AMERICAN INSTITUTE OF MINING AND METALLURGICAL ENGINEERS Technical Publication No. 2168

Class C and Class E, Metals Technology. June 1947

Thermodynamic Activities and Diffusion in Metallic Solid Solutions BY C. ERNEST BIRCHENALL* AND ROBERT F. M E H L , ~ MEMBER AIME

(New York Meeting, March 1947)

APPLICATION of ditIusion laws in the for solid solutions for which reliable customary form to experimental studies diffusion work has been done. The literature in binary metallic solid solutions has provides such information only for the shown the dausion coefficient to vary interstitial solution of carbon in gamma with concentration for all systems in- iron (austenite) and for the substitutional vestigated. The factors determining this copper-zinc solutions (alpha brass). variation have not been ascertained. Activity data for carbon in austenite However, the question of whether the dif- a t 8o0°C and rooo°C are available from fusion coefficient would become a constant, the recent work of Smith6 on the equilib- independent of concentration throughout rium of hydrogen-methane and carbon a single phase if the concentration gradient monoxide-carbon dioxide mixtures with

i in the diffusion equation were replaced gamma iron. Combining these with the by an activity gradient has been an at- diffusion studies of Wells and Meh16 tractive subject for speculation. makes possible a quantitative test for

Wagner' derived an expression for the this interstitial solid solution. force acting on a diffusing particle in The vapor-pressure determinations for terms of the chemical potential. This zinc over brass by Hargreaves7 allowed was also the approach chosen by Jost2 the calculation of activities in this system in hiis monograph. From a somewhat over a wide range of temperatures and different point of view, Eyringa and his concentrations. Extensive investigation of co-workers have used activity gradients diffusion in this substitutional solid solution when dealing with diffusion in concen- has been carried out by Rhiies and Mehl.8 trated nonideal solutions. The only actual test with experimental data is that given ACTIVITIES IN BRASS^ by ark en,^ though this is more qualita- If the vapor phase of zinc is assumed tive than quantitative. Many others to behave as an ideal gas, any solid or opposed the application in any way of liquid solutions that are in equilibrium thermodynamic concepts to kinetic phe- with the same partial pressure of zinc nomena, including diffusion processes. a t the same temperature have the same

A rigorous examination of this principle thermodynamic activity of zinc. In order requires good data expressing the relation- to assign a numerical value to the activity ship between activity and concentration azn, it is necessary to choose some standard

Manuscript received at the of the reference state to which an activity value Institute Jan. 13. 1947. of unity is arbitrarily assigned. I t is

Member of Staff. Metals Research Labora- tory. Carnegie Institute of Technology, Pitts- customary to choose the liquid or solid burgh. Pennsylvania. state of the pure material. Then az, is t Director, Metals Research Laboratory. and Head, Department of Metallurgical Engi- defined numerically by neering. Carnegie Institute of Technology.

References are given at the end of the paper. azn = P5/PzD0 111 Copyrieht. 1947. by the American Institute of Mining and Metallurgical Engineers. Inc.

Printed in USA

2 THERhtODYNAhUC ACTIVITIES AND DIFFUSION IN METALLIC SOLID SOLUTIONS

where Pzn is the partial pressure of zinc over the solution under consideration and P50 is the vapor pressure of pure zinc a t the same temperature in the chosen standard state.

If the mol fraction of zinc in the solution is N 5 , the activity coefficient, y%, is defined by the equation

If this were an ideal solution of zinc, where the interactions of the zinc atoms with the other components are the same as in pure zinc, the partial pressure of zinc in equilibrium with the solution would be proportional to the mol fraction of zinc atoms in solution. In this limiting case yz. would be unity. Greater attraction of the zinc atoms for the surroundings will be characterized by a decrease in activity, or yzn less than unity. These cases are designated as negative deviations from Raoult's law. Positive deviations, y a greater than unity, correspond to increased activity or weaker attraction between zinc and its surroundings than would exist in an ideal solution.

For a binary system, in this case copper and zinc, the Gibbs equation

makes possible the calculation of the activities of one component when the activities of the other are known.

Employing these equations, the vapor- pressure data for zinc in brass given by Hargreaves, and the equations of Maierlo for the vapor pressure of pure zinc, the activities and activity coefficients for zinc and copper in brass were calculated over a wide range of temperatures and concentrations. A few additional values were obtained from the paper of Schneider and Schmid,I1 who based their calculations of activity on the data for pure zinc from the Landolt-Bornstein tables determined by Braune, Heycock, Lamplough, and

Egerton. When recalculated using Maier's values, the same results were obtained.

Table I gives the activities and activity coefficients for zinc in brass as a function of concentration and temperature referred to liquid zinc as the standard state. These data are plotted on the brass-equilibrium diagram in Figs I and 2. Table 2 contains 1 the activities and activity coefficients for copper obtained by graphical integration of the Gibbs equation.

Examination of the data reveal the fol- 1,

lowing: I. For the copper-rich end of the

diagram the activity coefficients of zinc are much less than unity, increasing toward unity with both increased tem- perature and zinc concentration. Above 80 pct Zn and 700°C the activity coeffi- cient of zinc is practically unity.

2. The activity coefficients of copper in brass are everywhere less than unity, I increasing toward unity with increasing copper concentration and temperature. Above 80 pct Zn and 700°C the activity coefficient of Cu is practically constant.

The tendency for activity coefficients to shift toward unity with increasing temperature is normally to be expected in nonideal solutions.

Dijusion ThewylZ

The customary form of Fick's first I

diffusion equation is

for the one-dimensional case; that is, flow in only the x direction. P is the quantity of material flowing through a unit area per unit time when the con-

ac centration gradient is I t is called the

permeability. Customarily, G is taken as the mass per unit volume, and the quantity of material is measured in mass. However, it will be convenient here to measure quantity as a number of atoms.

C. ERNEST BIRCHENALL AND ROBERT F. MEHL

FIG ,I-ACTMTY OF ZINC IN BRASS.

4 THERMODYNAMIC ACTMTIES AND DIFFUSION IN METALLIC SOLID SOLUTIONS

C. ERNEST BIRCHENALL AND ROBERT P. MEHL 5

As long as it is understood that we always distance dx apart and examines the rate refer to unit volume, c may be expressed of accumulation of diffusing material as an atomic fraction. If only one com- within this volume and the planes are ponent moves, the conversion from the considered to be infinite so that no end

TABLE ~--Acti¶ity of Zinc in Brass (with Respect to Liquid Zinc)

number of atoms to the mass of this effects enter, the second Fick equation number of atoms is simply multiplication may be deduced from the first.

I by the atomic mass. a p ac ac D, the diffusion coefficient, is usually P - ( P + d x ) = a dx = - D -

I ax expressed in units of square centimeters

I per second. For diffusion in most metallic ac a

+ D ~ + ~ ( D ~ ) ~ ~ ISI solid solutions it has been observed that the diffusion coefficient is not a constant

ac = & ( D :;) [61

over a range of concentration but varies : with concentration. the second equation of Fick, where t is time.

6 THERMODYNAMIC ACTIVITIES AND Dl [FFUSION I N METALLIC SOLID SOLUTIONS

TABLE 2-Activity o j Copper in Brass

Atomic Cu 1 700° 1 800" 1 830° 1 850° 1 900'

Gamma Cu

- --

Activity Cu

interested are usually

BoltzmannlS showed that all solutions satisfying these boundary conditions will contain C as a function of a single variable A, where

Transforming Eq 6 we have

x ac ---- a x - [7l

which integrates to

To solve for D(c), the experimentally

determined c-x curve is plotted for a given time and the appropriate interface* chosen. D is then evaluated as a function of con- centration by graphical integration.

If, as has been suggested, the rate of diflusion is controlled by an activity gradient rather than a concentration gradient, the following analysis, due to W. A. Johnson,16 may be applicable. He rewrites the iirst Fick equation in the form

where, by deiinition, a = yc. But

Following the same reasoning as in the previous case, we iind

* The Grube method involves integra- tion of Eq 6 assuming that D is independent of concentration, choosing an interface at which the concentration is midway between the concentrations of the two original halves of the couple, then solving for D for the experi- mental points on the curve. Since D is then found to vary with concentration, the method is undesirable.

Matano" chose an interface corresponding to the transfer of equal amounts of one com- ponent in one direction and the other com- ponent in the opposite direction. Then we have

The interface usually is determined graphically by measuring the area under the x-c curve. This makes possible graphical determination of D as a function of concentration using Eq 8.

C. ERNEST BIRCHENALL AND ROBERT F. m H L 7

Comparison of Eqs 8 and 11 reveals that

If the activity gradient is then the con- trolling factor in dsusion, Dl should be constant over the range of concentration corresponding to a single phase for the case where only one component moves. Such a system would be the interstitial solution of carbon in iron.

Glasstone, Laidler, and Eyringa em- phasize that in a system where more than one component is involved in the rate-determining process of diffusion new factors containing the activities of the other components will also appear in the

equation. I f we regard the term (7 + c$)

as a probability factor* expressing the ( fraction of a given number of atoms

that will jump under existing concentration and activity conditions, the over-all probability when two different atoms must jump together will be a product of two such probability factors, one for each component.

In a binary substitutional solid solution the diffusion coefficient represents the behavior of both components. For this reason, any solution for this case must be symmetrical with respect to the two constituents. A modified form of the

, first Fick equation satisfying these condi- tionsis

This is completely symmetrical because dcl = 4 6 2 , where x is now measured in the reverse direction.

*Since y = eAr/BT. where AF is the differ- ence in free energy for a mol of component in real solution and ideal solution, this is a reason- able idea. For a more extended discussion along this line see reference 3.

Again following the same analysis as for the derivation of Eq 8 we h d

which gives the desired result:

This, then, is the equation that should apply to substitutional solid solutions; e.g., alpha brass.

COMPARISON WITH EXPERIMENT Self-diJusion.-Since the activity of

one isotope mixed with others of the same element should be unity regardless of concentration, diffusion should obey Eq 8. This is generally accepted, and no contradictory evidence has been reported.

Interestid Dijusion; Carbon in Austenite. Diffusion of the components in an inter- stitial solution should occur reasonably independently. The smaller interstitial atoms would be expected to migrate more rapidly under the gradient because of size and because many vacant inter- stitial positions will be waiting to receive them, and because large gradients are possible. The lattice atoms, on the other hand, must exchange with other lattice atoms or diffuse into the relatively small number of vacant lattice sites. Since the solutions are generally dilute in inter- stitial component, large gradients of the lattice component are not possible. Finally, of course, they are much larger and produce greater local distortion on moving.

I t would appear that ditrusion in interstitial solid solutions should obey two independent equations, one of type 12 and one of type IS, where

for "self-diffusion" of the lattice-point to Eq 12, using activities for carbon atoms. However, because of the limits on from the data given by R. P. Smith.6 the gradients possible for the lattice com- Cdculations.-The D values for 8oo°C ponent, the latter should approach very resulted from averaging the values given

closely to the behavior described by Eq 8. Interest, therefore, centers in the diffusion of the interstitial component and its de- pendence on concentration and activity.

The variation of dausion coefficients with concentration for carbon in austenite when calculated on the basis of the Grube and Matano solutions of Eq 8 has been determined by Wells and Mehl.6 Table 3 gives these values for D as well as those for Dl obtained by correcting D according

for Wells and Mehl samples 12, 13 and 14. At IOOOOC an average curve for D versus concentration was drawn for samples 8, 10, 35 and 36 and from this D was read off a t intervals of 0.1 wt pct carbon. The other D values are those given in the text, those in parentheses are results of the application of the Matano solution, the remainder results of the Grube method.

Activity data for 800' and IWOOC were taken directly from Smith's paper. At

C. ERNEST BIRCHENALL Ah'D ROBERT E. MEHL 9

TABLE 3 . 1 cliCity-diJ~isio)~ Data jor .I I I S ~ C I I ~ ~ C

I ( IJ X 10'

7 1 d r + 2 "1 (average)

I I - -- -~ - ~ .

8oo°C Samplcs 12, 13. 14 - ~~

~ ~~p ~~ ~ -- - - ~- ~~ - -- .

aa.60 6 .80 26.27 0 .31 0.012 i 0.0125 + 0.005 23.60 28.36 I 0.38 0.013

I - -- -- - -. - -

looo°C. Samples 8, xu, 35. 36 - - - - -- -- ~- .- ~ - .

~. - ~- - - - - - ~

0.1 1 4.89 0 0 3 I 1 8 9 1 12.9 3.64 I I

5.31 f . 04 6 .11 (16 .11 (a.511 5 .63 - . 70 7 . 5 2 (20.11 (2.67) ? . s f , i o . u l

I 19.0 2.53 : 6.62 3 .68

I 1 2 5 . 0 I 2.43 ~ -- -~~ ------ ~ -

925°C Sample 19 ~ -- ~~

0 . I I 13.88 ' 0 8 1 13.88 I

0.54 ! 13-91 0 .50 14.42 0 . 7 14.53 z . oh 16.59 (1.61

0.085 1 .0 1 15.19 17.53

i I rozaC. Sample 24

. ~ - ~- - -

~ ~ppp~ ~~ ~-pp

- - - - - - -- ~

0 . 1 6.90 0 .25 I 7.15 1 7.7 I 1.08 0.54 7.30 1.86 9.72 (9.21 (0.95) 0 . 7 7 .62 1 2.42 1 1 0 . 0 ~ , (10.91 (1.08) 1.07 + 0.08

1.00 1 0 8.42 1 2.94 11.36

- --

I 148'C. Sample 25 -. --

I I I I

1245°C. Sample 27

10 THERMODYNAMIC ACTIVITIES AND DIFFUSION IN METALLIC .SOLID SOLUTIONS

1200°C the equation for the carbon monoxide-carbon dioxide-graphite equilib- rium constant (Smith's K4) given by Chipman16 was graphically corrected to Smith's values a t 800' and 1000°C. and a new constant was read off for 1200°C. Graphical interpolation gave the data necessary for the four other temperatures. These latter naturally are not as reliable either in absolute value or internal con- sistency as the first. three.

Average Dl values have been calculated for each temperature, and the mean deviations show the consistency within each set of data. I t should be emphasized that these are not possible errors in the absolute values. Examination of the deviations in diffusion and activity data suggests that the total errors may approach 10 pct, although the 'several averaging processes employed may have decreased that considerably.

A plot of Dl against the reciprocal of the absolute temperature* gives a very good straight line in Fig 3. This gives for the constants in Eq 16:

Ql = 42000 cal per gram atom Do1 = 0.434 sq cm per sec.

These may be compared with Q equal to 32,000, given by Wells and Mehl.6 They give Do = 0.12.

The more reliable data are plotted as cir- cles, the rest as crosses.

.The variation of D with temperature is as- sumed to follow an Arrhenius type equation common to activated process. D = DoeQ/Br, where Q should be the energy of activation for the process. However. D, defined by Eq 6 , is not a constant for a given temperature. It has been customary to take D values for a chosen concentration and a series of temperatures to obtain Q. This Q is of doubtful value since D should really be compared at points correspond- ing to the same activity gradient, not the same concentration. Because of this, it would be bet- ter to write D = Ae-slT, reserving Do and Q for use with Dl and Dir.

Dl and 0 1 2 . defined by Eqs 12 and 15; re- spectively, are true constants for the cases investigated here and the values of Q1 and Q12 obtained from the Arrhenius equation should be true activation energies for the respective diffusion processes. The relation between Q and Qir is given by

We have seen that Eq 12 will account quantitatively for the ditrusion of carbon . in austenite. Replacing the concentration gradient with the activity gradient makes

a constant, dependent only on tem- perature and the nature of the com- ponents of the system. Therefore, it would appear to be more fundamental in the ,

kinetics of the process. We may speak of L

Dl and D12 as interstitial and substitutional diffusion constants to distinguish them

5

from the classical diffusion coejicient, which is a parameter depending on con- centration. I t also seems probable that all cases of interstitial ditrusion in solid solution will fo1lo.w this behavior.

Lattice+oint Diffusion.-In substitu- .

tional solid solutions, unless a vacant lattice site is present to receive a diffusing 1 atom, migration can occur only when two 1 atoms exchange positions, or by rotation of quadruplets, which is inherently un- likely. If two atoms of the same type change positions, this will not be detected by chemical analysis. Only interchange of unlike atoms will change the concentration gradient. Therefore, the chemically deter- mined diffusion coefficient will depend only on the probability that two unlike atoms will interchange lattice positions. This also involves the assumption that I an equal number of atoms diffuse in each direction. I n addition to this, two "self- I diffusion" coefficients for the individual components of the system may be deter- mined by radioactive tracer techniques.17

In order to describe the kinetics of diffusion for the entire binary substitutional solid system, three coefficients will be necessary. The interrelationship of these three will be discussed later with reference to the experiments of Johnson17 on the silver-gold system. First, however, the chemical diffusion coefficient in the alpha solid solution of brass will be examined.

Interdiffusion k Alpha Brass.-Average D values' for 750° and 840°C were read from Fig g o f Rhines and Mehl.8 Sample 41 was selected a t goo°C. The values for

C. ERNEST BIRCHENALL AND ROBERT F. MEHL I I

800°C were taken from the average curve in The variation in D12 at 800°C in the Fig 15. These were corrected first for the concentration range where the 750" and activity of zinc* to give the values listed 840°C data are consistent is probably a under Dl in Table 4. These were further result of the averaging process used by

corrected for the activity of copper to give the DI2 values according to the definition in Eq 15.

Examination of the data shows that twentyfold variations in D have been reduced to twofold variations and that these deviations occur a t the extremes of the penetration curves where errors in both the chemical and mathematical analysis would be greatest. Experimental deviations between individual diffusion samples show twofold discrepancies.

Rhines and Mehl to obtain the 800°C curve. The errors a t high concentrations would be distributed through a larger range.

Averaging Dl2 values for o to 18 at. pct Zn for the four temperatures and plotting Dl2 against the reciprocal of the absolute temperature did not give a straight line (Fig 4). However, inspection of the original data revealed that the same situation exists for D values cor- responding to the same concentration. The activation energy can be estimated only at 30,000 If: 50 pct cal per mol.

*Although pure solid zinc was used as the reference state for this calculation the same in- Despite the inconsistencies remaining ternalconsistency was found referring the activi- i, the data, there can be little doubt that, ties to liquid zinc. The choice of standard state does not affect the test of the equation. within the experimental error, correction

I2 THERMODYNAMIC ACTMTIES AND DIFFUSION IN METALLIC SOLID SOLUTIONS

TABLE 4-Activity-difluson Data for Alpha Brass

C. ERNEST BIRCHENALL AND ROBERT F. MEHL I3

of D for activities of both components has given a constant characteristic of the temperature a t which diffusion occurs and independent of the concentration. The activity correction for each com- ponent enters to the first power and in the same form. Therefore, substitutional

, diffusion is controlled by the activity gradients of both components.

, This indicates strongly that the rate- determining process of diffusion in binary substitutional solid solutions involves two atoms. The only such mechanism is the direct interchange of lattice sites by two neighboring atoms." The mechanisms of hole diffusion, previously favored on the basis of theoretical studies of idealized models,18 would seem to occur to only a minor extent on the basis of these results, at least for the alpha-brass system.

More precise diffusion and activity data for a number of similar systems allowing more accurate determination of activation energies are very desirable as a basis for further investigation of the detailed mechanism. Such studies are planned in tbis laboratory.

Interdiffusion and "Selj-diffusion" in Silver-gold Alloys.-If we assume the exchange mechanism, as mentioned above, in a binary substitutional solution AB, there are three possible processes: exchange of A with B, A with A, or B with B. I t is unlikely that, except for the case of solu- tions of isotopes of the same element, the probabilities of these processes will be equal. Only when the isotopes are present in the same ratio on both sides of the original interface will the chemical and radioactive tracer methods yield the same results. Ordinarily these con-

I t may be that some lattice imperfection (hole or dislocation) may be necessary in the immediate neighborhood to catalyze the ex- change. This will have no immediate effect on the discussion given here. Precise measurement of activation energies must ultimately dis- tinguish between this and direct interchange at the site of large displacements due to thermal motion in an otherwise normal lattice.

ditions are met. However, an experiment can be imagined where one side of the interface has composition, C,? containing stable A, and the other side has com- position, Cz, containing stable A plus some radioactive A.* The resultant D will be composite unless CI corresponds to no A a t all.

The chemically determined diffusion constant will measure the excess number of A-B exchanges per second per square centimeter along a unit activity gradient. If a tracer is added to one side so that the proportion of A* to A is not uniform throughout the diffusion couple the radio- active analysis will measure the chemical diffusion plus the additional diffusion of A* along the isotope concentration gradient.

If the activity gradient is eliminated by employing alloys of .the same concentra- tion for both sides of the couple, but with one half of the couple containing radioactive A*, the diffusion of A* along the isotope concentration gradient may be determined alone. However, it is composite in nature, consisting of the excess exchanges of A*-A moving A* forward and the equivalent pair of ex- changes A*-B moving A* forward while A-B is moving A backward. If the prob- ability of either A-A or B-B exchange is much less than that of A-B, nearly all the movement of A or B may occur by the latter process.

Johnson17 studied interdiffusion and "self-diffusion" of silver and gold in gold- silver alloys of approximately 50 atomic per cent silver. The interdiffusion was carried out with such a small concentration difference that D could be taken inde- pendent of concentration. The "self- diffusion " experiments were performed on alloys of the same composition so that no interdiffusion would occur. For convenience, let Dab be the chemical diffusion constant for gold-silver, D, the "self-diffusion" constant for silver in the

14 THERMODYNAMIC ACTMTIES AND DIFFUSION IN METALLIC SOLID SOLUTIONS

alloy, Db the "self-diffusion" constant and "self-diffusion" of gold in the alloys for gold in the alloy, Dk the self-diffusion is the same. Indeed, at any temperature constant for silver in pure silver, and in the range studied D.b = @b. AS stated DAu the self-diffusion constant for gold before, Db measures the total rate of two

vr x 10.

FIG 5-D VERSUS 1/T FOR INTERDIFFUSION AND SELP-DIPFUSION I N GOLD-SILVER ALLOYS.

in pure gold. In order of decreasing magnitude

Dab > D k > D. > DD > > DA.

Fig 5 shows the straight lines obtained by plotting D., Db, and Dab against the reciprocal of the absolute tempeiature. D, and Dab have the same slope; that is, the activation energy for interdiffusion

processes, while D d determines the rate of Ag-Au interchange uniquely.

Johnson observes that, while "self- diffusion" of silver in the alloy is nearly as fast as in pure silver, "self-diffusion" of gold is about twenty times as rapid in the alloy as in pure gold. It is not unreasonable to assume that transport of gold in the alloy occurs chiefly by the

C. ERNEST BIRCHENALL AND ROBERT F. htEHL I5

mechanism of double exchange with silver and only to a negligible extent by exchange with itself.

The average displacement in diffusion* is given by

However, in the double interchange mecha- nism our experiment measures only half of the total displacement, the half moving radioactive isotope forward, and neglects the half moving stable isotope backward. So we have

Dab = 4Db, showing that our hypotheses are consistent with experiment.

Since the rate of transfer of radioactive silver by this mechanism must be the same as for gold, the difference D. - Db is the diffusion coefficient for the "self- ditIusion" of silver by exchange with itself.

Use of the Various Diffusion Constan1s.- The classical diffusion equation based on a concentration gradient and the constants DO and Q (or A and b ) t derived from data treated in this way give a satisfactory empirical description of the diffusion process for a given system. I t is possible to interpolate and extrapolate these equations with the degree of accuracy with which the relationship between D and concentration is known experimentally. In the study of kinetic processes involving diffusion, where the interest lies in the rate of transport of the components, but not in the detailed nature of the diffusion process itself, the older treatment

, will provide a quite adequate description. I Examples of such problems are the dif-

I fusion in clad materials and the rate of

1 decomposition of austenite to pearlite.

I The treatment in this paragraph is due to Prof. R. Smoluchowski.

t See footnote,:page TO.

For better understanding of the diffusion process itself, it would seem imperative to use Dl and Dl2 in the appropriate cases. The fundamental factors to be determined for a given system are Do, and Q1, if diffusion is interstitial, and DolZ and Qla if diffusion is substitutional. A more detailed physical and chemical inter- pretation may then result by comparing these quantities for different systems and correlation with other properties of the systems. Except for the case of self- diffusion* in pure metals, all such com- parisons in the past have been based on Do and Q; reexamination is in order.

I. I t has been quantitatively demon- strated that the activity gradient is more fundamental than the concentration gra- dient in the process of diffusion in the copper-zinc and iron-carbon systems, and probably in general. The driving force is provided by a difference in free energy.

2. The process of solid metallic diffusion has been examined in detail. I t appears that diffusion in interstitial solid solutions can be described by two independent diffusion constants, each determined by a single activity gradient. Such a constant for the interstitial component Dl is related to the D(c) calculated from a concentration gradient by the equation

DitIusion in substitutional solid solution can be described apparently by three independent exchange probabilities yield- ing two " self-diff usion " constants and one interdiffusion constant. The self- ditIusion constants should depend on only

;1n the case of self-diffusion the older equa- tion applies without modification, for D = Dl2

ar and Q = Q12 since 7 is unity and = o. That

one activity gradient, but the inter- diffusion constant D12 is related to D(c) by

This strongly favors a rate-determining step involving direct exchange of atoms on adjacent lattice points as opposed to a hole-diffusion mechanism.

Chemical and radioactive methods for determining dzusion rates in substitutional solutions will not measure the same processes if the ratio of radioactive A * to stable A is not the same on both sides of the original interface. Two mechanisms appear possible for the diffusion of radio- active isotopes, direct exchange with a stable isotope of its own kind or double exchange involving one stable. isotope of its own kind and two atoms of the other kind.

This work was done under a contract

N. J.; and Professors R. Smoluchowski and R. J. Duffin, of the Camegie Institute of Technology, for helpful discussions.

REFERENCES I . C. Wagner: Zlsch. Physik. Chem. (1933)

B21, 25. 2. W. Jost: Diffusion und Chemische Reak.

tion in Festen Stoffen, Dresden and Leipzig, 193 7. Steinkopf.

3. S. Glasstone. K. Laidler and H. Eyring: The Theory of Rate Processes. New York and London. 1941. McGraw-Hill Book Co. Contains references to orieinal -- papers.

4. L. S. Darken: Trans. A.I.M.E. (1942) 150. 157.

5. R. P. Smith: Jnl. Am. Chem. Soc. (1946) 68,1163.

6. C. Wells and R. F. Mehl: Trans. A.I.M.E. (1940) 140, 279.

7. R. Hargreaves: Jnl. Inst. Metals (1939) 64, I 15.

8. F. N. Rhines and R. F. Mehl: Trans. A.I.M.E. (1938) 128, 185.

9. J. H. Hildebrand: Solubility of Non- Electrolytes. New York. 19 6 Reinhold.

10. C. G. Maier: U. S. Bureau of Mines Bull. 324 (1930).

11. A. Schneider and H. Schmid: Ztsch. Elektrochem (1942) 48, 627.

12. R. M. Barrer: Diffusion In and Through Solids, Cambridge. 1941.

13. L. Boltzmann: Ann. Phys. (1894) 53,959. 14. C. Matano: Jap. Jnl. Phys. (1930-3) 8, . .

109. with the Office of Research. The 15. W. A. Johnson: Private communication. authors also wish to thank Dr. W. A. 16. J. Chipman: 2nd. and Eng. Chem. (1932)

24, 1013. Johnson, the I,. W. A. Johnson: Trans. A.I.M.E. (1od2l . - . -

147, 3-31. Oak Term'; L' Darken' U. S' 18. H. B. Huntington and F. Seitz: Phvs, Rev.# Steel Research Laboratories, Keamy, (1942) 61. 315.