Aspects of the thermodynamics of metallic solutions

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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�

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,

Un travail récent sur la thermodynamique des solutions métalliques ^{a donné des}

informations qui ^ne peuvent être obtenues à partir ^d’une analyse des différents diagrammes ^de phases seulement. Certains systèmes de solutions binaire sont discutés en vue d’illustrer l’influence

sur les propriétés thermodynamiques d’une différence de taille, d’électronégativité et la valence

caractéristique des deux corps ^en solution. Une attention particulière ^{sera donnée à} l’effet de valence sur l’enthalpie ^de mélange pour les solutions des métaux du groupe B et ^au problème général ^de l’entropie ^de mélange ^{dans les} systèmes des solutions métalliques.

Abstract.

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. Selected

binary ^solution systems will be discussed with a view towards illustrating the influence on the

excess 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 of valence on the enthalpy ^of mixing ⁱⁿ terminal solutions of group B metals, and to the general problem ^{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 of} information has become available on the thermo--

dynamic properties of metallic solutions. This work originally ^was stimulated through ^{Gibbs’ dis-}

covery of thelphase ^{rule. Even} today ^{the most}

extensive body ^of thermodynamic information on

alloy systems ^{is contained in the} literature ^on phase equilibria. However, ^{it should be} recognized ^that

this type ^of information, ^{valuable as it} is, ^leaves

much to be desired _{in any} detailed discussion of solution thermodynamics. ^{While the} phase ^dia-

grams of ^course reflect the dependence of the total free energy of the system ^on composition ^{and on} temperature, they usually ^{do not} permit ^{an unam-} biguous ^numerical evaluation of the theoretically interesting ^excess thermodynamic quantities. ^In

the 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 ^of mixing, ^SB [1]. ^These quantities ^{are related} through ^the fundamental thermodynamic ^relation

GB = HB -TSB.

We shall not hère consider the various experi-

mental techniques ^{used in} detailed studies of the

thermodynamic properties ^{of metallic solution} [2].

Among ^{these the most important are} high tempe-

rature galvanic cell and vapor pressure methods for free _energy déterminations, ^and high tempe-

rature reaction calorimetry ^{for the} mixing ^enthal- pies. ^{The best} entropy ^data usually ^{are obtained} through combination of equilibrium ^free energy values 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 usual} thermodynamic measurements. Therefore, ^obser-

vations made on a given solid solution specimen

may ^not in fact relate to a state of thermodynamic equilibrium. ^Note also that the number of binary systems with extensive solid solubility ^is quite ^res-

tricted.

Some of the complications associated with studies of solid solutions are eliminated in work on liquid alloys. ^In fact, thermodynamic ^{data on} liquid

metallic solutions are in certain respects ^more extensive and also ^more reliable than corres-

ponding ^{data on} solid solutions. For this reason,

we shall in the present paper consider examples

selected both from solid and from liquid ^metallic

solution systems. Most of the available infor- mation relates to binary alloys involving Group ^B

metals. We shall restrict our discussion to this

type ^of system.

Idealized models for metallic solutions.

During ^recent decades, impressive strides have been taken in the statistical mechanics of mole- cular solutions [3]. However, ^{most of this work}

has been based on models which bear very little resemblance to the metallic state. For example,

a characteristic feature of the important practical

solution theories (i.e. ^{those that} lend themselves to detailed numerical calculations) ^{is the} assumption

that the total cohesive energy may be approxi-

mated by ^a summation of pair interactions. It has become increasingly ^{obvious in recent years}

tliat this approach ^is wholly unsatisfactory ^for

metallic systems. Nevertheless, among- the ^sta-

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

tistical theories the so-called lattice theories of solutions merit special ^{comments. These have}

contributed substantially ^{toward our understan-} ding ^{of the} order-disorder phenomenon, and pro- vide the basis for the usual theoretical estimates of

departures from random mixing ⁱⁿ metallic solu- tions..

In the simplest ^version of the lattice theory ^of

solution it is assumed that the total cohesive _energy arises from nearest neighbor interactions or

"

bonds. Each bond has a certain energy, v, ^the nature of which is unspecified. This energy is assumed to be independent ^{of the other bonds}

formed 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 nearest

neighbor bonds, ^{each of fixed} energy vAB, ^vAA and vBB, respectively.

In statistical calculations based on this model

one adopts ^a latthe frame of reference, ^{with a coor-}

dination number z. It is then found that all the molar excess thermodynamic properties ^{of the con-}

sidered mixture can be expressed in terms of a

single 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 assumed}

that À is sufficiently ^small (compared ^to RT) ^so

that the mixing ^{of the two} components ^{is essen-} tially random. In this (Bragg-Williams) ^{case we}

obtain

On general thermodynamic grounds ^{we know}

that for positive ^{values of HE} (and ^of À) ^this system

must segregate ^{at low} temperatures ^{into an A-rich}

and a B-rich phase. ^{In the} present approximation

the critical mixing temperature ^{is 7’c =} X12R.

In a similar way ^we arrive at the conclusion that for negative ^{values of HE} (and X) ^the equilibrium

state at low temperature ^{must involve the for-}

mation of an ordered phase ^AB. Again, ^{in the con-}

sidered approximation, the critical (Curie) point

for the ordering process is

b) ^In higher approximations ^{of this} theory ^{it is}

taken ^into account that for any ^non-zero value of the interaction parameter ^{À there will be some} departure from random mixing. ^{For À} > 0,

i.e. vAB > 1 /2 (VAA + VBB), ^a larger than random number of AA and BB bonds in predicted (" ^clus- tering "). Similarly, ^{for X 0 one} predicts ^some preference 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 the} ensuing ^{loss in} entropy. For moderate valu"es of À

we obtain for this configurational ^excess entropy

In view of this formula, ^{it is indicated that above}

the critical temperature (T > À/27?) ^these negative configurational entropy contributions should be

quite ^small (- 0. 1 cal/degree/mole ^or less). ^Often they may be neglected compared ^{to other and}

much larger thermal contributions (See below).

In recent _years attempts have been made to make calculations of the thermodynamic properties

of metallic solutions on the basis of solid state

theory. ^So far, ^these theories have not approached

the solution problems ^{from the more compre-}

hensive point ^of view of molecular solution theory.

Instead they ^have focused their attention on some

particular problem which lends itself to theoretical treatment. The most successful of these theories is Friedel’s treatment of the relative valence effect in terminal solutions [5]. We shall have more to say about this below.

Selected examples ^of real metallie solutions.

It is well known from the extensive°work on binary phase diagrams, particularly by Hume-Rothery

and his school, that these diagrams ^{reflect the} working of ^at least three important " factors " the

"

size " factor, the " electro-chemical " (or ^better the " chemical affinity " ^{or "} electronegativity ")

factor and the " valence " factor. It is obvious that these factors are interrelated, ^{and that to con-}

sider them separately represents ^{at best a crude} approximation. Nevertheless, ^{it serves as a useful} starting point ^{in a} survey of the properties ^{of real}

metallic 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 size

gives ^{rise to} positive contributions to H-, ^FE and ^SE Similarly, ^a difference in electronegativity gives

rise to negative contributions. Finally, ^{it is found}

that a difference in the characteristic valence of the two solution partners ^often produces ^an asymmetry in the excess thermodynamic functions, ^{i.e. a pro-}

nounced difference between the solutions of A in B and those of B in A. We shall illustrate these .

general observations by considering ^{some selected}

examples ^of binary ^solution systems.

765

a) ^{SYSTEMS WITHOUT} VALENCE EFFECTS. - The

alloys ^{of silver and} gold ^{are sometimes} uncritically

referred to as " ideal ". Thus this system ^{has a}

very simple phase diagram ^{with a} complete ^range

of solid solutions (see fig. 1), the lattice parameter

FIG. 1.

Phase diagram ^{and excess} thermodynamic ^data

for 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 ^at

low temperatures.

It is well known that the atomic radii of gold

and silver are very similar. Thus the " size f 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 ^’of figure 1, ^{shows a} very small liquidus-solidus ^sepa- ration, ^which points ^{in the same direction.}

Detailed thermodynamic measurements, the results of which have beeri summarized by White, ^{Orr and} Hultgren [7] ^confirm this, ^as illustrated in figure ^1.

The rather large negative enthalpies ^of mixing suggest ^{that this} system might ^be expected ^{to have}

an ordered intermetallic phase ^{at low} temperature.

So far, ^{this has not been observed.}

It is noteworthy ^{that this} system ^{also has a} negative ^excess entropy ^of mixing, ^{which amounts}

to about 0.3 cal/degree g. ^atom in the middle of the

system. Although this value is not accurately known, ^it certainly ^{is too} large ^{to be} explained by departures from random mixing. Therefore, ^{it is}

indicated that the excess entropy ^{in this} system ⁱⁿ

the main must be of thermal, i.e. in this case presu-

mably ^of vibrational origin. ^We have, ^{under the} assumption ^{that we can} eff ectively " ^{freeze in "}

the high temperature ^random configuration,

Here C§i represents ^{the deviation of the heat} capacity ^{from a linear} dependence ^on composition

( Kopj-Neumann ^{rule). Note that non-zero values}

of C,» of ^course contribute also to HB. However,

these contributions tend to be overshadowed by

the temperature independent ^terms.

The most significant ^thermal contributions to

,

the excess entropy usually ^{arise at} temperatures

well below the characteristic temperature ^{of the} alloy. ^But regrettably, ^{we do not have} low-tempe-

rature heat capacity ^{data for} silver-gold alloys

which permit ^a check of eqn. (3). On the other

hand, ^{if our} identification of the excess entropy ^as vibrational is correct, ^we should be able to turn the problem ^{around and use} the observed excess

entropies ^{to estimate the} Debye 6’s for the alloys.

For elevated temperatures (T » 6) ^{We have on}

the Debye ^{model the} following simple expression

for the vibrational excess entropy ^{of the} alloy

Here 0 is the Debye temperature ^{for the} alloy

and 6E

0 -[r0Au + (1-x) OAg]. The observed

negative ^excess entropies ^for gold-silver accordingly imply positive departures ^{of 0 from a linear} depen-

dence on composition (higher frequencies ^{of vibra-}

Fie. 2.

Young’s modulus for solid solutions in the

system silver-gold (ref. [8]).

766

tion). ^{If we} adopt values of 0 of 215° and 1800 for silver and gold, respectively, ^{we estimate from the} entropy ^{data a 6 of 209° for the} equi-atomic alloy.

This value is about 6 % larger ^{than the mean of}

the values for silver and gold. Our result is ^con- sistent with the data ^on Young’s modulus for this

system reported by Kôster and Rauscher [8], ^and presented ⁱⁿ figure ^{2. This} agreement ^demons- trates, ^{as was} originally suggested by ^Zener [9], ^that

data on the elastic constants for metallic solid solu- tions _may be used ^to advantage ⁱⁿ estimating ^vibra-

tional entropy contributions. Of course this holds

equally ^{well for} systems ^{with and without valence} effects.

As a second example ^of systems without valence effects let us consider copper-silver, ^{for which the} phase diagram ^{and some other} thermodynamic

data are presented ⁱⁿ figure ³ [10]. ^{Here the elec-}

FIG. 3.

Phase diagram ^{and excess} thermodynamic ^data

for silver-copper alloys (ref. [10]).

tronegativity différence is ^{zero on} Gordy’s scale,

while the size factor is fairly large. ^{Thus we} expect ^{and find} substantial positive ^deviations

from ideality, ^both in the solid and in the liquid

state. Note that the (positive) ^{excess fiée ener-} gies ^are significantly larger ^{in the} solid than in the

liquid solutions. This behavior is characteristic of

simple 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 the}

other hand, when the size difference is small, ^{as in} silver-gold (and ⁱⁿ silver-palladium, ^see below), ^it

may be found that the solid solutions have lowér

excess 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 ^to calculate excess thermodynamic quantities ^for binary solid solutions from elastic theory, ^{i.e. on}

the basis of the " misfit " between solvent and solute atoms. On the whole, ^{there is} only ^fair agreement ^between theory ^and experiment ^{in these}

calculations.

b) HUME-ROTHERY TYPE SYSTEMS.

Histo-

rically, ^{the most} important systems exhibiting

valence effects are the alloys ^{formed between the}

mono-valent metals copper, silver and gold ^and

many 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} periodic table, ^{such as the} alloys of copper with zinc, gal-

lium and germanium ^{and the} alloys ^{of silver with} cadmium, ^indium and tin. In these cases the size factors are moderate, and there is a relatively

small difference in electronegativity ^{between the}

two metals. Thus the setting ^is right ^{for a} display

of valence effects.

We give ⁱⁿ figure ^{4 the} equilibrium phase ^dia-

grams for the alloys ^of silver with cadmium, ^indium

and tin. These diagrams may be used to illustrate the well-known Hume-Rothery ^rules. However,

we shall restrict the present discussion to thermo-

dynamic information which cannot be derived from the phase diagrams ^{alone. In} particular, ^{we shall}

review the data on the enthalpies ^of mixing, ^which

are presented ⁱⁿ figure ⁵ [13, 14]. ^{These data} apply ^for alloys ^{which are stable at 450} OC, ^at

which temperature cadmium, ^{indium and tin are}

all liquid. ^This circumstance turns out to be a

fortunate _one, since it gives ^an insight ^{into the} thermodynamics ^{of these} systems ^{which can not} readily ^{be obtained} through study of the solid

phases ^alone.

The most striking ^{feature of} the data in figure ⁵

is the remarkable difference in the properties ^{of the}

terminal solutions in these systems, ^{i.e. between}

the solutions of cadmium, indium and ^{tin in} silver,

on the one hand, ^{and the solutions of} silver in the

liquid multi-valent metals on the other.

From a purely thermodynamic point ^{of view we}

may characterize the terminai solutions by ^two sets ^of quantities, namely ^the liniiting slopes ^and

the limiting curvatures of the excess thermodynamic

functions. In the special ^{case of the} enthalpy ^of

mixing ^the limiting slope represents ^the change ⁱⁿ

enthalpy associated with the transfer of a solute

767

FIG. 4.

Phase diagrams in the sys- tems silver-cadmium, silver-indium and silver-tin.

FiG. 5.

Excess enthalpy ^{data at}

450 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 general

this transfer will involve a very drastic change ⁱⁿ

the environment of the solute atom. Therefore, ^it

is obvious that a realistic calculation of this quan-

tity ^{is a} very formidable task. This was attempted by ^{Friedel on} thé basis of a suitable thermo-

dynamic cycle [5]. However, ^{in this} cycle ^{the heat}

of solution _appears as a différence between very

large numbers, ^and the results are of rather limited value.

More interest is attached ^to the limiting ^curva-

ture of the enthalpy ^of mixing. ^This quantity ^{is a}

measure of the interaction between the solute atoms in the matrix of the solvent, ^and was success-

fully calculated by ^Friedel [5]. ^{In the} present

paper ^we shall refrain from discussing ^{the funda-}

mental basis for this theory. ^{The most} important

conclusion reached is that if the complicating ^fac-

tors of a large différence in atomic size and of

strong ^chemical interaction between solute and solvent do not overshadow the valence effect, ^one might expect ^the limiting ^{curvature of the} enthalpy

of mixing ^to be determined by ^the difference in valence between solvent and solute. Thus, ^{if the}

solute is of higher valence than the solvent, ^there

should be an effective repulsion ^{between the solute} atoms, ^{i.e. a} positive curvature. If the situation is

reversed, ^there should be attraction and a negative

curvature. Furthermore, for différent solutes in the same solvent, ^{it is} predicted that there should be 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 for the solid solution of cadmium, indium and tin in silver and, ^{what is more} remarkable, negative ^cur-

vatures for silver in liquid ^{indium and tin. The}

data 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 have}

a positive ^{rather than a} negative limiting ^curvature.

In order to attempt ^{a more} quantitative ^check

of the stated predictions ^we give ⁱⁿ figure ^{6 a sui-}

table graph ^{for the solid solutions} of cadmium and indium in silver. Note that the slope ^of HE lx

versus x is a measure of the curvature of the

enthalpy ^of mixing. ^The figure ^{indicates that the}

curvature for indium is about twice that for cad- mium. Very ^recent calorimetric work by Kleppa

and King [15] ^on the solid solutions of zinc and

gallium in copper shows comparable agreement, ^as

demonstrated in figure ^7. However, it should be

recognized that the data in figures 6 and 7 do not

cover the very dilute range where the theory ^is

most applicable. Therefore, ^{one should not over-} emphasize ^this agreement ^with theory.

c) ^{ALLLOYS OF GROUP II B METALS.} -Many of

the Group ^{B metals} fall within a fairly ^restricted

range with respect ^{to atomic size and electro-} negativity. Therefore, pursuing ^{the line of rea-} soning advanced in the present paper, ^one might expect ^{that the} alloys ^formed between the divalent metals zinc, ^{cadmium and mercury, on the one} hand, ^{and metals of} higher valence, ^{on the} other,

should also display ^valence correlated solution _pro-

perties. ^{In this context it is} interesting ^{to note}

the suggestion ^made by Raynor [16], ^{that the} very ’

169 low solid sofubilities of multivalent metals in zinc

and. cadmium might ^{be related to certain details of}

the band structure of these solvent metals. Apart

from 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 excess} thermodynamic properties ^{for some of the} liquid alloys ^{of zinc and cadmium is more} revealing.

This is illustrated in figure 8, ^{where we} present

some 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 reflect

both a différence in size and a différence in valence between the solution partners. Nevertheless, ^the

correlation between excess* properties and valence différence 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} differential

excess enthalpies ^of liquid ^{zinc as a solvent for a}

wide range of other _{group B} metals. In figure ¹⁰

we present ^{a similar} graph which shows the diffe- rential excess entropies plotted against ^électron

concentration. 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 clear}

corrélation between excess enthalpy and valence difference. However, ^{it should be noted that the}

"

valence effect 1) for solutions in liquid ^{zinc (and} cadmium) ^is opposite ⁱⁿ sign ^{to that} predicted by

the Friedel theory [18 A).

.

We have suggested ^elsewhere [19]7,that ^{the ther-} modynamic properties ⁱⁿ these terminal solutions may possibly ^{be related to} departures ^{from free}

electron behavior ni liquid. ^{zinc and cadmium.}

There ^is some support for this in the anomalous temperature 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-In} only. ^{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 liquid alloys of mercury with metals of higher ^valence

show a much more complex pattern of thermo-

dynamic ^{behavior than the} corresponding ^{zinc and}

cadmium systems [21].

(d) ^{ALLOYS INVOLVING} TRANSITION METALS ; ^THE

SILVER-PALLADIUM SYSTEM. --At the present ^time

the bulk of the detailed thermodynamic ^infor-

mation on metallic solutions pertains ^to alloys ^for-

med by non-transition metals. Therefore, ^there

is as yet ^{no basis} for any general systematic ^dis-

cussion of the solution properties of transition metal

alloys. On the other hand, ^reliable thermodyna-

mic data have become available in recent years for

a number of binaries involving transition metals.

In some cases these exhibit features not found in the examples considered above, ^{as we} shall illustrate

by 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 it} might ^be compared, ^for example, ^to silver-gold.

Thus, ^{we estimate from the course of the} liquidus

curve that GE(s)

G’(1) ^for silver-palladium

FIG. 11.

Phase diagram ^{and excess} thermodynamic ^data

for silver-palladium alloys (refs. [23], [24]).

should be about -150 cal/g; ^{atom in} the middle of the system. ^The comparable figure ^{for silver-} gold ^{is about}

⁵⁰ cal/g. ^atom [22]. ^{On the other} hand, ^the larger liquidus-solidus separation ^indi-

cates a more positive (or ^less negative) departure

from ideality ^{than in} silver-gold.

It was only ^{on the} completion ^{of the recent}

e. m. f. study by ^Pratt [23], ^{and the} calorimetric work 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 a

brief look at the thermodynamic ^data given ^{in the}

lower 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/degree

g. ^{atom at} 1 000 OK. This figure ^{should be com-} pared ^{with the ideal} entropy ^of mixing ^of + ^1.38

cal/degree. ^{Thus we find that at elevated} tempe-

ratures the formation of an essentially ^{random solid}

solution 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 thermodynamic

effects which are either absent, ^{or are} present only

771 to a very limited extent, ^{in the} types. ^of systems

discussed ^so far. Presumably, these eff ects _are of electronic and magnetic ^{as well as} of vibrational

origin. ^{To a first} approximation ^it probably ^is justified ^{to consider} these three effects separately,

and they ^{will of course} contribute to all the various

excess thermodynamic properties of the mixture.

However, they ^{are most} readily recognized ^{in the} entropy ^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 ^excess entropy ^of silver-palladium alloys ^{can be made}

from the helium-range ^heat capacity ^{data of Hoare}

and Yates [25]. ^Values of 6 and _y obtained in their 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 linearly

on 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, ^we estimage ^{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 metals indicate that the electronic contributions to the heat capacity ^{at elevated} temperatures probably

are somewhat smaller than yT. Therefore, ^this

estimate 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 the}

nature of the magnetic properties of the compo- nents. In silver-palladium pure silver is diama-

gnetic, palladium ^is paramagnetic, ^{and the} alloys

are paramagnetic ^{for xpd} > 0.5 [26]. Clearly ^the d-electrons, presumably ^{localized on the} palladium atoms, ^are spin-paired ^at high ^silver contents, ^with

an ensuing ^{loss in} entropy.

This problem ^has been discussed in a recent paper by ^{Oriani and} Murphy [27]. Adapting ^an approach ^advanced by ^{Weiss and Tauer} [28], ^these

authors assume that in the diamagnetic ^{range the}

magnetic contribution to the entropy ^of mixing

may be represented by - xpd R ^In (03BCpa + 1).

Here _03BCpd is the effective atomic moment of palla-

dium (1.44 ^Bohr magnetons). Assuming that the alloy ^is fully diamagnetic ^{at xid ==} 0.4, they suggests ^{that the} magnetic ^excess entropy ^{in the} paramagnetic range will also vary linearly ^with composition (going ^{to zero at} pure palladium). ^In

this 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. Àlthough

this result is numerically ^somewhat larger ^{than the} experimental value, ^{it is of} comparable magnitude.

Acknowledgements.

^{This work has} been sup-

ported by ^the Office of Naval Research under contract 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 Yideal

where 0394 Y is the molar change ^{in the function Y on} mixing. ^{Note that}

0394Gideal = RT(x ^{In x} + (1 2014 x) ^In (1

x)).

Here x and (1 2014 x) ^{are the} mole fractions of ^the

two components. Similarly, ^{for the} entropy ^of mixing

0394Sideal = 2014 R(x ^{In x +} (1 2014 x) ^In (1 2014 x)),

i.e. the entropy ^{of random} mixing. On the other hand we have for enthalpy, internal energy, volume and heat capacity

0394Hideal

0394Eideal = 0394Videal

0394Cp ^ideal

^0.

772

[2] Detailed discussions of these methods are presented ⁱⁿ

the National Physical Laboratory Symposium No. 9,

The Physical Chemistry of Metallic Solutions and Intermetallic Compounds. H. M. S. O., London,

1959.

[3] See e.g. PRIGOGINE (I.), ^{The Molecular} Theory ^of Solutions, North Holland Publishing Company, Amsterdam, 1957.

[4] An authoritative discussion of the lattice theories of solution 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 quantities

of the solvent are related in a simple ^{manner to the}

curvature of the integral, ^excess quantity. (See ^also

ref. [21].)

[18A] Note added in proof. Very recent theoretical work

by Blandin and Deplante reported during ^{this col-} loquium, represents ^an improvement on the earlier Friedel theory, ^and appears ^to account for this change

in 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 Research

Laboratory, 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.

Les défauts ponctuels dans les métaux ont d’abord été introduits pour expliquer les phépomènes ^de diffusion, ^{et le} succès des modèles est généralement mesuré par le succès dans la 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 limites de la validité des modèles théoriques ^à la lumière des études expérimentales ^{de la} variation, ^en fonction de la température, ^{de la} pression ^{et de la masse,} de la diffusion dans un ensemble de métaux purs et de solutions solides.

Abstract.

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 diffusion measurements. 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 ^of

pure metals and solid solutions.

LE JOURNAL DE PHYSIQUE ^{ET LE RADIUM TOME} 23, ^OCTOBRE 1962,

1. Introduction.

Diffusional phenomena ^are

basic to reactions in metallic systems. ^Diffusion limits ^the rate of phase transformations, solubility,

creep, grain growth, ^and recrystallization. ^Dif-

fusion rates dictate whether a material will be useful 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 ^been lzigh.

From a purely scientific viewpoint, ^{the most} important problems ^have been associated with deli-

neating specific mechanisms for diffusion which

permit ^{the observed} large ^{flues of matter without} perturbing ^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 ^most viable. Since point ^{defects were} essentially

invented " ^to explain diffusion, ^{it is} perhaps appropriate ^{to consider} how diffusional measu-

rements have been useful _as, a tool for studying point defects in various systems,.

In homogeneous systems, the diffusion coefficient,