Abstract The Gibbs free energy of substitution binary alloys containing lattice
vacancies is derived by using the statistical moment method . The equilibrium concentration of vacancies in substituion binary alloys is studied for large temperature
interval up to the melting temperature taking into account of the anharmonicity of
lattice vibrations. The Gibbs free energy and equilibrium concentration of vacancies
are derived in closed forms. The numerical calculations are performed for Al atom
in AlNi, AlCr, AlCo alloys and they agree well with the experimental data.

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Journal of Science of Hanoi National University of Education
Natural sciences, Volume 52, Number 4, 2007, pp. 64- 71
Equilibrium Concentration of vacancies
in substitution binary Alloys
Hoang Van Tich
Faculty of Physics, Hanoi University of Education
Abstract The Gibbs free energy of substitution binary alloys containing lattice
vacancies is derived by using the statistical moment method . The equilibrium con-
centration of vacancies in substituion binary alloys is studied for large temperature
interval up to the melting temperature taking into account of the anharmonicity of
lattice vibrations. The Gibbs free energy and equilibrium concentration of vacancies
are derived in closed forms. The numerical calculations are performed for Al atom
in AlNi, AlCr, AlCo alloys and they agree well with the experimental data.
1 Introductions
It is well known that vacancy is a type of lattice defect in metal and plays an important
role in the process of both self- and impurity - diffusions [1,8] . In order to understand the
diffusion and related phenomena from the theorical point of view, it is quite important to
investigate fundamental properties of vacancy, divacancy and small vacancy clusters. Hay-
dock et al, calculated the binding energy, atomic relaxation and electronic and vibrational
densities of states (DOS) on the atom near the vacancy in bcc transition metals using a
tight - binding (TB) type electronic theory [6,12].
The statistical moment method (SMM) has been widely used to investigate the thermo-
dynamic properties of anharmonic crystals, metals with defects and ordered binary alloys
[13-15]. In real crystals, the vacancy problems also have been received a great deal of in-
terests over the years by other methods [6-10].
In this paper, using the obtained results in the previous papers [13,18] we apply the SMM
to determine the expression of the Helmholtz free energy of the substitution binary alloys
with defects and the change of the Gibbs free energy in forming a single vacancy. We also
obtain the expression of the equilibrium concentration of vacancies in the substitution
binary alloys with the small concentration of substitution atoms . The obtained results
are applied to AlCr, AlNi, AlCo alloys. Our calculated results are compared with the ex-
perimental data.
64
Equilibrium concentration of vacancies in substttution binary alloys
2 Equilibrium concentration of vacancies in substitution bi-
nary alloys
Let us consider a system consisting NA of atoms A being on lattice knots and NB of atoms
B inserting in the distance between atoms A. The number of atoms in alloy is
N = NA +NB. (2.1)
The Helmholtz free energy of substitution binary alloy in approximation of the nearest-
neigbour distance of atoms is determined by
ψ =
[
N − (NB +
6∑
l=1
ml
]
ψA +NBψB +
6∑
l=1
mlψABl (2.2)
where l is the number of atoms B inserting in the neigbour of atom A andml is the number
of atom A belonging to the neigbour of l atoms B, ψA is the Helmholtz free energy of a
single atom in the perfect metal A, ψB is the Helmholtz free energy of a single atom B
surrounded by atoms A, ψAB is the Helmholtz free energy of a single atom A surrounded
by l atoms B. Wαβ denotes the probability of replacing to finding an atom α surrounded
by an atom β ( α, β can be atoms A, B or vacancy ν ). In the case of noninteraction, we
have the approximate expression
Wvα =
n
N
cα, cv =
n
N
, cα =
Nα
N
,
Wαβ = cα.cβ , cβ =
Nβ
N
, (α, β = A,B). (2.3)
From (2) and (3)(in the case
NB << NA
), the Helmholtz free energy of substitution binary alloys can be written in the approximate
form
ψ =
[
N −NB
(
1 +
m1
NB
)]
ψA +NBψB +m1ψAB1 (2.4)
where
m1 ∼= n1NBCA
. The number m1 contains a coefficient n1 depending a crystal structure (for the fcc metal
n1 = 6 and for the bcc metal n2 = 8 ) by consider a capacity inserting a single atom B
surrounded by atoms A in alloys.
Analogously, the average displacement of an atom in the alloy from equilibrium position
can be determined approximately by
y0 =
1
N
[(NA − n1NBcA) yA +NB (yB + n1yABcA)] (2.5)
where [14]
y2A =
2γθ2
3k3
A (T ) ; θ = kB .T ; (2.6)
65
HONG VN TCH
k =
1
2
∑
i
(
∂2ϕI0
∂u2iβ
)
cq
, γ = 112
∑
i
[
6
(
∂4ϕi0
∂u2iα∂u
2
iβ
)
cq
+
(
∂4ϕi0
∂u4iβ
)
cq
]
(2.7)
where φio is the interaction potential between zero-th atom and i-th atom (0-th atom is
selected as the origin) (α 6= β = x, y, z), yA is the displacement of a single atom in the
metal A [14], yB is the displacement of a single atom B surrounded by atoms A and yAB
is the displacement of a single atom A surrounded by an atom B. ( the expressions for yB
and yAB are determined as (6) )
The nearest- neigbour distance a is determined by
a = a0 + y0 (2.8)
where a0 is the nearest- neigbour distance at T = 0K . When alloy with defects containing
n vacancies on lattice knots , the Helmholtz free energy ψ in (4) is transformed into ψreal.
Combining with (3), we can easily determine the approximate form of . Thus, in order to
determine the number N1 of atoms A with the free energy ψ
1
A containing a single vacancy
on the first coordination sphere or N2 of atoms A with the free energy ψ
1
ABlcontaining
a single vacancy on the first coordination sphere and a single atom B sitting next to a
vacancy , we have to use the expression (3) and calculate the possibility of replacing an
atom A with an atom B being next to the a vacancy.
We have
N1 ∼= n′1
(
cA
n
N
)
NA = n
′
1c
2
An,
N2 ∼= n′1
(
cBcA
n
N
)
NA = n
′
1ncBc
2
A,
n
′
1 = 12. (2.9)
From (4) and (9), we can find the contribution in the Helmholtz free energy ψreal of atoms
A in alloys ( corresponding to the Helmholtz free energy ψA without a vacancy being on
the first coordination sphere )
(NA −m1NB −N1)ψA. (2.10)
Similarly, the number N3 of atoms B corresponding to the free energy ψ
1
B containing a
single vacancy on the first coordination sphere is determined by
N3 ∼= n1
(
cB
n
N
)
NA = n1ncBcA (2.11)
Finally, from (4) - (11), the Helmholtz free energy ψreal of the substitution binary alloys
has the form
ψreal =
[
NA − n1NBcA − n′1nc2A
]
ψA + n
′
1nc
2
Aψ
1
A + (NB − n1ncBcA)ψB+
+n1ncBcAψ
1
B +
(
NBn1cA − n′1ncBc2A
)
ψABl + n
′
1ncBc
2
Aψ
1
ABl + n∆
(2.12)
where ∆ is the change of the Helmholtz free energy when an atom leaves lattice knot to
form a single vacancy
∆ = (∗B − 1)ψA (2.13)
66
Equilibrium concentration of vacancies in substttution binary alloys
with
∗B is an parameter determined as in [18].
The change of the Gibbs free energy of an atom in forming a single vacancy ( n = 1)
gfv = G (P, T )−G0 (P, T ) (2.14)
here
G0(P, T ) = ψ0 + PV0
is the Gibbs free energy of perfect alloy containing N atoms in the volume V0 and
G(P, T ) = ψreal + PV
is the Gibbs free energy of alloy containing a single vacancy in the volume V .
From(4),(12) and (14), we can find
gfv = ψreal − ψ + P (V − V0).
gfv = −n′1c2AψA+n′1c2Aψ1A+∆+
[
n1cB(ψ
1
B − ψB) + n′1cBcA(ψ1ABl − ψABl)
]
cA+P (V −V0)
(2.15)
Using (15), we can find an equilibrium concentration of vacancies in binary alloy
p
{
−n1cBcA
θ
[
ψ1B − ψB
]
exp
{
−n
′
1cBc
2
A
θ
[
ψ1ABl − ψABl
]}
exp−P∆V
θ
}
(2.16)
where nAν is the equilibrium concentration of vacancies of the metal A given form as in
[18]
nAv = exp
−
[
−n′1c2AψA + n
′
1c
2
Aψ
1
A +∆
]
θ
(2.17)
( when cB = 0, nν = n
A
ν and cA = 1. Then, the expression (17) has form as in [18] in the
nearest - neigbour approximation ).
If neglecting influence of substitution atoms B on the Helmholtz free energy of atoms A
locating in their neigbour, the expression (4) has the form
ψ ∼= (N −NB)ψA +NBψB (2.18)
Thus, (16) will be changed into the simple form
nv = n
A
v exp
{
−n1cBcA
θ
[
ψ1B − ψB
] }
exp
{
− P (V − V 0)
θ
}
(2.19)
In the case, in the metal A, the defect type is the Frenkel defect (CA = 1 ).
When alloy only contains a type of atoms, corresponding with a single vacancy, we have
a single atom sitting next to a vacancy .These defects are called as the Frenkel defects.
The concentration of substitution atoms in (4) is approximately as the concentration of
vacancy in the metals with the Frenkel defects When
cB ∼= n
N
= nv
67
HONG VN TCH
, (19) will be changed into
nv = n
A
v exp
{
−n1nvcA
θ
[
ψ1B − ψB
] }
exp
{
− P (V − V 0)
θ
}
(2.20)
Because nν << 1 and at high temperatures, we obtain
n1nνA
θ
(
ψ1A − ψA
)
<< 1,
nν ∼= nAν exp
{
1− n1nν
θ
[
ψ1A − ψA
] }
exp
{
− P (V − V0)
θ
}
(2.21)
Considering a concentration equilibrium of vacancies of a system at the pressure P = 0,
we have
nv ∼= nAv
1[
1 +
n1(ψ1A−ψA)nAv
θ
]
(2.22)
For metals, the Helmholtz free energy of atoms always is negative and ψ1A − ψA > 0.
According to (22), we find
nv
nAv
< 1. It means that a concentration of vacancies of the
Frenkel defect always is smaller than that of Shotky defect.
Analogously, in the case when ψ1B − ψB > 0, from (19) we can realize that the decrease
of the concentration of vacancies in the substitution binary alloys corresponds with the
increase of the concentration of substitution atoms cB at determined temperature and
pressure .
3 Numerical calculation for AlCr, AlCo, AlNi alloys
From the expression of the free energy of a single atom in the metal A in [14]
ψA = 3
{
U0
6
+ θ
[
x− ln(1− e−2x)]} (3.1)
where U0 =
∑
ϕio (|ai|) .
Using the above expression to determine the free energies ψABl and ψB in the substitution
binary alloys
ψABl = 3
{
1
6
∑
i
ϕAiA (|ai|) +
1
6
ϕAB (|ai|) + θ
[
x1 − ln
(
1− e−2x1)]}
ψB = 3
{
1
6
∑
i
ϕAiB (|ai|) + θ
[
x2 − ln
(
1− e−2x2)]} (3.2)
Values of the free energies ψABl and ψB of alloy at the temperature T are absolutely
determined , when we use [16]
ϕAB(/ai/) =
1
2
[ϕAA(/ai/) + ϕBB(/ai/)]
68
Equilibrium concentration of vacancies in substttution binary alloys
with the attention
x =
~ω
2θ
, ω =
√
k
m
, x1 =
~ω1
2θ
, ω1 =
√
k1
m
;x2 =
~ω2
2θ
, ω2 =
√
k2
m
,
k1 =
1
2
∑
i
(
∂2ϕAiA
∂u2iβ
)
eq
+
(
∂2ϕAiB
∂u2iβ
)
eq
k2 =
1
4
∑
i
(
∂2ϕAiA
∂u2iβ
)
eq
+
(
∂2ϕBiB
∂u2iβ
)
eq
,
and numerical calculations of
yo, ai, ψ
1
A, ψABl, ψ
1
ABl...
are similar as in [13,18].
Using the form φio of interaction potential of atoms in metals Al, Cr, Co, Ni in [17] and the
obtained expressions in section 2, we easily find the values of equilibrium concentrations
of vacancies in AlCr, AlCo, AlNi alloys. The calculated results are presented in Table 1. In
the case of pure metals (cB = 0), the equilibrium concentration of vacancies of Al is 1,133.
10-4 near the melting temperature . This concentration is 9,4. 10-4 according to Simon [19]
and is 3. 10-4 according to Phyder and Novic [19]. At low temperatures, the equilibrium
concentration of vacancies is small .When the temperature increases, the concentration of
vacancies also increases strongly and it agrees with the experiment data .
For the AlCr, AlCo, AlNi alloys, when the concentration of substitution atoms in-
creases, the concentration of vacancies decreases it agrees with other calculated results
and experiments. Increasing the concentration of substitution atoms causes decreasing the
concentration of vacancies and stabilizing crystalline lattice.
69
HONG VN TCH
H¼nh 1: Table 1. Equilibrium concentration of vacancies in AlCr, AlCo, AlNi. alloys at
different temperatures and concentration of substitution atoms.
References
[1 ] Frenken. A.I., Staststical Physic , ( M ANSSSR ) (1948) )(in Russian).
[2 ] Frenken. A.I., Kinetic Theoryof Liquite, (Uzd ANSSSR ) (1945) 223.
[3 ] Frenken. A.I., Elementment Theory of Metals, M.Phyz- Machiz. (1958) (in Rus-
sian).
[4 ] Smirnov. A.A., JTF (1953)56-65.
[5 ] Smirnov. A.A., Kinetic Theory of Metals, M. Nauka (1966)(in Russian).
[6 ] Lomer. W.M., Vacancies and other defects in metals and alloys , London, Inst of
metals (1958)79-98.
[7 ] Smirnov. A.A., DAN .UNSSR, No 5, (1950)351-357(in Russian).
[8 ] Krivoklaz. M.A., Problemes of Metalophysicka and Metalurgia, No.7 (1956)95-104.
(in Russian).
[9 ] Nhescimenco E.G., Smirnov.A., Tamze. A., p.152-166.
[10 ] Popov I.I,., Starotchenkov.M.D.,Glasov I.S., Izv.Vuzov.Physica (1969)123-125.
[11 ] Smirnov. A.A., Metalophysicka,No. 4 (1990)8896, UFJ.No.9(1990)1354.
[12 ] Gusac A.M, FMM, Vo. 68 (1989)481-485.
[13 ] Tang. N. and Hung .V.V., Phys.Stat. Sol. 162 (1990)371-379.
[14 ] Tang. N. and Hung .V.V., Phys.Stat. Sol. (b) .149 (1988) 511, 161 (1990) 165.
[15 ] Tang.N., Khoa.H.D and Hung V.V., Communication in Physics, Vo.3, No.3 (1993)87.
70
Equilibrium concentration of vacancies in substttution binary alloys
[16 ] Mark Mostoller et all., Physical .Revew,Vol.19.No.8(1979)3938
[17 ] Mazomedov. M. M., J.Fiz Khimic, 61(1987)1003
[18 ] Hung .V. V., Communication in Physics, Vo .4, No.3 (1994)122.
[19 ] Ubelode .A ., Melting and Structura crystal, M (1969)303(in Russian).
TÂM TT
NÇNG Ë C
N BNG VACANCY CÕA HÑP KIM
TH HAI THNH PHN .
Ho ng V«n T½ch
B i b¡o n y tr¼nh b y vi»c sû döng ph÷ìng ph¡p thèng k¶ mæ men º t½nh £nh h÷ðng
cõa hi»u ùng phi i·u háa cõa dao ëng m¤ng l¶n nçng ë c¥n b¬ng vacancy cõa hñp kim
thay th¸ hai th nh ph¦n. Chóng tæi ¢ t¼m ÷ñc biºu thùc gi£i t½ch t½nh nçng ë c¥n b¬ng
vacancy v n«ng l÷ñng tü do Gibbs cõa hñp kim thay th¸ hai th nh ph¦n phö thuëc v o
nhi»t ë. C¡c k¸t qu£ ¢ ÷ñc ¡p döng t½nh sè vîi kim lo¤i thay th¸ l Al trong ba hñp
kim AlNi, AlCr, AlCo ð c¡c nhi»t ë kh¡c nhau.
71