The melting curve of BCC substitutional alloy MoNi with defects

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.

THE MELTING CURVE OF BCC SUBSTITUTIONAL

THE MELTING CURVE OF BCC SUBSTITUTIONAL

ALLOY MoNi WITH DEFECTS ALLOY MoNi WITH DEFECTS

Nguyen Quang Hoc 1 , Bui Duc Tinh 1 , Tran Dinh Cuong 1 , Le Hong Viet 2

1 Hanoi National University of Education 2

Tran Quoc Tuan University

Abstract:

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:

Keywords: Substitutional alloy, equilibrium vacancy concentration, absolute stability for

crystalline state, statistical moment method

Email: hocnq@hnue.edu.vn

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 T mby 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

temperature Firstly they determine the absolute stability temperature T Sat different

pressures by using the SMM and then carry out a regulation in order to find T m from T S 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 u 0Xand the crystal’s parameters kX, γ1X, γ2X, γXfor pure metal

X (X = A, B) with BCC structure in the approximation of two coordination spheres have the form

0X 8 XX( 1X) 6 XX( 2X),

( 4) (3) ( 2) (1)

(4) (2) (1)

2 1 2 1 3 1

where ϕXX is the pair interaction potential between atoms X-X, a1Xis the nearest neighbor distance, 2 2 1

3

=

∂

m

iX

a

a

ϕ ϕ

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

0 1

ω

ℏ

(6)

we can find the nearest neighbor distance a1X( ,0) P and then we can calculate the displacements of atom X from the following formula

2 3

2 ( ,0)

X

P

where θ = k TBo , kBois the Boltzmann constant and A PX( ,0) was given in [11]

From that, we derive the nearest neighbor distance a1X( , ) P T at temperature T and pressure P

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

Figure 1 The model of perfect BCC substitutional alloy AB

The mean nearest neighbor distance for alloy AB is determined by

1

.

X AB

X TX X

c B a a

c B

=

∑

where cXis the atomic concentration  cX 1 

=

∑ , BTXis the isothermal bulk modulus [11]

From the condition of absolute stability limit for crystalline state

0,

S

AB T T

P

a =

=

∂

and the equation of state of the substitutional alloy AB

0 1

3 , 6

X G AB

X X

AB X AB

u a

γ θ

∂

∂

where γG is the Grüneisen parameter

1

X X X

k a

ω γ

θ

∂

∂

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

,

S

TS T

MS

=

2 2

0

1

X

u

2

1

2

4

X

X

AB oB X

X X

k MS

a

k

∂

=

∂

∑

(13)

Solving equation (13) will give us the value of T S Note that TSand MSmust be calculated at T S

After that, because T S is not far from T mat the same physical condition, so we can carry

out a regulation in order to find T m from T S

2

2

, 18

X

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 T Pm( ) at pressure P [2]as follows

0

0

1 0

1

(0)

B m

m

B

T P

G

B B P

′

′

=

′ +

(15)

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

pressure, B0 is the isothermal elastic modulus at zero pressure, 0

0

T P

dB B

dP =

( )

T T

B = B 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

, ,

0

vX

R

X

vX P T n

G

n

=

∂

  , we can find the equilibrium vacancy concentration n vX in defective metal X as follows

f

v X vX

g n

θ

(16)

where f

v X

g is the change of Gibbs thermodynamic potential when a vacancy is formed

0

Figure 2

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

f

v

c g g

n

∑

(18)

At constant pressure and constant substitutional atom concentration, the melting temperature R

m

T of defective alloy is the function of the equilibrium vacancy concentration

n v In first approximation the melting temperature R

m

T can be expanded in term of n vas

2( )

( )

vAB vAB v

m

oB

n

T P

k

θ

∂

∂

∂

4

f

X

c

θ

= −

where αT 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

0 0

D

Table 1

Table 1 The potential parameters D, m, n, r 0 for materials Mo [12] and Ni [13]

Interaction D (eV) m n r 0 (10-10 m)

Approximately

Mo-Ni Mo-Mo Ni-Ni

1

2

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 T m of perfect alloy MoNi increases

from 1754 to 3183 Kand the melting temperature T m 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

example, at zero pressure when cNi increases from 0 to 1.8%, the melting temperature T m

of perfect alloy MoNi decreases from 3089 to 1754 K and the melting temperature T m of

defective alloy MoNi decreases from 2948 to 1703 K Thirdly, the melting temperature T m

of defective alloy MoNi is smaller than the melting temperature T m 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

Table 2 The melti temperature T m of 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]

EXPT [19] [20] [21] [22] [23] [24] 1754K 1703K 1622K 1616K 1619K 1643K 1623K 1635K 1633K

Figure 3

Figure 3 The melting curve of Mo from SMM, EXPT [14]

and the other calculations [15-17]

Figure 4

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 tắắắắt: t: t: 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 ừừ ừ khóa: khóa: khóa: Hợp kim thay thế, nồng độ nút khuyết cân bằng, bền vững tuyệt đối trạng thái

tinh thể, phương pháp thống kê mômen