Abstract. The analytic expressions of the thermodynamic quantities such as mean nearest
neighbor distance, free energy, free isothermal and adiabatic compressibilities, free
isothermal and adiabatic elastic moduli, thermal expansion coefficient, the heat capacities
at constant volume and at constant pressure and entropy of a ternary interstitial alloy with
a body-centered cubic (BCC) structure with a very small concentration of interstitial atoms
are all derived using the statistical moment method. The obtained expressions of these
quantities depend on temperature, concentration of substitution atoms and concentration
of interstitial atoms. In this study, the theoretical results are applied to the interstitial
alloy FeCrSi. The thermodynamic properties of main metal Fe, substitution alloy FeCr
and interstitial alloy FeSi are special cases for the thermodynamic properties of interstitial
alloy FeCrSi. The calculated results of the thermal expansion coefficient in the temperature
interval 100 to 1000 K and heat capacity at constant pressure in the temperature interval
from 100 to 700 K for main metal Fe are in good agreement with the experimental data.
Keywords: Ternary interstitial alloy, statistical moment method.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0033
Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 65-74
This paper is available online at
THERMODYNAMIC PROPERTIES OF A TERNARY INTERSTITIAL ALLOY
WITH BCC STRUCTURE: DEPENDENCE ON TEMPERATURE,
CONCENTRATION OF SUBSTITUTION ATOMS
AND CONCENTRATION OF INTERSTITIAL ATOMS
Nguyen Quang Hoc1, Dinh Quang Vinh1, Nguyen Thi Hang, Nguyen Thi Nguyet,
Luong Xuan Phuong, Nguyen Nhu Hoa, Nguyen Thi Phuc and Trinh Thi Hien2
1Faculty of Physics, Hanoi National University of Education
2Faculty of Natural Sciences, Hong Duc University, Thanh Hoa City
Abstract. The analytic expressions of the thermodynamic quantities such as mean nearest
neighbor distance, free energy, free isothermal and adiabatic compressibilities, free
isothermal and adiabatic elastic moduli, thermal expansion coefficient, the heat capacities
at constant volume and at constant pressure and entropy of a ternary interstitial alloy with
a body-centered cubic (BCC) structure with a very small concentration of interstitial atoms
are all derived using the statistical moment method. The obtained expressions of these
quantities depend on temperature, concentration of substitution atoms and concentration
of interstitial atoms. In this study, the theoretical results are applied to the interstitial
alloy FeCrSi. The thermodynamic properties of main metal Fe, substitution alloy FeCr
and interstitial alloy FeSi are special cases for the thermodynamic properties of interstitial
alloy FeCrSi. The calculated results of the thermal expansion coefficient in the temperature
interval 100 to 1000 K and heat capacity at constant pressure in the temperature interval
from 100 to 700 K for main metal Fe are in good agreement with the experimental data.
Keywords: Ternary interstitial alloy, statistical moment method.
1. Introduction
Thermodynamic and elastic properties of interstitial alloys are of special interest to many
theoretical and experimental researchers [1-5]. In this paper, we build a thermodynamic theory for
a ternary interstitial alloy with a body-centered cubic (BCC) structure using the statistical moment
method (SMM) [6] and we apply the obtained theoretical results to the alloy FeCrSi.
Received March 4, 2016. Accepted July 24, 2016.
Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn
65
N. Q. Hoc, D. Q. Vinh, N. T. Hang, N. T. Nguyet, L. X. Phuong, N. N. Hoa, N. T. Phuc and T. T. Hien
2. Content
2.1. Thermodynamic quantities of interstitial alloy ABC with BCC
structure
The interstitial alloy ABC (the substitution alloy AB with the interstitial atom C) with BCC
structure has N atoms (NA main atoms A, NB substitution atoms B and NC interstitial atoms C).
The concentration of alloy component are cA =
NA
N , cB =
NB
N and cC =
NC
N . The alloy ABC
satisfies the condition cC << cB << cA. In order to have the interstitial alloy ABC with BCC
structure we take the interstitial alloy AC with BCC structure (interstitial atom C stays in face
centers of cubic unit cell) and then the atom B substitutes the atom A in body center of cubic unit
cell. The free energy of interstitial alloy ABC has the form
ψABC = ψAC + cB (ψB − ψA) + TSACc − TSABCc ,
ψAC = (1− 7cC)ψA + cCψC + 2cCψA1 + 4cCψA2 − TSACc ,
ψX = U0X + ψ0X + 3N
{
θ2
k2X
[
γ2Xx
2
Xcoth
2xX − 2γ1X
3
(
1 +
1
2
xX coth xX
)]
+
2θ3
k4X
[
4
3
γ22XxX coth xX
(
1 +
1
2
xX coth xX
)
−2 (γ21X + 2γ1Xγ2X)
(
1 +
1
2
xX coth xX
)
(1 + xX coth xX)
]}
,
ψ0X = 3Nθ
[
xX + ln
(
1− e−2xX)] ,X = A,C,A1, A2, (2.1)
where ψAC is the free energy of interstitial alloy AC, ψA is the free energy of atom A in pure metal
A, ψC is the free energy of atom C in alloy ABC, ψA1 is the free energy of atom A1 (atom A in
body center of cubic unit cell), ψA2 is the free energy of atom A2 (atom A in peaks of cubic unit
cell), SACc is the configuration entropy of interstitial alloy AC, S
ABC
c is the configuration entropy
of interstitial alloy ABC, the quantities U0X , kX , γ1X , γ2X , xX are defined as corresponding
quantities in [6], θ = kBT and kB is the Boltzmann constant.
The mean nearest neighbor distance between two atoms in the interstitial alloy ABC with
BCC 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 =
4
√
3aAC
(
aAC
a0AC
)3
(
∂2ψAC
∂a2
AC
)
T
,
(
∂2ψAC
∂a2AC
)
T
=
[
∂2ψAC
∂r1A2(0, T )
]
T
≈ (1− 7cC)
(
∂2ψA
∂a2A
)
T
+ cC
(
∂2ψC
∂a2C
)
T
66
Thermodynamic properties of a ternary interstitial alloy with BCC structure...
+2cC
(
∂2ψA1
∂a2A1
)
T
+ 4cC
(
∂2ψA2
∂a2A2
)
T
,
(
∂2ψX
∂a2X
)
T
=
(
∂2ψA
∂r21X(0, T )
)
T
,
1
3N
(
∂2ΨX
∂a2X
)
T
=
1
6
∂2u0X
∂a2X
+
~ωX
4kX
[
∂2kX
∂a2X
− 1
2kX
(
∂kX
∂aX
)2]
.
(2.2)
The mean nearest neighbor distance between two atoms in the interstitial alloy ABC with
BCC structure at 0 K is given by
a0ABC = cACa0AC
B0TAC
B0T
+ cBa0B
B0TB
B0T
. (2.3)
The quantities in (2.3) is the same as in (2.2) but are determined at 0 K. The isothermal
compressibility and the isothermal elastic modulus of interstitial alloy ABC has the form
χTABC =
3
(
aABC
a0ABC
)3
a2
ABC
3VABC
(
∂2ΨABC
∂a2
ABC
)
T
=
3
(
aABC
a0ABC
)3
a2
ABC
3vABC
(
∂2ψABC
∂a2
ABC
)
T
=
(
aABC
a0ABC
)3
√
3
4aABC
1
3N
(
∂2ΨABC
∂a2
ABC
)
T
,
BTABC =
1
χTABC
,
(
∂2ψABC
∂a2ABC
)
T
≈
(
∂2ψAC
∂a2A
)
T
+ cB
[(
∂2ψB
∂a2B
)
T
−
(
∂2ψA
∂a2A
)
T
]
. (2.4)
The thermal expansion coefficient of interstitial alloy ABC has the form
αTABC =
kB
α0ABC
daABC
dθ
= −kBχTABC
3
(
a0ABC
aABC
)2 aABC
3VABC
∂2ΨABC
∂θ∂aABC
= −kBχTABC
3
(
a0ABC
aABC
)2 aABC
3vABC
∂2ψABC
∂θ∂aABC
,
∂2ψABC
∂θ∂aABC
≈ ∂
2ψAC
∂θ∂aAC
+ cB
(
∂2ψB
∂θ∂aB
− ∂
2ψA
∂θ∂aA
)
,
∂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
x2X
sinh2xX
+
6θ2
k2X
[
γ1X
3kX
∂kX
∂aX
(
2 +
x3X cothxX
sinh2xX
)
−1
6
∂γ1X
∂aX
(
4 + xX coth xX +
x2X
sinh2xX
)
−
(
2γ2X
kX
∂kX
∂aX
− ∂γ2X
∂aX
)
x3X cothxX
sinh2xX
]
. (2.5)
The energy of interstitial alloy ABC is determined by
EABC ≈ EAC + cB (EB − EA) , EAC = (1− 7cC)EA + cCEC + 2cCEA1 + 4cCEA2 ,
67
N. Q. Hoc, D. Q. Vinh, N. T. Hang, N. T. Nguyet, L. X. Phuong, N. N. Hoa, N. T. Phuc and T. T. Hien
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 . (2.6)
The entropy of interstitial alloy ABC is determined by
SABC ≈ SAC + cB (SB − SA) , SAC = (1− 7cC)SA + cCSC + 2cCSA1 + 4cCSA2 ,
SX = S0X +
3NkBθ
k2X
[
γ1X
3
(
4 + xX cothxX +
x2X
sinh2xX
)
− 2γ2X x
3
X coth xX
sinh2xX
]
,
S0X = 3NkB [xX coth xX − ln (2 sinhxX)] . (2.7)
The heat capacity at constant volume of interstitial alloy ABC is determined by
CV ABC ≈ CV AC+cB (CV B −CV A) , CV AC = (1− 7cC)CV A+cCCV C+2cCCV A1+4cCCV A2 ,
CV X = 3NkB
{
x2X
sinh2xX
+
2θ
k2X
[(
2γ2 +
γ1X
3
) x3X coth xX
sinh2xX
+
2γ1X
3
− γ2X
(
x4X
sinh4xX
+
2x4Xcoth
2xX
sinh2xX
)]}
, (2.8)
The heat capacity at constant pressure of interstitial alloy ABC is determined by
CPABC = CV ABC +
9TVABCα
2
TABC
χTABC
. (2.9)
The adiabatic compressibility and elastic modulus of interstitial alloy ABC has the form
χSABC =
CV ABC
CPABC
χTABC , BSABC =
1
χSABC
. (2.10)
2.2. Numerical results for interstitial alloy FeCrSi
In numerical calculations for alloy FeCrSi, we use the n-m pair potential
ϕ(r) =
D
n−m
[
m
(r0
r
)n
− n
(r0
r
)m]
, (2.11)
where potential parameters are given in Table 1 [7].
Table 1. The parameters m,n,D, r0 of materials Fe, Cr and Si
Material m n D
(
10−16erg
)
r0
(
10−10m
)
Fe 7 11.5 6416.448 2.4775
Cr 6 15.5 6612.960 2.495
Si 6 12 45128.34 2.295
68
Thermodynamic properties of a ternary interstitial alloy with BCC structure...
Our numerical results are described by figures from Figure 1 to Figure 24. When the
concentration cSi → 0 and the concentration cCr → 0, we obtain thermodynamic quantities of
Fe. Our calculated results in Table 2 and Table 3 are in good agreement with experiments.
Table 2. Dependence of thermal expansion coefficient αT
(
10−6K−1
)
on temperature for Fe from SMM and EXPT [8]
T(K) 100 200 300 500 700 1000
αT -SMM 5.69 10.9 12.74 14.62 16.12 18.61
αT -EXPT 5.60 10.0 11.7 14.30 16.3 19.2
Table 3. Dependence of heat capacity at constant pressure CP (J/mol.K)
on temperature for Fe from SMM and EXPT[8]
T(K) 100 200 300 500 700
CP -SMM 11.38 21.79 25.53 29.43 32.65
CP –EXPT 12.067 21.503 25.131 29.639 34.618
According to figures from Figure 1 to Figure 24, for alloy FeCrSi in the same temperature
and concentration of substitution atoms Cr when the concentration of interstitial atoms Si
increases, the mean nearest neighbor distance increases (for example at 10 K and cCr = 5% when
cSi increases from 0 to 5%, a increases from 2.42 to 2.492
o
A ) and other quantities decrease (for
example at 10 K and cCr = 5%when cSiincreases from 0 to 5%, χT decreases from 3.046.10−12 to
1.821.10−12 Pa−1, αT decreases from 4.537.10−6 to 1.426.10−6 K−1, CV decreases from 5.605
to 5.404 J/mol.K, CP decreases from 5.606 to 5.404 J/mol.K and χS decreases from 3.046.10−12
to 1,821.10−12 Pa−1). For alloy FeCrSi in the same concentration of substitution atoms Cr and
concentration of interstitial atoms Si when the temperature increases, all quantities increase (for
example at cCr = 5% and cSi = 5% when T increases from 50 to 1000 K, a increases from 2.494
to 2.528
o
A , χT increases from 1.836. 10−12 to 2.252. 10−12 Pa−1, αT increases from 1.426.10−6
at 10 K to 14.522. 10−6 K−1, CV increases from 5.404 to 32.417 J/mol.K, CP increases from
5.404 to 31.11 J/mol.K and χS increases from 1.837.10−12 to 2.251.10−12 Pa−1). For alloy
FeCrSi in the same temperature and concentration of interstitial atoms when the concentration
of substitution atoms increases, some quantities such as a, χT increase (for example at 1000
K and cSi = 1% when cCr increases from 0 to 10%, a increases from 2.471 to 2.473
o
A , χT
increases from 3.771. 10−12 to 3,775. 10−12 Pa−1) and some quantities such as αT , CV , CP
and χS decrease (for example at 1000 K and cSi = 1% when cCr increases from 0 to 10%, αT
decreases from 13.81.10−6 to 13.805.10−6 K−1, CV decreases from 33.549 to 33.545 J/mol.K,
CP decreases from 36.477 to 36.473 J/mol.K and χS decreases from 3.792.10−12 to 3.774. 10−12
Pa−1). At zero concentration of substitution atoms and zero concentration of interstitial atoms,
the thermodynamic quantities of interstitial alloy FeCrSi becomes the thermodynamic quantities
of metal Fe in [6]. The change of thermodynamic quantities with temperature for interstitial alloy
FeCrSi is similar to that for interstitial alloy FeSi [9]. The change of thermodynamic quantities
with temperature and concentration of substitution atoms for interstitial alloy FeCrSi is similar to
that for substitution alloy FeCr [10].
69
N. Q. Hoc, D. Q. Vinh, N. T. Hang, N. T. Nguyet, L. X. Phuong, N. N. Hoa, N. T. Phuc and T. T. Hien
70
Thermodynamic properties of a ternary interstitial alloy with BCC structure...
71
N. Q. Hoc, D. Q. Vinh, N. T. Hang, N. T. Nguyet, L. X. Phuong, N. N. Hoa, N. T. Phuc and T. T. Hien
72
Thermodynamic properties of a ternary interstitial alloy with BCC structure...
73
N. Q. Hoc, D. Q. Vinh, N. T. Hang, N. T. Nguyet, L. X. Phuong, N. N. Hoa, N. T. Phuc and T. T. Hien
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 moduli, the thermal expansion
coefficient, the heat capacities at constant volume and constant pressure and the entropy of
the ternary interstitial alloy with BCC structure with very small concentration of interstitial
atoms (from 0 to 5%). The obtained expressions of these quantities depend on temperature,
concentration of substitution atoms and concentration of interstitial atoms. At zero concentration
of substitution atoms, the thermodynamic quantities of the interstitial alloy ABC become that of
the interstitial alloy AC. At zero concentration of interstitial atoms, the thermodynamic quantities
of the interstitial alloy ABC become that of the substitution alloy AB. At zero concentrations
of substitution and interstitial atoms, the thermodynamic quantities of the interstitial alloy ABC
become that of the main metal in the alloy. The theoretical results are applied to interstitial alloy
FeCrSi. The thermodynamic properties of main metal Fe, substitution alloy FeCr and interstitial
alloy FeSi are special cases for the thermodynamic properties of interstitial alloy FeCrSi. The
calculated results of the thermal expansion coefficient in the temperature interval from 100 to
1000 K and the heat capacity at constant pressure in the temperature from interval 100 to 700 K
for main metal Fe are in good agreement with the experimental data.
Acknowledgements. This work was carried out with the 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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