Abstract. The analytic expressions of the free energy, the mean nearest neighbor distance
between two atoms, the elastic moduli such as the Young modulus, the bulk modulus,
the rigidity modulus and the elastic constants for interstitial alloy AC and interstitial
alloy ABC (substitution alloy AB with interstitial atom C) with FCC structure at zero
pressure are derived using the statistical moment method. We apply the theoretical results
to the interstitial alloys AuLi and AuCuLi at zero pressure and at different temperatures,
concentrations of substitution atoms and interstitial atoms. Calculated results for the main
metal in these alloys are compared with the experimental data and other calculations.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0031
Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 47-57
This paper is available online at
ELASTIC DEFORMATION OF BINARY AND TERNARY INTERSTITIAL
ALLOYS WITH FCC STRUCTURE AT ZERO PRESSURE: DEPENDENCE
ON TEMPERATURE, CONCENTRATION OF SUBSTITUTION ATOMS
AND CONCENTRATION OF INTERSTITIAL ATOMS
Nguyen Quang Hoc1, Bui Duc Tinh1, Le Dinh Tuan2 and Nguyen Duc Hien3
1Faculty of Physics, Hanoi National University of Education
2Faculty of Natural Sciences, Hong Duc University
3Mac Dinh Chi High School, Chu Pah District, Gia Lai Province
Abstract. The analytic expressions of the free energy, the mean nearest neighbor distance
between two atoms, the elastic moduli such as the Young modulus, the bulk modulus,
the rigidity modulus and the elastic constants for interstitial alloy AC and interstitial
alloy ABC (substitution alloy AB with interstitial atom C) with FCC structure at zero
pressure are derived using the statistical moment method. We apply the theoretical results
to the interstitial alloys AuLi and AuCuLi at zero pressure and at different temperatures,
concentrations of substitution atoms and interstitial atoms. Calculated results for the main
metal in these alloys are compared with the experimental data and other calculations.
Keywords: Binary and ternary interstitial alloy, statistical moment method, Young
modulus, bulk modulus, rigidity modulus, elastic constant.
1. Introduction
The elastic deformation of metals and alloys has attracted the attention of many researchers
[1-16]. In this paper the authors have used the statistical moment method (SMM) [1, 2, 16] to
derive the analytic expressions of the free energy, the mean nearest neighbor distance between
two atoms, the elastic moduli such as the Young modulus E, the bulk modulus G and the rigidity
modulus K, and the elastic constants C11, C12, C44 depending on temperature, concentration of
substitution atoms and concentration of interstitial atoms for interstitial alloy AC and interstitial
alloy ABC (substitution alloy AB with interstitial atom C) with an FCC structure at zero pressure.
We applied the theoretical results to the interstitial alloys AuLi and AuCuLi at zero pressure and
at different temperatures, concentration of substitution atoms and interstitial atoms. Some of the
calculated results for the main metals in the alloys are compared with the experimental data and
the other calculations.
Received March 1, 2016. Accepted July 25, 2016.
Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn
47
Nguyen Quang Hoc, Bui Duc Tinh, Le Dinh Tuan and Nguyen Duc Hien
2. Content
2.1. Elastic deformation of interstitial alloy AC with FCC structure
The bound energy of atom C (in the body center of a cubic unit cell) with atoms A (in the
face centers and peaks of the cubic unit cell) in the approximation of three coordination spheres
with a center C and radii r1, r1
√
3, r1
√
5 is determined by [1, 2]
u0C =
ni∑
i=1
ϕAC (ri) = 6ϕAC (r1) + 8ϕAC
(
r1
√
3
)
+ 24ϕAC
(
r1
√
5
)
, (2.1)
where ϕAC is the interaction potential between atom A and atom C, r1 ≡ r1C = r01C + y0A1(T )
is the nearest neighbor distance between interstitial atom C and metallic atom A at temperature T,
r01C is the nearest neighbor distance between interstitial atom C and metallic atom A at 0 K and
is determined from the minimum condition of the bound energy u0C , y0A1(T ) is the displacement
of atom A1 (atom A stays in the face center of cubic unit cell) from equilibrium position at
temperature T. The alloy’s parameters for atom C in the approximation of three coordination
spheres have the form [1, 2]
kC = ϕ
(2)
AC (r1) +
2
r1
ϕ
(1)
AC (r1) +
4
3
ϕ
(2)
AC
(
r1
√
3
)
+
8
√
3
9r1
ϕ
(1)
AC
(
r1
√
3
)
+4ϕ
(2)
AC
(
r1
√
5
)
+
8
√
5
5r1
ϕ
(1)
AC
(
r1
√
5
)
,
γC = 4 (γ1C + γ2C) ,
γ1C =
1
24
ϕ
(4)
AC(r1) +
1
4r21
ϕ
(2)
AC(r1)−
1
4r31
ϕ
(1)
AC(r1) +
1
54
ϕ
(4)
AC(r1
√
3)
+
2
√
3
27r1
ϕ
(3)
AC(r1
√
3)− 2
27r21
ϕ
(2)
AC(r1
√
3)
+
2
√
3
81r31
ϕ
(1)
AC(r1
√
3) +
17
150
ϕ
(4)
AC(r1
√
5) +
8
√
5
125r1
ϕ
(3)
AC(r1
√
5)
+
1
25r21
ϕ
(2)
AC(r1
√
5)−
√
5
125r31
ϕ
(1)
AC(r1
√
5),
γ2C =
1
2r1
ϕ
(3)
AC(r1)−
3
4r21
ϕ
(2)
AC(r1) +
3
4r31
ϕ
(1)
AC(r1) +
1
4
ϕ
(4)
AC(r1
√
2)
+
√
2
8r1
ϕ
(3)
AC(r1
√
2) +
7
8r21
ϕ
(2)
AC(r1
√
2)
− 7
√
2
16r31
ϕ
(1)
AC(r1C
√
2) +
4
25
ϕ
(4)
AC(r1
√
5) +
26
√
5
125r1
ϕ
(3)
AC(r1
√
5)
− 3
25r21
ϕ
(2)
AC(r1
√
5) +
3
√
5
125r31
ϕ
(1)
AC(r1
√
5), (2.2)
48
Elastic deformation of binary and ternary interstitial alloys with fcc structure at zero pressure...
where ϕ(k)AC(ri) ≡ ∂
kϕAC
∂rk
≤i
(k = 1, 2, 3, 4).
The bound energy of atom A1 (which contains interstitial atom C on the first coordination
sphere) with the atoms in a crystalline lattice and the corresponding alloy’s parameters in the
approximation of three coordination spheres with the center A1 is determined by [1, 2]
u0A1 = u0A + ϕAC (r1A1) ,
kA1 = kA + ϕ
(2)
AC (r1A1) , γA1 = 4 (γ1A1 + γ2A1) , γ1A1 = γ1A +
1
24
ϕ
(4)
AC(r1A1),
γ2A1 = γ2A +
1
4r1A1
ϕ
(3)
AC(r1A1)−
1
2r21A1
ϕ
(2)
AC(r1A1) +
1
2r31A1
ϕ
(1)
AC(r1A1), (2.3)
where r1A1 ≈ r1C is the nearest neighbor distance between atom A1 and atoms in the crystalline
lattice.
The bound energy of atom A2 (which contains the interstitial atom C on the first
coordination sphere) with the atoms in the crystalline lattice and the corresponding alloy’s
parameters in the approximation of three coordination spheres with the center A2 is determined by
[1, 2]
u0A2 == u0A + ϕAC (r1A2) ,
kA2 = kA +
1
6
ϕ
(2)
AC (r1A2) +
23
6r1A2
ϕ
(1)
AC (r1A2) , γA2 = 4 (γ1A2 + γ2A2) ,
γ1A2 = γ1A +
1
54
ϕ
(4)
AC(r1A2) +
2
9r1A2
ϕ
(3)
AC(r1A2)−
2
9r21A2
ϕ
(2)
AC(r1A2) +
2
9r31A2
ϕ
(1)
AC(r1A2),
γ2A2 = γ2A +
1
81
ϕ
(4)
AC(r1A2) +
4
27r1A2
ϕ
(3)
AC(r1A2) +
14
27r21A2
ϕ
(2)
AC(r1A2)−
14
27r31A2
ϕ
(1)
AC(r1A2),
(2.4)
where r1A2 = r01A2 + y0C(T ), r01A2 is the nearest neighbor distance between atom A2 and
atoms in the crystalline lattice at 0 K and is determined from the minimum condition of the bound
energy u0A2 , y0C(T ) is the displacement of atom C at temperature T. In Eqs. (2.3) and (2.4),
u0A, kA, γ1A, γ2A are the coressponding quantities in clean metal A in the approximation of two
coordination sphere [1, 2].
The nearest neighbor distances r1X(0, T )(X = A,A1, A2, C) in the interstitial alloy at
pressure P = 0 and temperature T are derived from [16]
r1A(0, T ) = r1A(0, 0) + yA(0, T ), r1C (0, T ) = r1C(0, 0) + yC(0, T ),
r1A1(0, T ) ≈ r1C(0, T ), r1A2(0, T ) = r1A2(0, 0) + yC(0, T ) (2.5)
r1X(0, 0)(X = A,A1, A2, C) is determined from the equation of state or the minimum condition
of bound energy. From the r1X(0, 0) obtained with the use of Maple software, we can determine
the parameters kX(0, 0), γX (0, 0), ωX (0, 0) at 0 K. After that, we can calculate the displacements
[1, 2]
y0X(0, T ) =
√
2γX(0, 0)θ2
3k3X(0, 0)
AX(0, T ),X = A,A1, A2, C, (2.6)
49
Nguyen Quang Hoc, Bui Duc Tinh, Le Dinh Tuan and Nguyen Duc Hien
where θ = kBT, kB is the Boltzmann constant and has the form as in [1, 2]. Then, we calculate
the mean nearest neighbor distance in interstitial alloy AC using the expressions as follows [16]
r1A(0, T ) = r1A(0, 0) + y(0, T ), r1A(0, 0) = (1− cC) r1A(0, 0) + cCr′1A(0, 0),
r′1A(0, 0) ≈
√
3r1C(0, 0),
y(0, T ) = (1− 15cC) yA(0, T ) + cCyC(0, T ) + 6cCyA1(0, T ) + 8cCyA2(0, T ), (2.7)
The Young modulus of interstitial alloy AC is determined by
EAC (cC , T ) = EA (cC , T )
1− 15cC + cC ∂
2ψC
∂ε2
+ 6
∂2ψA1
∂ε2
+ 8
∂2ψA2
∂ε2
∂2ψA
∂ε2
,
EA (cC , T ) =
1
π.r1 (cC , T )A1 (cC , T )
,
A1 (cC , T ) =
1
kA (cC , T )
[
1 +
2γ2A (cC , T ) θ
2
k4A (cC , T )
(
1 +
1
2
XA
)
(1 +XA)
]
,
XA ≡ xA cothxA, xA = ~ωA (cC , T )
2θ
,
∂2ψX
∂ε2
=
{
1
2
∂2U0X
∂r21
+
3
4
~ωX
kX
[
∂2kX
∂r21
− 1
2kX
(
∂kX
∂r1
)2]}
4r201+
+
(
1
2
∂U0X
∂r1
+
3
2
~ωXcthxX
1
2kX
∂kX
∂r1
)
2r01, xX =
~ωX
2θ
, ωX =
√
kX
m
, (2.8)
where ε is the relative deformation, cC is the concentration of interstitial atoms C, EA is the
Young modulus of pure metal A, ψA is the free energy of atom A in pure metal A, and ψX(X =
C,A1, A2) is the free energy of atom X in interstitial alloy and has the form in [1, 2], U0X =
Nu0X .
The bulk modulus and the rigidity modulus of interstitial alloy AC have the form
KAC (cC , T ) =
EAC (cC , T )
3(1 − 2νA) , GAC (cC , T ) =
E (cC , T )
2 (1 + νA)
. (2.9)
The elastic constants of interstitial alloy AC have the form
C11AC (cC,T ) =
E (cC,T ) (1− νA)
(1 + νA) (1− 2νA) ,
C12AC (cC,T ) =
E (cC,T ) νA
(1 + νA) (1− 2νA) , C44AC (cC , T ) =
E (cC , T )
2 (1 + νA)
. (2.10)
The Poisson ratio of interstitial alloy AC has the form
νAC = cAνA + cCνC ≈ νA, (2.11)
where νA and νC are the Poisson ratio of the materials A, C and cA is the concentration of main
atoms A.
50
Elastic deformation of binary and ternary interstitial alloys with fcc structure at zero pressure...
2.2. Elastic deformation of ABC interstitial alloy with FCC structure
The mean nearest neighbor distance between two atoms in the interstitial alloy ABC with
FCC structure at temperature T is given by
aABC = cACaAC
BTAC
BT
+cBaB
BTB
BT
, BT = cACBTAC+cBBTB, cAC = cA+cC , aAC = r1A(0, T ),
BTAC =
1
χTAC
, χTAC =
3
(
aAC
a0AC
)3
2a2
AC
3
√
2a3
AC
(
∂2ψAC
∂a2
AC
)
T
,
(
∂2ψAC
∂a2AC
)
T
=
[
∂2ψAC
∂r1A2(0, T )
]
T
≈ (1− 15cC)
(
∂2ψA
∂a2A
)
T
+cC
(
∂2ψC
∂a2C
)
T
+ 6cC
(
∂2ψA1
∂a2A1
)
T
+ 8cC
(
∂2ψA2
∂a2A2
)
T
,
(
∂2ψX
∂a2X
)
T
=
(
∂2ψX
∂r21X(0, T )
)
T
,
1
3N
(
∂2ΨX
∂a2X
)
T
=
1
6
∂2u0X
∂a2X
+
~ωX
4kX
[
∂2kX
∂a2X
− 1
2kX
(
∂kX
∂aX
)2]
.
(2.12)
The Young modulus of interstitial alloy ABC is determined by
EABC = EAB − (cA + cB)EA + EAC , EAB = cAEA + cBEB,
EAC = EA
1− 15cC + cC ∂
2ψC
∂ε2
+ 6
∂2ψA1
∂ε2
+ 8
∂2ψA2
∂ε2
∂2ψA
∂ε2
, (2.13)
where EAB is the Young modulus of substitution alloy AB and EAC is the Young modulus of
interstitial alloy AC.
The bulk modulus and rigidity modulus of interstitial alloy ABC have the form
KABC =
EABC
3(1− 2νABC) , GABC =
EABC
2 (1 + νABC)
. (2.14)
The elastic constants of interstitial alloy ABC have the form
C11ABC =
EABC (1− νABC)
(1 + νABC) (1− 2νABC) , C12ABC =
EABCνABC
(1 + νABC) (1− 2νABC) ,
C44ABC =
EABC
2 (1 + νABC)
. (2.15)
The Poisson ratio of interstitial alloy ABC has the form
νABC = cAνA + cBνB + cCνC ≈ cAνA + cBνB = νAB. (2.16)
where cB is the concentration of substitution atoms B and νAB is the Poisson ratio of substitution
alloy AB
51
Nguyen Quang Hoc, Bui Duc Tinh, Le Dinh Tuan and Nguyen Duc Hien
2.3. Numerical results and discussion for alloys AuLi and AuCuLi
For metallic crystal, we can apply the n-m pair potential in the form as follows [1]
ϕ (a) =
D
n−m
[
m
(r0
a
)n − n(r0
a
)m]
, (2.17)
where the potential parameters and the Poisson ratio for materials Au, Cu and Li are presented in
Table 1 [3].
Table 1. The n-m potential parameters and the Poisson ratio for materials Au, Cu, Li [3]
Material m n D (10−16 erg) r0 (10−10 m) ν
Au 5.5 10.5 6462.54 2.8751 0.39
Cu 5.5 11 4693.518 2.5487 0.37
Li 1.66 3.39 6800.502 3.0077
For the interaction between the main atom A and the interstitial atom C in the
interstitial alloy,
ϕAC(a) ≈ 1
2
[ϕAA(a) + ϕCC(a)] . (2.18)
According to Figures 1 and 2 at the same temperature when the concentration of interstitial
atoms increases, the mean nearest neighbor distance in alloy AuLi increases. In the same
concentration of interstitial atoms when the temperatures increases, the mean nearest neighbor
distance in alloy AuLi increases. The increase in concentration of interstitial atoms is stronger
than the increase with temperature (when the concentration of intersitial atoms increases from
0 to 5%, the mean nearest neighbor distance increases from 2.8402 to 2.9457 A˚ but when the
temperature increases from 50 to 1000 K, the mean nearest neighbor distance increases from
2.8402 to 2.8848 A˚). At zero concentration of interstitial atoms, we obtain the nearest neighbor
distance of pure metal Au in [1].
According to Figures 3 and 4 at the same temperature when the concentration of interstitial
atoms increases, the elastic moduli E, K and the elastic contants of alloy AuLi decrease. In the
same concentration of interstitial atoms, when the temperatures increases, the elastic moduli E, K
of alloy AuLi decrease. At zero concentration of interstitial atoms, we obtain the elastic moduli E
and K of pure Au in [1].
According to Figures 5 and 6 at the same temperature when the concentration of
interstitial atoms increases, the elastic contants C11, C12, C44 of alloy AuLi decrease. At the
same concentration of interstitial atoms when the temperatures increases, the elastic contants
C11, C12, C44 of alloy AuLi decrease. At zero concentration of interstitial atoms, we obtain the
elastic contants C11, C12 and C44 of pure Au in [1].
The calculated (CAL) results from the SMM and the other calculations and the experimental
data (EXPT) for the nearest neighbor distance, the elastic moduli and the elastic constants of main
metal Au in interstitial alloys AuLi and AuCuLi are shown in Tables 2 and 3. Some calculated
results from the SMM are in rather good agreement with experimental data and better than that
obtained using other calculation methods.
52
Elastic deformation of binary and ternary interstitial alloys with fcc structure at zero pressure...
Table 2. The nearest neighbor distance and elastic moduli of Au
at P = 0, T = 300 K from the SMM and the experimental data [5, 6]
Method a(A˚) E(1010Pa) K(1010Pa) G(1010Pa)
SMM 2.8454 8.96 14.94 3.20
EXPT[5,6] 2.8838 2.8838 16.70 3.10
53
Nguyen Quang Hoc, Bui Duc Tinh, Le Dinh Tuan and Nguyen Duc Hien
Table 3. The elastic constants of Au at P = 0, T = 300 K
from the SMM, other calculations [7-15] and experimental data [6]
Elastic
constant SMM
EXPT Other calculations
(1010Pa) [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
C11 1.92 1.92 1.92 1.83 1.79 2.09 1.36 1.50 1.97 1.84 2.00
C12 1.28 1.65 1.66 1.54 1.47 1.75 0.91 1.29 1.84 1.54 1.73
C44 0.32 0.42 0.39 0.45 0.42 0.31 0.49 0.70 0.52 0.43 0.33
According to Figures 7-10 for alloy AuCuLi at the same temperature and concentration
of substitution atoms when the concentration of interstitial atoms increases, the mean nearest
neighbor distance increases (for example at 50 K and cCu = 6%, a increases from 2.8069
đến 2.9162 A˚ when cLi increases from 0 to 5%). For alloy AuCuLi at the same concentration
of substitution atoms and interstitial atoms when the temperature increases, the mean nearest
neighbor distance increases (for example at cCu = 6%, cLi = 5%, a increases from 2.9162
54
Elastic deformation of binary and ternary interstitial alloys with fcc structure at zero pressure...
to 3.6067 A˚ when T increases from 50 to 1000 K). For alloy AuCuLi at the same temperature
and concentration of interstitial atoms when the concentration of substitution atoms increases, the
mean nearest neighbor distance decreases (for example at 1000 K, cLi = 0.6%, a decreases from
2.9743 to 2.9521 A˚ when cCu increases from 0 to 6%). At zero concentration of substitution atoms,
andinterstitial atoms, the mean nearest neighbor distance of interstitial alloy AuCuLi becomes
the mean nearest neighbor distance of metal Au in [1]. The change of mean nearest neighbor
distance with temperature for interstitial alloy AuCuLi is similar to that for interstitial alloy AuLi.
The change of mean nearest neighbor distance with temperature and concentration of substitution
atoms for interstitial alloy AuCuLi is similar to that for substitution alloy AuCu [4].
According to Figures 11-13 for alloy AuCuLi in the same temperature and concentration
of substitution atoms when the concentration of interstitial atoms increases, the elastic moduli E
and K decrease (for example at 300 K, E decreases from 11.4259.1010 to 6.3259.1010 Pa when
cLi increases from 0 to 5%). For alloy AuCuLi in the same concentration of substitution atoms
55
Nguyen Quang Hoc, Bui Duc Tinh, Le Dinh Tuan and Nguyen Duc Hien
and concentration of interstitial atoms when the temperature increases, the elastic moduli E and K
decrease (for example cCu = 10%, cLi = 5%, at E decreases from 6.7681.1010 to 5.2362.1010Pa
when T increases from 100 to 700 K). For alloy AuCuLi at the same temperature and concentration
of interstitial atoms, when the concentration of substitution atoms increases, the elastic moduli E
and K increase (for example at 300 K, cLi = 5%, E increases from 5.97.1010 to 6.8597.1010Pa
when cCu increases from 0 to 25%). At zero concentrations of substitution atoms and interstitial
atoms, the elastic moduli E and K of interstitial alloy AuCuLi become the elastic moduli E and
K of metal Au in [1]. The change of elastic moduli E and K in temperature for interstitial alloy
AuCuLi is similar to that for interstitial alloy AuLi. The change of elastic moduli E and K in
temperature and concentration of substitution atoms for interstitial alloy AuCuLi is similar to that
for substitution alloy AuCu [4].
According to Figures 14-16 for alloy AuCuLi at the same temperature and concentration
of substitution atoms, when the concentration of interstitial atoms increases, the elastic constants
C11, C12 decrease (for example at 300 K and cCu = 10%, C11 decreases from 22.4909.1010 to
12.4519.1010 Pa when cLi increases from 0 to 5%). For alloy AuCuLi with the same concentration
of substitution atoms and concentration of interstitial atoms, when the temperature increases, the
elastic constants C11, C12 decrease (for example at cCu = 10%, cLi = 5% , C11 decreases from
13.3223.1010 to 10.2952.1010Pa when T inceases from 100 to 700 K). For alloy AuCuLi at the
same temperature and concentration of interstitial atoms, when the concentration of substitution
atoms increases, the elastic constants C11, C12 increase (for example at 300 K, cLi = 5%, C11
increases from 11.9089.1010 to 13.2435.1010Pa when cCu increases from 0 to 25%). At zero
concentration of substitution atoms and interstitial atoms, the elastic constants of interstitial alloy
AuCuLi become the elastic constants C11, C12 of metal Au in [1]. The change of elastic constants
C11, C12 with temperature for interstitial alloy AuCuLi is similar to that for interstitial alloy AuLi.
The change of elastic constants C11, C12 with temperature and concentration of substitution atoms
for interstitial alloy AuCuLi is similar to that for substitution alloy AuCu [4].
56
Elastic deformation of binary and ternary interstitial alloys with fcc structure at zero pressure...
3. Conclusion
Our results in using the elastic theory for interstitial alloys AC, ABC with FCC structure
based on the SMM are applied to alloys AuLi and AuCuLi at zero pressure in the temperature
intervals from 100 to 1000 K, in the range of concentration of substitution atoms from 0 to 25%
and in the range of concentration of interstitial atoms from 0 to 5%. The calculated results for
main metal Au in the interstitial alloys are in rather good agreement with experimental data and
are compared with other calculated results.
Acknowledgements. This work was carried out thanks to the financial support provided
by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam
under Grant No. 103.01-2013.20.
REFERENCES
[1] V. V. Hung, 2009. Statistical moment method in studying thermodynamic and elastic property of
crystal. HNUE Publishing House, pp.1-231.
[2]