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
DANS LES METAUX ET LES ALLIAGES
ET DES LACUNES
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&tiondes laounes peut 6tre exprimQ presque pa~aitement
psr l’expression courante approchee, mais deansles 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 iEcourte 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 temperature, 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
LEERSTELLEN
IN METALLEN
UND LEGIERUNGEN
UND
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 rtusgedrticktwerden; 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 beschriebenwerden 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
vacancies
for the equi~brium
in metals
and alloys
to construct
concentration
have been given
various papers and text books.(l)
of
a theory
whioh takes into account
the
arrangement of atoms and vacancies in the alloys
in which an order or clusters can develop.
in
These expressions
data,
Stimulated by these needs, several models have
been suggested for the determination of concentration
and they have been used with some success for the
of vacancies in the alloys with a long range order.(2-5)
qualitative
or semi-quantitative
various
experimental
results;
Furthermore, Cheng et aZ.(6)developed some models
to estimate the effect of a short range order or clusters
are convenient
expressions
analyses
of experimental
are still unsatisfactory
they are derived
dist~bution
for
interpretation
of
nevertheless
these
in the sense that
under the assumption
of vacancies
on
of a random
and atoms on lattice sites.
t Laboratory of Iron and Steel, Department of Metallurgy,
Kyushu University, Fukuoka, Japan.
$ Now at: Department of Nuclear Engineering, Kyushu
University, Fukuoka, Japan.
1972
concentration
of
These models, however,
vacancies
in
do not practi-
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
* Received May 27, 1971.
VOL. 29, JANUARY
equilibrium
cally give any i~ormation
on the arrangement of
atoms coordinated directly with vacancies.
In the present paper we apply Cowley’s methodc7*s)
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
ACTA METALLURGICA,
the
binary alloys.
45
46
ACTA
into account.
duced
in
Four statistical
order
vacancies
to
METALLURGICA,
parameters
describe
the
and the configuration
are intro-
concentration
of atoms and vacan-
cies on lattice sites, and these parameters
to satisfy the four transcendental
as the result of thermal equilibrium.
ture
quantities,
of solidus
curve,
of
specific
heat.
Once
mental equations are obtained
use of an electronic computer.
Cu-AI,
as practical
examples
metals, the vacancy
Warren’s
parameter
short
ordeP,‘J
will
range
vacancies
following
into
of the vacant
sites or
by our theory
almost
familiar
by
introducing
x
’
P
1 -
a2 =
a’
,
(1)
Y
1 +x+-y
Pbv
a,=l-
’
Y
1+x$-y
where pij
vacancy)
is the probability
is coordinated
that an atom j
among its nearest neighbors.
These probabilities
interrelated by the three identities :
and the energy to
Pb,
-t
Pbb
%a
+pvb
+
Pbv
+P,,
=’
l,
=
‘.
(2)
Counting the number of bonds of particular
see that the following
of
alloys,
the
elementary
however,
interaction
the
vacancy
energies.
conditions
In
concentration
Pa,
=
xPba,
Pm = 1/P,,,
limited
region of temperature
and concentration
P bu
solute
atoms.
the
vacant
sites in concentrated
fraction
of
of
alloys is very unlikely
that an A or a B atom is coordin-
ated to a vacancy does not always vary monotonically
with temperature,
decreasing
$.
In
order
to
obtain
the
system only the interactions
internal
is expected.
One of the anomalous changes in the electrical resistivity of alpha Cu-Al may be attributed
to the
unusual behavior of the number of vacancy-atom
pairs.
neighbors
are considered
equations
we obtain the internal energy as
as usual.
OF THE
EQUATIONS
FUNDAMENTAL
E-E,,=
-
2c1 +Nz+
x(x +
Y +
the
Using the above
al)Vl - y(x + y + a2)V2
+
In equation
xY(l
E,, = &Nz{E,a
+ X(2E,, -
E,,
following
v, = 2E,, -
tEaa +
Vv, = 2~%, -
(E,, + E,,,)
v, = 2E,, -
(E,, + E,,,),
notations :
total number of A atoms;
total number of B atoms;
-
a3)v3}.
(4)
(4) E,,, E,, VI, V, and V, are given by
E, = 2E,, -
NW
of
y) (~(1 + x + YJE,
We consider a perfect crystal of an alloy consisting
of A and B atoms plus vacancies.
We adopt the
N, = xNa,
energy
between the first nearest
but in some alloys an increase after
or a decrease after increasing
2. DERIVATION
=
(3)
the
to be expressed by an Arrhenius type equation.
The probability
types we
must be satisfied:
does not always agree with Lomer’s
expression,
and it is shown that the latter is valid only in a
Furthermore,
are
i%,+pub+l)av=l~
cies to form a divacancy
terms
(or a
to an atom i (or a vacancy)
form one defect and the binding energy of two vacan-
dilute
the
1+x+y
classical
may be obtained uniquely in
the
to take
Pnb
and the configura-
of the latter.
to represent
be generalized
consideration
a,=l-
in pure
The
number.
three parameters :
the con-
in alloys with clusters,
agrees with the corresponding
expressions for those quantities,
Furthermore,
alloys and alpha
concentration
which is predicted
coordination
a
results of our numerical calculation are compared with
those of the classical formulas in order to examine
divacancies
2,
is applied to the
to estimate
In pure metals the fraction
total number of vacancies ;
total number of lattice sites;
by the
and those in alloys with a short range order.
the validity
N = (1 + x + y)N,,
N, = YN~,
for the funda-
and divacancies
tion of atoms and vacancies
of
1972
interaction
numerically
aluminum
of the vacancies
the struc-
energy
the
The general theory thus developed
centrations
various
20,
at the maximum
energies are given then the solutions
cases of pure aluminum,
from,
formation
and the temperature
equations
The interaction
for example,
the
single vacancy
anomalous
are shown
functional
energies are related to, and estimated
thermodynamical
of
VOL.
Ebb)
E,,)}
(5)
KINOSHITA
AND
ATOMS
EGUCHI:
AND
E, is the infraction energy between the
corresponding $ pair. The motivation for the introduction 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
where
VACANCIES
XV1
_- 2
IN
METALS
2
-
kT(ln(1
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)
In (da1 + x + ya3))- da2 + Y + ~4
X In Ma2 + y + xa3)) - 241 - alI
X
al)) -
2xz4
Zy(1 -
a& In (~(1 -
ad In fxY:y(l -
az))
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 determine these parameters as functions of temperature
in the state of thermal equilibrium :
“3((1 +
d2E2 - 41 - al)Vl
5 +
--(x2 + (1 + 2y + a2N + y2 i- 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
+ $3 In MY + a2 + a3x:)) -241
X In (x(1 -
aI)) + 2(1 + x)(1 -
X In Ml
ad) + 2x(1 + x)(1 -
X ln (x@
-
a%)) -
(1 i
x In (1 + 2 + y)“> = 0,
x + Y)~
-
aI)
a2)
a3)
2 ln Ml -
a,))] = 0,
a,))> = 0,
XV,
- kT( ln (x(x + a1 + May))+ ln MY
2
+ a2 + 6(3x))- 2 In (xy(1 -
aa))) = 0.
(9)
The above equations include as a special case the
case of thermal equilibrium in an A-B system
namely, assuming y = 0 and
without vacancy;
a2 = a3 = 0 we obtain an equation for a = a1
which coincides with the one considered by Cowley.t7)
In the following sections we shall solve equations (9)
numerically for more complicated cases and discuss
about the concentration of vacancies and the arrangement of atoms and vacancies.
3. VACANCY
CONCENTRATIONS
METALS
AND ALLOYS
x In (1 + x+ Y) - (1 + a,~ + a,~)
x In (1 -t al% + azv) - +x1 + x + ya3)
X In (x(1 -
2 ln (x(1 -
+ alx + a,y)
-/- ln My + x2 + a$x)) -
b0nds.
47
ALLOYS
kT( In (1 + six -j- oz&
+ ln (x(x + tcl + a3y)) zv2
-
AND
IN
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 +
w
Y)
+ 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)
ACTA
48
where Ef is the formation
and Es, is the binding
form a &vacancy.
Comparing
METALLURGICA,
energy of a single vacancy
energy
of two vacancies
to
VOL.
1972
(b) Vacancy concentration in dilute binmy alloys
LomeG)
vacancy
equations
20,
has
coincide
Ef = 3E,,
(16)
and
E,, = 6V,,
(17)
C, = CF)
obtained
and
(13)
are the
is valid
results
of
an
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
and
divacancy
obtained
of
concentrations,
from
equations
Newton-Raphson
FACOM 230-60
University,
of
for
the
with
the
are shown
using
have
the
an electronic
Computer
in Fig.
concentration
(18)
in the pure
set of the interaction
expression
energies
for the quantity
Assuming
can be
laij and y
to be much smaller than x, we find from equations
Center,
been
xcv,-
xv1
( (E2-
c, = exp 2;
For sufficiently
method
temperatures,
computer
m-
V,)
1)
(1+x)
.
(19)
with Lomer’s
dilute alloys and at sufficiently
we find that
expression
Kyushu
1, with which
(9)
that
The vacancy
which
(9)
EB’
( kT 1) ’
zC, + zC, exp
also from our theory.
in order to obtain the concen-
tration of vacancies from our equations.
1 -
With an appropriate
the corresponding
which
(
where Cp) is the vacancy
cubic metal.
(12)
expression
A metal, EB the binding energy between a vacancy
and a solute atom, C, the fraction of solute B atoms.
where we have taken z = 12, assuming a face centered
Equations
an
in dilute binary alloys :
(12) and (13) with equations
(14) and (15), we see that these equations
with each other if we put
approximation
proposed
concentration
high
(19) coincides
(18) if we take EB as
Es = t(V,
those
equation
+ V, -
V,).
(20)
obtained from equations (14) and (15) essentially coincide. The interaction energies are chosen for pure
that the vacancy
aluminum as Ef = 0.76 eV (8841”K)(g) and E,, = 0.17
solidus
eV
numerical value for the binding energy EB. Under
the same assumption and equation (19) the value of
(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,
temperature
range of practical
-2,
IO?0 7?0
H
VI + V, -
with the assumption
is constant
Sprusil
et aZ.03) have
V, may be obtained
along the
estimated
and compared
the
with
that of Es. The formation energy of a single vacancy
Ef in this case is given from equation (19) as
over the whole
x(V’, - v,,
(1 + 4
=
&,Tm@),
(21)
30;
where k, = Ef/T,(0),
E, =0.76 eV
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
$-6
g-8.
SlO
curve,
(18) combined
concentration
importance.
T (OK)
500
ST0
-4 -
From equation
v, +
-
v, -
- 4lcs
v,
=
___
T,‘(O)
2
where T,‘(O)
-12 -
represents
solidus curve at x = 0.
-14 -
numerical
-16 -18 -
LX
I.0
2.0
I/T
3.0
10-3
(I/OK)
Fra. 1. Equilibrium concentration of vacency or divrtoancy as * function of temperature in pure metals.
the tangential
(22)
slope of the
Thus we have seen that the
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
49
ALLOYS
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
B and EyP are the values theoretically given by Sprusil and Valvoda and the experimental ones, respectively
Solute atom
$(V,
+ 8,
(eY)
ET-“’ (eV)
-
Zn
0.11
0.06
V3)
Eexp
B (eV)
assumptions
Ag
Si
Sn
0.13
0.30
0.35
0.36
0.28
0.31
0.31
0.06
0.10
0.12
0.08
0.2,
0.15-0.25,
0.18
kO.01
0.1-0.4,
0.3-0.4
0.3
(9) are obtained
numerically
method explained in the last subsection.
The ratio of the vacancy concentration
alloys
CU
0.06,
are not valid, the most stable solutions
for equations
Mg
to the one in pure metals
by the
as a function
of
through the energy
E, with the other three variables
Considering
that this should be noticeable
in the high temperature
in solids.
region, it may not be realistic
The disagreement
of Lomer’s
temperature
is shown in Fig. 2, in which the curve (a)
with our theory is also observed
is obtained
from
apparent
Lomer’s
expression
equation
(16)
formation
with EB = 0.0215 eV (250”K),
and (b)-(d)
our
result from equations (9) with three different sets of
from equations
values
eV (500”K),
for
I’,
and
I’,,
0.0862 eV (1000°K).
keeping
The energy
V, +
vs -
I’s =
E, is taken to be
0.2539 eV (2947’K) as implied in the above analysis
(1) and alsosupported by an experimental result.@‘)
It is seen in Fig. 2 that the agreement
Lomer’s
between
expression and our theory is poor even in the
case of small values of
has been understood
EB, where Lomer’s expression
to be applicable.
An increase in
the concentration
energy
and
(18)
eV (-500°K)
with
Another
set of solutions
with different
and
numerically
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.
V, +
400
are obtained
values of x, as an example
The curves
(“K)
500
E, =
equations
E, = 0.0215 eV (250°K)
2 and 3, and
T
and
from
E, = 0.3539eV.
eV (2000°K).
700
against
the one obtained
and the other
temperature
1000
Ef is plotted
(9) and (14) with I’i = 0, V, = 0.431
vs = -0.431
0.3539eV (2947”K),
(14)
expression
in Fig. 3, where the
of solute atoms:
the ratio which is seen in the curve (b) in the higher
side, is because of the variable y couple
0.35,
6.4
in equations (9)) and can not be expected from Lomer’s
equation.
in dilute
0.3
‘v, -
Vs is taken
to be 0.1723
(f) and (g) in Fig. 4 are
obtained
for x = 0.001 from equation
value for
Es is taken to be 0.362 or 0.225 eV, so that
(16), where the
0.8
z-
Fo.3
u’
G
9
(b) VfO.0689
y-O.0173
(c) W-V3
=0.0431
(d) &=0.0173
‘&-0.0689
0.2-
eV
eV
eV
eV
qO.O43l
0.6
eV
i
“’t
(b)
/
/
(0) EB=0.0215 eV
_-_-----3,0 x to-3
2.0
I/T
(I/OK)
___
I .o
FIG. 2. The rstio of the vacancy concentration in dilute
alloys to the one in pure metals es e function of temperature.
The curves (b)-(d) are calculated from equations (9), and (a) from Lomer’s Equation (16).
4
, &T’K,
j
x10-3
0123456
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).
ACTA
50
METALLURGICA,
VOL.
20,
1972
vacancies
in alloys
authors.
Schapink c3) has calculated
homogeneous
has been considered
binary
alloy
by various
the
one
and concluded
in a
that
the
activation energy for formation of vacancy depends
upon the temperature.
Furthermore, Krivoglaz and
and Cheng et CA(~) have shown that at a
Smirnov,(4)
given temperature
a disordered
or a clustered
of vacancies
explained.
concentration
of our theory for the case will be
If y and
of vacancies
jai] are very
C, is given
(19) also in the case of a concentrated
formation
in
state.
The implication
now
the concentration
state is higher than that in an ordered
small,
alloy, and the
energy is given by equation (21).
if 1cci( are comparatively
the
by equation
large, equation
However,
(19) does no
more hold, and we have to obtain C, from the numerical 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
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
Thus it is seen that there is no assurance that the
concentration
of vacancies
equation
in dilute
the concentration
the
equation
C,exp
from
ones
is remarkable
(
2
1
expression
in
formation
determined
by the elementary
composition
of the alloy, if the distribution
is random,
is heterogeneous,
is uniquely
interactions
and the
of atoms
but that if the distribution
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
equation
vacancy
(24)
is a
concentration
<I.
of this sort.
crude
It appears that
approximation
in the alloys
for
the
with a rather
(23)
the
with the ones
cases
when
the
0.7
condition
(23) does or does not hold, we conclude that
S
.%
at least
this
w’
using Lomer’s
(19) and Fig. 5 we see that the
energy of a vacancy
to be valid:
the results of our theory
Lomer’s
in
Schapink(12) showed the necessary
for Lomer’s
Comparing
and
It seems rather
between the experimental
calculated
Table 1. Recently,
condition
on the temperature
of solute atoms.
casual that the agreement
and
follows
values of Es are obtained
various
in a dilute alloy depending
results
alloys
over a wide range of temperature,
and consequently
equation
activation
and vacancies
and lOOO”K, respectively.
Lomer’s
atoms.
From
condition
has to
expression
(16).
be remembered
in
0.6
(c) Vacancy concentration in concentrated alloys
In most cases the concentration
concentrated
alloy is formally
C, = exp
(
of vacancies
in a
expressed by
2
)
,
and the formation energy E, is determined essentially
by measuring C, as a function of temperature.
Theoretically
the equilibrium
concentration
of
FIG. 5. The composition dependence of the apparent
formation energy E, in equation (21) at various temperatures. The vacancy concentration is obtained from
the numerical solutions of equations (9).
KINOSHITA
small ~oneentration
value of V, extremely
form
EGUCHI:
AND
ATOMS
AND
VACANCIES
IN
METALS
AND
ALLOYS
51
of solute atoms and with a small
V,, except for the case when they are
small, where equation
of equation
it is concluded
(19).
(24) is valid in the
From
these
considerations
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
that the vacancy
concentration
as a result of ordering
concentration
it is also seen
decreases or increases
or clustering.
The vacancy
in the ordered or the clustered
relative to the one in the disordered
alloys,
state, is shown in
Fig. 6.
4. ATOMIC
In
the
200
AND VACANCY
ARRANGEMENT
IN BINARY
ALLOYS
foregoing
sections
we have
developed
a
theory for the determination
of the vacancy concenof atoms and vacancies.
From the theory we find that a short range order or
clusters can develop
may
find,
according
furthermore,
as V, < 0 or V, > 0.
that
the
value
of
u1
mainly depends on the value of V, but hardly on the
values of V, and V,: however that CQdepend 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
p,,
decrease
and p,,
value the probabilities
or increase
monotonically
with
temperature,
while if it is negative these probabilities
may
signs of their derivatives
change
at a certain
temperature.
/
/
6.
section we have obt,ained equation
tion is constant
along the solidus line.
is possible
estimate
to
the
Although
numerical
values
it
of
the interaction
energies E,, VI and V, -
phase diagram
of any binary system, in ‘view of the
approximate
certainty
nature
of equation
V, from the
(21) and the un-
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
which is implied from equation
of solidus
curves,
(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
in the regions of the composition
vicinities
of
convex
or
which are
AB and pure B,
x/(1 + x) = 0, 0.5
and
1.
expected
from equation
complete
solubility
(21), for the systems with a
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
I
I
develop,
VI, in which a short range order can
and the curves (cf, (d), (h), (i) and (j) are
those for the positive
I
7
0
I
2
--02
_
v, /qcT
The vacancy concentration, in the ordered or the
clustered alloys, relative to the one in the disordered
state.
FIG.
1200
Figure 8 shows the profiles of solidus curves, which are
I
-I
In the previous
or in the
ZJ
!(
-2
1000
(21) under the assumption that the vacancy concentra-
close to the pure A, stoichiometric
I
-O.O862eV,, I
/ /I
800
(OK)
FIG. 7. Differential pus vs. temperature curves illustmting 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.
concave
x =0.2
EF 0.254 eV
V,- 0.0431 eV
V3=-0.0431 eV
V, =-0.0173 eV
:
600
T
tration and the arrrangement
We
400
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
ACTA
52
METALLURGICA,
VOL.
3. In
v&f-V&O
atoms
20,
the
1972
alloys
with
and vacancies,
formation
of a vacancy
elementary
a random
distribution
the activation
interaction
of
energy for the
determined
uniquely
by the
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,
vary
v
0
0-
1.0
1.0
x/(1+x)
cases
x/(1*x)
is negative
in this system.
sign
behavior
of pus and p,,,.
observed
of
behavior
namely,
of
specimen
This
the
the resistivity
crystalline
Vs) may
electrical
of Cu-15
of the
phase
diagrams
explain
cooled
the
poly-
at. % Al decreases in
From the struc-
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
and an attempt
for a kinetic
or local ordering
in binary
kinetic
consideration,
theory
alloys
of clustering
with vacancies
is
5.
we
have
constructed
the equilibrium
divacancies
and
vacancies.
and
of statistical
a theory with which we may find
concentration
of
the configuration
vacancies
of atoms
concentrated
alloys,
than
those
concerning
centration
the
of
vacancies
atoms and vacancies
reached :
1. The
single
and
and
presented
give a more precise description on
of vacancies in pure metals, dilute
hitherto been suggested.
The following
conclusions
and
which
the
arrangement
have
conof
in metals and alloys have been
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).
but in some
decrease
if the interaction
curve in the phase diagram
The theoretical
upon
treatment
the assumption
the alloys
interact
neighbors.
after
energies
of the alloy in
metals.
developed
that atoms
mainly
with
This assumption
criticism
based
on
At present,
the
their
in
first nearest
be subject
may
electronic
however,
here is based
and vacancies
band
to the
theory
it is difficult
of
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
elabolate
theory than ours.
Furthermore,
concentration
order.
thermodynamics
The results of our calculation
here, we believe,
the concentration
or
even by a more
our theory describes only the vacancy
and the atomic
arrangement
states for the cases of clusters
short range order in the absence
CONCLUSIONS
the method
decreasing,
is expected,
equilibrium
now in progress.
Using
after
question.
resistivity;(15)
in the furnace
two stages in the course of heating.
ture
solidus
cause unusual
might
pe)uaand pub, do not always
with temperature,
satisfy the condition Vi( V2 - V,) < 0. The sign of
V,( V, - V,) may be predicted from the nature of the
As we have seen above a
V,( Vv, -
increase
increasing
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.
negative
monotonically
in the
and the
of the long range
An attempt for a kinetic theory of the vacancy
concentration
and
atomic
arrangement
in
binary
alloys is now in progress.
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p. 1. Gordon and Breech (1963).
2. L. A. GIRIFALCO, J. phys. Chem. Solids 24, 323 (1964).
F.
W. SCHAPINK, Phil. Mug. 12, 1055 (1965).
3.
4. M. A. KRIVOQLAZ and A. A. SMIRNOV, The Theory of
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Elsavier (1965).
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