Aspects of the thermodynamics of metallic solutions

O.J. Kleppa

To cite this version:

O.J. Kleppa. Aspects of the thermodynamics of metallic solutions. J. Phys. Radium, 1962, 23(10), pp.763-772. <10.1051/jphysrad:019620023010076300>. <jpa-00236677>

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763.

ASPECTS OF THE THERMODYNAMICS OF METALLIC SOLUTIONS

By O. J. KLEPPA, Institute for the Study of Metals and Department of Chemistry,

The University of Chicago, Chicago 37, Illinois.

Résumé, 2014 Un travail récent sur la thermodynamique des solutions métalliques a donné desinformations qui ne peuvent être obtenues à partir d’une analyse des différents diagrammes dephases seulement. Certains systèmes de solutions binaire sont discutés en vue d’illustrer l’influencesur les propriétés thermodynamiques d’une différence de taille, d’électronégativité et la valencecaractéristique des deux corps en solution. Une attention particulière sera donnée à l’effet devalence sur l’enthalpie de mélange pour les solutions des métaux du groupe B et au problèmegénéral de l’entropie de mélange dans les systèmes des solutions métalliques.

Abstract. 2014 Recent work on the thermodynamics of metallic solutions has produced infor-mation which cannot be obtained from an analysis of the various phase diagrams alone. Selectedbinary solution systems will be discussed with a view towards illustrating the influence on theexcess thermodynamic properties of a difference in the size, the electronegativity and the charac-teristic valence of the two solution partners. Particular attention will be given to the effect ofvalence on the enthalpy of mixing in terminal solutions of group B metals, and to the generalproblem of the excess entropy of mixing in metallic solution systems.

LE JOURNAL DE PHYSIQUE ET LE RADIUM TOME 23. OCTOBRE 1962,

Since the turn of the century a great deal ofinformation has become available on the thermo--dynamic properties of metallic solutions. Thiswork originally was stimulated through Gibbs’ dis-covery of thelphase rule. Even today the mostextensive body of thermodynamic information onalloy systems is contained in the literature on phaseequilibria. However, it should be recognized thatthis type of information, valuable as it is, leavesmuch to be desired in any detailed discussion ofsolution thermodynamics. While the phase dia-grams of course reflect the dependence of the totalfree energy of the system on composition and ontemperature, they usually do not permit an unam-biguous numerical evaluation of the theoreticallyinteresting excess thermodynamic quantities. Inthe present discussion we shall in the main be con-cerned with the (intégral) excess free energy, GB,the excess enthalpy, HB, and the excess entropy ofmixing, SB [1]. These quantities are related

through the fundamental thermodynamic relationGB = HB -TSB.We shall not hère consider the various experi-

mental techniques used in detailed studies of thethermodynamic properties of metallic solution [2].Among these the most important are high tempe-rature galvanic cell and vapor pressure methodsfor free energy déterminations, and high tempe-rature reaction calorimetry for the mixing enthal-pies. The best entropy data usually are obtainedthrough combination of equilibrium free energyvalues with éalorimetric enthalpies.

In considering data on metallic solutions repor-ted in thé literature, a word of caution is in order.It is particularly important to keep in mind that

all solid state rate processes are exceedingly slow,compared to the time involved in making the usualthermodynamic measurements. Therefore, obser-vations made on a given solid solution specimenmay not in fact relate to a state of thermodynamicequilibrium. Note also that the number of binarysystems with extensive solid solubility is quite res-tricted.Some of the complications associated with studies

of solid solutions are eliminated in work on liquidalloys. In fact, thermodynamic data on liquidmetallic solutions are in certain respects moreextensive and also more reliable than corres-

ponding data on solid solutions. For this reason,we shall in the present paper consider examplesselected both from solid and from liquid metallicsolution systems. Most of the available infor-mation relates to binary alloys involving Group Bmetals. We shall restrict our discussion to thistype of system.

Idealized models for metallic solutions. -

During recent decades, impressive strides havebeen taken in the statistical mechanics of mole-cular solutions [3]. However, most of this workhas been based on models which bear very littleresemblance to the metallic state. For example,a characteristic feature of the important practicalsolution theories (i.e. those that lend themselves todetailed numerical calculations) is the assumptionthat the total cohesive energy may be approxi-mated by a summation of pair interactions. Ithas become increasingly obvious in recent yearstliat this approach is wholly unsatisfactory formetallic systems. Nevertheless, among- the sta-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010076300

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tistical theories the so-called lattice theories ofsolutions merit special comments. These havecontributed substantially toward our understan-ding of the order-disorder phenomenon, and pro-vide the basis for the usual theoretical estimates ofdepartures from random mixing in metallic solu-tions..

In the simplest version of the lattice theory ofsolution it is assumed that the total cohesive energyarises from nearest neighbor interactions or" bonds. Each bond has a certain energy, v, thenature of which is unspecified. This energy isassumed to be independent of the other bondsformed by the atoms in question, i.e. it is inde-

pendent of composition. Under this assumption,the total internal energy of the alloy A-B is simply

Here nAB, ngg and nBB are the numbers of nearestneighbor bonds, each of fixed energy vAB, vAA andvBB, respectively.

In statistical calculations based on this modelone adopts a latthe frame of reference, with a coor-dination number z. It is then found that all themolar excess thermodynamic properties of the con-sidered mixture can be expressed in terms of asingle interaction parameter

where No is Avogadro’s number.For the purpose of the present brief discussion it

is useful to consider this theory in two approxi-mations [4].

a) Inthe" zeroth " approximation it is assumedthat À is sufficiently small (compared to RT) sothat the mixing of the two components is essen-tially random. In this (Bragg-Williams) case weobtain

On general thermodynamic grounds we knowthat for positive values of HE (and of À) this systemmust segregate at low temperatures into an A-richand a B-rich phase. In the present approximationthe critical mixing temperature is 7’c = X12R.

In a similar way we arrive at the conclusion thatfor negative values of HE (and X) the equilibriumstate at low temperature must involve the for-mation of an ordered phase AB. Again, in the con-sidered approximation, the critical (Curie) pointfor the ordering process is

b) In higher approximations of this theory it istaken into account that for any non-zero value ofthe interaction parameter À there will be somedeparture from random mixing. For À &#x3E; 0,

i.e. vAB &#x3E; 1 /2 (VAA + VBB), a larger than randomnumber of AA and BB bonds in predicted (" clus-tering "). Similarly, for X 0 one predicts somepreference for the AB configuration at all finite

temperatures (" short range order ").The theory permits estimates to be made of

these departures from random mixing, and of theensuing loss in entropy. For moderate valu"es of Àwe obtain for this configurational excess entropy

In view of this formula, it is indicated that abovethe critical temperature (T &#x3E; À/27?) these negativeconfigurational entropy contributions should bequite small (- 0. 1 cal/degree/mole or less). Oftenthey may be neglected compared to other andmuch larger thermal contributions (See below).

In recent years attempts have been made tomake calculations of the thermodynamic propertiesof metallic solutions on the basis of solid state

theory. So far, these theories have not approachedthe solution problems from the more compre-hensive point of view of molecular solution theory.Instead they have focused their attention on someparticular problem which lends itself to theoreticaltreatment. The most successful of these theoriesis Friedel’s treatment of the relative valence effectin terminal solutions [5]. We shall have more tosay about this below.

Selected examples of real metallie solutions. -It is well known from the extensive°work on binaryphase diagrams, particularly by Hume-Rotheryand his school, that these diagrams reflect theworking of at least three important " factors " the" size " factor, the " electro-chemical " (or betterthe " chemical affinity " or " electronegativity ")factor and the " valence " factor. It is obviousthat these factors are interrelated, and that to con-sider them separately represents at best a crudeapproximation. Nevertheless, it serves as a usefulstarting point in a survey of the properties of realmetallic solutions.These three factors, since they are recognized

readily in the appearance of many phase diagrams,are even more strongly reflected in the excess ther-modynamic properties. As might be expected,one usually finds that a différence in atomic sizegives rise to positive contributions to H-, FE and SESimilarly, a difference in electronegativity givesrise to negative contributions. Finally, it is foundthat a difference in the characteristic valence of thetwo solution partners often produces an asymmetryin the excess thermodynamic functions, i.e. a pro-nounced difference between the solutions of A in Band those of B in A. We shall illustrate these .general observations by considering some selectedexamples of binary solution systems.

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a) SYSTEMS WITHOUT VALENCE EFFECTS. - Thealloys of silver and gold are sometimes uncriticallyreferred to as " ideal ". Thus this system has avery simple phase diagram with a complete rangeof solid solutions (see fig. 1), the lattice parameter

FIG. 1. - Phase diagram and excess thermodynamic datafor silver-gold alloys (ref. [7]).

departs by less than 0.15 % from Vegard’s rule,and there is no convincing evidence for phase sepa-’ration or for the existance of any ordered phase atlow temperatures.

It is well known that the atomic radii of goldand silver are very similar. Thus the " sizef actor " is very favorable. On the other hand,these metals have a substantial différence in electro-

negativity (about 0.5 units on Gordy’s scale [6]).Thus, chemical reasoning suggests that " ionic "forms of the type Ag+ Au7- may be of some impor-tance in bonding between these metals. We accor-dingly anticipate negative departures from ideality.The phase diagram, given in the upper part ’offigure 1, shows a very small liquidus-solidus sepa-ration, which points in the same direction.Detailed thermodynamic measurements, the resultsof which have beeri summarized by White, Orr andHultgren [7] confirm this, as illustrated in figure 1.The rather large negative enthalpies of mixingsuggest that this system might be expected to havean ordered intermetallic phase at low temperature.So far, this has not been observed.

It is noteworthy that this system also has anegative excess entropy of mixing, which amountsto about 0.3 cal/degree g. atom in the middle of thesystem. Although this value is not accuratelyknown, it certainly is too large to be explained bydepartures from random mixing. Therefore, it isindicated that the excess entropy in this system inthe main must be of thermal, i.e. in this case presu-mably of vibrational origin. We have, under theassumption that we can eff ectively " freeze in "the high temperature random configuration,

Here C§i represents the deviation of the heatcapacity from a linear dependence on composition( Kopj-Neumann rule). Note that non-zero valuesof C,» of course contribute also to HB. However,these contributions tend to be overshadowed bythe temperature independent terms. ,The most significant thermal contributions to

, the excess entropy usually arise at temperatureswell below the characteristic temperature of thealloy. But regrettably, we do not have low-tempe-rature heat capacity data for silver-gold alloyswhich permit a check of eqn. (3). On the otherhand, if our identification of the excess entropy asvibrational is correct, we should be able to turnthe problem around and use the observed excessentropies to estimate the Debye 6’s for the alloys.

For elevated temperatures (T » 6) We have onthe Debye model the following simple expressionfor the vibrational excess entropy of the alloy

Here 0 is the Debye temperature for the alloyand 6E = 0 -[r0Au + (1-x) OAg]. The observednegative excess entropies for gold-silver accordinglyimply positive departures of 0 from a linear depen-dence on composition (higher frequencies of vibra-

Fie. 2. - Young’s modulus for solid solutions in thesystem silver-gold (ref. [8]).

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tion). If we adopt values of 0 of 215° and 1800 forsilver and gold, respectively, we estimate from theentropy data a 6 of 209° for the equi-atomic alloy.This value is about 6 % larger than the mean ofthe values for silver and gold. Our result is con-sistent with the data on Young’s modulus for thissystem reported by Kôster and Rauscher [8], andpresented in figure 2. This agreement demons-trates, as was originally suggested by Zener [9], thatdata on the elastic constants for metallic solid solu-tions may be used to advantage in estimating vibra-tional entropy contributions. Of course this holdsequally well for systems with and without valenceeffects.As a second example of systems without valence

effects let us consider copper-silver, for which thephase diagram and some other thermodynamicdata are presented in figure 3 [10]. Here the elec-

FIG. 3. - Phase diagram and excess thermodynamic datafor silver-copper alloys (ref. [10]).

tronegativity différence is zero on Gordy’s scale,while the size factor is fairly large. Thus weexpect and find substantial positive deviationsfrom ideality, both in the solid and in the liquidstate. Note that the (positive) excess fiée ener-gies are significantly larger in the solid than in theliquid solutions. This behavior is characteristic ofsimple alloy systems where there is a large diffe-rence in size between the two solution partners. It

reflects the fact that the size factor is far more cri-tical in the solid than in the liquid state. On theother hand, when the size difference is small, as insilver-gold (and in silver-palladium, see below), itmay be found that the solid solutions have lowérexcess free energies of mixing than the corres-ponding liquid alloys.Beginning with the works of Pines [11] and

Lawson [12] several attempts have been made tocalculate excess thermodynamic quantities forbinary solid solutions from elastic theory, i.e. onthe basis of the " misfit " between solvent andsolute atoms. On the whole, there is only fairagreement between theory and experiment in thesecalculations.

b) HUME-ROTHERY TYPE SYSTEMS. - Histo-rically, the most important systems exhibitingvalence effects are the alloys formed between themono-valent metals copper, silver and gold andmany metals of higher valence. Among these sys-tems the simplest ones are those where both solu-tion partners belong to the same row in the periodictable, such as the alloys of copper with zinc, gal-lium and germanium and the alloys of silver withcadmium, indium and tin. In these cases the sizefactors are moderate, and there is a relativelysmall difference in electronegativity between thetwo metals. Thus the setting is right for a displayof valence effects.We give in figure 4 the equilibrium phase dia-

grams for the alloys of silver with cadmium, indiumand tin. These diagrams may be used to illustratethe well-known Hume-Rothery rules. However,we shall restrict the present discussion to thermo-dynamic information which cannot be derived fromthe phase diagrams alone. In particular, we shallreview the data on the enthalpies of mixing, whichare presented in figure 5 [13, 14]. These data

apply for alloys which are stable at 450 OC, atwhich temperature cadmium, indium and tin areall liquid. This circumstance turns out to be afortunate one, since it gives an insight into thethermodynamics of these systems which can notreadily be obtained through study of the solidphases alone.The most striking feature of the data in figure 5

is the remarkable difference in the properties of theterminal solutions in these systems, i.e. betweenthe solutions of cadmium, indium and tin in silver,on the one hand, and the solutions of silver in theliquid multi-valent metals on the other.From a purely thermodynamic point of view we

may characterize the terminai solutions by twosets of quantities, namely the liniiting slopes andthe limiting curvatures of the excess thermodynamicfunctions. In the special case of the enthalpy ofmixing the limiting slope represents the change inenthalpy associated with the transfer of a solute

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FIG. 4. - Phase diagrams in the sys-tems silver-cadmium, silver-indiumand silver-tin.

FiG. 5. - Excess enthalpy data at450 DC for the systems silver-cadmium, silver-indium and silver-tin (ref. [13], [14]).

768

atom from its reference state (here the pure solid.

or liquid solute) into the pure solvent. In generalthis transfer will involve a very drastic change inthe environment of the solute atom. Therefore, itis obvious that a realistic calculation of this quan-tity is a very formidable task. This was attemptedby Friedel on thé basis of a suitable thermo-dynamic cycle [5]. However, in this cycle the heatof solution appears as a différence between verylarge numbers, and the results are of rather limitedvalue.More interest is attached to the limiting curva-

ture of the enthalpy of mixing. This quantity is ameasure of the interaction between the soluteatoms in the matrix of the solvent, and was success-fully calculated by Friedel [5]. In the presentpaper we shall refrain from discussing the funda-mental basis for this theory. The most importantconclusion reached is that if the complicating fac-tors of a large différence in atomic size and of

strong chemical interaction between solute andsolvent do not overshadow the valence effect, onemight expect the limiting curvature of the enthalpyof mixing to be determined by the difference invalence between solvent and solute. Thus, if thesolute is of higher valence than the solvent, thereshould be an effective repulsion between the soluteatoms, i.e. a positive curvature. If the situation isreversed, there should be attraction and a negativecurvature. Furthermore, for différent solutes inthe same solvent, it is predicted that there shouldbe a rough proportionality between valence diffe-rence and curvature.The enthalpy data presented in figure 5 are in

reasonable agreement with these predictions.Thus we find positive and increasing curvatures forthe solid solution of cadmium, indium and tin insilver and, what is more remarkable, negative cur-vatures for silver in liquid indium and tin. Thedata for silver in liquid cadmium are somewhat

FIG. 6. - Plot of HEjx for cadmium and indium in silver(ref. [14]).

FIG. 7. - Plot of HE /x for zinc and gallium in copper(ref. [15]).

uncertain. In this case it is probable that we havea positive rather than a negative limiting curvature.

In order to attempt a more quantitative checkof the stated predictions we give in figure 6 a sui-table graph for the solid solutions of cadmium andindium in silver. Note that the slope of HE lxversus x is a measure of the curvature of the

enthalpy of mixing. The figure indicates that thecurvature for indium is about twice that for cad-mium. Very recent calorimetric work by Kleppaand King [15] on the solid solutions of zinc andgallium in copper shows comparable agreement, asdemonstrated in figure 7. However, it should berecognized that the data in figures 6 and 7 do notcover the very dilute range where the theory ismost applicable. Therefore, one should not over-emphasize this agreement with theory.

c) ALLLOYS OF GROUP II B METALS. -Many ofthe Group B metals fall within a fairly restrictedrange with respect to atomic size and electro-

negativity. Therefore, pursuing the line of rea-soning advanced in the present paper, one mightexpect that the alloys formed between the divalentmetals zinc, cadmium and mercury, on the onehand, and metals of higher valence, on the other,should also display valence correlated solution pro-perties. In this context it is interesting to notethe suggestion made by Raynor [16], that the very ’

(

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low solid sofubilities of multivalent metals in zincand. cadmium might be related to certain details ofthe band structure of these solvent metals. Apartfrom this, there is little evidence in the phase dia-grams for the importance of valence effects.

It turns out that a detailed analysis of the excessthermodynamic properties for some of the liquidalloys of zinc and cadmium is more revealing.This is illustrated in figure 8, where we presentsome of the integral excess thermodynamic quan-

Fic. 8. - Some excess thermodynamic quantities for liquid zinc-cadmium, zinc-indium and zinc-tin (ref. [17]).

tities for the liquid àlloys of zinc with cadmium,indium and tin [17]. The data undoubtedly reflectboth a différence in size and a différence in valencebetween the solution partners. Nevertheless, thecorrelation between excess* properties and valencedifférence is apparent.A survey of the differential excess quantities for

various solutes in zinc is even more suggestive [181.For this purpose we give in figure 9 the differentialexcess enthalpies of liquid zinc as a solvent for awide range of other group B metals. In figure 10we present a similar graph which shows the diffe-rential excess entropies plotted against électronconcentration. We see that for all the considered,muttivatent solutes the exeess entropies are posai-

tive-and,-except in"the case of bismuth, of compa-rable magnitude at the same electron concentra-tion. Similarly, the enthalpy data indicate a clearcorrélation between excess enthalpy and valencedifference. However, it should be noted that the" valence effect 1) for solutions in liquid zinc (andcadmium) is opposite in sign to that predicted bythe Friedel theory [18 A).. We have suggested elsewhere [19]7,that the ther-modynamic properties in these terminal solutionsmay possibly be related to departures from freeelectron behavior ni liquid. zinc and cadmium.There is some support for this in the anomaloustemperature dependenae of thé electrical conàuc-tivity ouf thèse liquid metals. On the other hand,

770

FIG. 9. - Differential excess enthalpies for zinc’in zinc-rich liquid alloys (ref. [19]).

FiG. 10. - Differential excess entropies for zinc in zinc-rich liquid alloys (scale on left for Zn-Ag and Zn-Inonly. Other curves displaced by multiple of 0.1 cal/deg,ref. [19]).

recent Hall coefficient measurements by Esch [20]do not seem to support this interpretation.

Finally, it should be mentioned that the liquidalloys of mercury with metals of higher valenceshow a much more complex pattern of thermo-dynamic behavior than the corresponding zinc andcadmium systems [21].

(d) ALLOYS INVOLVING TRANSITION METALS ; THESILVER-PALLADIUM SYSTEM. --At the present timethe bulk of the detailed thermodynamic infor-mation on metallic solutions pertains to alloys for-med by non-transition metals. Therefore, thereis as yet no basis for any general systematic dis-cussion of the solution properties of transition metalalloys. On the other hand, reliable thermodyna-mic data have become available in recent years fora number of binaries involving transition metals.In some cases these exhibit features not found inthe examples considered above, as we shall illustrateby considering the. silver-palladium system.We present first in the upper part on figure 11

the accepted phase diagram for this system. In

any discussion based only on the phase diagram,

this systems would appear to be very simple, and itmight be compared, for example, to silver-gold.Thus, we estimate from the course of the liquiduscurve that GE(s) - G’(1) for silver-palladium

FIG. 11. - Phase diagram and excess thermodynamic datafor silver-palladium alloys (refs. [23], [24]).

should be about -150 cal/g; atom in the middle ofthe system. The comparable figure for silver-gold is about - 50 cal/g. atom [22]. On the otherhand, the larger liquidus-solidus separation indi-cates a more positive (or less negative) departurefrom ideality than in silver-gold.

It was only on the completion of the recente. m. f. study by Pratt [23], and the calorimetricwork by Chan, Anderson, Orr and Hultgren [24]that it became generally recognized that the ther-modynamic properties of silver-palladium are

really quite complex. This will be noted from abrief look at the thermodynamic data given in thelower part of figure 11. It is particularly note-worthy that silver-palladium has a very large nega-tive excess entropy of mixing, which for equi-atomic alloys amounts to about - 1.8 cal/degreeg. atom at 1 000 OK. This figure should be com-pared with the ideal entropy of mixing of + 1.38cal/degree. Thus we find that at elevated tempe-ratures the formation of an essentially random solidsolution of composition Ago.5Pdo.5 actually is asso-ciated with a net reduction in entropy. This some-what surprising result shows that the unfüled d-shell in this case gives rise to large thermodynamiceffects which are either absent, or are present only

771

to a very limited extent, in the types. of systemsdiscussed so far. Presumably, these eff ects are ofelectronic and magnetic as well as of vibrationalorigin. To a first approximation it probably isjustified to consider these three effects separately,and they will of course contribute to all the variousexcess thermodynamic properties of the mixture.However, they are most readily recognized in theentropy of mixing, and we shall confine our dis-cussion here to this quantity.

Numerical estimates of the vibrational and elec-tronic contributions to the high temperature excessentropy of silver-palladium alloys can be madefrom the helium-range heat capacity data of Hoareand Yates [25]. Values of 6 and y obtained intheir work are given in figure 12, along with values

FIG. 12. - 03B8 and y for silver-palladium alloys (refs. [25],[26]).

of 0 derived from elastic constant measurements

by Hoare, Matthews and Walling [26].By use of eqn. (4) above we estimate from these

data the vibrational contribution to the excess

entropy for an equi-atomic alloy to fall between

--- 0.8 cal/degree g. atom (from elastic constants)and 2013’1.2 cal/degree (from heat capacities).

If the electronic heat capacities depend linearlyon the absolute temperature we have for the elec-tronic excess entropies

where Y-e = y - [XAg YAg + XPd ypd]. On the

assumption that this applies up to 1. 000 OK, weestimage Sli for a 50-50 alloy to be about - 0.8 cal/degree g. atom. Actually, available high tempe-rature heat capacity data for pure transition metalsindicate that the electronic contributions to theheat capacity at elevated temperatures probablyare somewhat smaller than yT. Therefore, thisestimate of Sé may be high numerically.

Finally, we shall consider very briefly the ma-gnetic contributions to the entropy of mixing. In

général such contributions will depend on thenature of the magnetic properties of the compo-nents. In silver-palladium pure silver is diama-gnetic, palladium is paramagnetic, and the alloysare paramagnetic for xpd &#x3E; 0.5 [26]. Clearly thed-electrons, presumably localized on the palladiumatoms, are spin-paired at high silver contents, withan ensuing loss in entropy.

This problem has been discussed in a recentpaper by Oriani and Murphy [27]. Adapting anapproach advanced by Weiss and Tauer [28], theseauthors assume that in the diamagnetic range themagnetic contribution to the entropy of mixingmay be represented by - xpd R In (03BCpa + 1).Here 03BCpd is the effective atomic moment of palla-dium (1.44 Bohr magnetons). Assuming that thealloy is fully diamagnetic at xid == 0.4, theysuggests that the magnetic excess entropy in theparamagnetic range will also vary linearly withcomposition (going to zero at pure palladium). Inthis manner they arrive at an estimate of about- 0 . 7 cal/degree g. atom for the magnetic contri-bution to the excess entropy in Ago.5Pdo.5.The sum of the quoted vibrational, electronic

and magnetic entropy contributions amounts to-- 2.3 to --2.7 cal/degree g. atom. Àlthoughthis result is numerically somewhat larger than theexperimental value, it is of comparable magnitude.

Acknowledgements. - This work has been sup-ported by the Office of Naval Research undercontract No. Nori-2121 with the University of

Chicago.REFERENCES

[1] The integral excess thermodynamic quantities are de-fined through the relations

YE = 0394 Y 2014 0394 Yidealwhere 0394 Y is the molar change in the function Y onmixing. Note that0394Gideal = RT(x In x + (1 2014 x) In (1 2014 x)).Here x and (1 2014 x) are the mole fractions of the

two components. Similarly, for the entropy ofmixing0394Sideal = 2014 R(x In x + (1 2014 x) In (1 2014 x)),i.e. the entropy of random mixing. On the otherhand we have for enthalpy, internal energy, volumeand heat capacity

0394Hideal = 0394Eideal = 0394Videal = 0394Cp ideal = 0.

772

[2] Detailed discussions of these methods are presented inthe National Physical Laboratory Symposium No. 9,The Physical Chemistry of Metallic Solutions andIntermetallic Compounds. H. M. S. O., London,1959.

[3] See e.g. PRIGOGINE (I.), The Molecular Theory ofSolutions, North Holland Publishing Company,Amsterdam, 1957.

[4] An authoritative discussion of the lattice theories ofsolution is given in E. A. Guggenheim’s " Mixtures ",Oxford University Press, Oxford, 1952.

[5] FRIEDEL (J.), Adv. in Physics, 1954, 3, 446.[6] GORDY (W.) and THOMAS (W. J. O.), J. Chem. Physics,

1956, 24, 439.[7] WHITE (J. L.), ORR (R. L.) and HULTGREN (R.), Acta

Met., 1957, 5, 747.[8] KÖSTER (W.) and RAUSCHER (W.), Z. Meta!lk., 1948,

39, 111.[9] ZENER (C.), in Thermodynamics in Physical Metal-

lurgy, A. S. M., Cleveland, 1950.[10] HULTGREN (R.), Private communication.[11] PINES (B. J.), J. Physics, U. S,B S. R., 1940, 3, 309.[12] LAWSON (A. W.), J. Chem. Physics, 1947, 15, 831.[13] KLEPPA (O. J.), Acta Met., 1955, 3, 255. [14] KLEPPA (O. J.), J. Phys. Chem., 1956, 60, 846.[15] KLEPPA (O.,J.) and KING (R. C.), To Acta Met (in

press).[16] RAYNOR (G. V.), Progress in Metal Physics, 1949, 1, 1.

[17] KLEPPA (O. J.), KAPLAN (M.) and THALMAYER (C. E.),J. Phys. Chem., 1961, 65, 843.

[18] In the dilute range, the differential excess quantitiesof the solvent are related in a simple manner to thecurvature of the integral, excess quantity. (See alsoref. [21].)

[18A] Note added in proof. Very recent theoretical workby Blandin and Deplante reported during this col-loquium, represents an improvement on the earlierFriedel theory, and appears to account for this changein sign.

[19] KLEPPA (O J.) and THALMAYER (C. E.), J. Phys.Chem., 1959, 63, 1953.

[20] WILSON (E. G.), Private communication.[21] KLEPPA (O J.), Acta Met., 1960, 8, 435.[22] KLEPPA (O. J.), Acta Met., 1960, 8, 804.[23] PRATT (J. N.) Trans. Faraday Soc. 1960, 56, 975.[24] CHAN (J. P.), ANDERSON (P. D.), ORR (R. L.) and

HULTGREN (R.), 4th Tech. Report, Mineral ResearchLaboratory, Berkeley, Calif.,1959.

[25 HOARE (F. E.) and YATES (B.), Proc. Roy. Soc., 1957,A 240, 42.

[26] HOARE (F. E.), MATTHEWS (J. C.) and WALLING (J. C.),Proc. Roy. Soc., 1953, A 216, 502.

[27] ORIANI (R. A.) and MURPHY (W. K.j, Acta Met., 1962,10, 879.

[28] WEISS (R. J.) and TAUER (K. J.), J. Phys. Chem.Solids, 1958, 4, 135.

DIFFUSION STUDIES OF VACANCIES AND IMPURITIES

By DAVID LAZARUS, ,Department of Physics, University of Illinois, Urbana, Illinois, U. S. A.

Résumé. 2014 Les défauts ponctuels dans les métaux ont d’abord été introduits pour expliquerles phépomènes de diffusion, et le succès des modèles est généralement mesuré par le succès dansla correlation des résultats des mesures de diffusion. Dans cet article, on passe en revue l’utilisa-tion de la diffusion comme instrument d’étude des imperfections, et on cherche à définir les limitesde la validité des modèles théoriques à la lumière des études expérimentales de la variation, enfonction de la température, de la pression et de la masse, de la diffusion dans un ensemble demétaux purs et de solutions solides.

Abstract. 2014 Point defects in metals were first introduced to explain diffusional phenomena,and the success of the models is generally measured by the success in correlating results of diffusionmeasurements. In this paper, the use of diffusion as a tool to study imperfections will be reviewed,and an attempt made to assess the limits of validity of theoretical models in the light of experi-mental studies of the temperature, pressure, and mass dependence of diffusion in a variety ofpure metals and solid solutions.

LE JOURNAL DE PHYSIQUE ET LE RADIUM TOME 23, OCTOBRE 1962,

1. Introduction. - Diffusional phenomena arebasic to reactions in metallic systems. Diffusionlimits the rate of phase transformations, solubility,creep, grain growth, and recrystallization. Dif-fusion rates dictate whether a material will beuseful in a given environment, as in high tempe-rature reactors, under high flux conditions, or com-pletely useless, as in corrosive atmospheres. Tech-nical interest in the field, therefore, has always beenlzigh.From a purely scientific viewpoint, the most

important problems have been associated with deli-

neating specific mechanisms for diffusion whichpermit the observed large flues of matter withoutperturbing the essentially perfect lattice structure.Of the many mechanisms suggested to explain dif-fusion, the concept of mobile point defects, parti-cularly interstitials and vacancies, has proven mostviable. Since point defects were essentially

invented " to explain diffusion, it is perhapsappropriate to consider how diffusional measu-rements have been useful as, a tool for studyingpoint defects in various systems,.

In homogeneous systems, the diffusion coefficient,