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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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