The melting curve of BCC substitutional alloy MoNi with defects

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63

THE MELTING CURVE OF BCC SUBSTITUTIONAL
ALLOY MoNi WITH DEFECTS
Nguyen Quang Hoc1, Bui Duc Tinh1, Tran Dinh Cuong1, Le Hong Viet2
1
Hanoi National University of Education
2
Tran Quoc Tuan University
Abstract: The melting curve of defective substitutional alloy AB with body-centered cubic
(BCC) structure under pressure is derived by the statistical moment method (SMM). The
temperature of absolute stability for crystalline stateand the equilibrium vacancy
concentration have been used to calculate the melting temperature. In limit case, we
obtain the melting theory of main metal A with BCC structure. The theoretical results are
numerically applied for molybdenum-nickel alloy (MoNi) with using Mie-Lennard-Jones
potential. These results are in good agreements with experimental data and other
calculations.
Keywords: Substitutional alloy, equilibrium vacancy concentration, absolute stability for
crystalline state, statistical moment method.
Email:
Received 6 December 2018
Accepted for publication 18 December 2018

1. INTRODUCTION
The melting temperatureis very important physical characteristic of alloys. Study on
the effect of pressure and impurities on the melting point of crystal pays particular
attention to many researchers [1, 2]. On the experimental side, we have Simon equation to
describe the pressure- temperature relationship at the melting point in the case of low
pressure [3]. In the case of high pressure, we can find the melting curve of crystal by using

Kumari - Dass equation [4].
On the theoretical side, the melting happens when the Gibbs energy of solid phase is
equal to the one of liquid phase. However, we cannot find the explicit expression of
melting temperature Tmby solving this condition, so building a theory for defining the
melting properties of crystal is one of the most interesting research topics in materials
science. In aid of the statistical moment method (SMM), Nguyen Tang and Vu Van Hung
[5, 6] show that we absolutely only use the solid phase of crystal to determine the melting


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64

temperature. Firstly they determine the absolute stability temperature TSat different
pressures by using the SMM and then carry out a regulation in order to find Tm from TS.
The obtained results from the SMM are better than that from other methods in comparison
with experiments [7, 8].
Besides, in [9, 10] the authors proved that the point defects including the vacancieshas
the significantly contribution on thermodynamic quantities of crystals at high temperature.
But in most studies the melting theory is only applied for perfect crystal. For the reasons
above, we will use the SMM to consider the effect of pressure, the substitutional atoms and
the equilibrium vacancy concentration on the melting temperature of substitutional alloy
AB with body-centered cubic (BCC) structure. The melting curve of alloy MoNi has been
builded in this paper.

2. METHODOF CALCULATION
2.1. The melting of perfect BCC substitutional alloy AB
The cohesive energy u0Xand the crystal’s parameters k X , γ 1 X , γ 2 X , γ X for pure metal
X (X = A, B) with BCC structure in the approximation of two coordination spheres have
the form

u0 X = 8ϕ XX (a1 X ) + 6ϕ XX (a2 X ),

(1)

4 (2)
8 (1)
3 (1)
(2)
k X = ϕ XX
(a1 X ) +
ϕ XX (a1 X ) + ϕ XX
(a2 X ) +
ϕ AA (a2 X ),
3
3a1 X
a1 X

(2)

γ 1X =

1 ( 4)
2 (3)
2 ( 2)
2 (1)
ϕ XX ( a1 X ) +
ϕ XX ( a1 X ) − 2 ϕ XX
( a1 X ) + 3 ϕ XX
(a1 X ) +

54
9a1 X
9a1 X
9a1 X

+

1 (4)
3
3 3 (1)
(2)
ϕ XX (a2 X ) +
ϕ XX
(a2 X ) −
ϕ XX (a2 X ),
2
24
16a1 X
32a13X

1
9

(4)
γ 2 X = ϕ XX
( a1 X ) +

+

2 (2)

2 (1)
ϕ XX (a1 X ) − 3 ϕ XX
( a1 X ) +
2
3a1 X
3a1 X

3 (3)
9
9 3 (1)
(2)
ϕ XX (a2 X ) −
ϕ XX
(a2 X ) +
ϕ XX (a2 X ),
2
4a1 X
16a1 X
32a13X

γ X = 4(γ 1 X + γ 2 X ),

(3)

(4)
(5)


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where ϕ XX is the pair interaction potential between atoms X-X, a1X is the nearest neighbor
distance, a2 X

2
∂ mϕ XX (aiX )
( m)
=
a1 X , ϕ XX (aiX ) =
(m = 1, 2, 3, 4; i = 1, 2).
∂aiXm
3

By using the equation of state at 0 K and pressure P

 1 ∂u0 X ℏω X ∂k X
+
PvX = −a1 X 
 6 ∂a1X 4k X ∂a1 X


,


(6)

we can find the nearest neighbor distance a1 X ( P,0) and then we can calculate the
displacements of atom X from the following formula

2γ X ( P, 0)θ 2

y X ( P, T ) =

3k X3 ( P, 0)

AX ( P, 0) ,

(7)

where θ = k BoT , k Bo is the Boltzmann constant and AX ( P,0) was given in [11].
From that, we derive the nearest neighbor distance a1 X ( P, T ) at temperature T and
pressure P

a1 X ( P, T ) = a1 X ( P,0) + y X ( P, T ).

(8)

In the model of perfect BCC substitutionalalloy AB, the main atoms A stay in the
peaks and the substitutional atoms B stay in the body centers of cubic unit cell (Figure 1).

Figure 1. The model of perfect BCC substitutional alloy AB

The mean nearest neighbor distance for alloy AB is determined by

∑c B a
=
∑c B
X

a AB

TX

1X

X

X

.

(9)

TX

X



∑c


where c X is the atomic concentration 

X

X



= 1 , BTX is the isothermal bulk modulus [11].



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From the condition of absolute stability limit for crystalline state

 ∂P 
= 0,


 ∂a AB T =T

(10)

S

and the equation of state of the substitutional alloy AB

P=−

∂u
3γ θ
a AB
cX 0 X + G ,
∑
∂a1X

6v AB X
v AB

(11)

where γ G is the Grüneisen parameter

γG = −

a AB c X ∂k X
ℏω
x X coth x X , x X = X ,
∑
6 X k X ∂a1X
2θ

(12)

we can derive the absolute stability temperature for crystalline state in the form

TS =

TS
,
MS
2
a AB
TS = 2 Pv AB +
6

2
a AB
koB
MS =
4

c
∑X k X2
X

2
∂ 2u0 X ℏa AB
∑X cX ∂a 2 − 4
1X

2
c X ω X  1  ∂k X  ∂ 2 k X
∑X k  2k  ∂a  − ∂a 2
1X
X
 X  1X 


,


2

 ∂k X 


.
 ∂a1 X 

(13)

Solving equation (13) will give us the value of TS. Note that TSand MSmust be
calculated at TS.
After that, because TSis not far from Tmat the same physical condition, so we can carry
out a regulation in order to find Tm from TS

a −a
Tm ≈ TS + m S
kBoγ G

 Pv
 ∂ 2u0 X 
cX  ∂u0 X 
AB
+ ∑ 

 + aS  2 
X 18  ∂a1 X 
 ∂a1 X T =T
 aS
T =T

S

S

 
,
 

(14)

where am = aAB ( P, Tm ) , aS = aAB ( P, TS ) .
If we know the melting temperature Tm (0) at zero pressure, we have another way to
calculate the melting temperature Tm ( P) at pressure P [2]as follows

Tm ( P) =

Tm (0) B0
G (0)

1
B0′

.

G( P)
( B0 + B0′ P)

1
B0′

,

(15)

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where G(P)and G(0) respectively are the rigidity bulk modulus at pressure P and zero

 dBT 
 ,
 dP  P =0

pressure, B0 is the isothermal elastic modulus at zero pressure, B0′ = 

BT = BT ( P) is the isothermal elastic modulus at pressure P.
2.2. The melting of defective BCC substitutional alloy AB
From

the

minimum

condition

of

real

Gibbs

thermodynamic

potential

 ∂G XR 
= 0 , we can find the equilibrium vacancy concentration nvX in defective


 ∂nvX  P ,T ,nvX
metal X as follows

nvX

 g vf X
= exp  −
 θ



 ,


(16)

where g vf X is the change of Gibbs thermodynamic potential when a vacancy is formed
g vf X = −

u0 X
u
+ ∆ Xψ X0 + P∆v ≈ − 0 X .
2

4

(17)

Figure 2. The model of perfect metal (left) and defective metal (right) with BCC structure

The equilibrium vacancy concentration in defective substitutional alloy AB is
determined by

 g vf AB
nv = exp  −
 θ


 ∑ c X g vf X


 = exp  − X
θ





.




(18)

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At constant pressure and constant substitutional atom concentration, the melting
temperature TmR of defective alloy is the function of the equilibrium vacancy concentration
nv. In first approximation the melting temperature TmR can be expanded in term of nvas

 ∂T 
Tm2 ( P )
TmR ( P ) = Tm ( P ) +  m  nv = Tm ( P ) +
,
f
f
g vAB
∂g vAB
 ∂nv 
Tm ( P )
−
koB
∂θ
f
∂gvAB
a
∂u
= − 0 AB αT ∑ c X 0 X ,
∂θ
∂a X

4koB
X

(19)

where α T is the thermal expansion coefficient of perfect alloy AB.

3. NUMERICAL RESULTS AND DISCUSSION
For alloy MoNi, we use the Mie-Lennard-Jones pair potential where potential
parameters are given in Table 1

D   r0 
 r0 
ϕ (r ) =
m   − n  
n − m   r 
r
n

m


.


(20)

Table 1. The potential parameters D, m, n, r0for materials Mo [12] and Ni [13]
Interaction

D (eV)

m

n

r0 (10-10 m)

Mo-Mo

1.7042

1.93

7.68

2.72

Ni-Ni

0.3727

8.0

9.0

2.478

Approximately

ϕ Mo-Ni ≈

1
(ϕ Mo-Mo + ϕ Ni-Ni ) .
2

(21)

We have some comments about the melting temperature of alloy MoNi. Firstly, at the
same concentration of substitutional atoms when pressure increases, the melting
temperature of alloy MoNi also increases. For example, at cNi = 1.8 % when pressure P
increases from 0 to 80 GPa, the melting temperature Tm of perfect alloy MoNi increases
from 1754 to 3183 Kand the melting temperature Tm of defective alloy MoNi increases
from 1703 to 3023 K.Secondly, at the same pressure when the concentration of
substitutional atoms increases, the melting temperature of alloy MoNi also decreases. For


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example, at zero pressure when c Ni increases from 0 to 1.8%, the melting temperature Tm
of perfect alloy MoNi decreases from 3089 to 1754 K and the melting temperature Tm of
defective alloy MoNi decreases from 2948 to 1703 K. Thirdly, the melting temperature Tm
of defective alloy MoNi is smaller than the melting temperature Tm of perfect alloy MoNi
at the same physical condition. Maximum melting temperature decreases about 8.6 %. The
calculated results of melting temperature for alloy MoNi with defect are nearer with
experiments and the other theoretical results than that for ideal alloy.
Table 2. The melti temperature Tmof alloy Mo-1.8%Ni at zero pressure
from SMM, CALPHAD [18] and experimental data (EXPT) [19 – 24]

SMM
(perfect
alloy)

SMM
(defective
alloy)

CALPHAD
[18]

[19]

[20]

[21]

[22]

[23]

[24]

1754K

1703K

1622K

1616K

1619K

1643K

1623K

1635K

1633K

EXPT

Figure 3. The melting curve of Mo from SMM, EXPT [14]
and the other calculations [15-17]


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Figure 4. The melting curve of alloy Mo-1.8%Ni from the SMM

4. CONCLUSION
The melting temperature of defective substitutional alloy AB with BCC structure has
been studied by using SMM. The theoretical results are numerically applied for alloy
MoNi with using Mie-Lennard-Jones potential in the interval of pressure from 0 to 80 GPa
and in the interval of concentration of substitutionalatoms from 0 to 1.8%. Our calculated
results are in good agreement with experiments and the other calculations. That proved that
the concentration of equilibrium vacancies has the contribution on thermodynamic

quantities of substitutional alloy in high temperatures.

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ĐƯỜNG CONG NÓNG CHẢY CỦA HỢP KIM THAY THẾ MONI
VỚI CẤU TRÚC LPTK CÓ KHUYẾT TẬT
Tóm tắ
tắt: Rút ra đường cong nóng chảy của hợp kim thay thế AB có khuyết tật với cấu
trúc lập phương tâm khối (LPTK) dưới tác dụng của áp suất bằng phương pháp thống kê
mômen. Nhiệt độ bền vững tuyệt đối trạng thái tinh thể và nồng độ nút khuyết cân bằng
được dùng để tính nhiệt độ nóng chảy. Trong trường hợp giới hạn, chúng tôi thu được lý
thuyết nóng chảy của kim loại chính A với cấu trúc LPTK. Các kết quả lý thuyết được áp
dụng tính số cho hợp kim MoNi khi sử dụng Mie-Lennard-Jones. Các kết quả này phù
hợp tốt với số liệu thực nghiệm và các kết quả tính toán khác.
Từ khóa: Hợp kim thay thế, nồng độ nút khuyết cân bằng, bền vững tuyệt đối trạng thái
tinh thể, phương pháp thống kê mômen.