migration can occur. The predilection which various 
sputtered or evaporated metallic deposits have for aggregation 
into micro- or macro-crystals is illustrated by the collection 
of observations in Table 84. The evidence from X-ray, electron 
diffraction, and optical experiments reveals that the three- 
dimensional crystalhne state is very readily formed. One notes 
also that a rise in temperature can cause an orientation of 
crystallites to agree with that of the underlying solid lattice. 
Sometimes alloy systems may occur, evidencing mobility and 
interdifiusion of atoms, or certain crystal parameters may be 



340 MIGRATION IN SURFACE LAYER OF SOLIDS 

derived from those of the underlying solid, i.e. the micro- 
crystal may continue the pattern of its substrate. In a great 
many of the systems studied, of which those in the table are 
only a few(i7), the crystal grains grow very rapidly as the 
temperature is raised, due to more rapid migrations of atoms. 
Such a crystallisation of silver, for example, has been observed 
from 250° C. (18) to - 173° C. (i9). 

Table 84. Some observations {obtained by X-ray and electron- 
diffraction methods) on crystalline structure developed in 
thin metal films 



Nature of systems 


Observed properties 


Authors 


Mirrors of Ni, Fe, 


CrystaUites. Parameters same 


Gen, Zelmanov and 


Cd, Hg deposited 


as for metal in bulk. Heating 


Schahiikow O) 


from the vapour 


causes growth of crystallites 




on cold surfaces 






Various thin eva- 


CrystaUine. Regular orientation 


Kirschner(8) 


porated films 


of crystal grains to match 
orientation of substrate 




Sputtered Pt 


Crystalline. Sometimes . with 


Thomson, Stuart 




lattice planes parallel to surface. 


and MurisonO) 




sometimes irregularly deposited 






crystallites 




Ag on Au 


CrystaUine. Same orientation as 
underlying .sohd. Parameters 
as for bulk silver 


Famsworth(io) 


Ag evaporated on 


Crystals on Au, "amorphous" 


Deubnercii) 


to cold Cu and Au 


on Cu 




Sputtered films on 


Crystals, with orientation paral- 


Swamy (i2) 


quartz 


lel to quartz base; heat treat- 
ment increases grain size 




Evaporated Bi films 


Crystals. Orientation of 111 plane 
parallel to base 


Lane (13) 


Au, Ni, Co, Cu, Cr, 


Above characteristic tempera- 


Briick(i4) 


Pd and Ag films 


ture, mosaic of small crystallites. 




on rock-salt 


similarly oriented on crystalline 
base 




Al on Pt 


Crystalline, thick and thin films. 


Finch and 




Some thin films show para- 


QuarreU(io) 




meters corresponding to the Pt 






base 




Very thin films Pt, 


Films formed of crystalline aUoy 


NattaciG) 


Pd, Ag on Cu and 






Ag 







While the true criterion of film stabilitj'' is that the free 
energy of film formation should be greater than the free energy 
of crystallite formation, another approximate criterion is also 
useful. It is usually noted that if the heat of condensation of 



CONDENSATION AND AGGREGATION 



341 



a metal as a monolayer upon a substrate {^H^) is much less 
than its heat of condensation on itself {AH2), the metal fails 
to form a stable monolayer on the substrate, while if zl^^ is 
greater than AH^, the monolayer is stable. This relationship 
is illustrated for some unstable films in Table 85. On the other 

Table 85. A comparison of heats of condensation in uniform 
layers on a foreign substrate (AHj), and heats of condensa- 
tion of the metal on itself (AH^) 



Author 


System 


k.cal./atom 


k.cal./atom 


Estermaiin(20) 
Cockcroft(2i) 


Cd-glass 

Cd-Cu 

Cd-Ag 

Hg-Ag 

Cd-Cu) 

Cd-Ag[ 


3-5 

3-0 
5-0 
2-5 

5-7 


28 
28 
28 
18-5 

*28 



hand, the following metal-substrate systems give large heats 
of sorption (AH^), and form relatively stable monolayer 
systems: 



Cs-W(22), 

Cs-glass(23), 

Th-W(24), 

Tl,In,Ga-W(25), 



Na-OW(26), 

K-OW(27), 

CS-W03(28), 

CS-CS20(29). 



It is systems belonging to the second group which have 
provided a great deal of quantitative information on surface 
diffusion. The group will be considered later, and one may now 
discuss the behaviour of unstable films. 

Some properties of unstable films 

The essential instabihty of some metal film-soHd systems may 
be shared by films of organic and inorganic soUds, which 
crystaUise readily. The observations on metal films are, 
however, more numerous, and their behaviour has been 
studied on surfaces of metals, mica, quartz, and diamond. 

When silver was evaporated on to pohshed quartz (SO) in 
amounts corresponding to less than a monolayer, optical 



342 MIGRATION IN SURFACE LAYER OF SOLIDS 

. examination showed that the film was not homogeneous but 
consisted of small crystalline islands. Aggregation into these 
islands could only occur by migration over the quartz surface. 
Similarly, when a cadmium atom beam was directed on to 
a cadmium-sensitised copper surface (2i), and a wire was 
placed in the path of the beam, the shadow thrown by the 
wire on the surface had a diffuse edge. This phenomenon was 
attributed to the creeping of cadmium atoms along the under- 
lying surface. The same observation (3i) was made when the 
cadmium beam was replaced by a mercury atom beam. 

Surface mobility can explain the experiments of Ditch- 
burn (32) on the deposition by sputtering of cadmium films on 
glass and metal surfaces. He found that when the cadmium 
particles were directed on to the surfaces through fine slits, 
no deposit could be observed if the width of the slit was below 
5 X 10~2mm., a phenomenon which he attributed to a loss by 
surface diffusion of particles which could not be sufficiently 
rapidly replaced by condensation. 

An investigation of processes of aggregation (18) of thin films 
of silver and gold led to definite conclusions concerning the 
mobility of atoms on surfaces. The films were initially about 
fifty atomic layers thick, and were heated to various tem- 
peratures before optical examination. It was found that after 
heating at temperatures from 250 to 280° C. the thin silver 
films had gathered into small spherulites (Fig. 123), at 300° C. 
more spherulites formed, while at 345° C. the size of the 
particle increased, and a crystalline outline began to show 
(Fig. 124). At 500° C. a crop of small crystallites appeared in 
hitherto optically empty areas. It was concluded that the 
most freely mobile part of the film was the surface layer, which 
for a silver layer on a silver film (of 50 layers) was mobile at 
temperatures 700° C. below the melting-point of the metal. 
This observation suggests an activation energy for migration 
of the surface layer of atoms of a clean metal very much smaller 
than the latent heat of evaporation, or of the activation energy 
for the self-diffusion constant of silver. Gold films showed their 
first gathering into spherulites at 400° C. The nature of the 



CONDEISrSATIOlSr AND AGGREGATION" 343 

spherulites is not quite clear, but they were supposed to 
consist of an aggregate of uniaxial crystalline fibres radiating 




Fig. 123. Ag-particles, about l/n across, which developed on heating 
an Ag-fihn 50 atoms thick to 280° C. (Magnification 1000.) 




Fig. 124. Ag-particles showing their additional growth when the 
Ag-fihn was heated to 345° C. for 2 hr. (Magnification 1000.) 

from a centre. Evidence of lateral diffusion is supplied by 
measurements of the electrical conductivity of thin films (23). 
When an alkali metal is deposited upon thoroughly outgassed 



344 MIGRATION IN SURFACE LAYER OF SOLIDS 

pyrex at low temperatures (90° K.), a measurable conductivity 
is observed when the deposit is only 10 % of a monolayer. The 
conductivity rises as the deposit is increased, but when the 
deposition is stopped, the conductivity diminishes agam. 
However, at a certain critical thickness the conductivity rises 
rapidly from 10"^ to 10"^ of that of the metal in bulk to a figure 
of the same order (e.g. Hg on pyrex, Fig. 125). In all cases, large 
irreversible changes in the conductivity may be effected by 
warming. Zahn and Kramer (33) attempted to explain the high 
resistivity of very thin films by assuming that a glass-like 
non-conducting deposit of metal is formed, but Tammann(34) 
has adversely criticised this suggestion. An amorphous state 
seems improbable in view of the numerous studies (Table 84) 
which indicate a crystalline form developed by aggregation 
even in films of high resistivity, and with films of very small 
' ' nominal thickness " . * 

The probable explanation of the resistivity changes of thin 
films can be given in terms of surface tension forces, and of 
lateral mobihty. That a film of nominal thickness 10 % of a 
monolayer can conduct, suggests aggregation into rays of 
atoms, or islands of atoms, which touch other aggregates, and 
so provide a few bridges for conduction. The increase in 
resistivity of films on ageing, at 64-90° K., has been ascribed 
to a cracking of the film under surface tension forces, to give 
a discrete film. The disrupted film aggregates into- spheru- 
lites or crystallites, especially if warmed, by processes of 
surface migration. As the deposition continues, there comes 
a time when crystallites, developing under surface tension 
forces, lateral migration, and bombardment of the crystallite 
surfaces by the impinging atomic stream, begin to touch one 
another more and more frequently, so that the conductivity, 
at a certain critical "nominal thickness", rises rapidly to a 
value very nearly that for the metal in bulk. Fig. 125 shows 
the family of curves of nominal thickness against resistivity 

* By "nominal thickness" is meant the thickness calculated from the time of 
deposition of a calibrated atom stream, neglecting all subsequent aggregation, 
and supposing the deposit to be uniform. 



20° 60" 64° 70' 
20. 1 )\( 



a 



(K 



Deposition teinperature (°K.) 
90° 




Thichiess (A.) 

Fig. 125. The change in resistivity at a critical nominal film 
thickness for mercury on pyrex. ' 




Fig. 



20 40 60 

Deposition temperature (°i^.) 

126. The critical nominal-thickness as a function of 
temperature for mercury films on pyrex. 



346 MIGRATION IN SURFACE LAYER OF SOLIDS 

for mercury deposited on pyrex, while Fig. 126 shows the 
critical thickness at which the large decrease in resistivity 
indicated in Fig. 125 occurs, as a function of the temperature 
of deposition. This critical thickness depends on temperature, 
and increases in the same direction with temperature as 
crystallite size. 

Unfortunately, most of the evidence of surface mobihty 
inferred from these experiments is qualitative only, and while 
with development of the theory of the processes of de- 
formation and surface migration in the films one may hope 
that figures such as those above may give quantitative in- 
formation concerning surface diffusion, one must at present 
obtain this information from the study of types of film which 
give stable monolayers. It is, however, possible to make an 
estimate of the activation energies for diffusion in a number of 
these systems, if the heat of sorption is known. In cases where 
the energy of activation for migration has been measured (35, 36) 
or calculated (37, 38) it has been found to be one-third to one- 
sixth of the heat of vaporisation. Accordingly, for the systems 
given in Table 85 the activation energies for migration of 
cadmium on glass, copper, or silver, or of mercury on silver, 
are from 500 to 1500cal./atom, and mobility must persist to 
very low temperatures. Applying the same rule to the alkali 
metals, the activation energy for diffusion in the surface layer 
should be about 5000 cal. /atom. Andrade's(i8) data (p. 342) 
on multi-atomic silver and gold films show that the surface 
layer migrates at temperatures about 700° C. below the 
melting-point, so that this surface layer is again very mobile. 

In a multi-atomic film it would be natural to assume a 
different mobility in the layer next the solid substrate, a laj'-er 
in the middle of the film, and a la5^er on the surface of the film. 
The evidence of the previous paragraph suggests that this is 
true for the surface and interior. The interpretation placed by 
Dixit on his results (39) on the orientation of crystalline aggre- 
gates, was that the layer next the substrate may also be very 
mobile. By assuming that this layer on heating to moderate 
temperatures could assume the properties of a two-dimensional 



CONDENSATION AND AGGREGATION 347 

gas, Dixit was able to show how the orientation of crystallites 
occurred to conform to the pattern of the substrate. It is 
interesting that when zinc is deposited on molybdenum the 
film aggregates into crystallites already orientated at 10° C. 
If there are Ng adatoms/cm.^ of surface, and each adatom 
requires an activation energy E to become mobile, the number 
N^ which is mobile is given by 

•^ _ Ia p-E/RT 

where /^ Siudf^ are the partition functions for the mobile and 
immobile atoms. The fraction Nj^KNg — Nj) approaches unity 
at high temperatures, and the system behaves as a two- 
dimensional gas. At low temperatures Nj^/{Ng — Nj) tends to 
zero, and one has an immobile film. Since mercury still 
aggregates on pyrex at 20° K., according to Appleyard and 
Loveirs(23) electrical conductivity measurements (p. 344), 
mercury -on-pyrex must have a very small value of ^. It is 
interesting that mercury shows a high mobility on a crystal 
of mercury at — 63° C. (p. 337) and a very small sorption heat 
on silver (Table 85). The activation energy for diffusion of 
mercury on the surface of tin, as an amalgam (p. 369), was only 
1920 cal. /atom, and occurred rapidly at room temperatures. 

Measurements of surface migration in 
some stable films 

Properties of stable films 
When atoms of barium, caesium, potassium, thorium or 
similar metals are deposited on a surface of metaUic tungsten, 
stable films may be built up varying in nominal thickness 
from a fraction of a monolayer to many monolayers. The new 
composite surface has contact potentials and thermionic or 
photoelectric work functions different from those of the clean 
metal. The movements of atoms in these films can therefore 
be followed by the variation in the thermionic or photoelectric 
currents, i, which alter as the fraction 6 of the surface covered 
alters. It is important to find how the current i depends on 6. 



348 MIGRATION IX SURFACE LAYER OF SOLIDS 

The thermionic emission from the clean surface is given by 
the Richardson equation 

in which 6 x ^ is the energy needed (in cal./g-ion of electrons) 
to remove an electron from the metal into free space. Since 
this work function enters as an exponent, small changes in 
bx R cause large changes in i. At 1500° K., the value of i for 
a tungsten surface covered by a monolayer of thorium is 10^ 
times the value for clean tungsten (40). To determine the 
connection between d and i, Langmuir(40) originally supposed 
that the change in contact potential, V, was proportional to 
the electric moment Nfi per unit area, where iV is the number 
of sorbed atoms and /< is the electric moment per atom. He 
assumed that /< did not depend on N, and therefore that 

V = kd, 

and that the work function for a clean surface becomes for the 

composite surface , r> i ^ 

^ bxB + k\0, 

where A.^ may be either positive or negative. If Iq denotes the 
current from a clean surface, and ig the current from the 
surface when a fraction 6 is covered, one has 

ln^, = ln^T^-^^^i4±M). 

Similarly, if i,„ is the maximum electron emission, supposed 
to occur at or near 6 = 1 , a similar pair of equations may be 
written, and by combining the four, one obtains 

0> _logio(Wv) 
logio(*"oA"m) * 

In this expression 6' would be the fraction of the surface 
covered if the following assumptions held : 

(1) that /I does not depend on N, 

(2) that maximum emission occurs at ^ = 1. 



MEASUREMENTS OF SURFACE MIGRATION 349 

Langmuir showed that for caesium-on-tungsten i^ occurs 
when ^ = 0-67, so that the second assumption is not always 
true, while Becker (4i) found for barium-on-tungsten that 
Langmuir's relation was better replaced by 

logio(^^Ao) ^ [1.1(1 _e-2-oe)] (0<^<0-85). 

logio(*mAo) 

This type of deviation may be attributed to the variations 
in jii with 6 or N, so that the first assumption need be true 
only for dilute films. 

Becker and Brattain(24) condensed thorium at a constant 
rate upon a tungsten strip, and measured the growth of log i 
as a function of time t. If the time needed to build up a film 
showing maximum electron emission is i„j, the following 
relationship holds: t f) 

^ "^ T ^ W' 

(9^ = 1 if maximum emission occurs when the surface is just 
covered with a monolayer. When log i/i^ was plotted against/, 
the curve of Fig. 127 was found, which up to (9,„ may be 
expressed by log^Q^/iQ = a(l — e~^^), for l>/>0, and where 
a = 6-54 and c = 2-38. This curve may be regarded as typical 
of the variation of log i/i^ with (9 or / for most stable alkali, 
or alkali earth, metal films. The thermionic emission rises to 
a maximum and then falls asymptotically to its value for the 
metal in bulk. For thorium-on-tungsten the relation between 
Langmuir's 6' and Becker and Brattain's / is 

^' = M35(l-e2-38/). 

One thus sees that to measure 6 from log i one must first 
measure log i/iQ as a function of / = t/t^ = 0/6^' ^^^ then 
evaluate 6^ by allowing atoms to fall at a known rate on to a 
known area of tungsten surface, or alternatively by measuring 
the integrated positive ion current when the total deposit is 
-evaporated in ionic form (42). The specific surface of tungsten 
may with some measure of certainty be taken as 1-4 x the 
geometrical area. The surface, which is made up of crystal 



350 MIGRATION IN SURFACE LAYER OF SOLIDS 

faces somewhat tilted from the horizontal, thus consists of an 
intersecting pattern of the surfaces of tungsten crystallites. 
Unit area of a crystal contains 1-425 x 10^* tungsten atoms and 
it is thought that adatoms occupy the surface so that the ratio 

Number of surface tungsten atoms 
Number of surface adatoms 





VALUE FOR CLEAN TUNGSTEN RIBBON 

MINIMUM VALUE LOGlQ '/: 
OBTAINED " 

EMPIRICAL EQUATION 

PROBABLE CURVE FORf >0-80 



Fig. 127. The thermionic activity of a tungsten ribbon versus the 
time of deposition from thorium wire. 

is an integer. For example, the integer for thorium on tungsten 
is two (22), and for caesium on tungsten is four (42). 

To bring the filament into a condition suitable for ther- 
mionic and photoelectric studies of films, it must be flashed 
at 2800° K. in order to remove a tenaciously held monolaj'^er 
of oxygen (43, 44). Tungstic oxide itself distils off the surface at 



MEASUREMENTS OF SURFACE MIGRATION 351 

a much lower temperature, and 2200° K. is sufficient for its 
removal (26). Thus one may prepare clean tungsten surfaces, 
and surfaces of W-0, both of which lend themselves admirably 
to migration experiments. 

Methods used in measuretnents of surface diffusion 

A group of methods depending on thermionic or photoelectric 
properties has been worked out for the measurement of 
surface migration rates. These methods and the way the data 
are analysed will now be discussed, taking in order thermionic 
and photoelectric studies. 

A. Surface diffusion by yneasurement of thermionic emission. 

(i) The movements of caesium, barium, and thorium in 
films on tungsten (24, 45) have been followed by evaporating the 
metal so as to give a uniform deposit on a tungsten strip, 
suitably pre-treated (p. 350). From calibration curves giving 
logi/ij^ as a function of 6, oi f = djd.^ (Fig. 127), the move- 
ments of a deposited film down concentration gradients may 
be followed as a function of time. If all the film is originally 
on one side only of the strip, the growth of log*/i^ on the bare 
side and the decay of log iji^^ on the side where the film was 
deposited give a mean value of / or (9 as a function of time 
on each side. The strip, after the deposition of a film where 
2 >/> 0, is raised to a temperature where migration but not 
evaporation proceeds. To interpret the data it is necessary to 
ideahse the process of diffusion by assuming that the diffusion 
constant D is not dependent upon the concentration. One 
may then write ^. ,^^o ^2f \ 

— — D\ — ^ -I — — \ 

dt \dx'- dy^J 

Take the ?/-axis to be parallel to the tungsten ribbon and in 
the middle of its front face, so that df/dt — = D{d'Y/dy^) in 
the ^/-direction, since the strip is very long. Let x be any 
distance normal to the ^/-axis in the surface of the ribbon, 



352 MIGRATION IN SURFACE LAYER OF SOLIDS 

whose width is w, so that the mid-point of the back face is 
then Sit X — w. The boundary conditions are: 

(1) at ^ = 0:/ = /q for x = ^wto — \w,f = for x = ^wto w, 
and —^w to —iv; 

rif 

(2) for all values of i, ^ = at a; = and x = w. 

ox 

The solution of the Fick law is then 

/ = i/„ + ^ ""£" (1 e-m^z>.2,/,,2 ^^g rrmx ^^ \ 

and since the series converges rapidly one may compute /for 
any value of x and t. Brattain and Becker, for thorium on 
tungsten, computed f a,s a, function of x, and of t (for a; = 
and X — ^w), when the time t was expressed in units oiw^/Dn^. 
Then from the experimental log* versus / curve, and the 
computed/ versus x curve, they found log* as a function oix, 
and finally by graphical integration log* was found as a 
function of t for the whole front surface of the ribbon. Next 
values of w^jDn^ were chosen until the calculated log i versus t 
curve fitted the experimental one. 

(ii) Langmuir and Taylor (46) followed the movement of 
caesium along a tungsten wire by depositing a uniform film 
along the wire, and removing caesium from the central portion. 
They then measured the amount which flowed in from the 
ends by evaporating it, once again only from the central part 
of the filament, as a measured positive ion current. To carry 
out these measurements three metal cylinders were arranged 
along the length of the wire in series, the wire passing axially 
through them, parallel to their length. The caesium was 
deposited with the cylinders at + 22 V. The caesium from the 
wire in the central cylinder was withdrawn as positive ions 
by altering the potential of the cylinder to —44 V., after which 
the potential was again returned to + 22 V., and the wire held 
at a temperature suitably high for migration without evapora- 
tion. Finally the total caesium that had moved into the centre 
was estimated, as positive ion current, by again altering the 



MEASUREMENTS OE SURFACE MIGRATION 353 

potential of the central cylinder and flashing. From this 
quantity of caesium the authors evaluated the diffusion 
constant. 

(iii) Another method due to Langmuir and Taylor (46) is 
based on the discovery that caesium films on tungsten may 
exist as two phases, a and j^, in equilibrium. From the con- 
densed a-phase caesium escapes as atoms, and from the 
/?-phase as ions, both rates being equal in the equihbrium 
condition to the rates of arrival from the gas. At a given 
temperature there is only one critical pressure, j3q, of caesium 
vapour at which the two phases can coexist. When ^q is 
altered to jp-^, one phase must disappear, and the phase 
boundary moves along the wire with a velocity v. The resulting 
surface migration is balanced by differences in the rates of 
evaporation {v^ and v^ for atoms and ions) and of condensation, 
pb^. v^ and Vp are functions of T and 6, and /^^ of 6 and the 
vapour pressure of caesium. If j^ = (^'a + ^i?)? sis ordinate, is 
plotted against N, the number of atoms/cm.^, as abscissa, the 
curve rises from iV = OtoiV^= 7x 10^^ falls from A" = 7 x lO^^ 
to A = 43 X 10^^ and rises when A > 43 x 10^^, inversely as 
the p-v curve of an imperfect gas. The curve {v—pi) versus 
A behaves similarly and .intersects the A-axis at three 
points, thus enclosing two areas A-^^ and J-g. The condition for 
the stationary phase boundary (46,47) is A-^ = A^, while if /* 
becomes [i-^, the velocity v of the movement of the phase 
boundary along the wire is 

v = {DI2Af-{li^-l^), 

when D is regarded as independent of A and very small. The 
experiments did indeed suggest that at equilibrium A^ = A2 
and that v was proportional to (/^i — /*), thus giving a value of 
D = Qx 10-*cm.2sec.-iat 967°K., where A = 7-3 x IQi^. 

B. By the measurement of photoelectric emission. 

That the photoelectric effect could be used to follow surface 
migration was first suggested by work of Ives (48), but the 
subsequent developments of the method and technique were 

BD 23 



354 MIGRATION IN SURFACE LAYER OF SOLIDS 

carried out by various workers (26, 27,28). The method has 
provided some detailed information concerning the forces 
governing the diffusions. It has one great advantage over the 
previous thermionic method, in which one measiu?es only the 
integrated current over the whole filament. The photoelectric 
method, on the other hand, permits one to measure the actual 
concentration gradients along the surface, and their change 
with time, since a small well-defined spot of fight maybe made 
to traverse the surface on which the film is deposited. 

(i) Bosworth deposited a small patch of sodium (26) or potas- 
sium(27) on the centre of a tungsten strip, and by traversing 
the patch with a spot of light measured the concentrations and 
concentration gradients as a function of time. Three methods 
were used to interpret the data, based on solutions of the 
diffusion equation and on a calibration curve giving log^ as 
a function of N. The strip of tungsten, which had been 
out-gassed at a temperature sufficient to leave the oxygen 
monolayer but to remove oxides of tungsten (26), absorbed the 
sodium or potassium, a portion of which reappeared on the 
surface when the strip was heated to a suitably high tem- 
perature. The alkah metal thus migrated over the surface of 
crystalfites and then between grain boundaries down into the 
tungsten. The capacity of tungsten to absorb the alkafi metal 
was limited, and when the hmit was reached the deposit of 
alkali metal simply spread over the surface to give a uniform 
layer and Fick's law could be applied only to the centre of 
the .deposit. This spreading took 1 or 2 hr. at 300° K., or 
5-10 sec. at 800° K. The families of curves of Fig. 128 
show the processes of absorption of sodium in sodium-free 
tungsten. 

The family of concentration-time-distance curves in Fig. 
128 corresjDonds very nearly to the solution of the diffusion 
equation (26) when Cq is the amount of substance per unit 
area at time t = 0, concentrated at the zero plane, and 
when diffusion occurs in the direction +x, inwards down 
inter- cry staUine surfaces (Chap. I, p. 44). The diffusion 



MEASUREMENTS OF SURFACE MIGRATION 



355 



constant for this process is denoted by A, and the solution 
at a; = is ' ^ 




1-2 1-6 2-0 

Distance along- strip, cm 

Fig. 128. The absorption of sodium by a sodium-free tungsten strip. 
A. 5 mins. 



B. 10 

C. 20 

D. 40 
e! 60 



rat 295°-K. 



F. 
G. 



1 min. 
3 mins. 



H. 5 „ I 
I. 11 „ J 



at 415° K. 



Accordingly 1/C^ should be a linear function of t, as indeed 
was found to be the case (Fig. 129). The curves do not always 
pass through the origin because there is an uncertainty in 
fixing the time of the beginning of the experiment. 

(ii) The analysis of the data in which the diffusion laws 
could be applied only to the central part of the curves(27) 

23-2 



356 MIGRATION IN SURFACE LAYER OF SOLIDS 

(p. 367) was carried out by employing the operational notation 
of Heaviside, with p = djdt; V^ = d^jdx"^, in the case of one 
dimensional diffusion. 




20 40 

Time in minutes 

Fig. 129. Tlie concentration of sodium at the peak of the 
curves of Fig. 128, as a function of time. 

Curve (a) 415° K. Slope = 0-022. 
' Curve (6) 293° K. Slope = 0-00067. 



or 



Then, since Cq = f{x) only, 

pC-pCo = DV^C, 

p-DV^' 

c = e^^^%m 



and thus 



7)2/2 D^t"^ 

or C = C, + DtV^C, + '^W^C, + ... + -^V^-C,. 

The total amount of sodium which has diffused from within 



MEASUREMENTS OF SURFACE MIGRATION 357 

two ordinates a and b of concentration-distance curves such 
as shown in Fig. 128 is then 



jyo.,...^m-m 



If the derivatives of higher order than d^/dx^ are neglected 
and the ordinates a and b are chosen so that d^GQJdx^ is zero, 
the expression reduces to 



iv-.*="[(a.-(a]- 

where the bar implies an average value of dC/dx over the 
time interval t. This method suffers from one weakness — 
the assumption that the derivatives of Cq form a rapidly 
converging series. It gave values of D = DQe~^/^^ and of E 
which were, however, in fair agreement with those obtained 
by the previous method., 

(iii) A third method of analysis of data obtained by using 
the photoelectric exploration of the surface with a spot of 
light depends on the setting up of an equiUbrium state of film 
concentrations along a wire with a temperature gradient. In 
this gradient the cooler parts of the wire are more densely 
covered than the hotter parts. At equilibrium, 

dt dx\ dx 

d^dD j^d^C _ 
dx dx dx^ 



Also, since D = D^e-^'^, 



dx dx\Tj' 



giving by substitution in the first expression 
E dldxllogAd Cldx)] 



358 MIGRATION IN SURFACE LAYER OF SOLIDS 

The values of {dCjdx) were measured photoelectrically, and of 
T a,s a, function of x with an optical pyrometer. The results 
obtained by this method were comparable with those obtained 
by the other two methods. 

(iv) A further method of obtaining the activation energy E, 

which was independent of the amounts of metal deposited, or 

of the thickness of the substrate into which diffusion occurred, 

was employed by Frank (28). He used the simple Tick law 

1 dC d^C 

— ^— = ^— - and so neglected the variations of D with C. The 

D at ox^ 

method consisted in measuring the relative beam intensities 
Tij and 7^2 for which deposition-time curves taken at tem- 
peratures T^ and T2, coincide. Under these conditions 

so that one may readily compute E. The Fick law is to be 
solved for the conditions: 

(1) At i = 0, inside the substrate O = for all x. 

(2) At the surface x = 0, the concentration depends on 
time t as determined by the deposition curve: C^-^q =/(0- 
Also, the total deposit is given by 

d 

C{x,t)dx = nt, 


where d is the thickness of the substrate into which the film 
diffuses. At the temperatures 7\ and T^ one may then write 

(1) (2) 

C^{x,0) = Q, (72(.T,0) = 0, 

rd rd 

Ci{x,t)dx = n^t, C2{x,t)dx = n^t. 

jo jo 

Each solution of the equations ( 1 ) can be transformed into 



MEASUREMENTS OF SURFACE MIGRATION" 359 

a solution of the equations (2) hy simply reducing the time 
scale. If ar = t, by substitution in (2) one obtains 

1 aOa a^Ca 



aD2 dr dx^ ' 
C2(^,0) = 0, 

I C2{x,ocT)dx = (Xn2T, 
Jo 

and the two sets of equations are equivalent if 
cc = DJD2 = ?^lA^2. 
The methods based upon the thermionic and photoelectric 
properties of surfaces form an interesting study in themselves. 
The sequel will show that the surface diffusion constant D is 
not independent of the surface concentration, and this fact 
constitutes the major objection to these analyses. Only by 
using the Fick law in the form 

ao__a_/ ao 

dt dx\ dx 

can one allow for variations in D with G. This law cannot be 
satisfactorily apphed by using thermionic methods, which 
give only integrated effects over the filament. The photo- 
electric method, however, is potentially capable of appKcation 
to measure D as a function of C. If, by using the method A (ii) 
of Taylor and Langmuir (p. 352) , the central part of a tungsten 
filament were cleared of caesium, for example, and the caesium 
from the sides then diffused inwards to the centre, the actual 
concentration gradients could be measured by a photoelectric 
exploration of the wire with a spot of light. The concentration- 
distance curves can then be submitted to Matano's analysis 
(Chap. I, p. 47) of the equation 

ac__a_/ ac\ 

dt ~ dx\ dxj' 

to give D as a function of C. No such experiment has yet been 
made, and until this has been done the surface diffusion data 
must suffer in accuracy, although the main properties of 
surface flow are reasonably well established. 



360 MIGRATION IN SURFACE LAYER OF SOLIDS 

Migration in films of caesium on tungsten 

The composite surface Cs-W has been the subject of numerous 
studies (35, 45, 46). Tungsten after ageing at 2800° K. provides 
a surface, homogeneous save for about 0-5 %, on which a 
caesium monolayer is completed when N, the number' of 
atoms per cm.^, is 3-56 x 10^*, an atom density giving a ratio 
caesium : tungsten — 1 : 4 on the surface. The sorption heat of 
caesium on the inhomogeneous 0-5 % of the surface is 80 k.cal., 
compared with AH — 63-5 k.cal. for dilute films on the rest of 
the surface. All the properties of the Cs-W surface — urates of 
evajDoration of atoms, ions, and electrons, heats of sorption, 
velocities and activation energies of migration — depend very 
strongly upon the fraction of the surface covered. The electron 
emission reaches its maximum at ^ = 0-67 of a monolayer. 

The migration of caesium on the surface of tungsten was 
first observed by Becker (45). It is interesting to compare values 
of the surface diffusion constant D obtained by different 
methods. By the method which depends on freeing the centre of 
the filament from caesium (p. 352), Langmuir and Taylor (35, 46) 
found values of Z) in cm.^,sec.~^ at iV^ = 2-73 x 10^^ atoms/cm.^ 
and at temperatures of 654, 702, 746 and 812° K. which 
conformed to the equation 

logioi) = -0-70-3060/T. (1) 

The method depending upon the movement of the phase 
boundary between the dense and dilute surface phases (which 
can coexist at appropriate temperatures and pressures of 
caesium (p. 353)) gave a value of -D of 6 x 10~* cm.^ sec.~i when 
N = 7-3 x 1013 at T - 967° K., whilst the extrapolation of 
equation (1) gives D = 1-1 x lO^^cm.^sec.-i at 967° K. and 
with N ■= 2-73 X 10^^. The discrepancy may be due to the 
variation in D with surface concentration (p. 372) for when 
iV = 2-73 X 1013 and 1-74 x lO^^ respectively the first method 
gives X)8i20K. = 3"^ '^ 10-^ and 1-4 x lO-'^cm.^sec.-i. 

x\s the surface concentration of caesium increases the heat 
of sorption diminishes, being about 41,000 cal. /atom when a 



MEASUREMENTS OF SURFACE MIGRATION 



361 



monolayer is nearly completed. The formation of multilayers 
results in a further decrease in the heat of sorption, until the 
film assumes the properties of caesium in bulk, for which the 
heat of condensation is 18,240 cal./atom. The activation energy 
for migration in the first layer, when N = 2-73 x lO^^, is about 
20 % of the heat of sorption, so that if the same ratio exists 
in a multi-atomic film, or for the surface of the metal in bulk, 
the energy of activation would be 4600 cal./atom (35). Using 
tliis assumption, Taylor and Langmuir calculated the values 
of Dj and D^, the diffusion constants in a monolayer, and in 
the surface layer of a thick deposit, respectively (Table 86). 
The diffusion constant D^ is greater than the average diffusion 
constant in liquids at room temperature, and the assumed 
activation energy (4-6 k.cal.) compares with the energy of 
activation of 5-3 k.cal. when DgO diffuses into HgOcso). One 

Table 86. Surface diffusion constants for caesium in a inono- 
layer, and in the surface layer of a thick deposit, of caesium 
on tungsten 



Temp. ° K. 


Di (cm.2 sec.-i) 


D^ (cm.^ sec.-i) 


300 


1-2 X 10-11 


0-00034 


400 


4-3 X 10-8 


0-00134 


500 


1-5 X 10-' 


0-0022 


600 


1-6 X 10-6 


0-0027 


700 


8 X 10-« 


0-0032 



may thus regard the surface layer of a thick caesium deposit 
as being in a condition not very different from that of a liquid. 
On the other hand, the mobility of the caesium in the first 
layer is 10^-fold less at room temperature than the mobility 
in a liquid, and does not approach the mobihty of a liquid 
until a temperature of 700-800° K. is reached. 

Migration of thorium, on tungsten 

Thorium-coated tungsten filaments have a high thermionic 
emissivity and, hke caesium-coated tungsten filaments, have 
been widely studied (45,24,1, 22). It is possible to prepare fila- 
ments of tungsten with thoria incorporated, and it is with 



362 MIGRATION IN SURFACE LAYER OF SOLIDS 

thoriated filaments of this type that much of the available 
information has been obtained. These studies have con- 
siderably extended knowledge of the grain-boundary diffusion 
processes already discussed (Chap. VII, p. 327), and will now 
be considered in relation to surface mobihty. 

Quantitative measurements on the surface diffusion constant 
are somewhat scanty, the most complete analysis being that 
of Brattain and Becker (24), who followed the diffusion of 
thorium evaporated on to tungsten from the covered to the 
uncovered side of the tungsten strip, by measuring the changes 
in thermioni<3 emission with time. The theory of their method 
has been described earlier (p. 351). 



s 







-1535° K. - 




^ 


" 1655 K. i- 










• 




^ 












• 


^^^ 












• 


• 
















• 


/ 


















/ 








• EXPERIMENTAL POINTS 




/ 


















/ 















TIME OF FLASHING IN HOURS 

Fig. 130. Comparison of experimental and calculated migration 
curves for front side of ribbon. 



At the start of the migration experiment/ = <9/(9,„ (p. 349) 
was 1-77. The strip was then flashed at 1535 or at 1655° K., 
and its thermionic emission measured periodically at a lower 
temperature of 1261° K. Fig. 130 shows the observed values 
of logi/i^ plotted against time after flashing at 1535 and at 
1655° K. In the same figure the full curve has been calculated 
according to the equation (p. 352): 



•^ 2^ 7T 



1 \m 



e-m^nWllw^-^Qs'^af^^^j^ 



w 






Very clearly the experimental and theoretical curves are a 
poor fit, and lack of agreement must be ascribed to the 



MEASUREMENTS OF SURFACE MIGRATION 363 

variation in D with/ (p. 371). Since tlie initial value of / was 
1-77 on the front and zero on the back of the tungsten strip, the 
final value on back and front was/ = 0-885, as was verified 
by the thermionic emission. Brattain and Becker made a 
rough calculation of the energy of activation, E, for thorium 
over tungsten, arriving at £^ = 110 k.cal./atom, but this 
value cannot be accurate since / was not the same at both 
the temperatures used (1535 and 1655° K.). The values of D 
(in cm.^sec"^) were 

at 1535° K. D = 1-84 x 10-^, 

at 1655° K. B = 2-44 x lO-^, 

but perhaps half this variation had its origin in the diEferent 
/ values at which the observations were made. Using the 
Dushman-Langmuir equation D = {EINQh)dH~^l^^ as an 
empirical guide (Chap. VI, p. 298), one finds agreement 
between the observed and calculated values of i) at 1655° K., 
when E = 66-4 k.cal./atom. This is a much more likely value 
than 110 k.cal./atom, since it gives a satisfactory sequence 
with the expressions for lattice and slip -plane diffusion already 
given (Chap. VII, p. 328), as the following set of equations 
demonstrates: 

Volume diffusion: . logjo-D = 0-0-26,200/7'. 

Grain-boundary diffusion: logi^D = -0-13- 19,700/T. 

Surface diffusion: log^oD = - 0-33 - 14,500/2^. 

The extent to which D varies as the surface concentration 
of the thorium is increased will be discussed later when the 
spreading pressure in films is considered (p. 372). At the 
moment it will be sufficient to comment that when 

6^0, DjD^ = 1, 

d = 0-49, DIDq= 11-3, 

^ = 0-98, DIDq = 99-5, it Dq = D a,t 6 = 0, 

according to the calculations of Langmuir(22). 



364 MIGRATION IN SURFACE LAYER OF SOLIDS 

The mobility of sodium on tungsten-oxygen surfaces 

Boswortli (26) using the three photoelectric methods described 
on pp. 353-9 measured the migration of sodium over and into 
tungsten which had been out-gassed at 22Q0° K. and therefore 
retained a monolayer of oxygen. These methods were: 

(i) The rate of diminution of sodium concentration at the 
centre of an island of sodium deposited on sodium free 
tungsten, by one dimensional migration into the tungsten. 

(ii) The rate of diminution of sodium concentration in an 
island of sodium deposited on sodium-saturated tungsten, 
using Heaviside's solution of the diffusion equation, in terms 
of dCjdx and d^C/dx^, etc. 

(iii) The equilibrium distribution of sodium over the surface 
of a strip, with a temperature gradient along it. 

It is noteworthy that the values of D = D^e-^'^'^ or of E 
obtained by the three methods, treated as one-dimensional 
diffusions, were in moderate agreement, although the diffusion 
processes must have varied among themselves. For instance, 
in (i) the diffusion is of sodium into the body of the tungsten, 
and in (ii) and (iii) is a spreading over the surface. The inference 
would be that in the strip of tungsten used the impedance to 
migration down grain boundaries is comparable to the im- 
pedance to migration over the surface. This is not true of the 
analogous thorium-tungsten systems where Langmuir's inter- 
pretation of available data gave 

-^gram boundary = 90-0k.cal./atom, 
and ^surface = 66-4k.cal./atom. 

Typical values of E computed by the method (i) far a 
number of experiments between 290 and 455° K. are: 

6950; 6260; 5800; 5330; 6730; and 5800cal./atoni, 
the mean of these data being 6260 cal./atora. The values 
obtained for E vary considerably with temperature, being as 
low as 3200 cal. between 76 and 200° K., and as high as 



MEASUREMENTS OF SURFACE MIGRATION 365 

8600 cal./atom between 400 to 550° K. The variation in E with 
temperature could be regarded as a specific heat effect — the 
specific heat of the sodium in the mobile state being less than 
that in the immobile state. Finally, it should be observed that 
the data take no account of the variation of D with d. 

As soon as the body of the tungsten was fully charged with 
sodium, the spreading of sodium over the surface could be 
observed at room temperatures. Using the method (ii), and 
choosing two ordinates, a and 6, such that d^C^dx^ is zero 
where they cut the Cq, one has 



[ {CQ-C)dx = DtU^ I - 1 ^ I 



(dC\ _(dC\ 

Then one may read ofi" values of dCjdx from the graph, and 
find (Cq—C) dx graphically, and so compute D. The results 

J a 

are indicated by the following set of data: 









t—Z2 min. 


t=21 min. 


t = 60 min. 


\dxl„ 






79 


39 


60 


\dxff. 






-62 


-31 


-55 


\ / u 

/'2-15 

{Co- 

J 1-85 


-C) 


dx 


2-22 


0-68 


2-90 


-L'OOQO IT 






0-8 X 10-5 


0-6 X 10-5 


0-7 X 10-5 



For this set of data the surface concentration was 2-1 x 10^^ 
atoms/cm. 2 

In the manner outhned, and for initial concentrations 
ranging from 2-8 x 10^^ to 6-6 x 10^^ atoms/cm. ^ of apparent 
surface a mean value of i)2930K. — 0-8 x lO-^cm.^sec.-^ was 
computed. It was, however, noted that the larger the value 
of the surface concentration, the greater was D. The values 
of i) at a number of temperatures are given in Table 87. 
From the data of this table one may plot the curve log D/T^ 
against 1/T and from the slope of this curve compute the 
activation energy for diffusion as 5-5 k. cal./atom. 



366 MIGRATION IN SURFACE LAYER OF SOLIDS 

Method (iii), depending on the equilibrium distribution of 
sodium along the strip when a temperature gradient existed 
along it, gave values of E varying from 7-4 to 4-4k.cal./atom, 
a^nd which were therefore similar to the values of the activa- 
tion energy obtained by the other methods. 

Table 87-. Diffusion constants D for the inigration of sodium 
into and over tungsten with a monolayer of oxygen 





10^ X i) in 


Temp. ° K. 


cm.2 sec.-^ 


293 


0-8 


350 


3-2 


375 


6-0 


410 


13 


420 


20 


430 


30 


450 


34 


500 


50 


520 


77 


555 


128 


620 


200 


690 


270 


740 


310 


800 


330 



The mobility of potassium on tungsten-oxygen surfaces 

The same photoelectric methods were applied by Bosworth (27) 
to this system as he had earher used in the analogous sodium- 
oxygen-tungsten surface films (i.e.). The behaviour of the two 
systems was closely analogous. As for sodium, the first efi'ect 
observed was a uniform fading out of the photo-emission, 
corresponding to an absorption of the potassium film by the 
potassium-free tungsten. Then when the tungsten was filled 
with potassium, the potassium deposit spread over the surface, 
by a process analogous to two-dimensional evaporation rather 
than as a true diffusion, save in the central part of the original 
island of potassium. 

The application of the method (i) of the previous section 
led to a value of the activation energy of 6960cal./atom — a 
similar value to that observed for sodium, but as with the 



MEASUREMENTS OF SURFACE MIGRATION 



367 



sodium-tungsten system this figure depended upon the surface 
concentration. Method (ii) of the previous section employs the 

"""" " [m-m in. 



\\Co- 

J a 



C)dx 



where the ordinates a and b are so chosen that d^C^dx^ is 
zero in a concentration-distance curve (Fig. 128). The values 
of D obtained were, for very dilute films : 

Temp.° K. 480 510 590 710 780 

D cm2. sec.-i 0-57 x 10"^ 1-4 x 10"^ 10 x 10"^ 140 x IQ-s 280 x 10-^ 

When the curve logD versus IjT was plotted, the activation 
energy was found to be 15,300 cal./atom. Table 88 shows that 
the slope of the logD versus IjT curves changes as the surface 
concentration grows, and the numerical values of E are given 
for various values of surface concentration in this table. 

Table 88. The variation in activation energy E with N, 
the number of atoms/ cm. '^ 



N X 10-" 


E 


atoms/cm.^ 


cal./atom 


[0] 


[16,700] 


0-06 


16,000 


012 


15,500 


0-24 


14,600 


0-48 


13,700 


0-60 


13,200 


1-2 


12,100 


1-5 


10,900 


2-4 


8,100 


3-0 


7,700 


4-8 


6,750 



Mobility of caesium on tungstic oxide 
Frank (28) measured the changes in photoelectric emissivity 
of tungstic oxide (p. 358), as caesium in measured quantity 
was deposited, and then allowed to diffuse away. He found 
that at high temperatures the deposit decayed more slowly, 
but this result was. due in some way to the caesium which 
had collected below the surface by migration, during the 
deposition, or in earlier experiments. 



368 MIGRATION IN SURFACE LAYER OF SOLIDS 
The MIGRATION OF OTHER FILM-FORMING SUBSTANCES 

The method of Brattain and Becker (p. 351) lias been used 
to show that migration of barium and caesium can occur. 
When barium (or caesium) was deposited on one side of the 
tungsten strip, which was then raised to 1000° K., the therm- 
ionic emissions from the front and back slowly became equal. 
The value of /= djd^ on the front was initiallj^ 0-80, while 
after the flashing at 1000° K., the final value of/ on both 
back and front was 0-4. 

Becker (5i) remarked that while most of the results described 
relate to electropositive films, they should also apply to 
electronegative ones, such as oxygen, save that here the 
electron emission is decreased as the quantity of oxygen sorbed 
is increased. An experiment suggested that oxygen migrated 
very rapidly at 1400° K. These statements would merit further 
study, since the great readiness with which tungsten chemi- 
sorbs oxygen from the surroundings may vitiate many results. 

In a study of the properties of indium, thallium and gaUium 
films on tungsten oxide, Powell and Mercer (25) observed that, 
at temperatures 200° C. "below the temperature of evaporation 
of ions, there was a gradual decay with tune in the positive ion 
current (tested by momentarily raising the temperature of 
the system). These effects may be understood by assuming a 
migration into the oxide or over its surface. Similar obser- 
vations (29) were made on a Cs-FegOg system, in which it was 
shown that the photoelectric current decayed with time, and 
that an inward or lateral spreading of caesium in the oxide 
was a possible explanation.* 

Koller(29) deposited caesium on a silver surface, to give a 
multimolecular layer, and then exposed the composite surface 
to the action of oxygen. Simultaneous observations of the 
photoelectric properties of the Cs-CsgO-Ag surface showed that 
as fast as caesium oxide was formed it was covered by a 
polyatomic layer by processes of readjustment by diffusion in 

* In this case the contamination of the metal by gas, or its re-evaporation, were 
not positively excluded, so that the evidence is not conclusive. 



MIGRATION OF OTHER SUBSTANCES 369 

the film, so that the photoelectric properties remained almost 
unaltered until nearly all the caesium was used up. 

Two other methods of obtaining information concerning 
surface migration are worthy of mention. The first is the 
method of the radioactive indicator. If polonium is deposited 
on a silver foil, at one end only, and the temperature is raised 
to 300° C, a creeping of the polonium along the silver could 
be noted, the velocity of which increased as the temperature 
was raised (52). No volume diffusion of polonium through the 
foil took place up to 500° C, an interesting commentary upon 
the relative ease with which surface and volume diifusion 
occur. 

The second method is one which may have some general 
appHcability to amalgams. It consists in measuring the rate at 
which mercury will spread over metal surfaces. Spiers (53) 
found that a drop of mercury spreads over tin foil in circular 
or elliptical areas in which, when diffusion has ceased, there 
is a uniform mercury content (ll-8%Hg). There is thus a 
concentration discontinuity from 11-8 % to % mercury at 
the edge of the area, and the edge may be easily observed. 
Alty and Clark (54) made quantitative measurements on rates 
of spreading, which they found to be sensitive to the pre- 
treatment of the surface and the nature of the medium (water, 
oil, or air) in contact with it. The surface .diffusion was much 
more rapid than the volume diffusion, for after a surface 
diifusion of several centimetres the mercury had penetrated 
into a tin block by a fraction of a millimetre only. 

The spreading of mercury up the surface of tm rods dipping 
into the mercury was a one-dimensional diffusion which was 
assumed to obey the following conditions: 

dn ^d^n 

where n is the number of mercury atoms/cm. ^ at time t and 
height X above the surface of the liquid mercury. 

(ii) n = a,t t = and x > 0, n = tIq Sit x — for all t. 

BD 24 



370 MIGRATION IN SURFACE LAYER OF SOLIDS 

This gives as the solution of (i) 



n = n, 



^'^ijimi (^''^P-i'' 



and since at the upper edge according to Spiers n = n^ for 
all i where n-^ corresponds to 11-8 % of mercury, the progress 
of this boundary is given by 

Thus ^ = ""^^(2^50) = °'>'^'*"*' 

X 

and so , = constant, C. 

Accordingly D = x^l4:CH and x'^ is a Knear function of t. From 
the slopes of a:;^ — t curves at various temperatures, an activation 
energy for surface diffusion of 1920 cal./atom was calculated. 

A COMPARISON OF THE DATA 

The numerical values of the diffusion constants are compared 
at a few selected temperatures for some of the systems showing 
surface and intergranular diffusion, in the data following: 

Temp. ° K. 

500 550 600 650 700 800 

Na-OW Dcm'.lBec. 

= 59 X 10-5 102 X 10-5 177 x lO-s 232 x 10"^ 278 x 10-^ 330 x 10-^ 

All for (9 ^ 1 

K-OW I>cm.»;seo. 

= 1-1x10-5 4-6x10-5 19x10-5 66x10-5 126x10-5 340x10-5 

AU for 6^0 

Cs-W Dcm.'kec. 

=0-015 x 10-5 0-055 X 10-5 q-iO x 10-5 0-4 x 10-5 0-85 x 10-5 3-0 x 10-* 

All for (9 ~ 

Th-W Dcm.=/Bee. 

= 1-84 X 10-9 at 1535° K. for i9 =^ 1 

The data show considerable differences, those in the Na-OW 
system comparing with data calculated by Langmuir(i, 22) for 
a surface of caesium metal: 

Temp, °K. = 500 600 700 

Cs-Cs„eUl Ocm.Vscc. = " 220 X 10-5 270 X 10-5 320 X 10-5 



COMPARISON OF THE DATA 



371 



Again the potassium in the K-OW system at low temperatures 
is much less mobile than in the Na-OW system, as corresponds 
to the difference in 6^ ( ~ 0, and ~ 1 respectively) ; but the 
mobihty of the potassium passes that of the sodium at 800° K. 

Table 89. The variation in heat of sorption with surface 
concentration and valence 



System 


N (atoms/cm.2) 
f=dld„„ovd 


AH 

cal./atom 


Reference 


Cs-W 


N^O 


65,100 


Taylor and 




N=2-7S X 101* (/ = 1, 0=0-67) 


44,500 


Langmuir(35) 




^=3-56x10" (0 = 1) 


40,800 




Na-OW 


A^~0,/^0 


32,000 


Bosworth(26) 




/=0-2 


28,500 






/=0-4 


27,000 






/=0-6 


23,000 






/=l-0 


17,000 




Th-W 


A^ =0-87x101* 


178,000 


Langmuir(i,22) 




27=2-5x101* 


174,000 






.?/ = 5-0xl0i* 


172,000 





So great is the effect of surface concentration upon mobihty, 
however, that errors in estimating the concentration could 
cause the observed trends. Monovalent metals are more mobile 
than bivalent metals and bivalent metals than tetravalent 
metals. Thus mobility in alkah metal monolayers on tungsten 
can be observed at 300° K. ; barium migrates measurably only 
at 1000° K.; and thorium at 1500° K. The trend shown m E, 
both in respect to surface concentration and valency, is 
reflected in corresponding trends in the heats of sorption AH 
(Table 89). 



The variation in the diffusion constants with 
surface concentration 

The cause of the increase in D or decrease in E, as the surface 
concentration increases, is considered to be a powerful lateral 
interaction of the dipoles of moment /i, formed by each adatom 
and its electrical image. The lateral repulsion between two 
such dipole systems whose centres are a distance r apart is 
given by 



Force = {d/2)/i^/r^ 



24-2 



372 MIGRATION IN SURFACE LAYER OF SOLIDS 

Langmuir(42) considered a metallic surface covered with 
adatoms at a surface concentration N, which interacted wdth 
the force given above. He was then able to show that such an 
array of dipoles, if the short range forces of repulsion were also 
considered (so that no two adatoms can simultaneously occupy 
the same site), would obey the equation 

F = NkTI{l-d) + 3-34N'^/i^+ 1-53 x IQ-^N^T^ju,^!. 

Here / is an integral whose numerical value can be obtained 
from values of /i, N, and 6, and is never greater than 0-89, 
and F is the spreading force in dynes/cm. 

Where these repulsive forces operate in a system in which 
there is a concentration gradient, there is a nett force operating 
in the direction of the concentration gradient, a force which 
rises rapidly with surface concentration. Thus Langmuir was 
able to show that the force which operates on a given adatom, 
due to the concentration gradient, was 



'dN 



-F-NkTl 



This means that the height of the energy barriers for an atom 
moving in the dkection of increasing N is greater than that 
for an atom moving in the opposite direction, and leads to the 
expression r 2F 

where Dq is the diffusion constant when iV ~ 0. This analysis, 
originally made for caesium films (42), was applied also to 
thorium (22) films, both on tungsten. Values of F and D/Dq are 
given below for a number of values of 6, for Th-W systems: 

Table 90. Effect of the spreading force F upon 
diffusion constants 



'] 



e 


0-00 


0-05 


0-1 


0-2 


0-3 


0-4 


F (dynes cm.-^) 


0-0 


7-7 


16-8 


39-0 


65-6 


96-1 


D/Do (Th-W) 


1-0 


1-25 


1-46 


1-85 


2-20 


2-25 


6 


0-5 


0-7 


0-9 


1-0 


1-2 


1-4 


F (dynes cm.-^) 


132 


226 


377 


502 


1069 


9630 


D/D, (Th-W) 


2-86 


3-71 


5a3 


6-35 


12-0 


99-5 



DIFFUSION AND SURFACE CONCENTRATION 373 

An analysis of the spreading forces for sodium on oxygen- 
tungsten was made by Bosworth(55). There is a relation be- 
tween the spreading force F and the vapour pressure, p, 
obtained from Gibb's equation: 



F ^ 2-20ZkT{Ndlog'p. 



Bosworth then found the experimental connection between 
^l^m = / ^iid p to be 

7000 - 3200/ 



log:P = log/+ 0-71 -1-6/- 



T 



so that by inserting this value of log ^ in the first equation, and 
using numerical values, one finds 

F = 0-037 T.f+ 140/2(1 -0-00041^) dynes/cm., 

since F — when/ = 0. 

Topping (56) deduced the relationship F" = 4-51/*W^ for the 
electrical force between an array of dipoles. His equation was 
equated to the second term in the right-hand side of the above 
equation for F in terms of/. The first term is considered as 
due to Gibbsian thermal pressure, F'. The observed values of 
the spreading forces due to dipole interaction and those 
ca,lculated from the Topping equation are given in Table 91, 
in which a quite reasonable agreement is found. 



Table 91. Values of spreading force due to dipole repulsion 
from experimental data and from Topping's equation, for 
the system Na-OW 





F" = l4:0px 




/ 


(l-0-0004r) 


F"^4:-51fimi 


(dynes cm.-^). 


(degrees cm.-^) 




when T is small 




0-02 


0-06 


0-15 


01 


1-4 


3-6 


0-2 


5-6 


13-0 


0-4 


22-0 


42-0 


0-8 


90-0 


1100 


10 


140-0 


1500 



374 MIGRATION IN SURFACE LAYER OF SOLIDS 

A somewhat different method of evaluatmg the spreading 
force is based upon the variation in the energy of activation, 
E, with N. If the value of iS" at iV = is E^, one may write 

F = {Eq-E)N.\-^^x 10-12 dynes cm.-i. 

Bosworth's (27 ) data on K-OW lead to the following values of F : 

iV^xlQi* 0-06 0-12 0-24 0-48 0-60 1-2 1-5 2-4 3-0 4-8 

atoms/cm.^ 
i?' (dynes/cm.) 0-23 0-96 3-5 10-0 14-3 38 60 142 187 322 

The recorded values of F are for a range of surface concen- 
trations comparable with those in Table 91 for Na-OW. 

Phase changes in stable monolayers 

The equation of state of a gas shows that under suitable 
conditions gaseous and condensed states may coexist. When 
hydrogen gas dissolves in palladium, dilute and condensed 
phases may exist in equihbrium (57), or when a film of myristic, 
palmitic, or similar fatty acids is spread upon water, com- 
pressed and expanded states can occur together at suitable 
temperature or pK of the underlying hquid (58). It is therefore 
interesting to inquire whether two phases can occur on stable 
monolayers on tungsten. 

Using the idea of thermal and electrostatic spreading 
pressures developed in the previous section, one may write : 

F = thermal spreading force + electrostatic spreading force 

hT 

= -1 J- + 4- 51 iim^ 

A-Ao 

where A and Aq denote the areas occupied per adatom in a 
film of surface concentration N{N = IjA), and at saturation 
{N = Nq= 1/Aq) respectively. When Aq = 11-7 A.^ one may 
employ the values of the electrostatic spreading pressure in 
Table 91, and so plot F-A curves for the Na-OW system. 
Below 700° K. the curves obtained resemble those for a con- 
densible gas, so that the spreading forces calculated from 
experiment should lead to the formation of two phases. 
Probably the tAvo phases of caesium on tungsten detected 



PHASE CHANGES IN STABLE MONOLAYERS 375 

by Langmuir and Taylor (46, 35, 47) were of this kind (p. 353). 
Langmuir(47) attempted a kinetic analysis of conditions 
favouring equilibrium between two such phases. 

The phases discussed here are coexistent in a monolayer. 
Their constitution therefore differs from that of films of 
cadmium or mercury on glass (pp. 341 et seq.), where mono- 
layer systems rearrange themselves into three-dimensional 
aggregates. An exact analysis, employing statistical mechanics, 
of conditions yielding a two-dimensional condensed phase 
or a three-dimensional aggregate would be of importance. 
Preliminary studies of the behaviour of double layer films 
have been made by Cernuschi(59) and by Dube(60). 



Calculation of the surface diffusion constant 

The surface diffusion constant has been calculated by 
several authors (37, 61,35,62). Lennard- Jones's theory (37) was 
used by Ward (63) to interpret the slow sorption of hydrogen 
by copper. 

The most recent expression (6i) takes the form 

L) = 



In this expression a denotes a constant (| or ^), v denotes the 
average velocity of an adsorbed atom for the period t during 
which it is activated and t* is the time between successive 
activations. The attempt to calculate t and r* has been made 
for two cases: 

(1) When the activation energy is received by atomic 
vibrations from the underlying solid (64). It was concluded 
that, when RT ^ E,t was of the order IQ-^^ sec, and nearly 
independent of temperature. 

(2) When the activation energy is received by collisions 
between metaUic electrons from the adsorbent, and the 
adatom(65). It was again found that t was approximately 
10~^^ sec, and nearly independent of the temperature. Since 
metallic electrons may have as much as 50k.cal. of energy. 



376 MIGRATION IN SURFACE LAYER OF SOLIDS 

and there may be 10^^ collisions/second with the adatom, it is 
evident that an ample reservoir of energy is available, and 
that strongly bound adatoms may be activated. 

The corresponding values of t* are of course strongly de- 
pendent on temperature, since they contain the Boltzmann 
factor e~^l^^. It 'can also be shown (6i) that 

r* _ #* 

where i^* and F are the partition functions of migrating and 
vibrating states respectively. Thus 

These equations may as an example be applied to a simple 
type of potential energy field. The field is supposed to consist 
of cylindrical potential energy holes separated from each other 
by walls of height Eq occupying a fraction of the whole 
surface, whose area is A. Then 

and D = ^''^ 



'l-(p) + 4>e^olRT' 



This when Eq^ BT reduces to 



-EJRT 






8,nd a E^^RT to D = 



\ + cl>{E,jRTy 

Finally, when Eq tends tp zero, D = ccvh-, where r is the 
interval between successive collisions in a two-dimensional 
gas. In this case simple theories give to a the value \. Applica- 
tions of the theory to the diffusion of sodium on oxygen- 
tungvSten surfaces (26) led to a mean free path in the activated 
state of ~ 10"'^ cm. 



377 



ApPLICATIOlSrS OF SURFACE MOBILITY IN 
PHYSICO-CHEMICAL THEORY 



Assuming that certain adsorbed films are mobile, gas laws 
such as 

[A~A,-\=^RT 



A^ 



may be proposed, by analogy with three-dimensional equations 
of state. Here F denotes the surface pressure, A the area per 
molecule, and Aq the area per molecule at saturation. If F is 
proportional to p, the gas pressure, and A to Ijx, when x 
denotes the amount adsorbed, an adsorption isotherm may be 
derived rather like Langmuir's isotherm. Aq, like the "co- 
volume" in van der Waals's equation, may depend on tem- 
perature, and so explain the experimentally observed variation 
in the saturation value of the sorption with temperature. 
Langmuir's original isotherm could not do this. Similarly, the 
swelhng of charcoals when they sorb vapours (66) may be 
explained as a penetration due to two-dimensional pressure 
which thrusts apart the interpenetrating graphitic flakes. 
Maxted's(67) studies of the catalytic homogeneity of certain 
surfaces could be reconciled with Taylor's (68) theory of active 
points if it is assumed that a migration of the reacting atoms, 
or atoms of the catalyst poison, could take place over the 
surface. 

Various workers (69, 70,7i) have reported that when atom 
streams of cadmium, mercury, or similar metals are directed 
on to cooled surfaces, there is a critical stream density for 
each temperature below which aggregation into three-dimen- 
sional micro-crystals cannot occur. The application of an 
equation , , 

\F + ^^{A-A,) = RT 

to the sorbed atoms, which gives condensed phases, analogously 
to van der Waals's equation of state, has been used to explain 
the critical stream density and temperature. On this view the 
three-dimensional aggregate must build on the top of a 



378 MIGRATION IN SURFACE LAYER OF SOLIDS 



relatively immobile condensed phase in two dimensions, but 
not on the mobile gaseous phase. One should note, however, 
that the critical conditions are not sharply defined (72) so that 
aggregates may form over a range of stream densities of the 
impinging beam of atoms. This is illustrated in Fig. 131. 

60 



50 



40 



30 



20 



10 . 



ft 



>p P 



.n. 



-60 -80 -100 -120 -140 

Temperatui'e ("^ C.) 



160 



Fig. 131. Critical condensation phenomena for cadmium. 
\Zi, on glass; 0, on sulphur; O, on naphthalene. 

An interesting appUcation of the theory was made by 
Devonshire (73) in an attempt to explain the anomalous 
diffraction of helium beams at crystal surfaces observed by 
Frisch and Stern (74). It was found that at suitable angles of 
incidence impinging helium atoms need not be reflected from 
the surface, but could move along it in a mobile state for some 
distance before being emitted. It could also be shown that 



APPLICATIONS OP SURFACE MOBILITY 379 

when helium is sorbed on crystal surfaces (e.g. LiF), the zero 
point energy of the helium is so large that it can always pass 
over the energy barriers produced by the periodicity of 
the crystal surface, and so would be in the state of a two- 
dimensional gas even at 0° K. 

Calculations of the energy periodicity of the crystal surface 
for argon on KCl were made by Lennard-Jones and Dent (37). 
Similar calculations were made by Barrer(38) for sorption on 
the basal planes of graphite. These calculations showed that 
gases would be mobile at quite low temperatures, in the case 
of hydrogen even at Uquid air temperatures. More detailed 
calculations of the same kind were made by Orr (36), on argon- 
KCl, and argon- Csl systems, allowing for van der Waals's, 
repulsive, and electrostatic energies. In the case of KCl, the 
energy periodicity gave the following data for the fractional 
number of atoms rendered mobile: 

Temp. ° K. 10 20 40 60 80 

Nu^t^er mobile 0-000012 0-0017 0-0306 0-0933 0-1667 

Total number adsorbed 

On Csl the energy periodicity of the surface was far more 
marked, and there was a strong preferential adsorption above 
the centres of lattice cells of layers of caesium or iodine ions. 



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(67) Maxted, E. B. and co-workers. J. chem. Soc. p. 502 (1933); 

pp. 26, 672 (1934); pp. 393, 1190 (1935). 

(68) 'E.g. Taylor, H. S. Trans. Faraday Soc. 28, 247 (1932). 

(69) Knudsen, M. Ann. Phys., Lpz., 50, 472 (1916). 

(70) Wood, R. W. Phil. Mag. 30, 300 (1915); 32, 364 (1916). 

(71) Semenoff, N. Z. phys. Chem. IB, 471 (1930). 

(72) Chariton, J., Semenoff, N. and Schalnikow, A. Trans. Faraday 

Soc. 28, 169 (1932). 

(73) Devonshire, A. Proc. Roy. Soc. 156 A, 37 (1937). 

(74) Frisch, R. and Stern, O. Z. P%s. 84, 430 (1933). 



CHAPTER IX 

PERMEATION, SOLUTION AND DIFFUSION 
OF GASES IN ORGANIC SOLIDS 

Permeability spectrum 

Membrane-forming organic solids include waxes, fats, rubbers, 
proteins and protein derivatives, cellulose and cellulose 
derivatives, resins and alkyl sulphide polymers. Many syn- 
thetic polymers have valuable properties of plasticity, rigidity 
or elasticity. One finds every type of gas flow through them, 
and a diversity of jaermeabilities which receives a number of 
practical applications. In technological journals there are 
numerous studies of gas flow through rubbers (especially 
helium, air and hydrogen) ; of the flow of water, air and carbon 
dioxide through fruit andfood wrappmgs and cartons; and of 
air and water through gutta percha and paragutta insulators, 
leathers, and paint and varnish films. On the theoretical side 
these polymers provide material for the studj^ of diffusion 
kinetics, and of various types of gas flow in solids, such as 
activated diffusion, molecular flow, streamline flow, or oriflce 
flow. Of special interest are transition regions between one 
type of flow and another, as yet little studied. 

The process of diffusion in polymers may conveniently be 
discussed in two parts : flrst, the flow of gases and not easily 
condensible vapours (COg, SO2, NHg) in organic solids; and 
second, the flow of water and organic liquids and vapours 
through the membranes. This is because, as the sequel Avill 
show, gas diffusion obeys the simple law 

dt ~~ dx^ 

fairly rigorously, while vapour diffusion does not usually do so. 
It is possible to arrange the permeabilities of organic mem- 
branes to air, for example, in a permeability spectFum(i) as 
indicated in Fig. 132. 



PERMEABILITY SPECTRUM 



383 



The permeabilities in Fig. 132 are expressed iii c.c./sec./ 
cm.^/cm. of Hg pressure and are therefore not absolute, since 
thickness (which may vary from several miUimetres to a 
fraction of a millimetre) has not been included. Generally a 
given type of membrane will give somewhat different per- 
meabiUties for different specimens. Thus we have border-hne 
cases such as vegetable parchment hsted in sections B and C; 
or cellulose compounds most of which could be grouped in 
section A as well as section B, according to the variable degrees 
of permeability which may be encountered. 



A 


B 




C 


D 




Includes 


Includes 


japers, fibre- 


Textile 




semi-papers 


boards and leather 


fabrics 


Balloon 


Cellulose 


Filter paper 


Tagboard 




fabrics 


nitrate 


Blotting 


Bond paper 




Cellophane 


Cellulose 


paper 


Railroad 




Regenerated 


acetate 


Insulating 


board 




cellulose 


Cellulose 


board 


Solid binders 




Rubbers 


esters 


Newsprint 


board 




Neoprene 


Glassine 


paper 


Pressboard 




Polysulphides 


Vegetable 


Antique 


Vegetable 




Resins 


parchment 


book paper 


parchment 




Lacquers 


Asphalt 


Lined straw 


Leathers 




Paints 


saturated 
paper 


board 
Super- 
calendered 






' 




book paper 







-6 



-1 







2 



5 -4 -3 -2 
log (permeabiHty) 

Fig. 132. Showing the logarithm of the permeability towards 
air of groups of organic membranes. 

The experimental problem involved differs according to the 
section being studied. For section A, for example, where very 
small permeabilities are being encountered, various types of 
diffusion apparatus have been described, all of which have as 
their object the making and maintaining of gastight junctions 
between the membrane and the gas chambers in contact with 
its ingoing and outgoing surfaces. Dewar (2) in one form of cell 
fastened his rubber membranes by tying and then waxing. 
Daynes(3) found that, with glycerine lubricant between the 
flanges of the two chambers and the surfaces of the rubber 
membranes, the joints were satisfactorily airtight. Schumacher 



384 PERMEATION, SOLUTION, DIFFUSION OF GASES 

and Ferguson (4) described a mercury sealed cell (Fig. 133) 
which they considered suitable for any membrane. Rayleigh (5) 
used membranes supported between funnels clamped together 
and waxed along the flanges to prevent lateral diffusion. 

It is usual in these systems, in 
which diffusion occurs from the high- 
pressure side into a vacuum, to support 
the membrane with a metal gauze, or 
another rigid porous support, in order 
to prevent distortion of the mem- 
brane. Other types of cell have been 
employed (Edwards and Pickering (6)), 
in which the total pressure on either 
side was kept the same, or nearly the 
same, but on one side was hydrogen, 
whose permeability was to be studied, 
and on the other was air. The hydro- 
gen diffusing into the air stream was 
estimated by means of a Rayleigh 
interferometer. These types of appa- 
ratus exemplify the two methods of 
measuring permeability, i.e. diffusion 
into vacuum, with manometric estima- 
tion of the diffusing gas, and diffusion 
into an air stream, with interferometric gas analysis. 

The diffusion cells used in studying section A can readily be 
used to study section B of the permeabihty chart (Fig. 132). 
Similarly the apparatus for section C may be extended in its 
application to section B. 

Apparatus has been developed in the Bureau of Standards 
(F. Carson (i) ; also (7)) for measuring the permeability of papers, 
fibreboards, and leathers (section C of Fig. 132). Instruments 
have also been described (8) for measuring permeabilities of 
fabrics and textiles, of even smaller impedance to air flow. 
Fig. 134 illustrates the permeameter of Schiefer and Best (9). 
The fabric pressure gauge is inclined at a slope of 1 in 10, which 
allows the difference in pressure between the chamber A and 




Fig. 133. Schumacher and 
Ferguson's diffusion cell. 



PERMEABILITY SPECTRUM 



385 



the atmosphere to be measured to gw hich. The difference in 
pressure between the chambers A and B is registered by the 
air orifice gauge, and gives at once the rate of flow of gas 
through the cahbrated air orifice, and hence through the fabric. 
Air is drawn from the atmosphere through the fabric and air 
orifice by a suction fan. 

The instruments mentioned cover the groups A-D of the 
permeability spectrum (Fig. 132); and so cover a lO^^.fQi^i 
variation in air permeabihty shown by organic membranes. 



ScRCCf*'— -iK 




Fig. 134. Permeameter for textiles. 

Structures of membrane -forming substances 

Considerable progress has been made towards understanding 
the chemical and physical structures of many of the polymers 
or condensation polymers whose permeabihty will be dis- 
cussed. One method of attack, which has been purely chemical, 
has been to investigate the stages in which the polymer can 
be synthesised or broken down. It is found that one may have 
polymerisation or .condensation polymerisation into chains, 
plates, or three-dimensional networks. The family of hnear or 
chain polymers is a large one including 

Natural and synthetic rubbers. 

Certain proteins, e.g. silk, wool. 

Cellulose and cellulose esters, 

Fusible and soluble resins, e.g. novolaks, 

Polyesters, -amides, and -anhydrides, 

Poly sulphides. 

BD 25 



386 PERMEATION, SOLUTION, DIFFUSION OF GASES 

Among platy substances one may include such naturally 
occurring inorganic condensation polymers as 

Mica, Stilbite and heulandite, 

whose structures have been considered elsewhere (Chap. Ill, 
pp. 93 and 95), and which cannot yet be synthesised. Platy 
polymers among organic highly polymerised substances seem 
to be rare, since graphite can hardly be regarded as a polymer. 
The three-dimensional networks include polymers and con- 
densation polymers 'Such as 

Poly-p-divinyl benzene, 
Styrene-^-divinyl benzene interpolymer, 
Bakelites and similar infusible insoluble resins, 
Certain urea formaldehyde resins. 



Fusible Polysulphide Polyester 
Polystyrene resin chain chain (from chain (from 
chain - (novolaks) dihalides) hydroxy-acid) 



Polyamide 
chain (silk) 
Cellulose chain (R=alkyl 
skeleton group) 



(CH,)„ 


6 

1 


i 


(CH,)„ 


CO 

1 


s. 


1 


(CH.)„ 


(CH,)„ 


1' 


CO 





Y 


(CH^),. 


(CH,)„ 


CO 

1 


I 


i 

1 




(CH,)„ 



CH,OH 



CH,OH 



CH,OH 



NR 

I 
R 

I 

CO 

I 
NR 

I 
R 

CO 

NR 

A 

I 
CO 



CH.OH 



Fig. 135. Sotno linear polymers and condensation polymers. 



MEMBRANE-FORMING SUBSTANCES 



387 



In Figs. 135 and 136 are illustrated diagrammatically the 
manner in which certain linear and three-dimensional polymers 
are built up. These diagrams have of course nothing to say 
concerning the spatial relationships of the various chains with 



-CH.CH, 



-CH— CH» 



• CH— CH.— CH— CH„— CH— CH,CH . CH, — 



/^ 




-GH — CHo — CHp — CH, 



-CH— CH, 



Poly-divinyl benzene 




Possible structure for bakelite (infusible resin) 
Fig. 136. Some three-dimensional networks. 



25-2 



388 PERMEATION, SOLUTION, DIFFUSION OF GASES 

one another, for the elucidation of which one must employ 
the X-ray method. 

From the X-ray dififraction patterns of the polymers one 
can very often decide the spatial arrangement of the whole 
macro-molecule. The unit cell of cellulose is indicated in 
Fig. 137. When cellulose is nitrated (lO) it 
is found that the glucose rings of Fig. 137 
still he in the same parallel planes and 
that the dimensions along the chain are 
nearly unaltered. Normal to the length of 
the chain, however, there is a big increase 
in the distance between the chains (in the 
ratio 1'7 to 1) accompanying the nitration. 
Similarly, when cellulose is esterified the 
distance between the chains grows as the 
aliphatic side chains become longer (ii). 
This is reflected in a progressive lowering 
of the melting-point, as shown in Table 92, the interaction 
between chains becoming progressively less. 

Table 92. The physical properties of some cellulose esters 




Fig. 137. The unit cell 
of cellulose (12). 



Ester 


Specific 
weight 


Melting-point 

(°C.)(13) 


Tensile 
strength (14) 
(kg./mm.2) 


Acetate 

Propionate 

Butyrate 

Valerate 

Capronate 


1-377 

1-268 
1-178 
1-178 
1-110 


245 
239 
183 
160 

87 


9-12 

6-7 
5-6 

4r-5 

2-5 



The structure of certain fibrous proteins (wool and silk) has 
also been studied by the X-ray method. Fig. 138 giving 
Astbury's(i5) model for part of the unit cell in wool. Usually 
the proteins become denatured when one attempts to de- 
hydrate them, a fact which has prevented successful elucida- 
tion of protein structure in many instances, although in a 
recent study it was found that there is a great deal of 
crystalline order in certain highly hydi-ated proteins (16). 

The structures of cellulose, hair, wool, and silk are plainly 



MEMBRANE-FORMIlSrG SUBSTANCES 



389 



fibrous ones, and the tensile strengtiis miglit be expected to 
be tliose of chemical bonds, if the chains were continuous 
throughout the length of the fibre. These chain strengths may 
be calculated (17), and it appears that the tensile strengths are 
not of the anticipated magnitude. This, with other evidence, 
leads one to the micellar view of the structure in such fibres. 
According to this theory, bundles of molecular threads are 
supposed to compose a block, and the aggregate of a number 




Fig. 138. Portion of space model of the structure of wool or human hair(i2). 



of such blocks or micelles then builds up the visible thread. 
The micellar theory of soHds has been extended to include 
rubbers, most organic highly polymerised substances, and 
inorganic crystals. Thus one might a priori expect inter- and 
intracellular diffusion processes in organic sohds analogous 
to grain-boundary and intra crystalline diffusion in sihca glass 
(Chap. Ill, p. 125) or copper oxide (Chap. VII, p. 332). There 
are two types of micro-crystalline structure possible in organic 
polymers (18, 19, 20, 21, 22, 23) illustrated by Figs. 139 and 140. 



390 PERMEATION, SOLUTION, DIFFUSION OF GASES 

Fig. 139 shows the discontinuous or block structure for rigid 
membranes such as cellulose or inorganic solids; while Fig. 140 
gives a model for elastic long-chain polymers such as rubbers. 
The arrangement of chains is lattice-like, but the chains may 
be too far from or too near to neighbouring chains, and so the 
structure tends to be that of disorder. But in Fig. 140 ordered 
regions may be distinguished (marked by thicker lines), 
although they are not the self-contained units of Fig. 139. 
Stretching rubber by putting the carbon chains under tension 
tends to increase the degree of order by drawing them into 




Fig. 139. Discontinuous micellar structure postulated for cellulose (23). a, Haupt- 
valenzketten ; b, intramicellar regions ; c, intermicellar holes ; d, intermicellar 
long spaces. 




Fig. 140. Continuous micellar structure postulated for rubber(23). 

alignment. The stretched rubber is then crystalline, Avith 
eight parallel isoprene residues per unit cell, although there 
is still some doubt as to their exact relative positions (is, 24). 

Staudinger(25,26,27) characterised the chain length of hnear 
polymers by the effect upon them of suitable solvents. The 
characteristic behaviour was 

(1) Chain-length 50-250 A. Mol.wt. > 10,000. Such colloids, 
which dispersed easily to give true solutions, were called 
"hemicolloids". 



MEMBRANE-FORMING SUBSTANCES 391 

(2) Chain-length 250-2500 A. These colloids dispersed after 
swelling to give highly viscous solutions. They were described 
as "mesocoUoids". 

(3) Chain-length > 2500 A. Polymers of this chain length 
dispersed only after very intense swelling to give solutions of 
anomalous viscosity even at great dilutions. Staudinger called 
these polymers "eu colloids". 

The solubility may also be used to determine whether cross- 
linking of polymer chains to a three-dimensional network has 
occurred. A network cannot actually disperse, although it may 
undergo intense swelling. Polystyrene is a hnear polymer, and 
therefore soluble; but when a small amount of ^-divinyl 
benzene is added, the interpolymer, due to cross-linking of 
the polystyrene chains by ^-divinyl benzene bridges, becomes 
insoluble (27). 

Permeability constants of groups A and B of Fig. 132 

In 1866 Graham (28) studied the diffusion of gases through 
rubber. He regarded the permeation process as solution, 
diffusion and re-evaporation of the diffusing gas, a viewpoint 
which in essential details is held to-day. Wroblewski (29) in 1 879 
considered that Fick's laws of diffusion apphed. This was 
later verified by experiment in many instances. Wroblewski 
also made some of the earliest measurements of the solubihty 
of gases in rubber, and his work in this field was followed by 
that of Hiifner (30) and Reychler(3i). Kanata(32) extended the 
study o*f membrane permeability from rubbers to celluloid and 
gelatin. As the importance of the permeabihty of leathers, 
balloon fabrics, packaging materials and textiles was realised, 
more and more studies of membrane permeabihty were made. 
The literature is so scattered in technological journals that no 
previous attempts have been made to give a comprehensive 
survey of the available data from a theoretical standpoint. 

Attention has been directed in Chap. II to the possibility 
of defining various permeability constants. The permeability 
constant used in the present chapter has the dimensions 



392 PERMEATION, SOLUTION, DIFFUSION OF GASES 

Pxtx m~^, and denotes c.c. of gas at 1 atm. pressure and a 
standard temperature (293 or 273° K.) passing per second 
through a membrane 1 cm.^ in area, 1 mm. or 1 cm. thick, when 
the pressure difference is 1 cm. of mercury or 1 atm. When 
only comparative data are being considered, the origmal units 
may have been retained. The data presented in Tables 94-99 
represent some of the more trustworthy of the published data, 
and are suitable for reference material. 

Table 93. The chemical nature of rubber-like 
membranes used in Table 94 





Main 


Formulae of simple or 


Name 


constituent 


polymerised molecules 


Rubber (vulcanised) 


Polyisoprene, 
cross-linked 


CH2=C — CH=CB[2 




by sulphur 


CH3 


Rubber (unvulcanised) 


Polyisoprene 


„ 


"Neoprene" (vulcanised 


Polychloroprene, 


CH2=a-CH=CH2 


commercial) 


cross-linked 


1 




by sulphur 


CI 


"Neoprene" (unvulcanised 


Polychloroprene 


„ 


commercial) 






Polychloroprene (pure) 


— 


„ 


"Vulcaplas" 


Polysulphide 


R_S— S— R— S [ 

— S— R— S— S— R 


Butadiene-methyl- 


— 


\^ii2=^Kj — (_/ii^=0id.2) 


methacrylate inter- 




CH3 


polymer 








0H2=C— COOCH3 






CH3 ! 


Butadiene-acrylo- 


— 


CH2==C— CH=CH„, 1 


nitrile interpolymer 




1 

CH3 
CH2=CH— CN 


Butadiene-stjrrene 


— 


CH2==C — CH^CHj, 


interpolymer 




CH3 
Cells — CH=CH2 


Ethylene polymer 


Ethylene 


CH2 — CH2 — CHo 

— CH2 — CH2 — ■ ■ ■ " 



In Table 94 are given permeability constants for the rubber- 
like polymers of Table 93. The data summarised may be 
taken as typical of rubber-like membranes. There are minor 
variations only in permeability towards a given gas over the 
whole group, with the exception of the i)olysulphide rubbers, 



PERMEABILITY CONSTANTS 393 

which are much less permeable (~ 100-fold) than the hydro- 
carbon rubbers. Small amounts of cross-linking of the poly- 
mer chains by sulphur cause no appreciable changes in the 
permeability of neoprene or natural rubber. 

The question of the effect on the permeabihty of cross- 
linking of polymer chains by sulphur or oxygen still remains 
uncertain. Edwards and Pickering (6) studied the influence of 
ageing and of vulcanisation of rubber upon its permeability, 
using as membranes rubber- coated balloon fabrics. It was 
observed: 

(1) That ageing of the rublier was accompanied by a 
characteristic decrease in permeability, and usually by a 
decrease in the free sulphur. 

(2) In a series of experiments where the percentage of ■ 
combined sulphur varied from 0-3 to 2-5, no change in per- 
meability was found; but in a second series where the combined 
sulphur varied from 1'5 to 10% a decrease in permeability 
occurred. In each case the acetone extract — which gives a 
measure of resinification and oxidation — was about the 
same. 

As the amount of combined sulphur is further increased, the 
rubber becomes dark and hard and finally forms a compound 
with 32 % of combined sulphur. This polymer of high sulphur 
content is called ebonite, and it will be seen on referring to 
Table 96 that the permeabihty of ebonite to hehum is about 
^ of the permeability of rubbers; to hydrogen it is about ^; 
and to nitrogen < 2^^. Thus the increased rigidity has caused 
the permeabihty to become less, and much more selective. 

The impermeability (33, 34) of poly sulphide rubbers may be 
of technical importance. The permeability of polysulphide 
rubbers of the two following types (35) has been the subject of 
investigation by Sager(34): 

R — S — S — ^R — S — S — R 

R— S— S— R— S— S— R 

II II II II 

b S 00 



394 PERMEATION, SOLUTION, DIFFUSION OF iSASES 



Table 94. Permeability constants, P , for rubber-like poly'mers(B3) 

(c.c. at 293° K./sec./cm.^nim. thiek/cm. Hg pressure) 



System 


Temp. ° C. 


PxlO« 


He-"neoprene" 





0-0022 


(vulcanised and with 


30-4 


0-0078 


fillers) 


41-5 


0-0158 




57-0 


0-035 




73-0 


0-048 




101-3 


0-094 


He-"neoprene" (raw, 


I 21-6 


0-0039 


unvulcanised) 


37-6 


0-0116 
(material softening) 


- 


II 18-8 


0-0023 




340 


0-0055 
(material softening) 




III 


0-0006 




18-8 


0-0025 




24-6 


0-0036 


He-rubber (2 % S, 5 min. 


19-2^ 


0-0051 


vulcanised) 


30-8 


0-0078 


- 


42-8 


0-0116 




57-0 


0-0179 




62-5 


0-0195 




79-2 


0-0300 


He-rubber (2% S, 45 min. 


19-5 


■ 0-0086 


vulcanised) 






He-" vulcaplas " 


50-0 


0000174 




59-0 


0-000342 




68-5 


0-00045 


He-polyethylene rubber 


170 


0-0067 




22-6 


0-0082 




30-8 


0-0115 




38-5 


0-0184. 


Hj-neoprene (vulcanised 


17-5 


0-0085 


commercial) 


18-2 


0-0090 




26-9 


0-0128 




34-6 


0-0201 




44-4 


0-0302 




52-0 


0-0370 




63-7 


0-0534 


Hj-butadiene-acrylonitrUe 





0-0032 


interpolymer 


200 


0-0085 




29-0 


0-0128 




41-5 


0-0200 




50-2 


0-0315 




65-3 


0-055 




78-1 


0-075 


Hj-butadiene-methyl-metha- 


20 


0023 


crylate interpolymer 






H^-polyethylene rubber 


56 


0053 


, 


37-2 


0-021 




34-8 


0-019 




22-6 


0-011 



PERMEABILITY COaSTSTANTS 



395 



Table 94 {continued) 



System 


Temp. ° C. 


P X 10« 


Hg-polystyrene-butadiene 


I 19-9 


0-0084 


polymer 


II 21-0 


0-0112 


Hj-chloroprene polymer 


31-8 


0-0049 


(pure) 


39-7 


0-0086 




41-3 


0-0088 




49-9 


0-0116 




58-7 


0-0179 




69-5 


0-0214 




73-5 


0-0411 


N2-"neoprene" 


27-1 


. 0-00137 




35-4 


0-0023 




441 


0-0032 




541 


0-0058 




65-4 


00106 




84-7 


0-0222 


Nj-butadiene-acrylonitrile 


20-0 


0-00061 


interpolymer 


381 


00019 




48-5 


0-0029 




59-5 


0-0048 




70-5 


0-0070 




78-6 


0-0178 


Nj-butadiene-methyl- 


21-2 


0-0028 


methacrylate interpolymer 


44-6 


0-0057 




540 


0-0087 




61-9 


0-0132 




77-0 


0-023 


Nj-polystyrene- butadiene 


20-0 


0-0029 


interpolymer 


35-5 


00057 




50-0 


0-0102 




64-2 


0-0161 


A-"neoprene" 


361 


0-0068 




52-2 


0-0144 




61-8 


0-0224 




73-7 


0-0311 




86-2 


0-0655 


A-butadiene-methyl- 


20-0 


0-0059 


methacrylate interpolymer 


30-8 


0-0111 




39-2 


0-0162 




51-8 


0-027 




62-3 


0-0395 


A-poly styrene- butadiene 


I 19-5 


0-0109 


polymer 


30-3 


0-0195 




40-7 


0-0287 




51-2 


0-041 




64-6 


0-074 




II 64-0 


0-036 



396 PERMEATION, SOLUTION, DIFFUSION OF GASES 

Sager found that the rubber with four sulphur atoms per 
primary molecule was the less permeable of the two, but in 
agreement with Table 94 both were much less permeable than 
polyisoprene rubber (Table 95). Barrer(35a) extended his work 
on rubber-like polymers to rigid or inelastic membranes of 

Table 95. Hydrogen permeability of polysulphide* and 
polyisoprene rubbers at room temperature 

(P in c.c. at N.T.p./cm.^nim. thick/cm. Hg pressure) 



P (polyisoprene 
rubber) x 10' 



P (for polydisulphide 
rubber) x 10' 



P (for polytetrasulphide 
rubber) x 10' 



Sample 1 0-46 
Sample 2 0-53 
Sample 3 0-54 



Sample 1 
Sample 2 
Sample 3 
Sample 4 
Sample 5 



0-028 
0031 
0-041 
0-034 
0-033 



Sample 1 
Sample 2 
Sample 3 
Sample 4 
Sample 5 



0-010 
0-015 
0-018 
0-020 
0-017 



bakelite, ebonite and cellophane (Table 96). For these mem- 
branes the permeability is considerably smaller than that of 
elastic membranes, with the exception of vulcaplas or poly- 
ethylene sulphide rubbers. This result is true for rigid mem- 
branes of inorganic (SiO.2, BgOg, glass) as well as of organic 
substances (Table 97). Not only are rigid membranes usually 
less permeable, but they are also more selective in preventing 
the diffusion of inert gases of high molecular weight (Ng, O2, A), 
while allowing light gases (He, Hg, Ne) to diffuse (Chap. III). 
Sager (36) measured the hydrogen permeability of a large 
number of film-forming substances supported on a closely 
woven cotton fabric. The membranes included 



CH,OH 



Inelastic : 



Regenerated cellulose 



Polyvinyl alcohol CH— CHa— CH— CH^— CH- 

OH OH OH 








CH»OH 



* Prepared from 2, 2'-dichlorethyl sulphide and sodium polysulphide. 



PERMEABILITY CONSTANTS 
CHoOR 



397 



Cellulose esters- 



Poly vinyl alcohol acetate - 




—0- 



:!H20R 
-CH— CH,— CH— CH,— CH 



O.CO.CH3 O.CO.CH3 O.CO.CH3 
Poly vinyl acetal CH— CHg— CH— CHj— CH— CH,— CH 

0— C3H4— 6— C2H4— 

Polyhydric alcohol-polybasic acid resin 

HO . OC— R— CO— OCH2— CH2— 0— CO— R— COOCHij-CHa— ■ 

Elastic: 

Polychloroprene rubber (Table 93) 

Polyethylene sulphide rubber (CH2)Sx(CH2)2Sx(CH2)2Sx 

Polyisoprene rubber (Table 93) 

Table 96. TJie permeability, P, of organic membranes 
of various kinds to gases 







PxlO« 


System 


Temp. ° C. 


c.c. at 293° K./sec./cm.7 
mm. thick/cm. Hg 


Hg-bakelite 


75-5 


0-00052 




57-0 


0-00035 




45-0 


0-00026 




34-2 


0-000164 




20-0 


0-000095 


N2-bakelite 


75-0 


0-00027 




56-6 


0-000115 




47-2 


0-000083 




37-5 


0-000048 




36-1 


0-000047 




200 


0-0000095 




18-0 


0-0000085 


He-ebonite 


82-5 


0-0072 




67-0 


0-0047 




54-5 


0-00305 




43-7 


0-00255 




17-0 


0-00105 


Hg-ebonite 


83-0 


0-00192 




67-0 


0-00141 




49-0 


0-00091 




34-0 


0-00045 


N2-ebonite 


67-2 


0-000025 


He-cellophane 


85-0 


0-00050 




74-0 


000035 




52-5 


0-00012 



^1^ s; 

t» DO -a ' 

f-H O 1^ 



o . — . 



lo rr- lO o ^ 
o o rt 00 ■* 
o o o o o 
6 6 6 6 6 



«5 cc 

(NC000"*^tJ(tJHC0O 

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6666666 66 



OOO rH —I , 






?: 53 go. 



ftfl«.<N 



O +3 



lO » <» 05 00 O rt (M (M O 00 O f-H o 

OOOOO I— Ir-HrH^li— lOF-ti-Hi— I 

ooooo ooooooooo 

66666 666666666 



(M00lO00M*q00r-((M 
01020105C<li— it-CDOl 

OOOO^rHOOO 

ooooooooo 
666666666 



(M CO (M lO I> I> 
cq t~ (M OS O lO 

M rt rt o oo 
o o o o oo 
666666 



(MiMOO-HCO <M<MC<51C00CO'*Q0ffii 

r-Hr-H(MO0"^ r-Hp-HtJ-lCOlOi-HOCvlOO l£5';C';0(— 1<:iJ(^'^Trr^ CT^IQOOOD CO 

66666 6 6 6 6 6 rH r^ i-H i-H o cb ^^ w ob r^ i^- -^ 6 6666c<iio 

1— I rt i-H i-H (M rt 1-1 !N !N 



ipcOifflf-HOO-^"*I> 



CO CO 



O O 

c s 

cS cS 



o o 






(M (M rt r-4 

(S C6 C^ C^-+^+3-ti^ N 



^5 -!:? 



^ c3 

■Is :S 



cS 



o o o +; -*? 

Q Q D Oi 
■X3 '-=• r-H C' Ci 

Q^ c$ cd cd C^ 

g.a.g.g.a 

^^n ^ K^ ^ ^ 

■5 o o o o 

MI^PmPhPh 



=3 ^, -*^ 

-^ 8 i:^^ 8 >>-3 >> I >^ 

■3!:>0<D'5'sloajg®®aj 



•r| -r; .2 .2 
aj -u ^ ^ ^ JS 



-^ "5 S f^:2 2 f^ S *^ 2 



OD ■ 

® 05 <D 05 05 
m cS o3 cS o3 S 

O 4^ +2 += -*^ T? 

r^ 05 05 05 g 
13 o O 05 O ^ 

^ cS c3 c^ c6 *3 
® _ _ — , — I O 
" !>i!>i>iP^a5 

r^ g s g g g 

•S >i >i >i >i^ 



05 05 05 

-^ 1^ -s 

crt CD cc 



05 05 05 05 

CO QQ 03 OQ 



!3 S 3 

-.--- -050505 
PMCLiPLiPLiQQQQ! 



c^ o 'o'o "o ^ 



<1<5 



J3 ^ 05 
05 05 Pj 

:a;3qj 

05 3 3 2 ? 
5 05 05 S ii 
t5 05 05 o3 

g :S :g 2 2 

L, 05 05 05 !h 

S >>>,>>2 



„ O O O -t^ 



05 05 

S S 

U t-l 

+= -f^ 

05 CD 

05 05 

CC 00 

05 05 

O O 

s a 

02 0! 



^/ 



PERMEABILITY CONSTANTS 399 

Sager's data do not give a strict comparison or measure of 
" the permeabilities, since it was not possible to obtain films of 
identical thickness, nor were the actual thicknesses stated. 
The permeation rate defined as c.c./cm.^/sec./atm. at 25° C. is 
not an absolute measure of permeabiHty. If, however, one 
uses the quantity 

Permeation rate x weight of film per unit area, 

one has a good approximation to comparable and absolute 
data, since variations in density are less important. The true 
permeabihty constant P (in c.c. at N.T.p./cm.^/sec./cm, 
thick/atm. at 25° C.) is given by the relation 

Permeation rate x weight of film in g. per sq. cm. / 
Density of film (g. per unit volume) 

It will then be remembered that the units of Tables 94 and 97 
are different (cm. thickness for mm. thickness, and atm. for 
cm. pressure), and that the constant P of Table 94 must be 
multiplied by 7-6 to compare with the constant P of Table 97. 
The second column of the table gives the solvent used in 
forming the film on the cotton weave support. 

Sager (36) considered that the data showed that permeabihty 
was governed by the chemical nature of the films. The films 
included crystalline, fibrous and amorphous substances, and 
the degree of polymerisation varied over a wide range, but 
no simple connection could be observed between permeability 
and crystalline structure or degree of polymerisation. This 
aspect of the permeabihty requires examination of a limited 
number of chemically different film types and a controlled 
degree of polymerisation. There is, however, a general corre- 
spondence between the hydrogen permeabihty in a given ^/ 
chemical type, and the hydrogen solubility in analogous un- 
polymerised molecules. The permeability of materials rich in 
hydroxyl groups (ceUophane, cellulose, polyvinyl alcohol, and 
hygroscopic resins) is very low, corresponding to the small 
solubUity of hydrogen in glycerol or water. As soon as ester 
groups are introduced (polyvinyl and cellulose esters), the 



400 PERMEATION, SOLUTION, DIFFUSION OF GASES 

permeability towards hydrogen increases, to parallel the 
higher solubility of hydrogen in esters. Similarly, the hydrogen 
solubility in liquid hydrocarbons is considerable and so one 
finds a large permeability of rubber to hydrogen. Hydrogen 
is very sparingly soluble in carbon disulphide; therefore it 
should diffuse with difficulty through poly sulphide rubbers. 
This analogy should not be carried too far, since the per- 
meabihty is conditioned both by the solubility of the gas 
and by its diffusion constant within the polymer. 

De Boer and Fast (37) studied the hydrogen permeability of a 
number of derivatives of cellulose. The permeabihty constants 
at 0° C. (c.c. at N.T.p./sec/cm.^nim. thick/cm. Hg) were: 

Regenerated cellulose P x 10' = 0-000047 

Triacetyl cellulose PxW = 0-0140 

Nitro-cellulose P x 10' = 0-0092 

Celluloid Px 10' = 0-0164 

These permeability constants are consistent with those in 
Table 97, and emphasise again the small permeability of 
cellulose and its derivatives. From the data of Table 98 one 
may construct a table showing the range of permeabilities to 
hydrogen of different chemical types of polymer. 

Table 98. Range in relative permeability in 
various polymers 



Type of substance 


Relative permeability 

towards hydrogen 

at 20° C. 


Rubbers (natural and synthetic) 
Polyvinyl and cellulose esters 
Polyhydric alcohol-polybasic acid resins 
Polysulphide resins (Thiokol A) 
Polysulphide rubbers 
Cellophane, polyvinyl alcohol and 
acetal, and regenerated cellulose 


1-0-0-6 

1-8-0-2 

(1-9-1-3) X 10-1 
(1-0-0-7) X 10-1 
(6-3-3-1) X 10-2 
(8-0-0-05) X 10-2 



There is a lO^-fold variation among the permeability con- 
stants, the rubbers being among the most permeable of the 
membranes. 



PERMEABILITY CONSTANTS 



401 



i 



5ft 









5s> 
5sj 



5£ 



'^ 






?iH 



Oi 





m" 


8, 


1 1 


1 








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^ ' ' ' ' 


1 1 


1 










<» <^ . . , 












O 


op lO 


1 1 


1 








O 


M (N 


1 1 


1 


















•5" 




o 


1- 1 1 1 


1 1 


1 








Q 


1 6 1 ' 1 


1 1 


1 




a 

-s 

. . 00 


















"^^ , , , 








^ « 

^-^ 

SrC 




O 


■^ M 


1 i 


1 








66 ' ' ' 








c6 ^ 


§ 












05 M 


C5 
























C+H 






?0 ^ O ,00 


^ t~ 






« O 




^ 


'Tr^'r 9 


^ cq 


1 




ft ® 

V > 

K -2 




OOO o 


6 6 


















, 05 02 , , 


« »o 






o M 




< 


T^"^ 


<M o 


1 




t*H CS 




' 66 ' ' 


6 6 


1 




a^5^ 


























o 














M 






(M Ort (N , 




lO 








CO CO «5 (N 


1 1 


c- 




o 1 1 




6666 


1 1 


6 




6 ' ' 














CO CO 






ooooo 


o o 


o 




, coco 




w 


OOO oo 


o o 


o 




oo 












' 66 




g 


g i 




g 






g 


t a 




(D 






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c. 

0°C. 
, 20° C. 
C. 
nterpol 


CD 'o 
^ 1 




g 
1 

to 


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b g 
It 


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g 


ed), 25° 
Qised), 2 
canised) 
mer, 20° 
nitrile i 




d 

o 

(M 


^ft 

o © 

ft-^ 




.S ce 'q >^o 


tot 


g 






S S ^Cub 


ene-me 
ler, 20° 
ene-pol 


>^ 




■;„'3 S 




r (vuk 
r (unv 
rene" 
prene 
ene-ac 


'o 

ft 


ft 


perature): 
"Vulcaplai 
Sager's dis 
Sager's tet 




llfii?i|i? 


>> 


1 



26 



402 PERMEATION, SOLUTION, DIFFUSION OF GASES 

Another interesting comparison can be made by considering 
the relative permeabilities in a given membrane of a series of 
gases. The relative permeation velocities bear no simple 
relation to the molecular weights of the diffusing gases. The 
ratio of the permeation velocities (6, 28, 38, 2) for COg and Hg 
prove to be almost independent of the sample of polyisoprene 
rubber employed, since for nine samples the average ratio 
of PcoJ^B.t ^^^ 2-76, and the extreme variations were from 
3-03 to 2-48. The permeation rate ratios for each of a series of 
gases in rubbers of different chemical types are found to show 
minor variations only (Table 99). 

Table 100. Relative permeation velocities in membranes 
(Og as standard) 



Gas 


Membrane 


CeUuloid 


Rubber 


Gelatin 


CDs 
SO2 
NH3 


2-47 
100 

8-98 

61-6 


2-86 
1-00 
9-06 
25-0 
36-0 


1-00 
1-00 
4-13 
31-9 
95-2 



The same absence of any connection of relative velocities 
with molecular weights (as required in effusion or molecular 
streaming) is found when gases diffuse through widely differing 
membranes such as celluloid, rubber and gelatin (32 ) (Table 100). 
Once more rather striking regularities in relative velocities are 
revealed, although irregularities and specific effects are be- 
ginning to creep in. 

The variables in permeation through membranes 
of groups A and B 

The pressure, area of membrane, thickness of membrane, and 
the temperature are the possible variables in the permeation 
kinetics. If (as is shown later, p. 413) Hem-y's law, S = kp, 
applies to the solubility S of gas in the polymer substance, 



and the Fick law 



dc 
dt 



Z) --- applies to the diffusion process, 



PERMEABILITY CONSTANTS 



403 



within the material, in the steady state of flow, one would 
find that the permeation rate was proportional to the pressure 
difference, the area of the membrane, and inversely to the 
thickness of the membrane. 

The jpressure difference 

In the steady state the velocity of diffusion for a given con- 
stituent of a gas mixture is proportional to the difference in 
pressure of that constituent between the ingoing and outgoing 





/ 




/ 




/ 


^ 


At 0°C 


^^ 


---is**'''^ — ' 


.,=ii;^I^===^ 


■ vT 



Pressures in Atmospheres 



Fig. 141. The effect of high pressure upon permeation rates. 



surfaces of the membra.ne (33, 37, 6,39,40). This law holds for 
pressures which are not too great (2), and is a characteristic of 
non-specific types of activated diffusion (see Chap. III). The 
velocity of diffusion is the same when hydrogen diffuses into 
a vacuum or into an atmosphere of air(4i). Similarly, when a 
mixture of oxygen and nitrogen diffuses into hydrogen, the 
rate of permeation of each gas is practically unaltered by the 
presence of the other gases (4i). At high partial pressure 
differences (2) (up to many atmospheres) the rates of diffusion 
through pure rubber both for hydrogen and carbon dioxide 
increased with pressure, the increases being shown in Fig. 141. 
The departure from Hnearity does not appear, in the case of 
hydrogen at any rate, to be due to breakdown of the perfect 

26-2 



404 PERMEATION, SOLUTION, DIFFUSION OF GASES 

gas laws at these pressures, since Wiistner (see Chap. Ill) found 
linear permeation rate-pressure curves for hydrogen and silica 
up to 800 atm. Rather the effect might be ascribed to an 
influence of pressure upon the membrane itself (compression 
or distension) and suggests that deformation can modify 
permeabilities. 




0-002J 0-0030 C-0033 0-0036. 

^ {Tsmperature in "Z^). 

Fig. 142. Influence of temperature upon the permeability constants of 

elastic polymer-gas systems. 

Curve 1. He- vulcanised rubber 

Curve 2. Hg- vulcanised neoprene. 

Curve 3. He-vulcanised neoprene. 

Curve 4. Argon-vulcanised neoprene. 

Curve 5. Na-vulcanised neoprene. 

Curve 6. He-vulcaplas (polysulphide rubber). 

The thickness of the membrane 
In the steady state of diffusion the velocity of diffusion is 
inversely proportional to the thickness of the membrane. 
This is shown by the results of Edwards and Pickering (6) who 
allowed hydrogen to diffuse through rubber membranes of 
varying thickness, and plotted the reciprocal of the per- 



PERMEABILITY CONSTANTS 



40^ 



meability (called the impedance) against the thickness. Their 
graph was linear. 

The influence of temperature 

The most interesting relationships were revealed by the study 
of the influence of temperature on the permeation rate. It was 
found by Shakespear (40) that the temperature coefficient of 
the permeation rate was the same for aU differences in 




2-7 



3-3 



Fig. 143. Influence of temperature upon the permeability constants of 
inelastic polymer-gas systems. 

partial pressure, and all thicknesses of membrane. Graham (28) 
long ago showed that the temperature coefficient was ab- 
normally large. This is also clear from the data of Edwards 
and Pickering (6), Taylor, Herrmann and Kemp (42), and 
Dewar(2). 

It may be shown that the permeabihties conform to the 
equation ^ ^ P^e-^/ier, 

a relationship first pointed out by Barrer(43). The exponential 
temperature coefficient is shown in Figs. 142 and 143(33,43), 



406 PERMEATION, SOLUTION, DIFFUSION OF GASES 

for elastic and for inelastic membranes. The temperatm-e 
coefficients of the permeability constant in cal./mol. of gas 
are collected in Table 101. It is interesting to find that 
these coefficients are not very different for the impermeable 
inelastic membranes and the permeable elastic membranes. 
The main differences in the permeabihty must therefore 
come from the temperature independent factor, Pq, in the 
equation 

As Table 101 shows, Pq is very much smaller for cellulose and 
its derivatives than for elastic polymers. The difference in Pq 
for rubbers and for cellulose must be due partly to the 
larger solubihty of hydrogen in rubber. It is then possible 
that in cellulose, because of the low solubility, diffusion 
is predominantly of grain-boundary type, while in rubber 
it can occur readily inside the micelles composing the 
coUoid. 

The air permeability of group C of Fig. 132 

Previously we have considered coherent membranes without 
pin-holes or capillary channels. Most papers, fibreboards, and 
leathers have capillaries down which streamline (44) or 
Knudsen(45,46) flow may occur, or pin-holes which may act as 
orifices for orifice flow (47) or effusion (48) (Chap. II). Stream- 
line flow conforms to the equation 

^iPi = 'V2V2 = Y^{Vi-pl) = -^p \ , 

where v^ = the volume flowing per unit time into the tube at 
pressure p^, 

V2 = the volume flowing per unit time into the tube at 
pressure j?2j 

^P=Pi-P2, 
7j = the viscosity of the gas, 
I = the length of the pore, of radius r. 



AIR PERMEABILITY 



407 



K; 



o o 
o o 

lO 00 






ffi- K W 



W 



o o o o o 
o >o o o o 

O CO lO CO <N 

CO as 00 00 



oo oo oooo 

OK5 OOIOOOO 

coo I— llffl |i£>l>lO<X> 

0000 lOtO K5lOt>t- 



w 



oooo 

0-* o o 

CO , -TOO 00 



o <^ 






. — . IB (D 

T3 .2r3 
© c f-i 

'3 " "c 

cS'a o 

^ © • 

— ^ © s 
1^ b.2 i 



5 o f-^-:; 



2 >3 2 



^1 5 
o a cs 



* 2 






o ® 



4) 









•J3 as 

cS cS 

ism 

a 



J. cST3?a >,S 

•2 P<-2 g ® fi 
S o ^ 2 S £, 



'-' CO Gh 



W W M 



408 PERMEATION, SOLUTION, DIFFUSION OF GASES 

For orifice flow to occur the capillary must be short enough 
to act as a jet or nozzle. Under these conditions the equation 
of flow becomes 



-^ = constant ^/jT 



Ap 



The constant depends upon the area of the nozzle, and not 
its length. 

The properties of streamline and orifice flow are summarised 
in Table 102(49), and can easily be verified by reference to the 
equations of flow. The very complete series of experiments by 



Table 102. Characteristics of high-pressure gas flows 



Poiseuille capillary flow 


Orifice flow 


v/Ap is constant for low values 
of Ap 

v/Ap is proportional to the reciprocal 
of length (or membrane thickness) 

The rate of flow decreases slightly 
as the temperature increases, be- 
cause the viscosity of the gas 
increases as T increases 

V, for small constant values of Ap, 
is independent oi Pi 


v/Ap decreases as Ap increases. 

v/Ap is independent of the length 
(or membrane thickness) 

The rate of flow increases slowly 
with temperature 

V, for constant values of Ap, in- 
creases slowly as p^ decreases 



Carson (49) has tested these features for numerous papers. For 
the thinnest tissues the behaviour with respect to Ap sug- 
gested that flow might be intermediate between streamline 
and orifice flow. Within the uncertainties of using different 
samples an inverse proportionahty existed between the per- 
meation velocity and the thickness, while a small change in 
temperature had a negligible influence upon the rate of flow. 
The results are on the whole more in conformity with Poiseuille 
flow than with orifice flow. The effect of altering the absolute 
pressure p^, for a constant Ap, however, was not in accord with 
either theory, and was attributed by the author to elastic 
deformations of the membrane under pressure. In addition, 
an approximate proportionality between area and difl^usion 



AIR PERMEABILITY 



409 



rate was observed, and it was shown that the relative humidity 
of the air bore no simple relationship to the permeation 
velocity. In order for Poiseuille flow to occur in these mem- 
branes, the length of the capillaries must be much greater than 
their diameter. It is therefore probable that the actual 
thickness of the membranes is less than the length of the 
capillaries. 

In Table 103 Carson's (49) data have been converted to 
absolute permeabilities, and show the range of permeability 
constants covered by the paper section of group C of Fig. 132. 

Table 103. The air permeabilities of some membranes 



Kind of material 


Permeability 
c.c./sec./cm.^/cm. 




Hg/mm. thickness 


Insulating board 
Filter paper 
Blotting paper 
Newsprint paper 
Antique book board 


36-5 
3-51 
1-51 
0-047 
0-0298 


Lined strawboard 


0-0177 


Supercalendered book paper 
- Tagboard 


0-0059 
0-0102 


Bond paper 
Railroad board 


0-00141 
0-00116 


Solid binders board 


0-000465 


Press board 


0-0000714 


Vegetable parchment 


0-0000018 



It is necessary to add that other types of flow than that of 
PoiseuiUe (e.g. molecular streaming, or even activated 
diffusion) may be occurring in the case of the least permeable 
membranes cited. The sorting out of flow processes is not yet 
at aU complete. 

Leathers 

The air permeability of leather shows the same wide variations 
that v/ere observed for papers and fibreboards, and some of 
the associated phenomena have been ascribed to Poiseuille 
flow (50, 51). The" leather may be regarded as a network of 
capillaries joining larger cavities within the material. These 
cavities and capillaries are not constant in size. Thus while in 



410 PERMEATION, SOLUTION, DIFFUSION OF GASES 



Table 104 



A. Sole leathers 



Type of leather 


Thickness 
cm. 


PermeabiMty x 10'^ 
c.c./sec./cm.Vmm. thick/cm. Hg 


G*^F 


F^G 


English bend 

Waterproofed Eng- 
lish bend 

French bend 


0-608 
0-438 
0-490 
0-420 
0-390 

0-536 
0-415 

0-350 
0-370 
0-420 
0-382 
0-624 


9-10 
5-76 
7-40 
4-65 
4-06 

4-76 
1-63 

4-42 
1-42 
1-97 
1-85 
3-94 


7-30 
5-40 
7-80 
4-74 
3-29 

5-34 
1-50 

4-14 
1-32 
2-21 

1-85 
4-08 


B. Upper leathers 


Box calf 

Gorse calf 

Willow calf 

Glace kid 

Heavy chrome 
tanned upper 

Heavy vegetable 
tanned upper 
(stuffed) 

Russia calf (vegetable 
tanned) 

Patent leather 


0-140 
0-130 
0-120 

0-195 
0-160 
0-210 

0-095 
0-130 
0130 

0-060 
0-070 
0-050 

0-312 
0-310 
0-280 
0-320 

0-240 
0-252 
0-240 
0-230 

0-220 
0-160 
0-180 

0-125 
0090 
0-120 


15-7 
14-2 
20-7 

13-0 
11-0 

28-2 

7-0 
28-3 
15-9 

0-734 

1-78 

1-22 

4-05 
2-78 
1-21 
7-42 - 

00128 
0-0176 
0-00428 
0-00845 

59-5 
37-9 

62-8 






11-1 

7-55 
17-65 

6-04 
. 600 
24-60 

2-95 
22-6 

8-52 

0-360 

1-24 

0-60 

2-47 

0-485 
0-422 
4-36 

0-00413 
0-0176 
0-00418 
0-0128 

62-5 
38-5 
65-1 







* G denotes "grain side"; F denotes "flesh side." 



AIR PERMEABILITY 411 

certain cases the rate of flow through the leather was pro- 
portional to the pressure difference (5i), in others the constant 
of proportionahty changed with pressure, or with the treat- 
ment given the membrane (50). Sometimes the rate of flow 
from the flesh to the grain side is different from that in the 
converse direction (52) (see Table 104). 

In Table 104 are given some absolute permeabilities 
for a number of leathers, calculated from data given by 
Edwards (50). 

Bergmann and Ludewig's(5i) figures for sole leather agree, 
when converted to similar units, with those found by Edwards 
in range and order of magnitude. Wilson and Lines (53) 
made a study of the air and water permeabihty of leather in 
which they found a parallelism between these permeabilities 
when the leather contained neatsfoot oil, or was finished 
with collodion or casein. 

Membranes in series 

When the leather membranes are placed in contact (5i), and 
the permeabilities of the separate membranes are i^ and P^, 
it was found that the resultant permeability, P, was within 
6 % given by 111 

The reciprocal of the permeability is called the impedance, so 
that the above expression means that the impedance is 
additive. The same property has been observed for gas-rubber 
diffusion systems (3). 



The solution and diffusion of gases in 

ELASTIC polymers 

The passage of a gas through a membrane is governed by the 
equation P = —D{dC/dx), where D is the diffusion constant, 
and dC/dx the concentration gradient. When the permeability 
P is known, and the solubility, the diffusion constant may be 



412 PERMEATION, SOLUTION, DIFFUSION OF GASES 

calculated. Two methods have been successfully employed to 
measure D\ 

(1) The solubUity has been measured directly, and then the 
permeabihty constant. Use of the Fick law in the form 



D 



I 



{I denotes the thickness, and C^ and C2 are the concentrations 
at the ingoing and the outgoing faces respectively) then allows 
the evaluation of D. For measuring the solubihty the same 
methods may be employed as in studies of sorption equi- 
libria (54), and any sensitive apparatus described for that 
purpose could be employed. 

(2) Another method may be used which allows one to 
measure P, D, and the solubility, in one experiment. This 
method was introduced by Daynes(3) and developed by 
Barrer(33,55,56). It will be seen, by referring to Chap. I, p. 18, 
that if one has a membrane with dissolved gas initially at a 
concentration Cq throughout, and with constant concentra- 
tions Ci and C2 at the faces x = and x — I respectively, the 
rate of flow of gas through the membrane takes some time to 
settle down to a steady state. If one plots the rate of increase 
of concentration on the low pressure side against time, the 
C-t curve approaches asymptotically a line which intersects 
the ^-axis at the point 



Da 



1 [C.C, Cq-] 

\-C^l6^3 2j' 



provided the increase in concentration is small. When Cq is 
zero, the intercept is 



lJ' ' 



DO^-C^ 



B4]^ 



and when both Cq and C2 are neghgible compared with C^, the 
intercept is L = l^/6D. Thus by a single experiment which 
follows both non-stationary and stationary states of flow one 



SOLUTION AND DIFFTJSION OF GASES 413 

measures P andD, and then calculates the solubihty, Ci. The 
equations j% 

P = i)^(Fick'slaw), 
C^ = kp (Henry's law), 




Time (mins.) 

Fig. 144. The time lag in setting up a steady state of flow. 0, argonin butadiene- 
methyl- methacrylate interpolymer; x, nitrogen in butadiene- methyl- 
methacrylate interpolymer. 



give also the relation 



PL 



kip 
"6~" 



The predictions of aU these expressions were verified by 
Daynes (3). The intercepts L were, for membranes about 1 mm. 
thick, suitably large for evaluating D, as Fig. 144 (33) indicates. 
Most early measurements of solubUity employed the static 
method (1), Wroblewski(29) found that Henry's law was 
obeyed: C-^^^ kp. The temperature dependence of the solu- 



414 PERMEATION, SOLUTION, DIFFUSION OF GASES 

bility does not agree with some more recent data (p. 418) for 
hydrogen and air, although the actual solubilities are similar. 
Hlifner (30) found that rubber will sorb its own volume of carbon 
dioxide at room temperature, but could not measure the 
absorption of hydrogen, oxygen, or nitrogen. Reychler(3i) 
found that rubber sorbed 1-06 vol. of CO2 at 18° C, and 
26 vol. of SOg. The most complete and satisfactory measure- 
ments of solubility by the static method were made by Venable 
and ruwa(57). They showed that Henry's law was obeyed for 



120 
110 
100 

90 

. 80 

i'° 
l'° 

50 
40 
30 
20 

to 









































































\ 


































\ 


s 


































\ 




































\ 




































\ 


































\ 


\, 


































s 


N 


































\ 


\ 


































\ 


k 




































s. 



















































r60 



20 40 60 80 too 120 140 

c.c.gas 3t N.P. T. per lOOc.c. rubber 
Fig. 145. The influence of temperature upon the solubihty of CO, in rubber. 



COg-rubber systems, and also that the solubihty of ethylene 
and carbon dioxide decreased with temperature (Fig. 145). 
The data plotted as log (solubility) against l/:7^ give exothermal 
heats of solution of 3300 for COg and 2700 for ethylene. 
Venable and Fuwa also showed that adsorption was not 
important in determining the amount of gas taken up, since 
the observed uptake was not altered by increasing the total 
rubber surface. 

Tammann and Bochow(58) measured the solubility of 
hydrogen in a number of rubbers and at pressures as high 
as 1150 kgm,/cm.2 Even at these pressui'es Heniy's law 
was approximately valid, and the solubilities extrapolated to 



SOLUTION AND DIFFUSION OF GASES 



415 



1 atm, pressure were of the same order as the solubilities given 
in Tables 106 and 107. When the pressure on such a hydrogen- 
filled rubber was released, the rubber swelled to several times 
its original volume. 

Oxygen may dissolve in rubber in two ways. Physical 
solution can be followed by chemical reaction. Chemical 
reaction occurs autocatalytically(59), and as resuiification 
proceeds the rubber becomes first tacky, and ultimately hard 
and brittle. When the surface area is small the rate of 
reaction with oxygen may depend upon the extent of the 
surface, but as the surface is increased diffusion becomes no 
longer a rate controlUng step, and the reaction velocity is 
independent of the surface area. 

The most complete and satisfactory data on solubiHty and 
diffusion in rubbers have been obtained by the second method 
of p. 412, depending on the time lag in setting up the stationary 

Table 105. A comparison of absorption coefficients 
measured by the methods (1) and (2) of p. 412 





^21° c. (c.c. at 


^17° 0. (c.c. at 




N.T.p./c.c. rubber) 


N.T.P./c.c. rubber) 


Gas 


by sorption 


by non-stationary 




method (Venable 


flow method 




and Fuwa) 


(Daynes) 


Ha 


<0-01 


0-040 


O2 


0-073 


0-091 


NH3 


9-30 


41-0 


CO2 


0-99 


0-86 


Air 


0-045 


0-043 



state. Where comparison is possible, the solubility constants 
k (from G — kp) agree well when obtained by the two methods 
(Table 105). Agreement is not good for ammonia which, 
however, approaches its equilibrium sorption value very 
slowly The agreement for the other gases provides support for 
the assumption made in the time lag method that the con- 
centrations just inside the ingoing and outgoing surfaces of 
rubber are governed by Henry's law, C = kp. 

Barrer(33) used the flow method (2) to measure P, D, and k. 



V 



416 PERMEATION, SOLUTION, DIFFUSION OF GASES 

It was found that the intercepts L and the permeabilities 

obeyed the relationships 

L = L^e^l^^' (Fig. 146), 

P = P^e~^ilRT (Figs. 142, 143). 

Thus since 



L-^" 
^-W' 



one finds 



D = D^e-^lR^, 




0-O0ZY5 O-O030 O-00325 0-OOSZ 

-^ (^Temperature in °K). 
Fig. 146. Curves showing that log (L) is a linear function of l/Tos). 

Curve 1. A-neoprene. 

Curve 2. Ng-neoprene. 

Curve 3. Ng-styrene-butadiene interpolymer. 

Curve 4. A-styrene-butadiene interpolymer. 

Curve 5. Hg-neoprene. 

Curve 6. Hg-butadiene-acrylonitrile interpolymer. 

so that diffusion in polymers is activated. From the Fick law 

one obtains k — k^e'^^"^^^!^'^ = kQe^^l-^^, 

where AH denotes the heat of solution. Although activated 
diffusion occurs in the polymers, it is non-specific, sinc6 any 
gas of suitably small molecular dimensions will diffuse.* It 



* And also large molecules which distend the rubber as they diflfuse. 



SOLUTION AND DIFFUSION OF GASES 



417 



differs from the diffusion of gases in metals, which is specific 
to- certain gases and certain metals with which the gas can 
react chemically (Ng-Fe), or form an alloy (Hg-Pd). 

In Tables 106-108 are given data on the solubility and the 
diffusion constants of gases in polymers. The heats of solution 

Table 106. Solubility, permeability and diffusion constants in 
vulcanised rubber at 25° C. {calculated from data of 
DaynesiS) and Edwards and Pickering {&)) 



Gas 


Solubility 

c.c. at N.T.P./C.C. 

rubber/atm. 

pressure 


Permeability constant 
(25° C.) c.c. at N.T.p./ 
sec./cm.2 TTiTTi./om. Hg 


Diffusion constant 

(25° C.) 

cm.2 sec.-i 


CO2 


0-040 
0-070 
0-035 
0-90 


0-045 X 10-8 
0-020 X 10-8 
0-0071 X 10-8 
0-132x10-8 


0-85 x 10-5 
0-21 x 10-5 
0-15 X 10-5 
0-11x10-5 



in Table 107 are not as accurate as the actual solubilities, 
since the latter decrease only very slowly as the temperature 
rises. It is also to be noted that the solubihty of argon is 
greater in all cases than that of nitrogen, which accounts in 
part for the higher permeability of rubbers to argon (Tables 107 
and 94). 

The solution of gases in rubbers occurs exothermally even 
in constant volume systems (33, 6O), with the possible exceptions 
of hehum and hydrogen. This must be contrasted with the 
endothermic heats of solution of many gases in organic 
liquids (61). The solubility constant k for a perfect gas is 
related to the standard free energy of solution, A Gq : 



But since 



AGQ = -BTlnk. 
AG = AH-TAS, 



one may calculate both the free energy and entropy of the 
solution process (33). The solubility constant of gases in organic 
liquids is practically the same as the solubility constant in 
organic polymers, and therefore the difference in the heats of 
solution is due to the entropy j;erm. When comparable entropy 

BD 27 



418 PERMEATION, SOLUTION", DIFFUSION OF GASES 

Table 107. Solubilities of gases in organic polymers 



Polymer 



Gas 



Temp. 

°C. 



Solu- 
bility k 
(cc/cc. 
rubber/ 

atm.) 



log,,k^\ogko-^:^^j^^ 



"Neoprene" 
(vulcanised) 



"Neoprene" 
(vulcanised) 



"Neoprene" 
(vulcanised) 



Chloroprene 
polymer 



Butadiene-acrylo- 
nitrile inter- 
polymer 



Butadiene-acrylo- 
nitrile inter- 
polymer 



Butadiene-methyl- 
methacrylate 
interpolymer 

Butadiene-niethyl- 

methacrylate 
■ interpolymer 



Butadiene-styrene 
interpolymer 
(sample I) 



Butadiene-styrene 
interpolymer 
(sample II) 



H, 



N« 



H, 



N, 



H, 



N, 



N, 



(A 
A 



(Na 



0-0 
17-0 
27-0 
36-1 
46-5 

36-1 
52-2 
61-8 
73-7 
86-2 

27-1 
36-4 
44-1 
54-1 
65-4 
84-7 

31-8 
41-3 
49-9 
60-8 
69-5 
73-5 

17-0 
38-1 
48-5 
59-5 
70-5 


20 
29 
41-6 
50-2 

39-5 
55-0 
66-0 
78-0 

20-0 
30-8 
39-2 
51-8 
52-3 

20-0 
35-5 
50-0 
64-2 
640 

19-5 
30-3 
40-7 
51-2 
64-6 
64-6 



0-065 
0-051 
0-053 
0-051 
0-050 

0-155 
0141 
0-117 
0-106 
0102 

0-054 
0-050 
0-047 
0-044 
0-040 
0-038 

0-115 
0-103 
0-097 
0-090 
0-083 
0-082 

0-063 
0-050 
0-048 
0-040 
0-038 

0-040 
0-037 
0-036 
0033 
0-036 

0-084 
0-075 
0-071 
0-060 

0-134 
0-125 
0112 
0110 
0-099 

0-094 
0-086 
0-082 
0-080 
0-168) 

0-218 
0-210 
0-196 
0-186 
0-170 
0-101) 



970 

logioA--l-97+^^g^ 



W„A;= -1-86 + 



los,.k= -2-28-1- 



w„i-= -2-09 -J- 



,k= -2-44-f 



.w„A-= -1-80-t- 



log,„fc= -2-45-f 



logiofc= -1-95-f 



loo;,„A=-l-87-t- 



W,„A'= -1-63-t- 



1630 
4-60r 



1400 
4-60T 



1600 
4-60r 



1700 
4-60!r 



500 
4-60T 



2000 
4-60r 



1450 
4-60r 



1000 
4-60r 



1100 
4-60r 



SOLUTIOlSr AND DIFFUSION OF GASES 419 

data are obtained for liquids (62) and polymers (33), it appears 

that there is a larger entropy decrease by 4 or 5 units for the 

process ^ ,. , . . , 

Gas dissolving m polymer 

than for the process 

Gas dissolving in liquid. 

This observation must be interpreted by supposing that more 
restrictions are imposed upon the movement of the gas 
molecules in the polymer than in the hquid. This shows itself 
in a higher energy of activation (about lOk.cal.) needed for 
diffusion in polymers (Table 108) than for diffusion in liquids 
(3-5k.caL). 

The activation energy for diffusions in rubbers is as large 
for hehum and for hydrogen as the corresponding energies in 
rigid inorganic membranes of sUica glass (Chap. III). The 
energy of activation rises as the molecular weight of the 
diffusing molecule increases, although the increases in the 
energy with molecular weight are very much smaller than 
those found in rigid membranes of silica glass. The influence 
of molecular weight upon the activation energy can be 
illustrated by reference to pubhshed data upon gas-siLica(62a), 
gas-heulandite(63), gas-ceUulose (35a, 37), and gas-neoprene(33) 
diffusion systems (Table 109). 

One feature which has to be explained is the magnitude of 
the activation energy in rubbers, since the internal elasticity 
on a molecular scale might be expected to reduce the energy 
barrier involved in migration. When one calculates the energy 
needed to cause a gas atom to pass through an elastic two- 
dimensional crystal, it can be shown (33) that the potential 
energy barrier becomes very smaU indeed with only minor 
elastic displacements of the components of the two-dimensional 
lattice. Barrer advanced the theory that the major part of 
the energy of activation was the energy needed to create 
' ' holes ' ' in the three-dimensional rubber network. * The energy 

* The theory of holes in the rubber substance agrees with the model of a 
rubber polymer given in Fig. 140. 

27-2 



/ 



420 PERMEATION, SOLUTION, DIFFUSION OF GASES 



Table 108. Diffusion constants in polymers 







D 






System 


Temp. 

°C. 


cm.2 sec.-i 
xlO^ 


cm.^ sec.~^ 


E 
cal./mol. 


Hj-" neoprene " 





0-037 


9-0 


9,250 


(vulcanised) 


■ 17 


0-103 








27 


0-180 


^ 






36-1 


0-297 








46-5 


0-481 






A-"neoprene" 


36- 1 


0-033 


54-6 


11,700 


(vulcanised) 


52-2 
61-8 
73-7 

86-2 


0-078 
0145 
0-253 
0-484 






N2-"neoprene" 


27-1 


0-019 


79 


11,900 


(vulcanised) 


35-4 
44-1 


0-034 
0-055 






' 


54-1 
65-4 

84-7 


0-096 
0-180 
0-450 






Hj-chloroprene 


31-8 


0-33 


39-4 


9,900 


polymer 


41-3 
49-9 
60-8 
- 69-5 
73-5 


0-56 

0-91 

1-44 

2-1 

2-4 






Hj-butadiene-acrylo- 





0-061 


54-4 


8,700 


nitrUe interpolymer 


20-0 
29-0 
41-5 
50-2 


0-177 
0-27 
0-46 
0-66 






Ng-butadiene-acrylo- 


17 


0-0066 


28-1 


11,500 


nitrUe interpolymer 


38-1 

48-5 
59-5 
70-5 


0-029 
0-044 
0-088 
0-14 






Nj-butadiene-metliyl- 


39-5 


0041 


38 


11,500 


methacrylate 


550 


0-092 






interpolymer 


66-0 


0-16 






(sample I) 


78-0 


0-29 






A-butadiene-methyl- 


20 


0-034 


15-1 


10,300 


methacrylate 


30-8 


0-062 






interpolymer 


39-2 


0-112 






(sample II) 


51-8 
62-3 


0-186 
0-309 






A-butadiene-styrene 


19-5 


0-038 


1-84 


9,000 


interpolymer 


30-3 


0-068 






(sample I) 


40-7 
51-2 
64-6 


0111 
0175 
0-304 


• 




N^-butadiene-styrene 


20-0 


0-0237 


0-93 


8,900 


interpolymer 


35-5 


0-0506 






(sample II) 


500 
64-2 


0-095 
0-153 







SOLUTION AND DIFFITSION OF GASES 421 

of hole formation would then consist of the van der Waals's 
energy of cohesion absorbed when a number of cohering > CH2 
groups in adjacent hydrocarbon chains were separated. The 
fluctuations of thermal energy in the rubber miceUe would 
always be sufficient to maintain a certain number of these holes 
in the rubber substance, because of the existence of which the 
rubber can dissolve gases. There must then be a drift of dis- 
solved gas in the direction of decreasing concentration gradient, 
by way of these holes. The energy absorbed in producing a hole 
large enough to accommodate a hehum atom is a httle less than 
the energy needed to produce a hole large enough to accommo- 
date the bigger argon atom, and so -E'argon > ^helium- 

Table 109. The influence of molecular weight upon the 
activation energy for diffusion in membranes 





E (neoprene) 
cal./mol. 


E* (SiOa 

glass) 
cal./mol. 


E (heulandite) 
cal./mol. 


E* (cellulose 

compounds) 

cal./mol. 


He 
A 


8,000 

9,250 

11,900 

11,700 

6,900 

(in rubber) 


5,600 
10,000 
26,000 
32,000 


5400 
(J_ 201 face) 


7600 to 5600 



* These values of E are approximate since they include the smaU temperature 
coefficient of solubility. 



It was also suggested that plastic deformation of a rubber 
under stress was due to the same causes. Sections of hydro- 
carbon chain become separated from other chains by thermal 
agitation, and under the shearing force become somewhat 
displaced before coming together again. The temperature 
coefficient for the viscosity of rubber has been given as 
10,000 cal. (64), in agreement with the values 8700-11,900 
cal./mol. observed (Table 108) when gases diffuse through 
rubbers. Permanent displacements of the hydrocarbon chains 
relatively to one another are only possible, however, if the 
chains are not strongly cross-linked by sulphur or oxygen 
bonds. 



422 permeation, solution,' diffusion of gases 

Models for diffusion in rubber 

A number of formulae have been derived for D, the diffusion 
constant, based upon various models of the diffusion process. 
The formulae all depend upon the premise of a medium in 
which the diffusing particle is vibrating, and moving to 
successive positions of equilibrium when a sufficient activation 
energy has been acquired by the system particle-medium. 
The most general expression based upon kinetic theory (Chap. 
VI) is 

1 V I E Y'^-i) 



D = - ~-r-, -^^ dH-EiRT (Wheeler) (65). 

Here v = the vibration frequency of the diffusing particle in 
the medium, 

n =■ the number of degrees of freedom in which the 
activation energy E is stored, 

d = the mean free path of the diffusing particle in the 
activated state. 

When n = 2, ^ 

^ = i^ -^ dH-^l^T (Bradley) (66), 

and if ?2 = 1, D = \vd^e-^l^T^ 

The transition state theory of reaction velocities may be 
applied to diffusion systems, the expression for D being (67): 

D = '^y^d^e-^l^T, 

where F,^, F^ denote the partition functions of the system 
particle-medium in the normal and activated states re- 
spectively, excluding from the latter the partition function for 
the co-ordinate in which diffusion occurs. 

In Table 108 the diffusion constants have been expressed 
by the formula D = i)(,e-^/^^', and the values of Dq may be 
used to calculate values of the mean free path, d (Table 110). 



MODELS FOR DIPFTJSIOISr IS" ETJBBER 423 

For all formulae involving a few degrees of freedom only- 
Table 110 shows that the mean free paths are very much larger 
than one would expect. In Table 110 are given for comparison 
the mean free paths in some liquid-liquid diffusion systems 
in which it can be seen that the mean free paths are of 
normal molecular magnitudes (excepting water which usually 
shows anomalous properties). 

An explanation of this behaviour of gas-polymer diffusion 
systems, compatible with the diffusion process as pictured 
previously (p. 421) and involving only a small mean free path 
of the diffusing particle, is that the activation energy is dis- 
tributed through many degrees of freedom in the particle- 
medium system. Such a distribution of energy is needed for 
hole formation in the polymer. Thus one has in these systems 
an analogy with "fast" chemical reactions (for example, the 
unimolecular decomposition of large organic molecules). The 
numbers of degrees of freedom needed to give values of d of 
molecular magnitude are given in Table 111. 

Eyring's formula (67) relating the diffusion constant D and 
the velocity constant for passing over an energy barrier, k, is 

D = M2, 

where d as before denotes the mean free path in the activated 
state. If the formula of Wynne- Jones and Eyring(68) for a 
reaction velocity constant is used, one has 

,kT 



h 



or 



D = Dne-^/^2' = ^-AH±/RTQAS±IRL'^\d2 ^ ^-E^IRT ^ ^2 

so that ^o = e-^^-(x)^^(2V2 

where E = Eq — RT and E^ and AH- are identified. When a 
value oid of 5 x 10~^ cm. is assumed, the entropies of activation 
are those of Table 111. A value of ^ = 10 x 10^^ cm. would 
reduce aU these entropies by 2-76 units. The large increase in 



424 PERMEATION, SOLUTION, DIFFUSION OF GASES 



X 

6^ 



53J 

Si 






=S. 



O 

Si' 
5S 









H 



^ 








* 1 s 


< 




Tj^ tJ< O "^ 


<M 


o 


•^IClOO i-Hr^rHr^O 


^h 


00 


M 


I> 00 rt in rt 

-H M rt 


(M 


l* 






t^ 


* 1 s 






SSI 


fenl!^ 






II 








Q° 


^ 

'^ 




• 


% 








Erl 1^' 


< 




Ci 05 O "^ 


M 


O 


to 


rH O O O CO CO ii >o ec 


^ t- 


oo 




O nH 03 >C CO 




Tin 

II 


CO 


(?q -* ^ CO 


^ io 






II 


■TS 






O 








c^ 








1 


< 




rH l> -^ C^ 


rHtS 


O 


o 


cooocD ojdidbcbeo 


II 


03 


(M 


t^ lO t- ■* '» 


O 




00 


Ttl — 1 C0 03 


Q 


7 




r-H ^H 


- 








a 








(^ 


o 


oooo o o o >o o 


^ 


o 


lOOOO OiOOOCOTji 


"^^ 


t> 


lO_ 


'^^ '^^ ''i ^. ^^ ^^ '-i ^ ''^ 










cS 


00 


i-H 


OT ^ r^ r^ lO" CO CO CO i-H 


^ 
















Kl 
















IH 
1 








6 






c-T 7 T " 


<D 






05 00 


CQ 






^ r-H I— I (-H I-H F-H 


°^. 


CO 


00 


oiinoot- oxxxx 


a 


lO 


(M 


lO I> CO CO 00 CO 


o 






•^ F^ CO ^ 
















CO CO I-H o 


cf 














e3 






rS 








'3 


h i 


a 


o 
o 


o 

t 


^ S o § "S ^ -a qp^^ a 


-Jl 


0) 


01 
3 




to 


-D 






W ^ 


;5" 



MODELS FOR DIFFUSION IN RUBBER 



425 



AS"^ if it were all due to the diffusing molecule would corre- 
spond to more than its entropy of solution (53). Thus the 
medium itself must share in the entropy change, and so both 
kinetic theory and transition state theory lead to the same 
viewpoint: that the activation energy is shared in part or 

Table 111. Degrees of freedom in acquiring energy of activation 
for diffusion from Wheeler's equation, and the entropy of 
activation from Wynne- Jones and Eyring's equation 













AS± 


Diffusion system 


Do 

cm.^sec.-i 


E 
cal./mol. 


d* X 108 
cm. 


„ 


(entropy 
units) 


Hj-butadiene-^acrylo- 


56 


8,700 


24 


14 


17-8 


nitrile interpolymer 












Na-butadiene-acrylo- 


28 


11,500 


9-1 


12 


16-4 


nitrUe iaterpolymer 












Hg-neoprene 


9-4 


9,250 


6-9 


14 


14-2 


A-neoprene 


55 


11,700 


3-2 


13 


17-7 


Ng-neoprene 


78 


11,900 


3-7 


13 


18-4 


Ng-butadiene-methyl- 


37 


11,500 


3-3 


12 


16-9 


methacrylate inter- 












polymer 













* (^ is calculated from the equation 

^2 ^Doe'"-'' ^^^^—^ (- f^ ,, p/7i V"""> wbere v =2-5 x lO^^. 

V \Eoi,a.+(n-l) BTJ 



wholly in various degrees of freedom of the polymer or 
polymer-solute system, a viewpoint already implied in the 
"hole" theory of solution and diffusion in elastic polymers 

(p. 421). 

1 



Since 



JrT 
^0-^ h"^ 2-72^ 



it is seen that logDo is proportional to AS^. Barrer(33) 
collected the available diffusion data for a variety of systems, 
and plotted the number of diffusion systems having Dq within 
certain limits against logDo- The frequency cm-ve of Fig. 147 
was obtained, in which the full curve represents the periodicity 
curve for all diffusion systems, and the dotted curves for 
special types of system. It can be seen that the entropy 



426 PERMEATION, SOLUTION, DIFFUSION OF GASES 

of activation for gas-elastic polymer diffusion systems is 
greater than of liquid-liquid or gas- and solid-solid diffusion 
systems. The peak of the periodicity curve for Hquid-Hquid 
systems lies between the peaks for the curves for rubber and 
solid diffusion systems. The sohd diffusion systems cover the 




^1 



Fig. 147. Periodicity curve for Dq in ' D 
O Rubbers; x crystals and metals; 
diffusions. 



--3 -5 /o^ Oa . 

= Df,e^l^-^ for activated diffusions. 
+ glasses; liquids; Q] surface 



greatest range in values of Dq (lO'^'-fold). This may be because 
the greatest variety of structures occurs in crystals, metals, 
and glasses, involving special mechanisms of diffusion. 
Rubbers, and liquids, approximate to two single types, in 
which structural singularities are missing, so that narrow 
bands in the periodicity curve cover the observed range of 
values of Dq. 



MODELS FOR DIFFUSION IN RUBBEIl 427 

The values of Dq and ^q in the expressions 
D = D^e-^/RT, 

have been related by the relation (p. 423) 

D, = k,dK 

This equation permits a comparison of unimolecular reactions 
and activated diffusions. The distribution curve corresponding 
to Fig. 147 for unimolecular reactions (69) has its peak at 
A^o = 10^2~^*, and a spread of 10^^-fold. The corresponding 
curve for bimolecular reactions (70) has a peak at Ajq = lO^^ 
(Ktres X g. mol.~^ x sec.^^) and a spread of 10^^-fold. If a value 
of c? = 2-1 X 10~^ cm. is assumed, the maxima in Fig. 147 
occur at 

^Q = 3 X lO^^sec."^ for rubbers, 

^^o = 3 X lO^^sec."^ for liquids, 

A^Q = 1 X lO^^sec"! for crystals and glasses. 

One thus notes the similarity in activated diffusions and in 
unimolecular reactions. 



^ REFERENCES 

(1) Carson, F. Bur. Stand. J. Res., Wash., 12, 567 (1934). 

(2) Dewar, J. Proc. Roy. Instn, 21, 813 (1914-16). 

(3) Daynes, H. Proc. Roy. Soc. 97 A, 286 (1920). 

(4) Schiamacher, E. and Ferguson, L. J. Amer. chem. Soc. 49, 427 

(1927). 

(5) Rayleigh, Lord. Proc. Roy. Soc. 156A, 350 (1936). 

(6) Edwards, J. and Pickering, S. Sci. Pap. U.S. Bur. Stand. 16, 

327 (1920). 

(7) E.g. Benton, A. F. Industr. Engng Chem. 11, 623 (1919). 
Buckingham, E. Tech. Pap. Bur. Stand. 14 (T 183) (1920). 
Doughty, R., Seborg, C. and Baird, P. Tech. Ass. Papers, 15, 287 

(1932). 
Emanueh, L. Paper Tr. J. 85 (TS 98) (1927). 
GaUagher, F. Paper, 33, 5 (1924). 

(8) E.g. Barr, G. J. Text. Inst. 23, 206 (1932). 
Marsh, M. J. Text. Inst. 11 (T 56) (1931). 

(9) Schiefer, H. and Best, A. Bur. Stand. J. Res., Wash., 6, 51 ( 1931). 



428 PERMEATION, SOLUTION, DIFFUSION OF GASES 

(10) Mathieu, M. La Nitration de la Cellulose, Actualites Scientifiques, 

No. 316. Paris, 1936. 

(11) Trillat, J. J. C.R. Acad. Sci., Paris, 197, 1616 (1933). 

(12) McBain, J. W. Sorption of Gases and Vapours by Solids, Figs. 

110 and 111. Routledge, 1932. 

(13) Sheppard, S. E. and Newsome, P. T. J. phys. Chem. 39, 143 

(1935). 

(14) Hagedorn, M. and Moeller, P. Veroff. ZentLab. Anilin. photogr. 

Abt. 1, 144 (1930). 

(15) Astbtiry, W. and Woods, H. Nature, Lond., 126, 913, Fig. 1 

(1930). 

(16) Bernal, J., Fankuchen, I. and Perutz, M. Nature, Lond., 141, 

523 (1938). 

(17) E.g. de Boer, J. H. Trans. Faraday Soc. 32, 10 (1936). 

(18) Mark, H. and Meyer, K. Der Aufbau der Hochpolymeren 

Organischen Naturstojfe. Leipzig, 1930. 

(19) Meyer, K. Kolloidzschr . 53, 8 (1930). 

(20) Siefriz, W. Protoplasma, 21, 129 (1934). 

(21) Frey-Wyssling, A. Protoplasma, 25, 262 (1936). 

(22) Guth, E. and Rogowin, S. S.B. Akad. Wiss. Wien, iia, 145, 531 

(1936). 

(23) Clews, C. B. and Scho'sberger, F. Proc. Roy. Soc. 164 A, 491 (19.38). 

(24) Sauter, E. Z. phys. Chem. 36B, 405, 427 (1937). 

(25) Staudinger, II. Ber. dtsch. Chem. Ges. 62B, 2893 {1929). 

(26) Staudinger, H. and Husemann, E. Ber. dtsch. Chem. Ges. 68B, 

1691 (1935). 

(27) Staudinger, H., Heuer, W. and Husemann, E. Trans. Faraday 

Soc. 32, 323 (1936). 

(28) Graham, T. Phil. Mag. 32, 401 (1866). 

(29) Wroblewski, S. Ann. Phys., Lpz., S, 29 (1879). 

(30) Hiifner, G. Ann. Phys., Lpz., 34, 1 (1888). 

(31) Reychler, A. J. Chim. phys. 8, 617 (1910). 

(32) Kanata, K. Bull. Chem. Soc. Japan, 3, 183 (1928). 

(33) Barrer, R. Trans. Faraday Soc. 35, 628, 644 (1939). 

(34) Sager, T. P. Bur. Stand. Res., Wash., 19, 181 (1937). 

(35) Martin, S. and Patrick, J. Industr. Engng Chem. 28, 1144 

(1936). 
(35a) Barrer, R. Trans. Faraday Soc. 36, 644 (1940). 

(36) Sager, T. P. Bur. Stand. J. Res., Wash., 13, 879 (1934). 

(37) de Boer, J. H. and Fast, J. Rec. Trav. chim.. Pays-Bas, 57, 317 

(1938). 

(38) Kayser, H. Ann. Phys., Lpz., 43, 544 (1891). 

(39) Barr, G. Tech. Rep. Adv. Comm. Aero., Lond. (ref. by H. 

Daynes (3)). 

(40) Shakespear, G. Unpublished (ref. by H. Daynes (3)). 

(41) Shakespear, G., Daynes, H. and Larabourn. A brief account of 

some Experiments on the Permeability of Balloon Fabrics to A ir. 
Adv. Comm. for Aeronautics (ref. by H. Daynes (3)). 



(42 

(43 

(44 
(45 
(46 
(47 
(48 
(49 
(50 
(51 

(52 
(53 
(54 

(55 
(56 
(57 
(58 
(59 
(60 
(61 

(62 
(62 
(63 
(64 
(65 
(66 
(67 
(68 

(69 
(70 



REFERENCES 429 

Taylor, R., Herrmann, D. and Kemp, A. Industr. Engng Chem. 

28, 1255 (1936). 
Barrer, R. M. Nature, Land., 140, 107 (1937); Trans. Faraday 

Soc. 34, 849 (1938). 
Meyer, O. Ann. Phys., Lpz., Ill, 253 (1866). 
E.g. Knudsen, M. Ann. Phys., Lpz., 41, 289 (1913). 
Clausing, P. Ann. Phys., Lpz., 1, 489, 569 (1930). 
E.g. Buckingham, E. Tech. Pap. Bur. Stand. T 183 (1920-21). 
E.g. Thomson, W. Sci. Papers, 2, 681, 711. Cambridge, 1890. 
Carson, F. Bur. Stand. J. Res., Wash., 12, 587 (1934). 
Edwards, R. J. Soc. Leath. Tr. Chem. 14, 392 (1930). 
Bergmann, M. and Ludewig, S. J. Soc. Leath. Tr. Chetn. IS, 279 

(1929). 
Bergmann, M. J. Soc. Leath. Tr. Chem. 12, 170 (1928). 
Wilson, J. and Lines, G. Industr. Engng Chem. 17, 570 (1925). 
McBain, J. W. The Sorption of Gases and Vapours by Solids. 

Routledge, 1932. 
Barrer, R. M. Phil. Mag. 28, 148 (1939). 

Trans. Faraday Soc. 36, 1235 (1940). 

Venable, C. and Fuwa, T. Industr. Engng Chem. 14, 139 (1922). 
Tammann, G. and Bochow, K. Z. anorg. Chem. 168, 263 (1928). 
Kohman, G. J. phys. Chem. 33, 226 (1929). 
Bekkedahl, N. Bur. Stand. J. Res., Wash., 13, 411 (1934). 
Horiuti, J. Sci. Pap. Inst. phys. chem. Res., Tokyo, 17, 125 (1931). 
Lannung, A. J. Atner. chem. Soc. 52, 68 (1930). 
BeU, R. P. Trans. Faraday Soc. 33, 496 (1937). 
a) Barrer, R. M. J. chem. Soc. p. 378 (1934). 
Tiselius, A. Z. phys. Chem. 169 A, 425 (1934). 
Ewell, R. H. J. appl. Phys. 9, 252 (1938). 
Wheeler, T. S. Trans. Nat. Inst. Sci: India, I, 333 (1938). 
Bradley, R. S. Trans. Faraday Soc. 33, 1185 (1937). 
Eyring, H. J. chem. Phys. 4, 283 (1936). 
Wynne-Jones, W. F. and Eyring, H. J. chem. Phys. 3, 492 

(1935). 
Polanyi, M. and Wigner, E. Z. phys. Chem. 139 A, 439 (1928). 
Moelwyn-Hughes, E. A. Kinetics of Reactions in Solutions, 

chap. IV, p. 439 (1933). 



CHAPTER X 

PERMEATION OF VAPOURS THROUGH, AND 
DIFFUSION IN, ORGANIC SOLIDS 

Diffusion of simple gases in many organic solids obeys Pick's 
laws and may therefore be regarded as showing the ideal 
behaviour. The diffusion of vapours, however, only approxi- 
mately follows these laws, which then become limiting con- 
ditions only. It is these non-ideal systems which now remain 
to be discussed, 

WATBR-ORGAlSriC MEMBRAjSTE DIFFUSION SYSTEMS 

The most numerous researches have been carried out uj)on 
water-membrane systems, for which a summarising paper by 
Carson (1) has outlined the main methods of measurement. To 
prevent lateral diffusion of water through the edges of the 
specimen, or between it and its supports, various devices 
have been adopted. The edges have been moulded in wax, or 
compressed with a mixture of beeswax and resin. Shellac, 
petrolatum, and rubber have been used for the same purpose; 
or the cell may be mercmy sealed. Films of lacquers and paints 
may be made by painting the lacquer on suitable surfaces, such 
as amalgamated tin-plate (2), and then jDceling off the dried 
membrane. 

A great number of the measurements on permeabihty have 
been made upon the complex membranes of Table 112, in which 
the chemical nature of the membranes is briefly indicated. 
Most of the membranes sorb water freely. Natural rubber, one 
of the least hydrophUic of these substances, may if exposed 
to water vary from a non-swelling non-sorbing medium to 
complete dispersion in the water, according to the carbo- 
hydrate and protein content (3). Nearly all cellulose or protein- 
containing substances swell and sorb water strongly (i), and 
the accompanying sorption is then influenced by tempera- 



WATER-ORGANIC MEMBRANE SYSTEMS 



431 



ture(5), vapour pressure, and markedly by saline (6, 7) sub- 
stances. This has led to the suggestion that sorption and 
sweUing represent a tendency of water to equahse the salt 
concentrations inside or outside the membrane (8). In so far as 
the permeabihty must depend on the concentration gradients 
of water within the material, the water permeability will 
depend upon the various factors which influence sorj)tion 
processes. 

Table 112 



Membrane 


Chemical constituents 


Technical rubber and its 
derivatives (gutta-percha, 
paragutta, insulating com- 
pounds) 

Paints, varnishes, lacquers 

Leathers 

Papers 

Textiles 


Rubber hydrocarbon (polyisoprene) 

Sulphur 

Resins "i 

FiUers 

Carbohydrates J- present as impurity 

Protein 

Salts J 

Synthetic resins (phenol formalde- 
hydes, glycerol phthalates, glycol 
sebacates, etc.) 

Nitrocellulose and cellulose esters 

Drying oils 

Paraffin, aromatic and terpene hydro- 
carbons 

Alcohols and esters 

PbCOg, Al, or C black 

Natural protein membranes 

Taimins 

Oils 

Salts 

Solids such as CrjOj, AI2O3, Fe.jOg 

Carbohydrates (cellulose, lignin) 

Sizing agents (alum) 

Protein 

Carbohydrates (cellulose, lignin) 

Proteins 

Dyes 

Mordants 



On the basis of chemical composition one may construct a 
permeability spectrum for water (Fig. 148). The permeabilities 
cover a 10^-fold range and may arbitrarily be regarded as 
comprising four main groups. In constructing the permeability 
chart the permeability constant was derived for a thickness 
of 1 mm., wherever possible. Unfortunately, in many references 
the thickness was not given. The range of permeabilities 



432 PERMEATION AND DIFFUSION OF VAPOURS 

(10^-fold) is 10^-fold less than the range of air permeabilities 
previously given (p. 383). A number of permeability data will 
be given later (p. 440) and the comparison with air or hydrogen 
permeabilities made in more detail (p. 443). 



Wax 


Rubber 


Moisture-proofed 


Paints, varnishes, 


Cellulose 


Rubber 


Asphalt 


or waxed fibre- 


lacquers 


TextUes 






board, cellulose, 


Cellulose acetate 


Metal 






papers, waxed 


Cellulose ethers 


gauzes 






glassine 


Cellulose nitrates 


Leathers 






Paints, varnishes, 


Parchments, 


Open 






lacquers 


glassine 


cells 






Cellulose acetate 


Leathers 
Cellulose 





-7 -6 -5 -4 -3 

log permeability 

Fig. 148. A water-permeability spectrum. 
(Permeability in c.c./sec./cm.^cm. Hg.) 



Dependence of permeability upon thickness and 
vapour-p'essure difference 

Membranes which sorb little water (e.g. purified polystyrene (9)) 
behave at low relative vapour pressure as indicated by Fick's 
law, and the permeation velocity is proportional to l/l 
(l denotes the membrane thickness). However, Fig. 149 in- 
dicates that at high humidities the permeation velocity 
constant rises with increasing thickness. Certain paint and 
varnish films (io,ii) also obeyed the law p oc l/l; but m other 
cases marked divergences arise (12,13). 

Similarly a number of researches(i4,i5,i6, 17, 18,19,20 ) at low 
humidity showed that the permeation velocity was pro- 
portional to the vapour -pressure difference. The more hygro- 
scopic the substance, however, the lower is the humidity above 
which this relationship breaks down. Pure rubbers and waxes 
sorb little water, and the permeability "constants" calculated 
from Fick's law are little affected by variations of humidity at 
the surfaces of the membrane. Many rubbers, however, contain 
hydrophilic impurity and sorption occurs freely. Figs. 150 (fi) 
and 151(9) show that the permeation rate and the sorption 
isotherms follow similar courses. 



WATER-OEGANIC MEMBRAlSrE SYSTEMS 



433 



Cellulose and cellulose compounds are more hydrophilic 
than rubbers, and departures from the relation Permeation 
velocity oc Vapour pressure difference quickly appear. In 

























■^ 


"^ 


















^ 


^^ 


p, = 23-8 MM.Hg 
P2= 0-0 MM.Hg 












J 


^ 


^' 














L^ 


■^ 






















^ 


y' 


















































































































p ; = 7-66 MM. Hq 
P2 = 0-0 MM.Hg 




~ 










...J-- 







•0 


4 


•0 


e 


'1 


2 


•: 


J 


•20 


■2^ 





CE->T!MiT£RS 



Fig. 149. Permeatrion rate constants at Mgli and low 
humidities as a function of thickness 0). 




1-2 1-6 2-0 

pe:<? cent water 



'0 0-4 

Fig. 150. Absorption of water vapour by gutta percha at 25° C. (6). 



membranes made of resins, paints, and varnishes, the permea- 
tion velocity again follows roughly the sorption isothermal (16) 
(Fig. 152). Leathers show a very great range in their water 



28 



434 PERMEATION AND DIFFUSION OF VAPOURS 

and air permeability, but again on the whole conform to the 
general principle of increased permeability "constants" at 
high humidity. 

















i 
































J 














^ 










t 


= 250°C. 


1 


1 


1 


P2 = 

1 1 


0-OMM.Hg 


1 


1 



8 12 16 2 

VAPOR PRESSURE p, IN MM.Hg 

0-20 0-40 0-60 080 

RELATIVE VAPOR PRESSURE 



Pig. 151. Permeation rate constant as a function 
of vapour pressure for rubberO). 



<-30 
o 
I 
5-25 

1-20 

I— 
< 

^•15 

Ui 

Z 
u> 

UJ 

<r 

5 -05 

o 











































/ 








































^A 








































/ 




































A 












Moisture absorption,-—. 








»- 


A 


L^ 


/ 


























o-- 


\ 


"' 










/ 






















r 


' 












A 


y 




























1 






^ 


































^ 




































»^ 


-V 


-Moisture penetration.] 


















^ 




" 






























^ 


^ 


\t 
































<i\ 


1> 


jr 








i 






















Jr' 












TT-i" 

















•015 2 

010 i 

u. 
03 

•005 o 

CO 

S 



20 40 50 50 70 3-3 90 100 

Relative humidity at eo'e -Percent 
Fig. 152. The influence of varying the water- vapour pressure upon the quantity 
sorbed and upon the permeation velocity through aluminium paints (nj). 

The concentration gradients established in rubber 
during permeation 

In the steady state, if the Fick law were valid, linear gradients 
would be established. By using thin laminae (9) of rubber 
joined in series, one can determine the actual concentration 
gradients existing in the membranes. That there is not a linear 
gradient at high relative humidities is shown by Fig. 153, in 



WATER-ORGANIC MEMBRANE SYSTEMS 



435 



which one sees an actual concentration gradient for soft 
vulcanised rubber under diffusion equilibrium. The dotted 
curve gives the concentration gradient calculated from the 
concentration-pressure isothermals of Lowry and Kohman(6). 
The two curves lie close together and indicate that while 
Henry's law holds at low pressure, an abnormal sorption occurs 
at higher humidities. . . 



HI 

11 




U— THESE DOTTED LINES REPRESENT LAYERS 




n 




1 
-1 


IN 

1 


THE EXPERIMENTAL SAMPLE 

1 ! i| h 




— 






1 


1 
1 

rUAL D 


1 

STRIBUI 


! 

ION OF 


WATER 


1 

1 




— 


\L 


' — ' 


(EXPERIMENTAL VALUES) 






V 

\ 


1 


1 
1 
1 


1 
1 

i . 


1 

1 


1 
1 




1 
1 








l\ 


^^ CALCULATED FROM PRESSURE VS. 










CONCENTRATION DATA OF 








'YC" 


LOWRY AND KOHMAN 










^ 


1 

1 


1 
1 
1 


1 

1 




1 
1 

1 








1 
1 

1 


1 

1 


1 
1 


1 
1 


1 
1 


ICftss: 


1 


^^£38 





eo too 120 

THICKNESS IN MILS. 



1.1 I 



I I I 



VAPOR PRESSURE IN MM.«g 



WET SIDE 
OF SAMPLE 



Fig. 153. 



CRY sioe 

OF SAMPLE 



The influence of temperature upon the 
permeability constoMts 

The data of previous sections indicates that at low humidities 
one may use the linear Fick law* as an approximation which 
breaks down as the humidity increases. The influence of 
temperature upon permeability constants is two-fold. There 
is first the effect upon the diffusion constant and secondly the 
effect upon the absorption coefficient, k. Sometimes in colloidal 
systems the coefficient k increases with temperature even when 

* P =D {Ci- 02)11 = Dk (p^-p^) II, where k denotes the a,hsoT-ptiorxcoeiG.cient, 
and the other symbols have their usual significance. 



436 PERMEATION AND DIFFUSION OF VAPOURS 

sorption is exothermal, due to irreversible changes in the 
colloid. The data on the influence of temperature upon 
permeability constants may he summarised as follows: 



Impermeable membranes 


Permeable membranes | 


Rubbers, silk, and rubber-like 
polymers (9) gave permeability 
constants which increased with 
temperature, often exponentially 

Gelatin latex gas-cell fabrics (2) 
gave permeability constants in- 
creasing rapidly with temperature 

Synthetic resins gave permeability 
constants increasing exponenti- 
ally with temperature 
1 


Cellulose and cellulose compounds 
gave permeability constants which 
did not depend on temperature in 
many cases(M.i5,i6,]s, 19,22) 

The permeability constants for 
porous leathers did not depend on 
temperature (10) 

Paiat films (16) containing alu- 
minium gave permeability con- 
stants not appreciably depending 
on temperature 



A rule which was found valid in rubber-gas diffusion systems 
again emerges, that the permeabihty constants of the least 
permeable membranes are highly sensitive to temperature 
changes, but that in porous membranes the constants are 
independent, or slightly dependent upon, temperature. 

Some data giving permeabihty constants and permeation 
rates at different temperatures are presented in Tables 113 
and 114, for systems with high temperature coefficients, and 
neghgible coefficients respectively. For a number of resin, and 
rubber, membranes the curves of log P {P — permeabihty 
constant) against l/T are linear, and from the slopes one may 
evaluate E in the equation P = PqE'^I^'^ . Some of these values 
of E are collected in Table 115; their values are on an average 
somewhat smaller than E values for the permeability of gas- 
rubber systems. Diffusion of water in rubber is activated 
(p. 445); and it seems that diffusion is also activated in the 
resin membranes of Table 115. 



Miscellaneous effects in permeation rate studies 

Soluble salts, either in the membrane or in the solution in 
contact with the membrane, alter the amount of water 
absorbed, and therefore the concentration gradients and 
velocities of permeation. The addition of salts to water in 



WATER-ORGANIC MEMBRANE SYSTEMS 



437 



Table 113. The influence of temperature upon permeability 
constants for water diffusion (9) 



Substance 


Temp. 

°C. 


Permeability 

constant x 10^ 

g./cm.7cm./ 

hr./mm. Hg 


Polystyrene 
Varnished silk 
Vulcanised rubber 


2M 
35-0 

30-0 
350 

0-0 
21-0 
25-0 
35-0 


4-0 
4-5 

3-4 

4-7 

5-0 
6-9 
7-3 

8-5 



Table 114. The influence of temperature upon permeation 
rate through leather (20) 



Temp. 


Permeation rates 
mg. H20/24hr./l-267 cm.^ 


Ratio 
of rates 


Through 
leather 


Through 
, air space 


5 
20 
25 
30 
35 
40 
45 


71 

192 
236 

328 
430 
560 

765 


98 
286 
360 
505 
660 
863 
1214 


0-72 
0-67 
0-66 
0-65 
0-65 
0-65 
0-63 



Table 115. Values of E in P = P^e^l^T (^ in cal.lmol.] 



Composition of membrane 


E 


Non-volatile 


Volatile (solvents) 


Bakelite and glyceryl phtha- 
late(2) 

Cellulose nitrate, glycol seba- 
cate, glyceryl phthalate(2) 

BakeUte, China-wood oil, 
castor oil and linseed oil (2) 

Glyceryl phthalate(2) 
Rubber (9) 


Butyl alcohol and hydro- 
carbons 

Esters, alcohols and aromatic 
hydrocarbons 

Butyl alcohol, turpentine, di- 
pentine, mineral spirits 

Aliphatic and terpene hydro- 
carbons 


8200 

4700 

7500 

5800 
6200 
6400 

4300 

2800 



438 PERMEATION AND DIFFUSION OF VAPOURS 

which leather was immersed decreased the amounts of water 
sorbed(7) in the order: 

SO4 > citrate > acetate > CI' > NO3 > CNS', 

and . Na+ > K+ > Li+. 

It was similarly estabhshed (21 ) that salt solutions decreased 
the permeation rates. The leather was almost impermeable 
at sodium chloride concentrations of 2-5 normal. 

Coatings and fillers also influence permeability. Waxing 
diminished the permeability of celluloses, cellulose compounds, 
and synthetic resins (2 ). Collodion and casein coatings decreased 
the air and water permeabilities of leather (20). Sager (23) found. 
that carbon black incorporated in polysulphide rubbers 
increased their permeabihty to hydrogen. On the other 
hand (11), the water permeability of paint films was diminished 
by incorporating aluminium. 

Sometimes irreversible sorption effects arise. For example, 
the permeability of a certain rubber sample was twice as great 
in contact with hquid water as when saturated vapour was in 
contact with the membrane (24). Occasionally, diffusion does 
not occur so readily in one direction as in the converse 
direction (20). 

The permeabii^ity constants to water of 
various membranes 

Based on the tables of Carson (i), Table 116 provides numerical 
values of the permeability constants, in c.c./sec./cm.^/cm. Hg, 
and per mm. thick, when the thickness has been given. The , 
final column of Table 116 states whether or not the thickness 
was known. The values of the permeability constants are not 
strictly comparable unless this is so, especially since the thick- 
ness of paper and of cellulosic membranes, paint films, or 
varnish fihns will usually be less than 1 mm. There are, how- 
ever, sufficient references for which thicknesses were given to 
permit comparisons to be made. Taking Levey's (28) value for 
paper(29 x 10~^ c. c. /sec. /cm. ^/mm. thick/cm. ofHg)asanindex 



PERMEABILITY CONSTANTS 439 

figure, one finds values of the permeability of the same order 
or less for cellulose ethers and esters, or regenerated cellulose. 
It is noteworthy that certain textiles offer a resistance to the 
flow of water vapour comparable with that for cellulosic com- 
pounds. Moisture-proofed and waxed cellulosic substances are 
about one hundred times as impermeable (when comparable in 
thickness) as paper; and the paint, varnish and lacquer films 
vary from one five-hundredth to one-quarter of Levey's value 
for paper. Wax is more than one thousand times less permeable, 
as are certain members of the rubber group. The permeability 
is greater the more hydrophihc the membrane substance. One 
notes the extremely low permeabihty of the rubber -like 
polymer polyethylene tetrasulphide, only four times the 
permeabihty of wax, and four hundred times less permeable 
than paper. The synthetic hydrocarbon polymers, such as 
polystyrene, show permeabilities between those of soft 
vulcanised and of hard rubber, the permeabihty of the latter 
being one-quarter that of the former. 

It has Already been observed (p. 432) that the range of 
values of 10^-fold for water permeability is much less than the 
corresponding range of values for air of lO^^-fold. At one end 
of the permeabihty spectrum of Fig. 148, that of the least 
permeable substances in Table 116, the water permeabihty 
is 10- to 1000-fold greater than the air permeabihty. Similarly, 
for some members of the ceUulosic group (e.g. ordinary re- 
generated ceUulose) the water permeability is lO^-fold greater 
than the air permeability. The hydrogen permeability, like the 
air permeability, is smaller than that for water, there being 
a 30-100-fold difference for rubber, chloroprene, and poly- 
ethylene sulphide; but a 3000-1500-fold difference for the 
cellulosic substances ceUophane and ceUulose acetate. Once 
more it is the rule that hke is more permeable to hke. Hydrogen 
by analogy with its solubility in hydrocarbons should dissolve 
more freely in rubber, and water less freely. Typical data 
are coUected in Table 117. On the other hand, at the high 
permeabihty end of the spectrum (the leathers and textiles) 
the air permeability may be 10-1000 times greater for 





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PERMEABILITY CONSTANTS 



443 



leathers, and even lO^-fold greater for textiles, than the 
water permeabihty. 

Table 117. A comparison of some hydrogen and water 
permeabilities of membranes 





10® X approximate* 


10® X water 


Membrane 


hydrogen permeability 


permeability 


at 25° C. (c.c./cm.V 


(c.c./cm.2/sec./ 


■ 


sec./mm./cm. of Hg) 


mm-./cm. of Hg) 


Cellophanef 


0-00008 


0-12-0-24 at 38° C. 


Cellulose acetate 


0-018 


54-0 at 25° C. 


Polyethylene sulphide 


O'OOl 


0-076 at 21-1° C. 


Polychloroprene 


0-011 


0-91 at 21-1° C. 


Rubbers 


0-019 


0-52 at 25° C. 




(smoked sheet rubber) 


(hard rubber) 




0-019 


2-3-2-7 at 25° C. 




(ether soluble rubber) 


(soft vulcanised rubber) 



* In computing the hydiogen permeability a density .of unity had to be 
assumed, since Sager's(4J) original paper did not give his film densities, needed 
to compute the actual thickness, of his membranes. 

t Cellophane according to Hyden(30) is manufactured in thicknesses of 
0-025-0-05 mm. His permeabilities, for which the membrane thickness was not 
stated, have been computed on this basis. 



Fick's linear law 



Sorption kinetics in organic solids 

dC 



P = -D 



dx 



has been shown to be a hmiting law only, for vapour-membrane 

dC d'^G 

diffusion systems. This is also true of the law -^ = D-^-^ 

when applied to systems which swell during sorption of 
vapours, and give sigmoid isotherms. Such isotherms are 
given by cotton (43), wool (44), leather (4i), or wood (45). More 
generahsed laws such as 

dt dx\ dx 

(see also p. 47) should be used in such systems. However, it 
has been customary to employ the simpler law, and to use 
slabs of leather (42), rubber (2), bakehte(48) or cellulose deri- 



444 PERMEATION AND DIFFUSION OF VAPOURS 



vatives(46) in computing the diffusion constants D. A suitable 
solution is then 






1— ,(e-^ + 



' + : 



-25(9 



+ ...), 



where 6 = 7T^Dt/4:P, I = the thickness of the slab, and Q, Q^y 
denote the amounts sorbed at time t and at equihbrium 
respectively. 

When water is sorbed by rigid non-swelling membranes of 
bakelite, the solution given above holds with some accuracy (48). 



C« 0-60 
0-50 



0-10 







































— f 


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EASING 






























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Z ACETATE 


























EFFECT OF THICKNESS 
ON HEOUCeO ABSORPTION RATE 














III 






































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, 




, 
















1 — 

















20 40 



100 120 UO 160 180 200 220 24-0 260 280 300 320 340 360 380 
TIME (MIN.) 



Tig. 154. Curves of relative sorption against time for water-acetyl cellulose 
systems (46). The form of the curve ~ against t is dependent on the mem- 
brane thickness. 

On the other hand, in non-rigid swelling membranes of 
rubber (47) or cellulose acetate (Fig. 154) the departures were 
considerable. 

The sorption velocity of water in rubber samples increases 
as the temperature rises. The amount sorbed after a period of 
20 hr. (^20) is given below for a typical case: 



Temp. ° C. 

24 

60 

70 

100 



Amount sorbed in 20 hr. (^20) 
0-0110 
0-0265 
0-0390 
0-079 



SORPTION KINETICS IN ORGANIC SOLIDS 



445 



Assuming that the parabolic diffusion law [QIQ^f- = ^DtjnP' 
appKes, the slope of the curve log ^(20) versus IjT {T denotes 
°K.) gives an activation energy of 6900 cal./mol., a figure 
which may be compared with 6300 cal./atom for He-rubber 
to 11,900 cal./mol. for Na-neoprene. Smoked rubber, para- 
rubber, and pale crepe rubber similarly gave values of the 
activation energy of 7000, 9800 and 9600 cal./mol. of water 
respectively. 

When rubber swelled in benzene (49), or when paraffin (50 
was absorbed by cured rubber and by rubber gum, the positive 
temperature coefficients of the sorption velocity suggested 
that in these and other organic sohds vapours are sorbed by 
processes of activated diffusion. The occurrence of activated 
diffusion has already been estabhshed for gas-rubber (52), 
cellulose, -bakelite and similar systems (Chap. IX), In liquids 
also, diffusion constants (53) may be expressed by the equation 
I) = DqC-^/^^, although the values of E are usually smaller. 

Table 118. Diffusion constants in miscellaneous systems 



System 


D (cm.^ sec.~^) 


Temp. ° G. 


HgO - cellulose acetate od 


0-17-1-7 X 10-8 


25 


H20-leathers(4:) 


0-41-19-5 X 10-' 


25 


D20-H20(46) 


2-5 X 10-5 


25 


Hj-rubber 


0-85 X 10-5 


25 


Ng-rubber 


011x10-5 


25 


CH3OH-C2H5OH 


2-7 x 10-5 


19 


C2H2Cl4-C2H2Br4 


0-56 x 10-5 


19 



The data in Table 118 permit a qualitative comparison of 
the diffusion constants of water, gases, and liquids in organic 
polymers and liquids. 



A MODIFIED DIFFUSION LAW FOR SORPTION OF 
WATER BY RUBBERS 

The presence of salts in rubbers and leathers alters the sorption 
of water by these colloids (p. 436). For this reason Daynes(8) 
advanced a theory of diffusion as an osmotic phenomenon, the 
water being sorbed to equalise salt concentration differences 



446 PERMEATION AND DIFFUSION OF VAPOURS 

inside and outside the membrane. The law was then for- 
mulated as o^ 02 7 

where h denotes the humidity of the atmosphere which would 
be in equihbrium with the solution at x. The relation between 
C and h is given by the sorption isothermal. The diffusion 
equation may now be written 

dh_/dh \dVi, 

So long as dC/dh is a constant (i.e. at low humidities), a simple 

D^wp] may be used. The tempo of 

412 
diffusion is then governed by a single parameter d = -^ , 







- 




^.^ 


z 






^^-^^ 









^^ 


'""""''^ 




0. 




A,,.^^"^ 






^ 










1 


^ 






(/> 




^^^ 






0) 

< 


\ x^ 










y 


8 








A^ 


i ! i 



Fig. 155. Absorption and desorption rate curves in rubbered?). 

. , . n, 4Z2 ac 

but in the more complex case by the parameter a = — =- -^ . 

Because dC/dh, and therefore 6', increase rapidly at high 
humidity, sorption must proceed more and more slowly as 
humidity increases, or saturation is approached. In desorption, 
as humidity decreases, the converse is true, i.e. the desorption 
velocity steadily increases. The first prediction was fulfilled 
when water was sorbed by vulcanised and unvulcanised 
rubber (54, 8), gutta-percha, and paragutta(54); and the second 



VAPOURS OTHER THAN WATER 



447 



was indicated by the data of Fry (55) and of Cooper and 

dC d^C 

Scott (56) (Fig. 155 (57)). The simple law -^ = D -^ would on the 

other hand require the ordinates of the asymptotes to the 
curves A and B to stand in the ratio 2:1, while the ciu^ves 
A and C should be symmetrical. 



The passage of vapours other than water 
through membranes 

Although few studies have been made of the permeability of 
membranes to vapours other than water, the data are of 
considerable mterest theoretically and practically. Typical, 
data are those of Dewar(58) and Edwards and Pickering (27) 
(Table 119). 

Table 119. Permeability of rubber to vapours at 
room temperature 



Vapour 


10^ X permeability 

constant (c.c./sec./cm.^/mm. 

thick/cm. Hg pressure) 


Ratio of permeability 
constants for vapour 
and for hydrogen (23) 


C2H5OH 

CH3CI 

C2H5CI 


2-41 (23) 
1-28 (56) 
1-14 (56) 
0-81 (23) 

8-75 (23) 


55 
29-1 
25-9 
18-5 
198 



Kahlenberg (60) showed that if a cell HgO/rubber /ethyl 
alcohol was set up the alcohol diffuses into the water more 
rapidly than water into alcohol. At 15° C. the vapour pressures 
are 12 mm. for water and 32 mm. for alcohol, so that, with the 
permeability constants of Dewar in Table 119, one could 
predict the direction of diffusion actually observed by Kahlen- 
berg. As usual the permeation velocities bear no simple 
relationship to molecular size or mass, the large molecule, 
ethyl chloride, being transmitted ten times as fast as the 
smaller molecule, methyl chloride. The solubihty of the 
diffusing substance in the membrane is one important factor 
in diffusion. Another must be the extent to which the mem- 



448 PERMEATION AND DIFFUSION OF" VAPOURS 

brane swells. The swelling, which increases as the amount 
sorbed increases, causes the polyisoprene chains to separate 
more and more. One might then imagine that the looser the 
structure becomes the less its resistance to diffusion would be. 

Payne and Gardner (59) also showed that the permeabihty 
of a membrane to a vapour depended upon the solvent 
capacity of the membrane for the vapour. They used smooth 
cellulose paper of thickness 0-1 mm. as support for the films, 
which in nearly every instance, in the form of a suitable 
varnish, penetrated the paper completely. The paper itself had 
only a shght impedance to diffusion of the vapour. Fig. 156 
gives a summary of their relative permeability data for a large 
number of films, and diffusing substances. 

The data of Fig. 156 give the following series for the relative 
permeabilities of a number of vapours: 

Glue [water-soluble, but with macropores] : 
H2O > CeHg > C2H5OH > CH3COCH3 > C2H2OCOCH3. 

Rubber [soluble in CgHg, and partially ia CH3COCH3]: 
CeHfi > CH3COCH3 > C2H5OH > H2O. 

Paraffin wax [soluble in CgHg] : 

CfiHe > CH3COCH3 > C2H5OCOCH3 > C2H5OH > H2O. 
Cellulose nitrate [soluble in CH3COCH3, C2H5OCOCH3]: 

CH3COCH3 > C2H5OH > C2H5OCOCH3 > CeHg > H2O. 
Raw linseed oil [soluble in CH3COCH3, C2H5OCOCH3]: 

C2H5OH, CH3COCH3 > C2H5OCOCH3 > CgHg > H2O. 

Remarks on the permeation and diffusion processes 

The nature of the movement of water, organic vapours or 
gases depends primarily upon the manner in which the diffusing 
substance is held inside the solid, and on the nature of the 
channels. The sorption may be: 

A. Van der Waals's sorption, 

B. Dipole sorption in a monolayer, 




PER MEABILITY 



1. Raw linseed oil 

2. Bodied linseed oil 

3. Rosin spar varnish 

4. Phenolic varnish 

5. Shellac 

6. Rosin 

7. Alkyd resin 

8. Modified alkyd 

9 Resin-oil modified alkyd 



10. Rubber 

It. Paraffin waj 

13. Glue 

14. Gelatin 

15. Cellulose nitrate 

16. Plasticized cellulose 

17. Ester gum lacquer 

18. .Aluminum lacquer 

19. Porous lacquer 



Fig. 156. Comparison of permeabilities of films. 



29 



450 PERMEATION AND DIFFUSION OF VAPOURS 

C. Multilayer sorption at higher humidities, with possible 

orientation, 

D. Capillary condensation in pores. 

The sorbed substance "may be in true solution within the 
solid, or it may be non-homogeneously distributed or per- 
sorbed. Very often, for example in cellulose (6i), van der Waals 
sorption, or dipole sorption is superseded at higher pressures 
by capillary condensation. Occasionally, as in certain proteins, 
chemical effects are suspected (62). The mercerising of natural 
cellulose is thought to cause a re-orientation of the pyranose 
chains (see Fig. 137, Chap. IX, for a diagram of the unit .cell) 
in the manner indicated in Fig. 157 (63), the hydration causing 




Fig. 157. A. Section of cell of native cellulose. 
B. Section of cell of hydrated cellulose. 

The 6-axis is perpendicular to the plane of the diagram. 

a contraction parallel to the cell direction and an expansion 
in directions normal to it. Ordinary sorption does not alter 
the cell, but may still affect membrane permeabihty since 
it was observed that while desiccated cellulose membranes 
were impermeable to au', the same membranes after sorbing 
water became permeable to air (64), 

The dispersion of rubber in organic liquids is dependent upon 
the extent to which the isoprene chains are cross-linked by 
sulphur. Excessive vulcanisation causes a complete netting of 
rubber, and as would be anticipated, the extent of vulcanisa- 
tion conditions the extent of sorption of benzene (65, 66) or a 




PERMEATION AND DIFFUSION PROCESSES 451 

similar organic liquid (Fig. 158). The limiting case is repre- 
sented by a membrane of polymerised ^-divinyl benzene 
which neither swells in nor sorbs benzene appreciably. These 
phenomena must influence profoundly 
the permeabihty of the membrane, for 
it is not likely that netted membranes 1 ^] isv-iiiii^i^^x- 
will be able to transmit large organic 
vapoiirs readily, unless the transmission 
is intermicellar. Experiments on these 
aspects of permeability do not appear ^ ^^ ^^ ^^ 

to have been made. ^""^ '" '^"""^ 

The pecuhar nature of some of the Kg. 158. Dependence of the 
^ amount and velocity 

water-protein systems is best brought of sorption upon the 
out by considering X-ray data. While <iegree of vulcanisa- 
certain natural proteins such as wool, 

silk,- or horn are obtained, when desiccated, in crystalline 
form (67), others become denatured completely if they are 
dehydrated. They then fail to give X-ray patterns corre- 
sponding to a crystaUine arrangement; but the X-ray patterns 
of the same forms when highly hydrated may indicate a 
remarkable degree of crystalline order (68). Thus the water 
is not only sorbed but sometimes plays a vital part in the 
protein structure. The diffusion of water in such gelatinous 
membranes might have interesting features owing to the 
unique role of the water in the gel. 

It has been noted that the stronger the sorption of a vapour 
by an organic colloid the more quickly does the permeation 

dC dC d^C 

process deviate from the Fick laws P = D^ and -^ = D -^— ^, 

as the humidity increases. The classification of Wilson and 
Fuwa(69), which gives a rough measure of the capacity of 
various types of organic sohd to sorb water, may therefore be 
used as a guide to the degree of apphcabihty which the Fick 
laws might be expected to have. Some of the substances hsted 
(finely divided inorganic solids, carbon black, sihca gel, etc.) 
are permeable by hydrodynamic flow, but other membranes 
which have been considered in this chapter show permeation 

29-2 



452 PERMEATION AND DIFFUSION OF VAPOURS 

rates which are independent of the hydrostatic head of 
pressure, and therefore do not transmit liquids by hydro- 
dynamic flow. 



REFERENCES 

(1) Carson, F. Misc. Publ. U.S. Bur. Stand. M. 127 (1937). 

(2) Kline, G. Bur. Stand. J. Res., Wash., 18, 235 (1937). 

(3) Boggs, C. and Blake, J. Jndustr. Engng Chem. 18, 224 (1926). 

(4) For a summary of the data see J. W. McBain, The Sorption of 

Gases and Vapours by Solids, chap. xii. Routledge, 1932. 

(5) McBain, J. W. The Sorption of Oases and Vapours by Solids, 

p. 372. Routledge, 1932. 

(6) Lowry, H. and Kohman, G. J. phys. Chem. 31, 23 (1927). 

(7) Kubelka, V. Kolloidzschr. 51, 331-6 (1930); also M. Bergmann, 

J. Soc. Leath. Tr. Chem. 14, 307 (1930). 

(8) E.g. Daynes, H. Trans. Faraday Soc. 33, 531 (1937). 

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28, 1255 (1936). 

(10) Schumacher, E. and Ferguson, L. Industr. Engng Chem. 21, 159 

(1929). 

(11) Wing, H. Industr. Engng Chem. 28, 786 (1936). 

(12) Wray, R. and van Vorst, A. Industr. Engng Chem. 25, 842 (1933). 

(13) Wosnessenski, S. and Dubnikow, L. M. Kolloidzschr. 74, 183 

(1936). 

(14) Abrams, A. and Chilson, W. Paper Tr. J. 91, T.S. 193 (1930). 

(15) Charch, W. and Scroggie, A. G. Paper Tr. J. 101, T.S. 201 

(1935). 

(16) Edwards, J. and Wray, R. Industr. Engng Chem. 28, 549 (1936). 

(17) Hunt, J. and Lansing, D. Industr. Engng Chem. 27, 26 (1935). 

(18) Staedel, W. Pa^^er i^afer. 31, 535 (1933). 

(19) StUlwell, S. Tech. Pap. For. Prod. Res., Lond., 1 (publ. 1926-36). 

(20) WUson, J. and Lines, G. Industr. Engng Chem. 17, 570 (1925). 

(21) Bergmann, M. J. Soc. Leath. Tr. Chem. 13, 161 (1929). 

(22) Hartley, J. Tech. Pap. For. Prod. Res., Lond., 2 (publ. 1926-36). 

(23) Sager, T. P. Bur. Stand. J. Res., Wash., 19, 181 (1937). 

(24) Schroeder, P. N. Z. phys. Chem. 45, 75 (1903). 

(25) Daynes, H. Proc. Roy. Soc. 97 A, 286 (1920). 

(26) Bergmann, M. and Ludewig, S. J. Soc. Leath. Tr. Chem. 13, 279 

(1929). 

(27) Edwards, J. and Pickering, S. Sci. Pap. Bur. Stand. 16, 327 

(1920). 

(28) Levey, H. Plastic Prod. 11, 52 (1934). 

(29) Birdseye, C. Industr. Engng Chem. 21, 513 (1929). 

(30) Hyden, W. Industr. Engng Chem. 21, 405 (1929). 



REFERENCES 453 

Fabel, K. Kunstseide, 15, 383 (1933). 

Tressler, D. and Evers, C. Paper Tr. J. 101, T.S. 113 (1935). 

Abramg, A. and Brabender, G. Paper Tr. J. 102, T.S. 204 (1936). 

Harvey, A. Paper Tr. J. 78, T.S. 256 (1924). 

Thomas, C. and Reboulet, H. Industr. Engng Chem. Anal, ed., 

2,390(1930). 
Gregory, J. J. Text. Inst. 21, T. 66 (1930). 
TrUlat, J. and Matricon, M. J. chim.Phys. 32, 101 (1935). 
Edwards, R. J. Soc. Leath. Tr. Chem. 16, 439 (1932). 
Sale, P. and Hedrick, A. Bur. Stand. Tech. Paper, 18, 540 (1924). 
Barr, G. Second Rep. Fabrics Co-ord. Res. Committee, D.S.I. R. 

Gt Brit. p. 113 (1930). 
Sager, T. P. Bur. Stand. J. Res. J., Wash., 13, 879 (1934). 
Bradley, H., McKay, A. and Worswick, B. J. Soc. Leath. Tr. 

Chem. 13, 10, 87 (1929). 
Masson, O. and Richards, E. Proc. Roy. Soc. 78 A, 412 (1907). 
Hedges, J. Trans. Faraday Soc. 22, 178 (1926). 
Tech. Pap. For. Prod. Res., Lond., 1 and 2. 
Sheppard, S. and Newsome, P. J. phys. Chem. 34, i, 1160 (1930). 
Andrews, D. and Johnston, J. J. Amer. Chem. Soc. 46, 640 

(1924). 
Leopold, H. and Johnston, J. J. phys. Chem. 32, 876 (1928). 
Kirchof, F. Kolloidchem. Beih. 6, 1 (1914). 
Lundal, A. E. Ann. Phys., Lpz., 66, 741 (1898). 
Ostwald, W. Orundiss der Kolloidchonie, p. 370. 
Barrer, R. M. Trans. Faraday Soc. 35, 628 (1939). 

Trans. Faraday Soc. 35, 644 (1939). 

Daynes, H. India Rubb. J. 84, 376 (1932). 

Fry, J. India Rubb. J. 73, 513 (1927). 

Cooper and Scott. Lab. Circ. Res. Ass. Brit. Rubber Manuf. 

no. 65 (1930). 
Daynes, H. Rubber Tech. Conf., London, May 23-5, 1938. 
Dewar, J. Proc. Roy. Instn, 21, 81S {1914^16). 
Payne, H. and Gardner, W. Industr. Engng Chem. 29, 893 (1937). 
Kahlenberg, L. J. phys. Chem. 10, 141 (1906). 
Sheppard, S. Trans. Faraday Soc. 29, 77 (1933). 
Bancroft, W. and Barnett, C. J. phys. Chem. 34, 449, 753, 1217, 

2433 (1930). 
Meyer, K. and Mark, H. Ber. dtsch.-chem. Qes. 61, 593 (1928). 
Alexejev, A. and Matalski, V. J. chim. Phys. 24, 737 (1927). 
Kirchof, F. Kolloidzschr. 35, 367 (1924). 
Stamberger, P. Kolloidzschr. 45, 239 (1928). 
AstbTory, W. and Woods, H. J. Nature, Lond., 126, 913 (1930); 

127, 663 (1931). 
Bernal, J., Fankuchen, I. and Perutz, M. Nature, Lond., 141, 

528 (1938); 
WUson, R. and Fuwa, T. Industr. Engng Chem. 14, 915 (1922). 



AUTHOR INDEX 



Abrams, A. and Chilson, W., 432, 
436, 440, 441, 442 

— and Brabender, G., 440, 441, 442 
Ackerel, T., 57, 58 

Adzumi, H., 60, 65, 66 

Alexejev, A. and Matalski, V., 450 

Alexejew, D. and Polukarew, O., 145, 

204 
Alty, T., 117, 142, 375 

— and Clark, A., 369 
Andrade, E., 316, 340, 342, 346 

— and Martindale, J., 316 
Andrews, D. and Johnston, J., 444 
Andrews, M., 229, 329, 333 

— and Dusbman, S., 329 
Appleyard, E., 314, 341, 343 

— and LoveU, A., 341, 343, 347 
Arkel, A, van, 241 

Arnold, J. and M'William, A., 239 
Arrbenius, S., 240 

Astbury, W. and Woods, H., 388, 451 
Aten, A. and Zieren, M., 145, 203 

Baedeker, K., 263 

Bancroft, W. and Barnett, C, 450 

Bangbam, D. and Fakboury, N., 377 

Bansen, H., 71 

Barnes, C, 25 

Barr, G., 384, 403, 442 

Barrer, R. M., 41, 87, 93, 105, 106, 
117, 121, 122, 125, 126, 127, 129, 
130, 131, 137, 162, 168, 174, 177, 
178, 179, 180, 181, 182, 184, 188, 
194, 196, 199, 201, 203, 217, 218, 
221, 231, 233, 318, 333, 334, 346, 
379, 393, 394, 396, 401, 403, 405, 
407, 412, 413, 415, 416, 417, 419, 
425, 445 

— and Rideal, E. K., 87 
Bartell, F., 73 

— and Carpenter, D. C, 73 

— and MiUer, F., 73 

— and Osterbof, H., 73 
Barwich, H., 79 

Baukloh, W. andGutbman, H., 196, 234 

— and Kayser, H., 192, 193, 196, 198 
Baumbacb, H. v. and Wagner, C, 212, 

249, 267, 312 
Becker, J. A., 44, 349, 351, 360, 361, 368 
Beebe, R., Low, G., Wildner, E. and 

Goldwasser, S., 230 



Beetz, W., 240 

Bekkedabl, N., 417 

BeU, R. P., 419 . 

BeUati and Lussana, 144, 145 

Benton, A. F.', 384 

— and Elgin, J., 230 

— and Wbite, T., 230 
Beran, O. and Quittner, F., 325 
Berg, W., 338 

Bergmann, M., 411, 438 
-^ and Ludewig, S., 409, 411 
Bernal, J., 292 

— Fankuchen, I. and Perutz, M., 
388, 451 

Berthelot, M., 117 

Birdseye, C, 440, 441, 442 

Blasins, H., 57, 58 

Blytbswood, Lord, and Allen, H. S., 88 

Bodenstein, M., 144, 201 

— and Kranendieck, F., 117 
Boer, J. H. de, 320, 389 

— and Fast, J., 157, 158, 212, 400, 
403, 419 

Boggs, C. and Blake, J., 430 
Boltzmann, L., 48 
Borelius, G., 291 

— and Lindblom, S., 161, 162, 168, 
171, 176, 178, 201, 202 

Bose, E., 240 

Boswortb, R. C, 44, 245, 341, 351, 

354, 355, 364, 366, 371, 373, 374, 376 
Braaten, E. 0. and Clark, G., 120, 126, 

129, 168 
Bradley, H., McKay, A. and Worswick, 

B., 443 
Bradley, R. S., 115, 299, 305, 422 
Bragg, W., Sykes, C. and Bradley, A., 

291 
Bramley, A., 164, 208, 224, 225, 241, 

245, 275 

— and AUen, K., 226 

— and Beeby, G., 208, 224, 225, 241, 
245, 275 

— and Heywood, F., 241, 245, 275 

— Heywood, F., Cooper, A. and 
Watts, J., 208, 224, 225, 241, 245, 
275 

— and Jinkings, A., 208, 224, 225, 
241, 245, 275 

— and Lawton, G., 208, 224, 241, 
245, 275 



AUTHOR INDEX 



455 



Bramley, A. and Lord, H., 208, 209, 
224, 241, 245, 275 

— and Turner, G., 208, 224, 241, 245, 
275 

Brattain, W. and Becker, J. A., 341, 

349, 351, 361, 362, 363, 368 
Braunbek, W., 297, 305 
Braune, H., 269, 272, 275, 285, 301 

— and Kahn, 0., 274 
Bremond, P., 72 

Brick, M. and Philips, A., 241, 275 

Bridgman, P., 315 

Brillouin, M., 55, 56 

Brower, T., Larsen, B. and Schenk, 

W., 241 
Briick, L., 340 
Bruni, G. and Meneghini, D., 241 

— and Scarpa, G., 266 
Bryce, G., 230 

Buckingham, E., 53, 58, 66, 384, 406 
Buerger, M. J., 318 
Bugahow, W. and Rybalko, F., 278, 
327, 331, 333 

— and Breschnewa, N., 278 
Burmeister, W. and Schloetter, M., 227 
Burton, E., Braaten, E. 0. and 

Wilhelm, J. 0., 121, 129, 334 

CaiUetet, L., 144, 145 

Capron, P., Delfosse, J., Hemptinne, 

IVL de and Taylor, H. S., 79, 80 
Carslaw, H., 4, 10, 15, 20, 35, 36, 38, 40 
Carson, F., 382, 384, 408, 409, 413, 430, 

438 
Cernuschi, F., 375 
Chalmers, J., Taliaferro, D. and 

Rawlins, E., 74 
Chapman, S., 80 . 

Charch, W. and Scroggie, A. G., 432, 

436, 440, 442 
Chariton, J., Semenoff, N. and 

Schahiikow, A., 378 
Charpy, G. and Bonnerot, S., 144 
Chilton, J. and Colbourn, A., 74 
Cichocki, J., 245, 297, 305 
Clausing, P., 53, 64, 82, 85, 86, 406 
Clews, C. B. and Schosberger, F., 389, 

390 
Clews, F. and Green, A., 70, 71 
Clusius, K. and Dickel, G., 80, 82 
Cockcroft, J., 341, 342 
Coehn, A. and Jiirgens, H., 212 

— and Specht, W., 212, 220, 221, 268 

— and Sperhng, K., 213, 221, 268 
Cooper and Scott, 447 

Cremer, E. and Polanyi, M., 188 



Damkohler, G., 87 
Darwin, C. G., 318 
Davnes, H., 217, 383, 411, 412, 413, 

417, 431, 445, 446, 447 
Deming, H. and Hendricks, B., 168, 

191, 334 
Deubner, A., 340 
Deville, H. and Troost, L., 144 
Devonshire, A., 378 
Dewar, J., 383, 402, 403, 405, 415, 447 
Diergarten, H., 239 
Diet], A., 88 
Ditchburn, R., 342 
Dixit, K. R., 346 
Doehlemann, E., 182 
Donnan, F., 65 
Dorn, J. and Harder, 0., 297 
Doughty, R., Seborg, C. and Baird, P., 

384 
Dube, G., 375 
Duhm, B., 220 
Dunn, J., 241, 329, 332, 333 
Dunwald, H. and Wagner, C, 212, 

250, 252, 267, 312 
Durau, F. and Schratz, V., 115 
Dushman, S., Dennison, D. and 

Reynolds, N., 275, 329, 333 

Eason, A. B., 57, 58 
Edwards, C, 162, 198, 202, 203, 222 
Edwards, J. and Pickering, S., 384, 
393, 402, 403, 404, 405, 417, 440 

— and Wray, R., 432, 433, 434, 436, 441 
Edwards, R., 409, 411, 442 
Eilender, W. and Meyer, 0., 225 
Einstein, A., 268 

Elam, C, 276 

Eltzin, I. and Jewlew, A., 228 

Emanueli, L., 384 

Emeleus, H. and Anderson, J., 147, 

148, 149 
Engelhardt, G. and Wagner, C, 182 
Estermann, J., 341 
Euringer, G., 214, 222 
EweU, R. H., 421 
Eyring, H., 274, 302, 305, 422, 423 

— and Sherman, J., 188 

— and Wynne-Jones, W., 274, 303, 305 

Fabel, K., 440, 442 

Fajans, K., 295 

Fancher, G. and Lewis, J., 73, 74, 76, 77 

Faraday, M., 240 

— and Stodart, 239 
Farkas, A., 78, 93, 185, 186 

— and Farkas, L., 184 



456 



AUTHOR INDEX 



Farnsworth, H. E., 340 
Feitknecht, W., 329, 332 
Finch, G. and QuarreU, A, G., 340 
Fonda, G., Young, A. and Walker, A., 

275, 327, 328, 333 
Foussereau, 257 
Frank, L., 341, 354, 358, 367 
Franzini, T., 213 
Frazer, J. and Heard, L., 231 
Frenkel, J., 85, 248, 289, 293, 294, 

305, 311 
Frey-Wyssling, A., 389 
Frisch, R. and Stern, O., 378 
Fritsch, C, 323 
Fry, J., 447 

Gaede, W., 53, 55, 64 

GaUagher, F., 384 

Gehrts, A., 328, 333 

Gen, M., Zelmanov, I. and Schalnikow, 

A., 340 
Giess, W. and Liempt, J. van, 333 
Giolotti, F. and Tavanti, G., 239 
Goethals, C, 322, 333 
Goetz, A., 317 
Graetz, L., 240 

Graham, T., 144, 334, 391, 402, 405 
Gray, A., Mathews, G. and MacRobert, 

T., 36 
Green, H. and Ampt, G., 76, 78 
Gregory, J., 442 
Griffith, R. and Hill, S., 231 
Grimshaw, L., 239 
Grinten, W., 80 

Groh, J. and Hevesy, 6. v., 241 
Groth, W. and Harteck, P., 82 
Grube, G. and Jedele, A., 245, 246 
Guillet, L. and Roux, A., 227 
Guth, E. and Rogowin, S., 389 
Gyulai, Z., 262, 274, 323, 326, 333 

— and Hartley, D., 325 

Haber, F. and Tolloczko, St., 266 
Hagedorn, M. and Moeller, P., 388 
Hagenbach, E., 56 
Hagg, G., 145, 157, 158, 210 

— and Kindstrom, A. C 250 

— and Sucksdorff J., 250 
Ham, W., 161, 168, 173, 178 

— and Rast, W., 198 

— and Sauter, J. D., 170, 193, 197, 234 
Hammett, P. and Brunauer, S., 230 
Hanawalt, J. D., 210 

Harkness, R. and Emmett, P., 230 
Harmsen, H., 78, 79 

— Hertz, G. and Schutze, W., 78, 79 



Harvey, A., 440 

Hass, G., 340 

Hedges, J., 443 

Hemptinne, M. de and Capron, P., 79, 

80 
Hendricks, B. and Ralston, R., 334 
Herbert, J., 115 
Herbst, H., 88 - 
Hertz, G., 78, 79 

Herzfeld, K. and Smallwood, M., 54 ■ 
Hessenbruch, W., 227 
Hevesy, G. v., 241, 248, 269, 272, 274, 

275, 287, 289, 292, 300, 322, 327, 333 

— and Obrutcheva, A., 241 

— and Paneth, F., 241, 242 

— and Seiiih, W., 244, 258, 269, 274, 
275, 276, 277 

— Seith, W. and Keil, A., 244, 258, 
269, 274, 275, 277, 328 

Hey, M., 87, 103, 106, 108 

Hicks, L., 246 

Hilsoh, R., 108, 112, 250 

— and Pohl. R., 250, 316, 319, 320, 
321 

Hollings, H., Griffith, R. and Bruce, 

R., 231 
Holt, A., 170 
Horiuti, J., 417 
Hiifner, G., 391, 414 
Hunt, J. and Lansing, D., 432, 441 
Hyden, W., 440, 441, 443 

lijima, S., 230 

IngersoU, L. and Zobel, 0., 4, 38 

Ives, H., 353 

— and Olpin, A., 353 

Jacquerod, A. and Perrot, F., 117 

Jeffreys, H., 356 

Jette, G. and Foote, F., 250 

Joffe, A., 277, 315, 326 

Johnson, F. and Larose, P., 164, 168 

Johnson, J. and Burt, R., 126, 137 

Jost, W., 10, 12, 248, 254, 256, 263, 

268, 274. 275, 287, 289, 293, 294, 

295, 305, 311 

— and Linke, R., 268 

— and Nehlep, G., 254, 256, 287 

— and Widmann, A., 184, 221, 233 
Jouan, R., 168, 184 

Kahlenberg, L., 447 
Kalberer, W. and Mark, H., 85 
Kanata, K., 391, 402 
Kanz, A., 70, 71 
Kawalki, W., 14 



ATJTHOR INDEX 



457 



Kayser, H., 402 

Ketelaar, J., 249, 263 

Ketzer, R., 323 

Kirchof, F., 445, 450 

Kirschfeld, L. and Sieverts, A., 159 

Kirschner, F., 340 

Klaiber, F., 263 

Kline, G., 430, 436, 437, 438, 441, 443 

Klose, W., 53, 55, 64 

Knauer, F. and Stern, 0., 342 

Knudsen, M., 53, 54, 55, 63, 64, 377, 

406 
Koch, E. and Wagner, C, 256 
Kohbausch, W., 240 
Kohlschutter, H., 231 
Kohman, G., 415 
KoUer, L., 341, 368 
Konigsberger, J., 258 
Korsching, H. and Wirtz, G., 80, 82 
Kraft, H., 116 
Kramer, J., 344 

Kriiger, F. and Gehm, G., 145, 210, 241 
Kubelka, V., 431 

Lacher, J., 150, 152, 154, 189, 374 

Lane, C. T., 340 

Lange, H., 340 

Langmuir, I., 245, 275, 328, 333, 337, 

341, 348, 349, 350, 353, 361, 363, 

370, 371, 372. 375 

— and Dushman, S., 298, 305 

— and Kingdon, H., 349, 350, 372 

— and Taylor, J. B., 44, 352, 353, 
360, 375 

— and Villars, D., 350 
Lannung, A., 417 

Le Blanc, M., 323 
Lehfeldt, W., 264, 265, 324, 333 
Lennard- Jones, J. E., 117, 142, 234, 
346, 375, 379 

— and Dent, B., 314 

— and Goodwin, E., 375 

— and Strachan, C, 375 
Leopold, H. and Johnston, J., 443, 444 
Levey, H., 438, 440, 441, 442 
Lewkonja, G. and Baukloh, W., 196, 

234 
Leypunsky, 0., 230 
Liempt, J. van, 215, 229, 275, 285, 

301, 305, 330, 333 
Liepus, T., 340, 342 
Linde, J. and BoreHus, G., 145, 210, 

241 
Lombard, V., 164j 168 

— and Eiohner, C, 169, 170, 175, 
182, 196 



Lombard, V., Eichner, C. and Albert, 
M., 164, 168, 191, 194, 196 

LoveU, A., 314, 341, 343 

Lowry, H. and Kohman, G., 431, 432, 
433, 435 

Lundal, A. E., 445 

Magnus, A. and Sartori, G., 230 

Manegold, E., 60, 78 

March, H. and Weaver, W., 22, 24, 39 

Mark, H. and Meyer, K., 389, 390 

Marsh, M., 384 

Martin, S. and Patrick, J., 393 

Martley, J., 436 

Masing, G., 239 

— and Overlach, H., 239 
Masson, 0. and Richards, E., 443 
Matano, C, 47, 49, 275, 279, 280 
Mathewson, C, Spire, E. and MiUigan, 

W., 157 
Mathieu, M., 388 

Maxted, E. B. and co-workers, 377 
Mayer, E., 126 
McBain, J. W., 145, 161, 313, 334, 388, 

389, 431 
McKay, H., 244, 275 
Mehl, R., 239, 275, 280, 283, 285, 287, 

328, 330, 333 
MelviUe, H., 188 

— and Rideal, E. K., 168, 179, 186, 
188 

Merica, P. and Waltenburg, R., 157 
Meyer, K., 389 
- — and Mark, H., 450 
Meyer, L., 210 
Meyer, 0., 406 
Moelwyn-Hughes, E. A., 427 
MoU, F., 338 

MoUwo, E., 108, 110, 111, 250 
Morosov, N. M., 230 
Morris, T., 200 
Mott, N. F., 320 
Muskat, M. and Botset, H., 76 

Nagelschmidt, G., 93, 98 

Narayanamurti, 0., 78 

Natta, G., 340 

Nehlep, G., Jost, W. and Linke, R., 

268 ^ 
Nernst, W., 240, 268 

— and Leasing, A., 144, 145 
Norton, A. and MarshaU, F., 227, 229 

Orowan, E., 313, 317 
Orr, W. J. C, 346, 379 

— and Butler, J., 361 



458 



AUTHOR INDEX , 



Ostwald, W., 445 

Owen, E. A. and Jones, J., 210 

Pace, J. and Taylor, H. S., 231 
Paneth, F. and Peters, K., 122 
Paschke, M. and Hauttmann, A., 226, 

275 
Payne, H. and Gardner, W., 448 
Phipps, T., Lansing, W. and Cooke, 

T., 258 
Pming, N. and Bedworth, R., 329, 332 
Piutti, A. and Boggiolera, E., 126 
Pohl, R., 316, 319, 320, 321 
Polanyi, M. and Wigner, E., 298, 427 
Post, C. and Ham, W., 161, 168, 169, 

173, 191 
Poulter, T. and Uffelman, L., 199, 315, 

333 

— and Wilson, R., 314 
Preston, E., 69, 71 
Prins, I., 318 

Rahlfs, P., 249 
Raisch, E., 78 

— and Steger, H., 78 ' • - 
Ramsay, W., 144 

— and CoUie, N., 65 

Rasch, E. and Hinrichsen, F., 258 

Rawdon, H., 239 

Rayleigh, Lord, 116, 119, 132, 384 

Reis, A., 295 

Renninger, M., 247, 316, 317, 318 

Reychler, A., 391, 414 

Reynolds, 0., 57, 144, 240 

Rhines, F. and Mathewson, C, 157 

— and Mehl, R., 275, 280, 282, 283 
Richardson, 0., 169 

— Nicol, J. and ParneU, T., 144, 164, 
168 

— and Richardson, R. C, 117 
Richter, M., 338 

Rideal, E. K., 374 
Roberts, J. K., 350 
Roberts-Austen, W., 239 
Roeser, W., 130, 333 
Rogener, H., 108, 250 
Rontgen, P. and Braun, H., 158 

— and Moller, H., 227 
Runge, B., 209, 224, 226 
Ryder, 168 

Sager, T. P., 393, 396, 398, 399, 438, 

443, 445 
Sakharova, M., 333 
Sale, P. and Hedrick, A., 442 
Sameshima, J., 66 



Sauter, E., 390 

Scherr, R., 81 

Schiefer, H. and Best, A., 384 

Schlichter, C, 73 

Schmidt, G., 170 

Schottky, W., 248, 249 

Schroeder, P. N., 438 

Schumacher, E. and Ferguson, L., 

384, 432, 436, 440 
Schwartz, K., 369 
Seelen, D. v., 297, 322, 325 
Seith, W., 245, 250, 259, 260, 261, 264, 

266, 267, 275, 277, 278, 324, 327, 333 

— and Etzold, H., 268, 275 

— Hofer, E. and Etzold, H., 275 

— and Keil, A., 275, 328 

— and Kubaschewski, 0., 268 

— and Peretti, E., 241, 258, 259, 275, 
287 

Semenoff, N., 377 

Sen, B., 288 

Seybold, A. and Mathewson, C, 157 

Shakespear, G., 403, 405 

— Daynes, H. and Lambourn, 403 
Sheppard, S., 450 

— and Newsome, P. T., 388, 444, 445 
Shishacow, N. A., 93, 125 

Siefriz, W., 389 
Sie verts. A., 159 

— and Bruning, K., 157, 159, 212 

— and Danz, W., 149, 189 

— and Hagen, H., 159, 212 

— and Hagenacker, J., 155 

— and Krumbharr, W., 158 

— and Zapf, G., 149, 189, 190 

— Zapf, G. and Moritz, H., 158, 189 
Simons, J. H., 155 

Smekal, A., 258, 263, 273, 274, 275, 
295, 313, 316, 321, 323, 326, 333 

Smith, C, 276 

Smith, D. and Derge, G., 200, 318 

SmitheUs, C, 145, 151, 157, 158, 159, 
161, 162, 191, 223 

— and Fowler, R. H., 151, 152, 154 

— and Ransley, C. E., 122, 164, 168, 
169, 171, 175, 176, 179, 195, 201, 
227, 228, 229 

Smoluchowski, M. v., 53, 55, 56 

Spencer, L., 168 

Spiers, F., 279, 369 

Spring, W., 239 

Staedel, W., 432, 436, 440, 441, 442 

Stamberger, P., 450 

Staudinger, H., 390 

— Heuer, W. and Husemann, E., 390, 
391 



AUTHOR INDEX 



459 



Staudinger, H. and Husemann, E., 390 
Steacie, E. and Johnson, F., 155 

— and Stovel, H., 231 
Stefan, J., 14 

Steigman, J., Shockley, W. and Nix, 

F., 244, 275 
Stepanow, A., 326, 333 
StiUweU, S., 432, 436 
Stodola, A. and Lowenstein, L., 57, 58 
Strock, L., 249, 263 
Swamy, R. S., 340 , 
Swan, E. and Urquhart, A., 88 
Sykes, C. and Evans, H., 291 

Takei, T. and Murakami, T., 159 
Tammann, G., 344 

— and Bochow, K., 414 

— and Schneider, J., 220, 221 

— and Schonert, K., 224, 226 

— and Veszi, G., 322, 325 
Tanaka, S. and Matano, C, 241 
Taylor, H. S., 232, 377 

— and Diamond, H., 231 

Taylor, J. B. and Langmuir, I., 346, 

360, 361, 371, 375 
Taylor, N. W. and Rast, W., 130 
Taylor, R., Herrmann, D. and Kemp, 

A., 405, 432, 433, 434, 436, 437, 440, 

441 
Taylor, W. H., 92, 93 
Thomas, C. and Reboulet, H., 441 
Thomson, G. P., Stuart, N. and 

Murison, C. A., 340 
Thomson, S. P., 240 
Thomson, W., 406 

Tiselius, A., 87, 96, 97, 101, 106, 276, 419 
Tompkins, F. C, 115 
Toole, F. and Johnson, F., 159 
Topping, J., 373 

Tressler, D. and Evers, C, 440, 441 
Trillat, J., 388 

— and Matricon, M., 442 
Troost, L., 144 

T'sai, L. S. and Hogness, T:, 121, 126, 

129, 137 
Tubandt, C, 245, 250, 260, 261, 266 

— Eggert, S. and Schibbe, G., 245, 
250, 261, 263, 266 

— and Reinhold, H., 263, 324 

— Reinhold, H. and Jost, W., 269, 
271, 272, 274 

and Liebold, G., 245, 250, 261, 

266 

Urry, W., 116, 118, 120, 121, 126, 129, 
137, 142, 333, 334 



Venable, C. and Fuwa, T., 414 
Villachon, A. and Chaudron, G., 227 
Villard, P., 117, 122 
Volmer, M., 338 

— and Adhikari, G., 338 

— and Esterman, I., 337 
Voorhis, C. C. v., 126, 129 

Wagner, C, 182, 212, 248, 250, 252, 
253, 256, 263, 267, ^69, 271, 311, 
312 

— and Schottky, W., 248, 267, 311 
Waldmann, L., 80 

Wang, J. S., 179, 180 
Warburg, E., 53, 61 

— and Tegetmeier, F., 277 
Ward, A., 117, 230, 233, 375 
Watson, G., 36 

Watson, W., 117 
Wells, C, 276 

— and Mehl, R., 225, 226 
Wenderowitsch, A. and Drisina, R., 

325 
Wheeler, C, 300, 305 
Wheeler, T. S., 422 
Wicke, E., 87 
Wiedmann, E., 240 
Wiener, 0., 48 
Wilde, H. and Moore, T., 74 
WiMns, F. and Rideal, E. K., 329, 

332 
Williams, G. A. and Ferguson, J., 120, 

121, 126, 139, 140 
Williams, J. and Cady, L., 44 
Wilson, J. and Lines, G., 411, 432, 

437, 438, 442, 445 
Wilson, R. and Fuwa, T., 451 
Wing, H., 432, 438, 441 
Winkelmann, A., 144, 164, 170 
Wirtz, K., 80 
Wood, R. W., 377 
Wooldridge, D. and Smythe, W., 78 
Wosnessenski, S. and Dubnikow, 

L. M., 432, 441 
Wray, R. and Vorst, A. van, 432, 

441 
Wroblewski, S., 391, 413 
Wiistner, H., 117, 118, 120, 121, 126, 

139, 140, 175 
Wyckoff, R., Botset, H., Muskat, M. 

and Reed, D., 77 
Wynne- Jones, W. and Eyring, H., 423 

Zahn, H. and Kramer, J., 344 
Zwicky, F., 317 
Zwikker, C, 330, 333 



SUBJECT INDEX 



Absorption, see Sorption, Solution 

— of alkali metals by alkali balides, 
109-10, 111-15, 312, 316 

— of alkali metals in tungsten, 354-5 

— of ammonia, sulphur dioxide, and 
hydrochloric acid by alkali halides, 
115 

— and diffusion of gases in finely 
divided metals, 230-4 

— of halogens by alkali halides, 109- 
10, 111-14, 312 

— of hydrogen by alkali halides, 
109-10, 111-14, 312 

— of thaUium by alkali haUdes, 113 
Activated adsorption, 232-3 
Activated diffusion, 71, 73, 106, 121-3, 

132-3, 169, 207-8, 222-3, 382, 403, 

409, 416, 436, 445 
Activation energy for diffusion, 103, 

141, 221, 222, 223, 225, 274, 275, 420, 

421, 424, 425 
for structure-sensitive diffusion, 

327-9 
for surface diffusion, 346, 360-1, 

363, 364-6, 366-7, 370 
Activation energy, radius and polari- 

sability in alkali haKdes, 264-5 
Adatoms, 347, 350 
Ammoniate formation in natroHte, 

105-6 
Analogy between activated diffusion 

and chemical reaction, 427 
Anionic conductors, 267 

Base exchange, 240 
Berthollide compounds, 249-50. See 
also Interstitial compounds 

Calculation of conductivity, 293-7 

— of diffusion constant, 293-6, 298- 
303. 8ee also Models of conduction 
and diffusion processes, etc. 

— of energy periodicity, of crystal 
surfaces, 379 

— of surface diffusion constant, 375- 
,6 

Capillary condensation, 450 

Chemical composition of gases de- 

sorbed from metals, 227 
of glasses in relation to per- 
meability, 129-30, 137-9 



Classification of some diffusion sys- 
tems, 123 

Colour centres in alkali halides, 109- 
10, 319-21 

Condensation and aggregation of atom 
beams on surfaces, 339-47 

Conductivity of salts, 240, 247-50, 
251-3, 256, 257-65, 265-7, 268-72, 
272-3, 274, 276-8, 283, 289, 294-7, 
305. See also Structure-sensitive 
conductivity 

— of solid solutions of salts, 256, 
261-3, 323-5 

— and space-charge redistribution, 
325 

— of thin films, 343-6 

Criteria of stability in films, 339, 340-1 

Critical nominal thickness of thin 
films, 344-6 

Critical streaming density for aggrega- 
tion of atom beams on solids, 377-8 

Cross -linking and permeability of 
polymers, 393 

— and solubility of polymers, 391 

— and sorption by poljrmers, 450-1 
Crystal growth, 337-9, 339-41, 341-3, 

375, 377-8. ' 

Degassing of metals, 31, 226-30 

Degrees of freedom in diffusion pro- 
cesses, 299-300, 422-3, 425 

Dependance of activation energy for 
diffusion upon concentration, 282-3, 
367, 372, 374 

— ■ of tensile strength upon sorption, 
etching and fibre diameter, 315-16 

Derivation of diffusion constants from 
solutions of Fick's laws, 47-9, 50, 
97-8, 112-14, 214-19, 351-2, 355, 
356-9, 412-13. See also Measurement 
of diffusion constants 

Differential forms of the diffusion 
equation, 1-5 

Diffusion anistropy, 98, 102-3, 193, 
276-9, 327 

Diffusion equation when the diffusion 
constant depends upon concentra- 
tion, 47-9, 101, 443, 446 

— coupled with interface reactions, 
37-43, 174-5, 176-8, 179-83, 191 - 

— in cylinders, 31-7 



SUBJECT INDEX 



461 



Diffusion between finite layers, 14 
— - in finite solids, 13 

— of gases in glasses, 139-41 

— of gases in metals, 207-25 

— of gases and alkali metals in 
alkali halides, 108-16 

— of gases in organic membranes, 
411-27 

— and grain size, 328-30 

— of instantaneous plane source of 
solute, 44 

— of instantaneous point source of 
solute, 46 

— of instantaneous spherical surface 
source of solute, 47 

— of interstitial ions, 292, 293-6 

— of nascent hydrogen through 
metals, 144^5, 200^ 

— of non-metals in metals, 224-5 

— through a permeable membrane 
separating two stirred fluids, 24-8 

— by place exchange, 29, 96, 98, 
292-3, 300, 303 

— of salts through metal foils, 245 

— in semi-infinite solids, 11—12 

— across sharp boundaries, 8-9 

— in spheres, 28-31 

— by spontaneous gliding, 292-3 

— by spreading, 96. See also ZeoUtic 
diffusion 

— to and from a stirred fluid in con- 
tact with a quiescent medium, 21-4 

— with surface concentration a func- 
tion of time, 19-20, 35 

■ — in two different media, 10-11 

— of vacant lattice sites, 294, 297 

— through the wall of a hollow 
cylinder, 35-7 

— in wires when the surface concen- 
tration is fixed, 32-5 

Dipole adsorption, 449-50 
Displacement of oil from oil-bearing 
sand, 73 

Effect of diffusion on mechanical 
strength, 204, 334 

— of finishes and fillers on the 
permeability of organic membranes, 
411, 438 

— of gas pressure on the conductivity 
of crystals, 251-3, 267 

— of mechanical deformation on 
conductivity of crystals, 325-6 

— of soluble salts on the permeability 
to water of organic membranes, 436, 
438 



Effect of spreading pressure on diffu- 
sion constants, 363, 372-4 

— of temperature upon conductivity, 
257-65, 321-5 

— oftemperature upon diffusion, 115, 
133, 141, 207, 221-6, 258, 278, 288, 
328, 444-5. See also Activation 
energy for diffusion. Influence of 
temperature on permeability 

— of temperature on the solubility of 
gases in soMs, 111, 140, 150-2, 155- 
7, 414, 418 

— of temperature on time lag in 
estabhshing steady states of flow, 
203, 218, 413, 416 

Effusion, 53-4, 61, 65-6, 78, 406 
Electrolysis of sodium ions through 

glass, 96, 240 
Electronic conductors, 267 
Energy of disorder in crystals, 254-7 
Entropy of activation for diffusion, 

274, 276, 287, 423-6 
■ — of solution of gases in polymers, 

417, 419 
Equilibrium disorder ia crystals, 247- 

50, 254-7, 293, 311 
Equilibrium in a monolayer under a 

temperature gradient, 357-8, 364, 

366 
Evidence of smrface mobility from 

properties of unstable films, 339-47 

Factors governing relative permeabi- 
lity of metals to hydrogen isotopes, 
187-8 

— influencing diffusion constants, 
279-83, 283-91, 304 

Farbzentren, see Colour centres 

i^- centres, see Colour centres 

J"-centres, 320 

Fick's laws, see Differential forms of 

the diffusion equation 
Flow of fluids through capillary 

systems, 53-78, 82-8, 406-11 

— of gases in consolidated and un- 
consoMdated sands, 73-8 

through miscellaneous solids, 

78 

through porous plates, 65-9 

through refractories, 69-73 

Fractionation of gases by activated 

diffusion, 120-1, 393, 396 
Free energy of solution of gases in 

organic polymers, 417 
Frenkel disorder in crystals, 254, 257, 
293-5, 300, 303 



462 



SUBJECT INDEX 



Glass electrode, 240 

Grain-boundary diffusion, 127-30, 
131, 141, 197-200, 245, 311-12, 
327-34, 337, 389. See also Structure- 
sensitive diffusion 

Heats of adsorption by non-stationary 
streaming, 85-7 

— of condensation of metals on 
solids, 341 

— of solution of alkali metals in 
alkali haUdes, 110-11 

of gases in alkali halides, 110- 

11 

of gases in metals, 153 

of gases in polymers, 418 

— of sorption of metal monolayers 
on tungsten, 360-1, 371 

Hemicolloids, 300 

Hertz method of fractionating gas 
mixtures, 78^80 

Hysteresis effects in gas -metal sys- 
tems, 150, 192, 194-5, 199 

Identity of current carrying ions, 265- 
8 

Impedance, 405, 411 

Influence of acid strength, upon per- 
meation rate of nascent hydrogen 
through metals, 203 

— of concentration upon diffusion 
constants, 47, 101-2, 225-6, 279- 
83, 371-4, 443, 445-7 

— of current density upon permea- 
tion rate of nascent hydrogen 
through metals, 182-3, 201 

— of hydration upon cellulose struc- 
tiure, 450 

— of hydroxyl groups upon per- 
meability of organic membranes, 
399^00 

— of impm'ity upon diffusion con- 
stants, 234-6 

— of mechanical working upon diffu- 
sion constants, 328, 331-2 

— of phase changes upon permeabi- 
lity, 191-2 

— of pressure upon permeation 
velocity, 66-9, 74, 76, 120-1, 169- 
83, 403-4, 406, 408-9, 411, 432-4 

— of pre-treatment upon permeabi- 
Uty, 192-7 

— of temperature upon permeability, 
71-3, 120-5, 125-31, 126-9, 133-7, 
162-9, 202-3, 394-5, 397, 405-6, 
408, 435-6, 437 



Influence of thickness upon permea- 
tion rates, 120, 404-5, 408, 433 

Interdiffusion of metals, 8, 10, 31, 
44, 47, 49, 239, 241, 244-5, 245-7, 
272-6, 278-9, 279-91, 292, 298-305, 
327-34 

— of salts, 8, 240, 245, 272, 274. See 
also Conductivity of salts 

Internal surfaces, 313-19 

Interstitial compounds, 92, 96, 156-7, 
249-50, 292 

Interstitial ions, 92, 248-57, 293 

Intra -crystalline diffusion, see Grain 
boundary diffusion 

Investigations of gas flow in capillaries, 
61-5 

Irreversible permeation velocities 
through glasses, 125-31 

Irreversible sorption by organic col- 
loids, 438 

Kinetic theory of diffusion in zeolites, 

106-8 
Kinetics of sorption and desorption, 

86-8, 228-9, 230^, 443-5, 445-7 
Knudsen flow, see Molecular streaming 

Lag in establishing steady states of- 
flow, 18-19, 31, 37, 203, 217-19, 
221, 222-3, 412-13, 415-16 

Lateral contraction of surfaces, 314 

Lattice diffusion, 127-8, 130, 141, 
197-200, 331, 334, 389. See also 
Interdiffusion of metals, Inter- 
diffusion of salts, Volume diffusion 

Lifetime of adatoms in the activated 
state, 375 

— of atoms adsorbed on glass, 85-6 
Lineage structures, 318-19 

Linear polymers, 385-6 

Measurement of concentration gra- 
dients, 97, 111, 208-10, 241, 244, 
354-5, 359, 434-5 

— of diffusion constants, 47-9, 50, 97, 
111-14, 139-40, 208-19, 241-5, 272, 
351-9, 369-70, 411-13 

— of permeability, 117-20, 161-2, 
163, 383-5, 430 

— of sorption and desorption ve- 
locities, see Measurement of diffu- 
sion constants 

— of transport numbers in crystals, 
266, 267-8 

Mechanism of activa-tion of adsorbed 
atoms, 375-6 



SUBJECT INDEX 



463 



Mechanism of flow through metals, 
166-7, 170-1, 178-83 

— of flow through rubbers, 391, 420, 
421 

Membrane forming polymers, 382, 
385-91, 392, 396-7, 431 

Mesocolloids, 391 

Micellar structures of chain polymers, 
389-90 

Micrographic evidence of grain boun- 
daries, 314, 315, 317, 319 

Mixed conductors, 267 

Mobility of ions, 265, 268-72, 272 

Models of conduction and diffusion 
processes in crystals, 291-305 

— for diffusion in rubber, 422-7 
Molecular streaming, 53, 54-5, 60, 64, 

65, 69, 71, 82-8, 130, 131, 133, 142, 
402, 406 
Multilayer adsorption, 450 

Nature of hydrogen-metal systems, 
149-55 

— of metaUic crystals, 147-9 

— of permeation processes through 
organic solids, 448, 450-2 

Nernst lamp, 240 

Non-equilibrium disorder in crystals, 
247-8, 311-12, 313-21 

Non-stationary states of flow in 
capillary systems, 82-8 

Numerical values of conductivity con- 
stants, 264 

— -- — of diffusion constants j 98, 101, 
103, 104, 108, 113, 116, 141, 217, 
221, 222, 223, 224, 225, 229, 271, 
274-5, 278, 281, 284, 328, 329, 330, 
360, 361, 363, 365, 367, 370, 420, 445 

of permeability constants, 70, 

76-7, 116, 133-7, 138, 168, 394-5, 
396, 397, 398, 400, 409, 410, 417, 
440-2, 443, 447 

of solubility constants, 140, 

415, 417, 418 

Optical absorption of alkali halides, 
109-10, 111, 319-21 

Order-disorder transformation in me- 
tals, 290-1 

Orifice flow, 53, 58-60, 382, 406, 408-9 

Osmotic theory of diffusion in organic 
solids, 445-7 

Oxidation of metals, 332 

Periodicity curves for activated diffu- 
sion systems, 426 



Permeability constants, definitions 

and dimensions, 7, 60-1, 391-2 
Permeability of membranes in series, 

411 
Permeability spectrum, 383, 432 
Permeation rates at high pressures, 

117, 175-8, 403 
Phase boundary processes, 37-43, 177, 

179-83, 187, 218-19 
Phase changes in monolayers, 353, 

374-5 
Photochemical processes in alkaH 

halides, 108-10, 319-21 
Photoelectric emission, 347, 353-9, 

364, 366, 367, 368 
Physical properties of cellulose esters, 

388 
Platelike polymers, 385-6 
Poiseuille fiow, see Streamline flow 
Primary flaws in crystals, 313-14 
Properties of fllms on tungsten, 347- 

51 

Radioactive indicators, 241-5, 265, 
271, 272, 304, 369 

Relation between diffusion and con- 
ductivity constants, 268-72, 294-5 

— between permeability and chemi- 
cal nature of organic membranes, 
399-400 

— between types of molecular flow, 
131-3 

Relative activation energies for volume, 
grain-boundary and surface diffu- 
sion, 326, 337, 346, 363 

Relative behaviour of hydrogen iso- 
topes in solution and diffusion in 
metals, 183-91 

Relative permeabUities of membranes, 
137, 400, 402, 439, 443, 447-8, 449 

Removal of benzophenone by surface 
diffusion, 338 

Role of water in proteins, 450, 451 

Schottky disorder in crystals, 254, 257, 

294,300,303 
Selective adsorption on ionic crystals, 

379 
Self-diffusion, 244, 271, 273, 278, 290, 

300-1 
Sigmoid sorption isotherms, 443 
Slip-plane diffusion, 200 
Solution of gases in alloys, 158-61 

in glasses, 139-41 

in organic membranes, 411-19 

in metals, 145-61 



464 



SUBJECT INDEX 



Solution of gases and vapours in ionic 

crystals, 109, 110-11 
Solutions for the steady state of flow, 

5-6 
Sorption and desorption from a hollow 
cylinder, 35-7 

from a membrane, 14—17, 443-5 

from a solid cylinder, 33-5, 

214-15 

— from a spherical shell, 29 

from a solid sphere, 29 

— and diffusion in tungsten oxide,367 

— in swelling media, 93, 430, 432^, 
435, 443-5, 450-1 

Spreading pressure in films, 363, 372-4 
Streamline flow, 53, 55-6, 58, 60, 61, 

63, 64, 65, 69, 71, 73, 74, 130-1, 133, 

382, 406, 407-9 
Structure of cellulose, 388 

— of crystalline rubber, 390 

— of proteins, 388-9 

Structure -sensitive conductivity, 110, 

311-13, 321-6, 333 
Structure-sensitive diff'usion, 127-30, 

245, 311-12, 327-34, 389. See also 

Grain-boundary diffusion 
Structure-sensitive properties, 313 
Structures of some silicates and glasses, 

91-6 



Summary of reactivity of metals to 

gases, 145-7 
Surface diffusion, 3, 44, 245, 337-47, 

351-79 
Surface migration, see Surface diffusion 
Surface mobility, see Surface diffusion 
Sweep curves, 316-17, 318 

Thermionic emission, 327, 329, 347-53, 

359, 361-3, 368 
Thermo-diffusion method for frac- 
tionating gas mixtures, 80-2 
Three-dimensional polymers, 385-7 
Turbulent flow, 53, 57-8, 73, 74, 75 
Two-dimensional gas, 346, 377-9 
Types of concentration gradient, 210- 
11, 245-7, 434-5 

Unsolved diffusion problems, 50-1 

van der Waals adsorption, 230, 448, 

450 
Variation of electrical resistance with 

amount sorbed, 212, 329-30 
Volume diffusion, 311, 326, 363, 369. 

See also Lattice difiiision 

Zeolitic diffusion, 96, 223, 292, 300, 
303 



CAMBRIDGE: PRINTED BY W, LEWIS, M.A. AT THE UNIVERSITY PRESS