VACANCY CONCENTRATION AND ARRANGEMENT OF ATOMS AND VACANCIES IN METALS AND ALLOYS*

C. KINOSHITAtS and T. EGUCHIt

With a method of statistical thermodynamics fundamental equations are derived which describe the arrangement of atoms and vacancies in the alloys in the state of thermal equilibrium, where an ordering or a clustering can take place. The solutions of these equations give, among other things, a more precise description on the concentmtion of vacancies than the &s&al ones. The vacancy concentration in pure met& may almost precisely be expressed by the usual approximate expression, but in dilute alloys it is not always expressed by Lamer’s expression. The fraction of vacant sites in the alloys, furthermore, is very unlikely to be expressed by an Arrhenius type of equation, but it decreases or increases, according as a result of ordering or clustering. In binary elloys with short range orders or clusters the probability that one of the nearest neighbor sites of & vacancy is ocoupied by en atom of any particular kind does not always vary monotonically with temperature but in some alloys it increases after decreasing or decreases after increasing. A possibility for an interpretation of the anomalous behaviors of c&u-Al alloys is pointed out.

CONCENTRATION DES LACUNES ET ARRANGEMENT DES ATOMES ET DES LACUNES DANS LES METAUX ET LES ALLIAGES

Les equations fondamentales d&rivs;nt l’arrangement des atomes et des lrtcunes dans les alliages en gquilibre thermique, oh pout se produire la, formation dun &at ordonne ou la formation d’sgglomerats, sont obtenues per une method% de the~od~~mique stetistique. Les solutions de oes equations donnent, parmi d’autres r&mlt&s, une description de la concentration des laeunes plus precise que la. description classique. Dans les metaux purs, la ~onoentr&tion des laounes peut 6tre exprimQ presque pa~aitement psr l’expression courante approchee, mais deans les alliages d&n%, elle n’est pas toujours correctement exprimee par l’expression de Lamer. La proportion de sites v&cants dans les alliages, en outre, eat t&s imparfaitement exprimee par une equation d’Arrhenius, meis elle diminue ou augment0 par formation d’un &at ordonne ou d’agglomerats. Dsns les &ages binaires ordonnes iE courte distance ou presentant des agglomerats, la probabilite pour que l’un des sites premiers voisins d’une lacune soit ocoupe per un etome de n’importe quell% espece perticuliere ne varie pas toujours de fapon monotone avec la tempera- ture, mais drtns certains elliages elle augmente apres avoir diminue ou diminue apres avoir augment& Une possibilite d’interpretation des comportements anormaux des alliages G( Cu-Al est indiquee.

LEERSTELLENKONZENTRATION UND DIE ANORDNUNG VON ATOMEN UND LEERSTELLEN IN METALLEN UND LEGIERUNGEN

Mit einer Method% der st~tistischen The~od~Emik werden F~d&men~lgloioh~gen abgeleitet, die die Anordnung von Atomen und Leerstellen im thermod~~isehen Gleioh~ewieht in solchen Legierungen beschreiben, in denen Ordnung oder Cl~terbild~g mijglich ist. Die Liisungen dieser Gleichungen geben unter anderem eine genauere Beschreibung der Lee~tellenkonzentr&tion sls die Losungen der klassischen Gleichungen. Die Leerstellenkonzentration in reinen Met&en kann recht genau in der tibliohsn N&herung rtusgedrtickt werden; in verdiinnten Legierungen ist sio jedoch nicht immer durch den Lomer-Ausdruck gegeben. Es ist auDerdem sehr unwahrscheinlioh, daR in Legierungen der Anteil leerer Gitterpletze durch eine Arrhenius-Beziehung beschrieben werden kann; dieser Anteil nimmt je nach Ordnung oder Clusterbildung ob oder zu. In biniiren Legierungen mit Nahordnung oder Clustern variiert die Wahrsoheinlickkeit, da9 einer der niichsten Naohbarpliitze einer Leerstelle von einem Atom einer bestimmten Sorte eingenommen wird nicht immer monoton mit der Temper&m. Fur das anomale Verhalten einer cc-Cu-Al-Legierung wird auf eine miigliche Interpretation hingewiesen.

1. INTRODUCTION

Expressions for the equi~brium concentration of

vacancies in metals and alloys have been given in

various papers and text books.(l) These expressions

are convenient for analyses of experimental data,

and they have been used with some success for the

qualitative or semi-quantitative interpretation of

various experimental results; nevertheless these

expressions are still unsatisfactory in the sense that

they are derived under the assumption of a random

dist~bution of vacancies and atoms on lattice sites.

Vacancies play impo~ant roles in the kinetic

processes in metals and alloys. For the sake of a

microscopic description of the processes it is necessary

* Received May 27, 1971. t Laboratory of Iron and Steel, Department of Metallurgy,

Kyushu University, Fukuoka, Japan. $ Now at: Department of Nuclear Engineering, Kyushu

University, Fukuoka, Japan.

ACTA METALLURGICA, VOL. 29, JANUARY 1972 45

to construct a theory whioh takes into account the

arrangement of atoms and vacancies in the alloys

in which an order or clusters can develop.

Stimulated by these needs, several models have

been suggested for the determination of concentration

of vacancies in the alloys with a long range order.(2-5)

Furthermore, Cheng et aZ.(6) developed some models

to estimate the effect of a short range order or clusters

on the equilibrium concentration of vacancies in

binary alloys. These models, however, do not practi-

cally give any i~ormation on the arrangement of

atoms coordinated directly with vacancies.

In the present paper we apply Cowley’s methodc7*s)

of short range order to consider the concentration of

vacancies as well as the arrangement of atoms and

vacancies in a binary system. Besides the interaction

between the nearest neighbor atoms, those between

vacancies and between atoms and vacancies are taken

46 ACTA METALLURGICA, VOL. 20, 1972

into account. Four statistical parameters are intro-

duced in order to describe the concentration of

vacancies and the configuration of atoms and vacan-

cies on lattice sites, and these parameters are shown

to satisfy the four transcendental functional equations

as the result of thermal equilibrium. The interaction

energies are related to, and estimated from, various

thermodynamical quantities, for example, the struc-

ture of solidus curve, the formation energy of a

single vacancy and the temperature at the maximum

of anomalous specific heat. Once the interaction

energies are given then the solutions for the funda-

mental equations are obtained numerically by the

use of an electronic computer.

The general theory thus developed is applied to the

cases of pure aluminum, aluminum alloys and alpha

Cu-AI, as practical examples to estimate the con-

centrations of the vacancies and divacancies in pure

metals, the vacancy concentration and the configura-

tion of atoms and vacancies in alloys with clusters,

and those in alloys with a short range order. The

results of our numerical calculation are compared with

those of the classical formulas in order to examine

the validity of the latter.

In pure metals the fraction of the vacant sites or

divacancies which is predicted by our theory almost

agrees with the corresponding familiar classical

expressions for those quantities, and the energy to

form one defect and the binding energy of two vacan-

cies to form a divacancy may be obtained uniquely in

terms of the elementary interaction energies. In

dilute alloys, however, the vacancy concentration

does not always agree with Lomer’s expression,

and it is shown that the latter is valid only in a

limited region of temperature and concentration of

solute atoms. Furthermore, the fraction of the

vacant sites in concentrated alloys is very unlikely

to be expressed by an Arrhenius type equation.

The probability that an A or a B atom is coordin-

ated to a vacancy does not always vary monotonically

with temperature, but in some alloys an increase after

decreasing or a decrease after increasing is expected.

One of the anomalous changes in the electrical resis-

tivity of alpha Cu-Al may be attributed to the

unusual behavior of the number of vacancy-atom

pairs.

2. DERIVATION OF THE FUNDAMENTAL EQUATIONS

We consider a perfect crystal of an alloy consisting

of A and B atoms plus vacancies. We adopt the

following notations :

NW total number of A atoms;

N, = xNa, total number of B atoms;

N, = YN~, total number of vacancies ;

N = (1 + x + y)N,, total number of lattice sites;

2, coordination number.

Furthermore, Warren’s parameter to represent the

short range ordeP,‘J will be generalized to take

vacancies into consideration by introducing the

following three parameters :

a,=l- Pnb

x ’

1+x+y

a2 = 1 - P a’ , Y

(1)

1 +x+-y

a,=l- Pbv

Y ’ 1+x$-y

where pij is the probability that an atom j (or a

vacancy) is coordinated to an atom i (or a vacancy)

among its nearest neighbors. These probabilities are

interrelated by the three identities :

i%,+pub+l)av=l~

Pb, -t Pbb + Pbv =’ l,

%a +pvb +P,, = ‘.

(2)

Counting the number of bonds of particular types we

see that the following conditions must be satisfied:

Pa, = xPba,

Pm = 1/P,,, (3)

P bu = $.

In order to obtain the internal energy of the

system only the interactions between the first nearest

neighbors are considered as usual. Using the above

equations we obtain the internal energy as

E-E,,= 2c1 +Nz+ y) (~(1 + x + YJE,

- x(x + Y + al)Vl - y(x + y + a2)V2 + xY(l - a3)v3}. (4)

In equation (4) E,,, E,, VI, V, and V, are given by

E,, = &Nz{E,a + X(2E,, - E,,)}

E, = 2E,, - E,,

v, = 2E,, - tEaa + Ebb)

Vv, = 2~%, - (E,, + E,,,)

(5)

v, = 2E,, - (E,, + E,,,),

KINOSHITA AND EGUCHI: ATOMS AND VACANCIES IN METALS AND ALLOYS 47

where E, is the infraction energy between the corresponding $ pair. The motivation for the intro- duction of the interaction in the bonds with vacancies is that the existence of a vacancy would induce a local distortion of the lattice, which woula give rise to an effective infraction between the members of the b0nds.

The entropy for mixing atoms and vacancies in the binary system under consideration is given by

S=klnW, (6)

where W is the number of complex ions within a given configurational energy and is obtained using Cowley’s method.@)

where -r\;iij represents the number of ij vectors. From equations (6) and (7) the entropy is thus

given by

s-x()= 1 +y+ y ((1 + x + Y)

x In (1 + x+ Y) - (1 + a,~ + a,~)

x In (1 -t al% + azv) - +x1 + x + ya3)

X In (da1 + x + ya3)) - da2 + Y + ~4 X In Ma2 + y + xa3)) - 241 - alI

X In (x(1 - al)) - Zy(1 - a& In (~(1 - az))

- 2xz4 - ad In fxY:y(l - ad)), (8)

where 8, is a term independent of the four parameters. Finally the free energy of the system is given by

F = E - TS, and the equilibrium state is therefore obtained by minimizing the free energy with respect to the four parameters y, ar, u2 and a,. Thus we obtained the following four equations which deter- mine these parameters as functions in the state of thermal equilibrium :

“3 ((1 + 5 + d2E2 - 41 - al)Vl

--(x2 + (1 + 2y + a2N + y2 i-

of temperature

2y + a2) va

-I- 41 + X)(1 - a3)v3f + kT((oc, + a$

-l- a& ln (1 + a15 + a& + (a,x

+ a3xa - x2 - xq) In (x(x + a, + a,y))

+ (2?/ + a2 + a3x i- 2xy + azx + %x2

+ $3 In MY + a2 + a3x:)) -241 - aI)

X In (x(1 - aI)) + 2(1 + x)(1 - a2)

X In Ml - ad) + 2x(1 + x)(1 - a3)

X ln (x@ - a%)) - (1 i x + Y)~

x In (1 + 2 + y)“> = 0,

XV1 _- 2

- kT( In (1 + six -j- oz&

+ ln (x(x + tcl + a3y)) - 2 ln (x(1 - a,))] = 0,

zv2 - - kT(ln(1 + alx + a,y) 2

-/- ln My + x2 + a$x)) - 2 ln Ml - a,))> = 0,

XV, - - kT( ln (x(x + a1 + May)) + ln MY 2

+ a2 + 6(3x)) - 2 In (xy(1 - aa))) = 0. (9)

The above equations include as a special case the case of thermal equilibrium in an A-B system without vacancy; namely, assuming y = 0 and a2 = a3 = 0 we obtain an equation for a = a1

which coincides with the one considered by Cowley.t7) In the following sections we shall solve equations (9) numerically for more complicated cases and discuss about the concentration of vacancies and the arrange- ment of atoms and vacancies.

3. VACANCY CONCENTRATIONS IN METALS AND ALLOYS

In order to consider the case of pure metals with vacancies we let in equations (9) 2 = 0, a1 = a, = 0

and V, = V, = 0, and obtain dual simultaneous equations for y and ua, from which we can calculate the concentration of single vacancies C, and that of divacancies C,, by

C/L, 1-i-V

Q2, =_ s$s! = v(a2 + Y) w + g2 ’ (11)

The equations for cr, and y, in the case when they are very small, are soIved analytically and C, and C,, are given by

C, = exp (12)

(13)

On the other hand the equilibrium concentrations of vacancies and divacancies in a pure metal have usually been approximated by (l)

(14)

(15)

48 ACTA METALLURGICA, VOL. 20, 1972

where Ef is the formation energy of a single vacancy

and Es, is the binding energy of two vacancies to

form a &vacancy.

Comparing equations (12) and (13) with equations

(14) and (15), we see that these equations coincide

with each other if we put

Ef = 3E,, (16)

and

E,, = 6V,, (17)

where we have taken z = 12, assuming a face centered

cubic metal.

Equations (12) and (13) are the results of an

approximation which is valid only for small values

of 1%)) so that, if the conditions Ia21 << 1 and y < 1

do not hold, there is no choice other than to solve the

equations numerically in order to obtain the concen-

tration of vacancies from our equations. The vacancy

and divacancy concentrations, which have been

obtained from equations (9) using the method

of Newton-Raphson with an electronic computer

FACOM 230-60 of the Computer Center, Kyushu

University, are shown in Fig. 1, with which those

obtained from equations (14) and (15) essentially coin-

cide. The interaction energies are chosen for pure

aluminum as Ef = 0.76 eV (8841”K)(g) and E,, = 0.17

eV (1973°K),(r0) or E, = 0.2539 eV (2947°K) and

I’, = 0.0283 eV (329°K) from equations (16) and (17).

Thus we see that the equilibrium concentrations of

vacancies and divacancies in pure metals can be

approximated within an accuracy of 0.1 per cent by

equations (14) and (15), respectively, over the whole

temperature range of practical importance.

T (OK) -2, IO?0 7?0 500 ST0 30;

-4 - E, =0.76 eV

$-6 g-8-

.

SlO -

H -12 -

-14 -

-16 -

-18 -

LX 10-3 I.0 2.0 3.0 I/T (I/OK)

Fra. 1. Equilibrium concentration of vacency or di- vrtoancy as * function of temperature in pure metals.

(b) Vacancy concentration in dilute binmy alloys

LomeG) has proposed an expression for the

vacancy concentration in dilute binary alloys :

C, = CF) (

EB’ 1 - zC, + zC, exp -

( 1) kT ’ (18)

where Cp) is the vacancy concentration in the pure

A metal, EB the binding energy between a vacancy

and a solute atom, C, the fraction of solute B atoms.

With an appropriate set of the interaction energies

the corresponding expression for the quantity can be

obtained also from our theory. Assuming laij and y

to be much smaller than x, we find from equations (9)

that

c, = exp 2; ( ( xv1 E2 - m-

xcv, - V,) 1) (1+x) .

(19)

For sufficiently dilute alloys and at sufficiently high

temperatures, we find that equation (19) coincides

with Lomer’s expression (18) if we take EB as

Es = t(V, + V, - V,). (20)

From equation (18) combined with the assumption

that the vacancy concentration is constant along the

solidus curve, Sprusil et aZ.03) have estimated the

numerical value for the binding energy EB. Under

the same assumption and equation (19) the value of

VI + V, - V, may be obtained and compared with

that of Es. The formation energy of a single vacancy

Ef in this case is given from equation (19) as

x(V’, - v,, (1 + 4

= &,Tm@),

(21)

where k, = Ef/T,(0), T,(x) being the temperature

on the solidus line of the alloy with a composition x.

Differentiating equation (21) with respect to x

and rearranging the terms, we obtain

v, + v, - v, =

- 4lcs ___ T,‘(O) (22)

2

where T,‘(O) represents the tangential slope of the

solidus curve at x = 0. Thus we have seen that the

numerical value for VI + V2 - V, can be estimated

from the equilibrium phase diagram. The results of

our analysis concerning aluminum alloys are given in

Table 1 together with Sprusil’s(13) values for EB and experimental ones.(14)

The fundamental equations (9) may be solved

analytically only in the case when the above conditions

hold. In the cases, however, when the preceding

KINOSHITA AND EGUCHI: ATOMS AND VACANCIES IN METALS AND ALLOYS 49

TABLE 1. Values of Vi + v, - F'S and the binding energy between a vacancy and a solute atom in various aluminum alloys. Values of V, + V, - V3 are calculated from the phase diagrams using equation (22). Ecal and EyP are the values theoretically given by Sprusil and Valvoda and the experimental ones, respectively B

Solute atom Zn Ag Mg CU Si Sn

$(V, + 8, - V3) 0.06 0.11 0.13 0.30 0.35 0.36

(eY) ET-“’ (eV) 0.06 0.10 0.12 0.28 0.31 0.31

Eexp (eV) B 0.06, 0.08 0.2, 0.15-0.25, 0.3 0.35,

0.18 kO.01 0.1-0.4, 0.3 6.4 0.3-0.4

assumptions are not valid, the most stable solutions

for equations (9) are obtained numerically by the

method explained in the last subsection.

The ratio of the vacancy concentration in dilute

alloys to the one in pure metals as a function of

temperature is shown in Fig. 2, in which the curve (a)

is obtained from Lomer’s expression equation (16)

with EB = 0.0215 eV (250”K), and (b)-(d) our

result from equations (9) with three different sets of

values for I’, and I’,, keeping V, + vs - I’s =

0.0862 eV (1000°K). The energy E, is taken to be

0.2539 eV (2947’K) as implied in the above analysis

(1) and also supported by an experimental result.@‘)

It is seen in Fig. 2 that the agreement between

Lomer’s expression and our theory is poor even in the

case of small values of EB, where Lomer’s expression

has been understood to be applicable. An increase in

the ratio which is seen in the curve (b) in the higher

temperature side, is because of the variable y couple

T (“K) 1000 700 500 400

z-

Fo.3 - u’ G 9

0.2-

(b)

(c)

(d)

VfO.0689 eV

y-O.0173 eV

W-V3 =0.0431 eV

i

&=0.0173 eV

‘&-0.0689

“’ t (b) /

(0) EB=0.0215 eV ___ _-_------

I .o 2.0 3,0 x to-3

I/T (I/OK)

FIG. 2. The rstio of the vacancy concentration in dilute alloys to the one in pure metals es e function of tem- perature. The curves (b)-(d) are calculated from equa-

tions (9), and (a) from Lomer’s Equation (16).

4

through the energy E, with the other three variables

in equations (9)) and can not be expected from Lomer’s

equation. Considering that this should be noticeable

in the high temperature region, it may not be realistic

in solids. The disagreement of Lomer’s expression

with our theory is also observed in Fig. 3, where the

apparent formation energy Ef is plotted against

the concentration of solute atoms: the one obtained

from equations (9) and (14) with I’i = 0, V, = 0.431

eV (500”K), vs = -0.431 eV (-500°K) and E, = 0.3539 eV (2947”K), and the other from equations

(14) and (18) with E, = 0.0215 eV (250°K) and

E, = 0.3539 eV.

Another set of solutions are obtained numerically

with different values of x, as an example of larger

values of Es, and the results are shown in Fig. 4.

The value for E, is the same as the one used in Figs.

2 and 3, and V, + ‘v, - Vs is taken to be 0.1723

eV (2000°K). The curves (f) and (g) in Fig. 4 are

obtained for x = 0.001 from equation (16), where the

value for Es is taken to be 0.362 or 0.225 eV, so that

0.8

0.6 q O.O43l eV

/ j , &T’K,

0123456 x10-3

x/(1+x)

FIQ. 3. The relation between the apparent formation energy of a vacency and the content of solute atoms in dilute alloys at various temperatures. The solid curves are obtained from equations (9) and (14), and the broken

ones from equations (14) and (18).

50 ACTA METALLURGICA, VOL. 20, 1972

I/T (I/OK)

FIG. 4. The same as Fig. 2 but with different values for z, V, and Vs, The curves (a)-(e) are obtained from

equations (9), and (f) and (g) from equation (16).

each curve may coincide with the curve (c) at T = 500

and lOOO”K, respectively.

Thus it is seen that there is no assurance that the

concentration of vacancies in dilute alloys follows

Lomer’s equation over a wide range of temperature,

and consequently various values of Es are obtained

in a dilute alloy depending on the temperature and

the concentration of solute atoms. It seems rather

casual that the agreement between the experimental

results and the calculated ones is remarkable in

Table 1. Recently, Schapink(12) showed the necessary

condition for Lomer’s equation to be valid:

C,exp 2 <I. ( 1

(23)

Comparing the results of our theory with the ones

from Lomer’s expression in the cases when the

condition (23) does or does not hold, we conclude that

at least this condition has to be remembered in

using Lomer’s expression (16).

(c) Vacancy concentration in concentrated alloys

In most cases the concentration of vacancies in a

concentrated alloy is formally expressed by

C, = exp 2 , ( )

and the formation energy E, is determined essentially

by measuring C, as a function of temperature.

Theoretically the equilibrium concentration of

vacancies in alloys has been considered by various

authors. Schapink c3) has calculated the one in a

homogeneous binary alloy and concluded that the

activation energy for formation of vacancy depends

upon the temperature. Furthermore, Krivoglaz and

Smirnov,(4) and Cheng et CA(~) have shown that at a

given temperature the concentration of vacancies in

a disordered state is higher than that in an ordered

or a clustered state.

The implication of our theory for the case will be

now explained. If y and jai] are very small, the

concentration of vacancies C, is given by equation

(19) also in the case of a concentrated alloy, and the

formation energy is given by equation (21). However,

if 1 cci( are comparatively large, equation (19) does no

more hold, and we have to obtain C, from the numeri-

cal solutions of equations (9). The apparent activation

energy thus calculated is shown in Fig. 5 as functions

of the temperature and the concentration of solute

atoms.

From equation (19) and Fig. 5 we see that the

activation energy of a vacancy formation is uniquely

determined by the elementary interactions and the

composition of the alloy, if the distribution of atoms

and vacancies is random, but that if the distribution

is heterogeneous, Ef depends not only on the con-

centration of solute atoms but also on the temperature.

In the cases exemplified in Fig. 5 there is a region of

the composition and temperature where the variation

of the apparent formation energy Ef is larger than

the uncertainty which is inherent to any experimental

data for the quantities of this sort. It appears that

equation (24) is a crude approximation for the

vacancy concentration in the alloys with a rather

0.7

S .%

w’

0.6

FIG. 5. The composition dependence of the apparent formation energy E, in equation (21) at various tem- peratures. The vacancy concentration is obtained from

the numerical solutions of equations (9).

KINOSHITA AND EGUCHI: ATOMS AND VACANCIES IN METALS AND ALLOYS 51

small ~oneentration of solute atoms and with a small

value of V, - V,, except for the case when they are

extremely small, where equation (24) is valid in the

form of equation (19). From these considerations

it is concluded that a caution must be paid when we

apply equation (24) for the vacancy concentration.

We should not be prejudiced that Ef should be a

constant.

From the analysis of our equations it is also seen

that the vacancy concentration decreases or increases

as a result of ordering or clustering. The vacancy

concentration in the ordered or the clustered alloys,

relative to the one in the disordered state, is shown in

Fig. 6.

4. ATOMIC AND VACANCY ARRANGEMENT IN BINARY ALLOYS

In the foregoing sections we have developed a

theory for the determination of the vacancy concen-

tration and the arrrangement of atoms and vacancies.

From the theory we find that a short range order or

clusters can develop according as V, < 0 or V, > 0.

We may find, furthermore, that the value of u1

mainly depends on the value of V, but hardly on the

values of V, and V,: however that CQ depend not only

on Vs and Va but also on 1;.

With typical sets of the interaction energies in

the cases of V,( V, - V,) > 0 and V1( V, - V,) < 0,

the variation of p,, with temperature is obtained and

shown in Fig. 7. The numerical values of z and

various interaction energies which have been employed

are given in each figure. It is interesting to note that

if V,( V, - l’s) has a positive value the probabilities

p,, and p,, decrease or increase monotonically with

temperature, while if it is negative these probabilities

may change signs of their derivatives at a certain

temperature.

V, =-0.0173 eV :

-O.O862eV,, I / /I

ZJ !(

/ / x =0.2 EF 0.254 eV V,- 0.0431 eV V3=-0.0431 eV

: I

I I

7 I

0

--02 I _ -2 -I

v, /qcT I 2

FIG. 6. The vacancy concentration, in the ordered or the clustered alloys, relative to the one in the disordered

state.

200 400 600 800 1000 1200 T (OK)

FIG. 7. Differential pus vs. temperature curves illustmt- ing its dependence on the interaction energy 8, - Vs. The values of 2, E, and V, we taken to be 0.2, 0.346 eV (4000°K) and -0.0172 eV (-200°K) for the alloys with

a short range order.

In the previous section we have obt,ained equation

(21) under the assumption that the vacancy concentra-

tion is constant along the solidus line. Although it

is possible to estimate the numerical values of

the interaction energies E,, VI and V, - V, from the

phase diagram of any binary system, in ‘view of the

approximate nature of equation (21) and the un-

certainty of the details of phase diagrams in many

cases, we do not try to estimate these, but investigate

the general relation between the signs of the energies

V, and Vz - V, and structures of solidus curves,

which is implied from equation (21).

From the first and second derivatives of equation

(21) with respect to x, we can see whether the solidus

curve is increasing or decreasing, and convex or

concave in the regions of the composition which are

close to the pure A, stoichiometric AB and pure B,

or in the vicinities of x/(1 + x) = 0, 0.5 and 1.

Figure 8 shows the profiles of solidus curves, which are

expected from equation (21), for the systems with a

complete solubility phase. The signs of the inter-

action energies are shown in the figures, and the

curves (a), (b), (e), (f) and (g) are those for the alloys

with negative VI, in which a short range order can

develop, and the curves (cf, (d), (h), (i) and (j) are

those for the positive VI, which results in clustering.

Consider the Cu-Al system as an example. The

well known phase diagram for the system indicates

that the solidus curve of its alpha phase is of the

structure of the curve (f) in Fig. 8, which corresponds

to the case when V,( V’s - V,) and V, are negative.

In fact a short range order can develop in c&u-Al,

which is in conformity with our conclusion that VI

52 ACTA METALLURGICA, VOL. 20, 1972

v&f-V&O

v 0 - 1.0 0 1.0 x/(1+x) x/(1*x)

FIQ. 8. Profiles of solidus curves, which are expected under the assumption that the concentration of vacancy is constant along the solidus cnrve. The cnrves (a)-(d) are those for the alloys with V,( V, - Va) > 0, and the

curves (e)-(j) for V1(F7, - V,) < 0.

is negative in this system. As we have seen above a

negative sign of V,( Vv, - Vs) may cause unusual

behavior of pus and p,,,. This might explain the

observed behavior of the electrical resistivity;(15)

namely, the resistivity in the furnace cooled poly-

crystalline specimen of Cu-15 at. % Al decreases in

two stages in the course of heating. From the struc-

ture of the phase diagrams similar behaviors are

expected also in c&u-Zn and uAg-Cd alloys. More

detailed analysis of the experimental fact along this

line is difficult without any kinetic consideration,

and an attempt for a kinetic theory of clustering

or local ordering in binary alloys with vacancies is

now in progress. 5. CONCLUSIONS

Using the method of statistical thermodynamics

we have constructed a theory with which we may find

the equilibrium concentration of vacancies and

divacancies and the configuration of atoms and

vacancies. The results of our calculation presented

here, we believe, give a more precise description on

the concentration of vacancies in pure metals, dilute

and concentrated alloys, than those which have

hitherto been suggested. The following conclusions concerning the con-

centration of vacancies and the arrangement of

atoms and vacancies in metals and alloys have been

reached : 1. The single and divacancy concentrations in

pure metals may almost precisely be described by the

familiar approximate expressions equations (14) and

(15), respectively. 2. For the vacancy concentration in a dilute alloy

the agreement between Lomer’s expression and the

result of our theory is poor. With this reservation

in mind a careful attention must be paid in using

Lomer’s expression equation (18).

3. In the alloys with a random distribution of

atoms and vacancies, the activation energy for the

formation of a vacancy determined uniquely by the

elementary interaction energies and the composition.

But in the alloys with a short range order or clusters

it depends not only on the concentration of the

solute atoms but also on the temperature.

4. The concentration of vacancies in alloys de-

creases or increases as a result of ordering or clustering.

5. The probabilities, pe)ua and pub, do not always

vary monotonically with temperature, but in some

cases increase after decreasing, or decrease after

increasing is expected, if the interaction energies

satisfy the condition Vi( V2 - V,) < 0. The sign of

V,( V, - V,) may be predicted from the nature of the

solidus curve in the phase diagram of the alloy in

question.

The theoretical treatment developed here is based

upon the assumption that atoms and vacancies in

the alloys interact mainly with their first nearest

neighbors. This assumption may be subject to the

criticism based on the electronic band theory of

metals. At present, however, it is difficult to treat

the present problem by the band theory because of the

lack of any dependable ones for alloys, and no matter

how our treatment is simple, its essential features

described above would be unaltered even by a more

elabolate theory than ours.

Furthermore, our theory describes only the vacancy

concentration and the atomic arrangement in the

equilibrium states for the cases of clusters and the

short range order in the absence of the long range

order. An attempt for a kinetic theory of the vacancy

concentration and atomic arrangement in binary

alloys is now in progress.

1.

2. 3. 4.

5.

6.

7. 8. 9.

10.

11.

12. 13. 14.

15.

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