Abstract. Thermodynamic quantities such as mean nearest neighbor distance, free energy,
isothermal and adiabatic compressibilities, isothermal and adiabatic elastic modulus,
thermal expansion coefficient, heat capacities at constant volume and constant pressure,
and entropy of binary interstitial alloys with a body-centered cubic (BCC) structure
with a very small concentration of interstitial atoms are derived using the statistical
moment method. The obtained expressions of these quantities depend on temperature and
concentration of interstitial atoms. The theoretical results are applied to the interstitial alloy
FeSi. In the case where the concentration of interstitial atoms of Si is equal to zero, we have
the thermodynamic quantities of the main metal and the numerical results for the alloy
FeSi give the numerical results for Fe. The calculated results of the thermal expansion
coefficient and heat capacity at constant pressure in the interval of temperature from 100
to 700 K for Fe are in good agreement with the experimental data.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0044
Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 146-155
This paper is available online at
THERMODYNAMIC PROPERTIES OF BINARY INTERSTITIAL ALLOYS
WITH A BCC STRUCTURE: DEPENDENCE ON TEMPERATURE
AND CONCENTRATION OF INTERSTITIAL ATOMS
Nguyen Quang Hoc, Dinh Quang Vinh, Bui Duc Tinh, Tran Thi Cam Loan,
Ngo Lien Phuong, Tang Thi Hue and Dinh Thi Thanh Thuy
Faculty of Physics, Hanoi National University of Education
Abstract. Thermodynamic quantities such as mean nearest neighbor distance, free energy,
isothermal and adiabatic compressibilities, isothermal and adiabatic elastic modulus,
thermal expansion coefficient, heat capacities at constant volume and constant pressure,
and entropy of binary interstitial alloys with a body-centered cubic (BCC) structure
with a very small concentration of interstitial atoms are derived using the statistical
moment method. The obtained expressions of these quantities depend on temperature and
concentration of interstitial atoms. The theoretical results are applied to the interstitial alloy
FeSi. In the case where the concentration of interstitial atoms of Si is equal to zero, we have
the thermodynamic quantities of the main metal and the numerical results for the alloy
FeSi give the numerical results for Fe. The calculated results of the thermal expansion
coefficient and heat capacity at constant pressure in the interval of temperature from 100
to 700 K for Fe are in good agreement with the experimental data.
Keywords: Binary interstitial alloy, statistical moment method, coordination sphere.
1. Introduction
Thermodynamic and elastic properties of interstitial alloys are of special interest to many
theoretical and experimental researchers, for example [1-3].
In [4, 5] the equilibrium vacancy concentration in BCC substitution and interstitial alloys
is calculated taking into account thermal redistribution of the interstitial component in different
types of interstices. The conditions where this effect gives rise to a decrease or increase in vacancy
concentration are formulated.
Coatings based on interstitial alloys of transition metals have a wide range of applicability.
However, interest in synthesizing coatings from new materials with requisite service properties is
limited due to the scarceness of data on their melting temperature. In [6], the authors considered
calculating the melting temperature for interstitial alloys of transition metals based on the
characteristics of intermolecular interaction.
In [7], the authors attempt to present a survey of the order-disorder transformations of the
interstitial alloys of transition metals with hydrogen, deuterium and oxygen. Special attention
Received August 19, 2015. Accepted October 26, 2015.
Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn
146
Thermodynamic properties of binary interstitial alloys with a BCC structure...
is given to the formation of interstitial superstructures, the stepwise process of disordering and
property changes attributed to order-disorder. Four groups of interstitial alloys are considered:
(1) TO, ZrO, HfO, (2) VO, (3) VH, VD and (4) TaH, TaD. Characteristic features of the
phase transformations in each group and each system are presented and discussed in comparison
with others.
In this paper, we build the thermodynamic theory for binary interstitial alloy with a BCC
structure using the statistical moment method (SMM) [8] and applying the obtained theoretical
results to the alloy FeSi.
2. Content
2.1. Thermodynamic quantities of binary interstitial alloys with a BCC
structure
The cohesive energy of atom C (in face centers of cubic unit cell) with atoms A (in body
centers and peaks of cubic unit cell) in the approximation of three coordination spheres with center
C and radii r1, r1
√
2, r1
√
5 is determined by
u0C =
1
2
ni∑
i=1
ϕAC (ri) =
1
2
[
2ϕAC (r1) + 4ϕAC
(
r1
√
2
)
+ 8ϕAC
(
r1
√
5
)]
= ϕAC (r1) + 2ϕAC
(
r1
√
2
)
+ 4ϕAC
(
r1
√
5
)
, (2.1)
where ϕAC is the interaction potential between atom A and atom C, ni is the number of atoms
on the ith coordination sphere with radius ri(i = 1, 2, 3), 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 cohesive energy u0C , y0A1(T ) is the displacement
of atom A1 (atom A stays in the body center of the cubic unit cell) from the equilibrium position
at temperature T. The alloy’s parameters for atoms C in the approximation of three coordination
spheres have the following form
kC =
1
2
∑
i
(
∂2ϕAC
∂u2iβ
)
eq
= ϕ
(2)
AC (r1) +
√
2
r1
ϕ
(1)
AC
(
r1
√
2
)
+
4
5r21
ϕ
(2)
AC
(
r1
√
5
)
+
16
5
√
5r1
ϕ
(1)
AC
(
r1
√
5
)
, γC = 4 (γ1C + γ2C) ,
γ1C =
1
48
∑
i
(
∂4ϕAC
∂u4iβ
)
eq
=
1
24
ϕ
(4)
AC(r1) +
1
8r21
ϕ
(2)
AC
(
r1
√
2
)
−
√
2
16r31
ϕ
(1)
AC
(
r1
√
2
)
+
1
150
ϕ
(4)
AC(r1
√
5) +
4
√
5
125r1
ϕ
(3)
AC(r1
√
5),
γ2C =
6
48
∑
i
(
∂4ϕAC
∂u2iα∂u
2
iβ
)
eq
=
1
4r1
ϕ
(3)
AC(r1)−
1
2r21
ϕ
(2)
AC(r1) +
5
8r31
ϕ
(1)
AC(r1)
147
N. Q. Hoc, D. Q. Vinh, B. D. Tinh, T. T. C. Loan, N. L. Phuong, T. T. Hue and D. T. T. Thuy
+
√
2
8r1
ϕ
(3)
AC(r1
√
2)− 1
8r21
ϕ
(2)
AC(r1
√
2) +
1
8r31
ϕ
(1)
AC(r1
√
2) +
2
25
ϕ
(4)
AC(r1
√
5)
+
3
25
√
5r1
ϕ
(3)
AC(r1
√
5) +
2
25r21
ϕ
(2)
AC(r1
√
5)− 2
25
√
5r31
ϕ
(1)
AC(r1
√
5), (2.2)
where ϕ(i)AC(ri) = ∂
2ϕAC(ri)/∂r
2
i (i = 1, 2, 3, 4), α, β = x, y, z, α 6= β and uiβ is the
displacement of the ith atom in direction β
When atom A1, which contains interstitial atom C on the first coordination sphere, is taken
as the coordinate origin, the cohesive energy of atom A1 with the atoms in crystalline lattice and the
corresponding alloy’s parameters in the approximation of three coordination spheres mentioned
above is determined by
u0A1 = u0A + ϕAC (r1A1) ; kA1 = kA +
1
2
∑
i
(∂2ϕAC
∂u2iβ
)
eq
r=rA1
= kA + ϕ
(2)
AC (r1A1) +
5
2r1A1
ϕ
(1)
AC (r1A1) , γA1 = 4 (γ1A1 + γ2A1) ,
γ1A1 = γ1A +
1
48
∑
i
(∂4ϕAC
∂u4iβ
)
eq
r=rA1
= γ1A +
1
24
ϕ
(4)
AC(r1A1) +
1
8r21A1
ϕ
(2)
AC(r1A1)−
1
8r31A1
ϕ
(1)
AC(r1A1),
γ2A1 = γ2A +
6
48
∑
i
( ∂4ϕAC
∂u2iα∂u
2
iβ
)
eq
r=rA1
= γ2A +
1
2r1A1
ϕ
(3)
AC(r1A1)−
3
4r21A1
ϕ
(2)
AC(r1A1) +
3
4r31A1
ϕ
(1)
AC(r1A1), (2.3)
where u0A, kA, γ1A, γ2A are the coressponding quantities in clean metal A in the approximation
of two coordination spheres [8] and r1A1 ≈ r1C is the nearest neighbor distance between atom A1,
atom A stays in body of cubic unit cell and atoms in the crystalline lattic.
When atom A2 (atom A stays in peaks of cubic unit cell) which contain interstitial atom C
on the first coordination sphere is taken as the coordinate origin, the cohesive energy of atom A2
with the atoms in crystalline lattice and the corresponding alloy’s parameters in the approximation
of three coordination spheres mentioned above is determined by
u0A2 = u0A + ϕAC (r1A2) , kA2 = kA +
1
2
∑
i
(∂2ϕAC
∂u2iβ
)
eq
r=rA2
= kA + 2ϕ
(2)
AC (r1A2) +
4
r1A2
ϕ
(1)
AC (r1A2) , γA2 = 4 (γ1A2 + γ2A2) ,
148
Thermodynamic properties of binary interstitial alloys with a BCC structure...
γ1A2 = γ1A +
1
48
∑
i
(∂4ϕAC
∂u4iβ
)
eq
r=rA1
= γ1A +
1
24
ϕ
(4)
AC(r1A2) +
1
4r1A2
ϕ
(3)
AC(r1A2)−
1
8r21A2
ϕ
(2)
AC(r1A2) +
1
8r31A2
ϕ
(1)
AC(r1A2),
γ2A2 = γ2A +
6
48
∑
i
( ∂4ϕAC
∂u2iα∂u
2
iβ
)
eq
r=rA2
= γ2A +
1
8
ϕ
(4)
AC(r1A2) +
1
4r1A2
ϕ
(3)
AC(r1A2) +
3
8r21A2
ϕ
(2)
A2C
(r1A2)−
3
8r31A2
ϕ
(1)
A2C
(r1A2), (2.4)
where r1A2 = r01A2 + y0C(T ), r01A2 is the nearest neighbor distance between atom A2 and atoms
in crystalline lattice at 0 K and is determined from the minimum condition of the cohesive energy
u0A2 , y0C(T ) is the displacement of atom C at temperature T.
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
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)
where r1X(0, 0)(X = A,A1, A2, C) is determined from the equation of state or the minimum
condition of cohesive energy. From the obtained r1X(0, 0) using 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 [8]
y0X(0, T ) =
√
2γX(0, 0)θ2
3k3X(0, 0)
AX(0, T ).X = A,A1, A2, C,
AX = a1X +
5∑
i=2
(
γXθ
k2X
)i
aiX , kX = mω
2
X , xX =
~ωX
2θ
, a1X = 1 +
XX
2
,
a2X =
13
3
+
47
6
XX +
23
6
X2X +
1
2
X3X , a3X = −
(
25
3
+
121
6
XX +
50
3
X2X +
16
3
X3X +
1
2
X4X
)
,
a4X =
43
3
+
93
2
XX +
169
3
X2X +
83
3
X3X +
22
4
X4X +
1
2
X5X ,
a5X = −
(
103
3
+
749
6
XX +
363
3
X2X +
733
3
X3X +
148
3
X4X +
53
6
X5X +
1
2
X6X
)
,
a6X = 65 +
561
2
XX +
1489
3
X2X +
927
2
X3X +
733
3
X4X +
145
2
X5X +
31
3
X6X +
1
2
X7X ,
XX ≡ xX cothxX . (2.6)
Then, we calculate the mean nearest neighbor distance in interstitial alloy AC by the
expressions as follows:
149
N. Q. Hoc, D. Q. Vinh, B. D. Tinh, T. T. C. Loan, N. L. Phuong, T. T. Hue and D. T. T. Thuy
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− 7cC) yA(0, T ) + cCyC(0, T ) + 2cCyA1(0, T ) + 4cCyA2(0, T ), (2.7)
where r1A(0, T ) is the mean nearest neighbor distance between atoms A in interstitial alloy AC
at P = 0 and temperature T, r1A(0, 0) is the mean nearest neighbor distance between atoms A in
interstitial alloy AC at P = 0 and 0 K, r1A(0, 0) is the nearest neighbor distance between atoms
A in clean metal A at P = 0 and 0 K, r′1A(0, 0) is the nearest neighbor distance between atoms
A in the zone containing the interstitial atom C at P = 0 and 0 K and cC is the concentration of
interstitial atoms C.
The free energy per mole of interstitial alloy AC is determined by
ψAC = (1− 7cC)ψA + cCψC + 2cCψA1 + 4cCψA2 − TSc,
ψX = U0X + ψ0X + 3N
{
θ2
k2X
[
γ2XX
2
X −
2γ1X
3
(
1 +
1
2
XX
)]
+
2θ3
k4X
[
4
3
γ22XXX
(
1 +
1
2
XX
)
− 2γ21X + 2γ1Xγ2X
(
1 +
1
2
XX
)
(1 +XX)
]}
,
ψ0X = 3Nθ
[
xX + ln
(
1− e−xX)] . (2.8)
where SC is the configuration entropy.
The isothermal compressibility of interstitial alloy AC has the form
χTAC =
3
(
aAC
a0AC
)3
a2
AC
3VAC
(
∂2ΨAC
∂a2
AC
)
T
=
3
(
aAC
a0AC
)3
2a2
AC
3
√
2a3
AC
(
∂2ψAC
∂a2
AC
)
T
=
(
aAC
a0AC
)3
√
2
3aAC
1
3N
(
∂2ΨAC
∂a2
AC
)
T
,
(
∂2ψAC
∂a2AC
)
T
=
[
∂2ψAC
∂r1A2(0, T )
]
T
= (1− 7cC)
(
∂2ψA
∂a2A
)
T
+ cC
(
∂2ψC
∂a2C
)
T
+2cC
(
∂2ψA1
∂a2A1
)
T
+ 4cC
(
∂2ψA2
∂a2A2
)
T
,
1
3N
(
∂2ΨX
∂a2X
)
T
=
1
6
∂2u0X
∂a2X
+
~ωX
4kX
[
∂2kX
∂a2X
− 1
2kX
(
∂kX
∂aX
)2]
,
(
∂2ψX
∂a2X
)
T
=
1
2
∂2u0X
∂a2X
+
3~ωX
4kX
[
∂2kX
∂a2X
− 1
2kX
(
∂kX
∂aX
)2]
, (2.9)
where ΨAC = NψAC , aAC = r1A(0, T )anda0AC = r1A(0, 0).
150
Thermodynamic properties of binary interstitial alloys with a BCC structure...
The thermal expansion coefficient of interstitial alloy AC has the form
αTAC =
kB
α0AC
daAC
dθ
= −kBχTAC
3
(
a0AC
aAC
)2 aAC
3VAC
∂2ΨAC
∂θ∂aAC
= −kBχTAC
3
(
a0AC
aAC
)2 aAC
3vAC
∂2ψAC
∂θ∂aAC
,
∂2ψAC
∂θ∂aAC
= (1− 7cC) ∂
2ψA
∂θ∂aA
+ cC
∂2ψC
∂θ∂aC
+ 2cC
∂2ψA1
∂θ∂aA1
+ 4cC
∂2ψA2
∂θ∂aA2
,
∂2ψX
∂θ∂aX
=
3
2kX
∂kX
∂aX
Y 2X +
6θ2
k2X
[
γ1X
3kX
∂kX
∂aX
(
2 +XXY
2
X
)
−1
6
∂γ1X
∂aX
(
4 +XX + Y
2
X
)− (2γ2X
kX
∂kX
∂aX
− ∂γ2X
∂aX
)
XXY
2
X
]
,
YX ≡ xX
sinhxX
. (2.10)
The energy of interstitial alloy AC is determined by
EAC = (1− 7cC)EA + cCEC + 2cCEA1 + 4cCEA2 ,
EX = U0X + E0X +
3Nθ2
k2X
[
γ2Xx
2
X coth xX +
γ1X
3
(
2 +
x2X
sinh2xX
)
− 2γ2X x
3
X coth xX
sinh2xX
]
,
E0X = 3NθxX coth xX . (2.11)
The entropy of interstitial alloy AC is determined by
SAC = (1− 7cC)SA + cCSC + 2cCSA1 + 4cCSA2 ,
SX = S0X +
3NkBθ
k2X
[γ1X
3
(
4 +XX + Y
2
X
)− 2γ2XXXY 2X] ,
S0X = 3NkB [XX − ln (2 sinhxX)] . (2.12)
The heat capacity at constant volume of interstitial alloy AC is determined
CV AC = (1− 7cC)CV A + cCCV C + 2cCCV A1 + 4cCCV A2 ,
CV X = 3NkB
{
Y 2X +
2θ
k2X
[(
2γ2 +
γ1X
3
)
XXY
2
X
+
2γ1X
3
− γ2X
(
Y 4X + 2X
2
XY
2
X
)]}
. (2.13)
The heat capacity at constant pressure of interstitial alloy AC is determined by
CPAC = CV AC +
9TVACα
2
TAC
χTAC
. (2.14)
The adiabatic compressibility of interstitial alloy AC has the form
χSAC =
CV AC
CPAC
χTAC . (2.15)
151
N. Q. Hoc, D. Q. Vinh, B. D. Tinh, T. T. C. Loan, N. L. Phuong, T. T. Hue and D. T. T. Thuy
2.2. Numerical results for interstitial alloy FeSi
In numerical calculations for alloy FeSi, we use the n-m pair potential
ϕ(r) =
D
n−m
[
m
(r0
r
)n − n(r0
r
)m]
, (2.16)
where potential parameters are given in Table 1 [9].
Our numerical results are described by figures in Figures 1-14. When the concentration
cSi → 0, we obtain thermodynamic quantities of Fe. Our calculated results in Table 2 are in rather
good agreement with the experimental data.
At the same temperature, when the concentration of interstitial atoms increases, the
thermodynamic quantities of alloy decrease. When the concentration of interstitial atoms remains
the same and temperature increases, the thermodynamic quantities of alloy increase.
Figure 1. r1FeSi(T ) at P = 0, cSi = 0 - 5% Figure 2. r1FeSi(cSi) at P = 0, T =100 - 1000 K
Figure 3. χTFeSi (cSi) at P = 0, T =100 - 1000 K Figure 4. χTFeSi (T ) at P = 0, cSi = 0 - 5%
152
Thermodynamic properties of binary interstitial alloys with a BCC structure...
Table 1. Parameters m, n, D, r0 of materials Fe, Si
Material m n D(10−16 erg) r0(10−10 m)
Fe 7 11.5 6416.448 2.4775
Si 6 12 45128.340 2.2950
Table 2. Dependence of the thermal expansion coefficient on temperature for Fe
T (K) 100 200 300 500 700
αT
(
10−6K−1
)
This paper 5.69 10.90 12.74 14.82 16.12
Expt [10] 5.6 10.0 11.7 14.3 16.3
Figure 5. αTFeSi (T ) at P = 0, cSi = 0 - 5% Figure 6. αTFeSi (cSi) at P = 0, T =100 - 1000 K
Figure 7. CV FeSi (T ) at P = 0, cSi = 0 - 5% Figure 8. CV FeSi (cSi) at P = 0, T = 100 - 1000
153
N. Q. Hoc, D. Q. Vinh, B. D. Tinh, T. T. C. Loan, N. L. Phuong, T. T. Hue and D. T. T. Thuy
Figure 9. CPFeSi (cSi) at P = 0,
T =100 - 1000 K
Figure 10. CPFeSi (T ) at P = 0,
cSi = 0 - 5%
Figure 11. χSFeSi(T ) at P = 0,
cSi = 0 - 5%
Figure 12. χSFeSi(cSi) at P = 0,
T =100 - 1000 K
Figure 13. SFeSi(cSi) at P = 0, T =100 - 1000 K Figure 14. SFeSi(T ) at P = 0, cSi = 0 - 5%
154
Thermodynamic properties of binary interstitial alloys with a BCC structure...
3. Conclusion
From the SMM, the minimum condition of cohesive energies and the method of three
coordination spheres, we find the mean nearest neighbor distance, the free energy, the isothermal
and adiabatic compressibilities, the isothermal and adiabatic elastic modulus, the thermal
expansion coefficient, the heat capacities at constant volume and at constant pressure and the
entropy of binary interstitial alloy with BCC structure with very small concentration of interstitial
atoms. The obtained expressions of these quantities depend on the temperature and concentration
of interstitial atoms. At zero concentration of interstitial atoms Si, thermodynamic quantities of
interstitial alloy FeSi become ones of main metal Fe. At zero concentration of interstitial atoms
Si, our calculated results for the thermal expansion coefficient and the heat capacity at constant
pressure of interstitial alloy are in rather good agreement with experimental data. We have only
considered the interstitial alloy FeSi in the interval of temperature of 100 to 1000 K where the
anharmonicity of lattice vibrations has a considerable influence.
Acknowledgements. This work was carried out with financial support from the National
Foundation for Science and Technology Development (NAFOSTED) of Vietnam under Grant No.
103.01-2013.20.
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