Abstract. From the model of substitutional alloy AB with interstitial atom C and FCC
structure and the condition of absolute stability for crystalline state we derive analytic
expressions for the temperature of absolute stability for the crystalline state and the melting
temperature together with the equation of the melting curve of this alloy by way of applying
the statistical moment method. The obtained results allow us to determine the melting
temperature of alloy ABC at zero pressure and under pressure. In limit cases, we obtain the
melting theory of main metal A, substitutional alloy AB and interstitial alloy AC with FCC
structure. The theoretical results are numerically applied for alloys AuCuSi and AgCuSi

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59
JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2017-0032
Mathematical and Physical Sci. 2017, Vol. 62, Iss. 8, pp. 59-71
This paper is available online at
STUDY ON THE MELTING OF SUBSTITUTIONAL ALLOY AB
WITH INTERSTITIAL ATOM C AND FCC STRUCTURE UNDER PRESSURE
Nguyen Quang Hoc and Dinh Quang Vinh
Faculty of Physics, Hanoi National University of Education
Abstract. From the model of substitutional alloy AB with interstitial atom C and FCC
structure and the condition of absolute stability for crystalline state we derive analytic
expressions for the temperature of absolute stability for the crystalline state and the melting
temperature together with the equation of the melting curve of this alloy by way of applying
the statistical moment method. The obtained results allow us to determine the melting
temperature of alloy ABC at zero pressure and under pressure. In limit cases, we obtain the
melting theory of main metal A, substitutional alloy AB and interstitial alloy AC with FCC
structure. The theoretical results are numerically applied for alloys AuCuSi and AgCuSi.
Keywords: Interstitial and substitutional alloys, limiting temperature for absolute stability of
crystalline state, statistical moment method
1. Introduction
Alloys in general and interstitial alloys in particular are commonly used materials in
material technology and science. Studies on interstitial alloys have been carried out by many
researchers.
The melting temperature (MT) of materials under pressure is a very important problem in
solid state physics and material science [1, 2]. The MT of crystal usually is determined from the
Simon experimental equation
0
0( ) 1,
cm
m
P P
T T
a
(1.1)
where mT is the MT, mP is the melting pressure, a and c are constants, 0P and 0T respectively are
the pressure and the temperature of triple point on the phase diagram .
Normally, when the value of 0P is small, we can write (1.1) in the form
0( ) 1.
cm
m
P
T T
a
(1.2)
Received March 27, 2017. Accepted August 30, 2017.
Contact Nguyen Quang Hoc, email: hocnq@hnue.edu.vn
Nguyen Quang Hoc and Dinh Quang Vinh
60
However, eq. (1.2) can not describe the melting of crystal at high pressures. Kumari et al. [3]
introduce the following phenomenological equation
0
0 0
,
( )
m
m
m
T
A B P P
T P P
(1.3)
where temperatures
mT and 0T respectively are the MT at pressures mP and 0P , 0m mT T T ,
A and B are empirical constants. Eq. (1.3) allows us to determine the MT of crystal at high
pressures.
Theoretically in order to determine the MT of crystal it is necessary to apply the
equilibrium condition of solid phase and liquid phase. However, we cannot in this way find the
explicit expression of MT. According to some researchers, the sT temperature corresponding to
the absolute stability limit for crystalline state at some pressure is not far from the MT at the same
pressure. Therefore, according to authors in [4] the melting curve of crystal coincides with the
curve of absolute stability limit for crystalline state. With that idea, the self-consistent phonon
field method and the one-particle distribution function method are used to study the MT. However,
the obtained results are not in good agreement with experimental data. From that some scientists
had concluded that it is impossible to find the melting temperature by using the limit of absolute
stability only for the solid phase. Some other researchers apply the correlation effect to calculate
the temperature of absolute stability limit for crystalline state. The obtained results from this
regulation are more exact but are limited at low pressures.
Using the statistical moment method (SMM), Nguyen Tang and Vu Van Hung [4, 5] show
that we can use absolutely only use the solid phase of crystal only to determine the MT. First they
determine the sT absolute stability temperature at different temperatures by using the SMM and
then carry out a regulation in order to find mT from sT . The obtained results by the SMM are in
good agreement with the experimental data.
2. Content
2.1. Analytic result
In the model of interstitial alloy AC with the face-centured cubic (FCC) structure, the A
atoms of large size stay in the peaks and the face centers of cubic unit cells and the C interstitial
atoms of smaller size stay in the body center. In [6, 7], we derived the analytic expressions of the
nearest neighbor distance, the cohesive energy and the alloy parameters for atoms C, A, A1 (the
atom A in the face centers) and A2 (the atom A in the peaks).
The equation of state of the interstitial aloy AC with FCC structure at temperature T and
pressure P is described by
01
1 1
1 1
cth .
6 2
u k
Pv r x x
r k r
(2.1)
At 0K and pressure P, this equation has the form
0 01
1 1
.
4
u k
Pv r
r k r
(2.2)
Study on the melting of substitution alloy AB with interstitial atom C and FCC structure under pressure
61
Knowing the form of the interaction potential, Eq. (2.1) allows us to determine the nearest
neighbor distance 1 , 0Xr P 1 2, , ,X C A A A at 0K and pressure P. Knowing 1 , 0Xr P , we
can determine the parameters 1 2( ,0), ( ,0), ( ,0), ( ,0)X X X Xk P P P P at 0K and pressure P for
each case of X. The displacement
0 ( , )Xy P T of atoms from the equilibrium position at
temperature T and pressure P is determined as in [6, 7]. From that, we can calculate the neares
neighbor distance 1 ,Xr P T at temperature T and pressure P as follows:
11 1 1 1
( , ) ( ,0) ( , ), ( , ) ( ,0) ( , ),C C A A A Ar P T r P y P T r P T r P y P T
1 2 21 1 1 1
( , ) ( , ), ( , ) ( ,0) ( , ).A C A A Cr P T r P T r P T r P y P T (2.3)
The mean nearest neighbor distance between two atoms in interstitial alloy AC with FCC
structure is approximately determined by
1 1, ,0 , ,A Ar P T r P y P T
1 1 1 1 1( ,0) 1 ( ,0) ( ,0), ( ,0) 2 ( ,0),A C A C A A Cr P c r P c r P r P r P
1 2
, 1 15 , , 6 , 8 , .C A C C C A C Ay P T c y P T c y P T c y P T c y P T
(2.4)
The free energy of interstitial alloy AC with FCC structure and concentration condition
C Ac c has the form
1 2
1 15 6 8 ,ACAC C A C C C A C A cc c c c TS
2
2 1
0 0 22
2
3 1
3 2
X X
X X X X X
X
X
U N X
k
3
2 2
2 1 1 24
2 4
1 2 2 1 1 ,
3 2 2
X X
X X X X X X
X
X X
X X
k
2
0 3 ln(1 ) , coth .
Xx
X X X X XN x e X x x
(2.5)
where X is the free energy of an atom 1 2 , , ,X X A A A C and
AC
cS is the configurational
entropy of interstitial alloy AC.
In the interstitial alloy ABC with FCC structure (the interstitial alloy AC with the atoms A in
the peaks and the face centers and the interstitial atoms C in the body center and then the atoms
B substitutes for the atoms A in the face centers), the mean nearest neighbor distance between two
atoms at temperature T and pressure P is determined by
Nguyen Quang Hoc and Dinh Quang Vinh
62
1
( , , , ) , ,
1 1
, ( , ), , ,
TAC TB
ABC B C AC AC B B T AC TAC B TB
T T
AC A C AC A TAC TB
TAC TB
B B
a P T c c c a c a B c B c B
B B
c c c a r P T B B
3
0
2
2
, ,( ) ,
1
2
3
, ,
, 0
( )
( )
2
3 ( )
,
, ,
TAC C
AC
AC
AC
AC A
C
C
C C T
P T c
P
N
P T c
P c
a
a
P T aca
1 2
1 2
2 2 2
22 2
1
2 22
2 2 2
1 15
( , )
6 8 ,AC AC AC
AC A TT A T
A AC
C C C
C A AT T T
c
a ar P T
c c c
a a a
2
22 2
0
12 2 2
,
1 1 1
( , ).
3 6 4 2
XX X X X
X X
X X X X X XT
u k k
a r P T
N a a k a k a
2.6)
The mean nearest neighbor distance between two atoms at 0K and pressure P is
determined by
0 0
0 0 0
0 0
) .,( , , TAC TBABC B AC AC B B
T T
C
B B
a c a c a
B
P c c
B
T (2.7)
The free energy of interstitial alloy ABC with FCC structure and the concentration condition
C B Ac c c has the form [7]
,AC ABCABC AC B B A c cc TS TS
(2.8)
where B is the free energy of an atom B and
ABC
cS is the cofigurational entropy of alloy ABC.
The pressure is calculated by
.
3T T
a
P
V V a
(2.9)
Setting
Study on the melting of substitution alloy AB with interstitial atom C and FCC structure under pressure
63
1 2
1 1 2 2
1 1 2 2
1 1
coth
1 1
1 15 coth coth
6
1 1
coth coth6 8
B A
B B B
B B A
T ABC CA
G C A A C C C
A A C C
A A
C A A C A A
A A A A
k k
c x x
k a k a
a kk
c x x c x x
k a k a
k k
c x x c x x
k a k a
coth . (2.10)
A A
A
x x
Here,
T
G is the Grüneisen parameter of alloy ABC. Then,
1 2
1 2
0 0 0 01 15
6
6 8ABC A CC C
ABC A C
A A
C C
A A
a U U
P c c
V a a
U U
c c
a a
0 0 3 . .
T
B A G
B
B A ABC
U U
c
a a V
(2.11)
From the condition of absolute stability limit
0
s
ABC T
P
V
hay 0
s
ABC T
P
a
(2.12)
we can derive the absolute stability temperature for the crystalline state in the form
1 2
1 2
2 22 2 2 2
0 00 0 0
2 2 2 2 2
, 2 1 15 6 8
6
A AABC A C B
s ABC C C C C B
A C A A B
U Ua U U UTS
T N PV c c c c c
MS a a a a a
22
2 22
0
2 2 2
1 15
2 4 2 4
A ABC A ABC ABC C C C ABCA A
B C ABC C ABC
A A A A A C C C C
U a a a k k ak k
c c a c a
a k a a k k a a k
1 1 1 2 2 2
1 1 1 1 2 2 2 2
2 2
2 2
2 2
6 8
2 4 2 4
A A A ABC A A A ABCABC ABC
C ABC C ABC
A A A A A A A A
k k a k k aa a
c a c a
k a a k k a a k
2 2
2 2
2 2
22
2 2 2
2 2
,
2 4 2 4
1 15 6
4 4
ABC B ABC ABC A ABCB B A A
B ABC B ABC
B B B B A A A A
ABC Bo ABC Bo C ABCA
C C C
A A C C
a a a ak k k k
c a c a
k a a k k a a k
a k a k k a kk
D c c c
k a k a
1
1 1
2
2
4
ABo
A A
k
k a
2
2 2
2 2 2
2 2 2
2 2 2
8 ,
4 4 4
AABC Bo ABC Bo ABC BoB A
C B B
A A B B A A
ka k a k a kk k
c c c
k a k a k a
(2.13)
Nguyen Quang Hoc and Dinh Quang Vinh
64
Because the curve of absolute stability limit for the crystalline state is not far from the
melting curve of crystal, the
sT temperature is usually large and coth 1X Xx x at sT therefore,
1
1 1
2
2 2
222
2 2 2
2 2 2
12 2 2
2 2 2
2 2 2
1 15 6
4 4 4
8
4 4 4
AABC Bo ABC Bo C ABC BoA
C C C
A A C C A A
AABC Bo ABC Bo ABC BoB A
C B B
A A B B A A
ka k a k k a kk
c c c
k a k a k a
ka k a k a kk k
c c c
k a k a k a
1 2
1 2
2 22 2 2 2 2
0 00 0 0 0
2 2 2 2 2 2
22
22
2
2 1 15 6 8
6
1 15
2 4 2
A AABC A C A A
ABC C C C C B B
A C A A A A
ABC A ABC ABC C CA A
C ABC C ABC
A A A A C C
U Ua U U U U
PV c c c c c c
a a a a a a
a a a k kk k
c a c a
k a a k k a
1 1 1 2 2 2
1 1 1 1 2 2 2 2
2
2 2
2 2
2 2
2
2
2
4
6 8
2 4 2 4
2
C ABC
C C
A A A ABC A A A ABCABC ABC
C ABC C ABC
A A A A A A A A
ABC B AB B
B ABC
B B B
a
a k
k k a k k aa a
c a c a
k a a k k a a k
a ak k
c a
k a a
1 2
1 1 2 2
2
2
24 2 4
2
1 1 1 1 1 1
1 15 6 8
BC ABC A ABCA A
B ABC
B A A A A
ABC
A ACA B A
Bo ABC C C C C B B
A A C C A A A A B B A A
a ak k
c a
k k a a k
V
k kkk k k
k a c c c c c c
k a k a k a k a k a k a
1 2
1 2
0 00 0 0 01 15 6 8 0.
6
A AABC A C B A
C C C C B B
ABC A C A A B A
U Ua U U U U
P c c c c c c
V a a a a a a
(2.14)
This is the equation for the curve of absolute stability limit for the crystalline state. Therefore,
pressure is a function of the mean nearest neighbor distance
( ).ABCP P a (2.15)
Temperature 0sT at zero pressure has the form
1 2
1 2
0 00 0 0 0(0) 1 15 6 8 ,
18
A AABC A C B A
s C C C C B BT
A C A A B AG B
U Ua U U U U
T c c c c c c
a a a a a ak
(2.16)
where the parameters 0, , TXAB G
X
U
a
a
are determined at (0).sT Temperature sT at pressure
P has the form
Study on the melting of substitution alloy AB with interstitial atom C and FCC structure under pressure
65
2
(0) .
3 ABC
ABC G
s s s
T a
Bo G
V P
T T T
Tk
(2.17)
Here,
Bok is the Boltzmann constant, , , /
T T
ABC G GV T are determined at .sT Approximately,
.m sT T . In order to solve Eq.(2.17), we can use the approximate iteration method. In the first
approximate iteration,
1
( (0))
(0) .
3 ( (0))
ABC s
s s
Bo G s
V T P
T T
k T
(2.18)
Here 0sT is the temperature of absolute stability limit for the crystalline state at pressure P in
the first approximate iteration of Eq. (2.17). Substituting Ts1 into Eq. (2.17), we obtain a better
approximate value 2sT of sT at pressure P in the second approximate iteration
1 1
2 12
1 1
( ) ( )
(0) .
3 ( ) 3 ( )
AB
ABC s ABC s G
s s s
Bo G s aBo G s
V T P V T P
T T T
k T Tk T
(2.19)
Analogouslty, we can obtain the better approximate values 3 4, ,...s sT T of sT at pressure P in the
third, fourth, etc. approximate iterations. These approximations are applyed at low pressures.
In the case of high pressure, the MT of the alloy at pressure P is calculated by
0
0
1
0
1
0 0
(0) ( )
( ) . ,
(0)
( )
B
m
m
B
T B G P
T P
G
B B P
(2.20)
where ( )mT P and (0)mT respectively are the MT at pressure P and zero pressure, ( )G P and
(0)G respectively are the rigidity bulk modulus at pressure P and zero pressure, 0B is the
isothermal elastic modulus at zero pressure, 0
0
, ( )T T T
P
dB
B B B P
dP
is the isothermal
elastic modulus at pressure P.
2.2. Numerical results for alloys AuCuSi and AgCuSi
For alloys AuCuSi and AgCuSi, we use the n-m pair potential
0 0( ) ,
n m
r rD
r m n
n m r r
(2.21)
where potential parameters are given in Table 1 [8].
Nguyen Quang Hoc and Dinh Quang Vinh
66
Table 1. Potential parameters
0, , ,m n D r of materials
Material m n 1610 ergD
10
0 10 mr
Au
Ag
5.5
5.5
10.5
11.5
6462.540
4589.328
2.8751
2.8760
Cu 5.5 11.0 4693.518 2.5487
Si 6.0 12.0 45128.340 2.2950
Considering the interaction between atoms Au(Ag) and Si and between atoms Au(Ag) and
Cu in the above mentioned alloys, we use approximations
Au(Ag)-Si Au(Ag)-Au(Ag) Si-Si Au(Ag)-Cu Au(Ag)-Au(Ag) Cu-Cu
1 1
,
2 2
(2.22)
and ignore the interaction between atoms Cu and Si
At 0.1 MPa, Au has a FCC structure with 104,0785.10 ma at 300K and the melting
point at 1337 K. The melting curve of Au is determined up to 1673 K and 6.5 GPa with the slope
dT/dP = 60 K/GPa [9,10] and up to 1923 K and 12 GPa [11]. The melting curve of Au is shown in
Figure 1 [12].
At 0.1 MPa, Ag has a FCC structure with 104,0862.10 ma at 300K and the melting
point at 1235 K. The melting curve of Au is determined up to 1563 K and 6.5 GPa with the slope
dT/dP = 60 K/GPa [9]. The melting curve of Au is shown in Figure 2 [12].
Figure 1. The melting curve of Au Figure 2. The melting curve of Ag
Our numerical results are summarized in tables from Table 2 to Table 5 and illustrated in
figures from Figure 3 to Figure 6. The concentration of substitution atoms changes from 0 to 10%,
the concentration of interstitial atoms changes from 0 to 5% and the pressure changes from
0 to 65 GPa.
Study on the melting of substitution alloy AB with interstitial atom C and FCC structure under pressure
67
Table 2. Dependence of melting temperature on pressure and concentration
of interstitial atoms for alloy Au-0.06Cu-ySi
P[GPa] y(%) 0 1 2 3 5
5
1559.69 1592.27 1632.67 1684.05 1844.11
10
1776.22 1804.72 1840.92 1840.92 2046.19
30 T(K) 2579.51 2588.15 2601.58 2623.19 2727.50
50
3316.22 3304.56 3293.49 3285.20 3310.00
65
3842.18 3815.44 3785.85 3754.39 3712.51
Table 3. Dependence of melting temperature on pressure and concentration
of substitution atoms for alloy Au-xCu-0.05Si
P[GPa] x(%) 0 6 8 10
5
1833.70 1844.11 1847.64 1851.20
10
2029.17 2046.19 2051.98 2057.82
30 T(K) 2686.19 2727.50 2741.58 2755.81
50
3246.91 3310.00 3331.50 3353.23
65
3634.30 3712.51 3739.16 3766.10
Table 4. Dependence of melting temperature on pressure and concentration
of interstitial atoms for alloy Ag-0.06Cu-ySi
P[GPa] y(%) 0 1 2 3 5
5
1561.96 1558.42 1554.42 1549.87 1539.06
10
1879.00 1878.42 1878.51 1879.87 1892.38
30 T(K) 3031.86 3009.25 2983.20 2953.26 2884.36
50
4075.18 4021.95 3957.92 3879.66 3661.14
65
4817.91 4741.12 4647.55 4531.13 4188.98
Table 3. Dependence of melting temperature on pressure and concentration
of substitution atoms for alloy Ag-xCu-0.05Si
P[GPa] x(%) 0 6 8 10
5
1544.21 1539.06 1537.14 1535.15
10
1913.53 1892.38 1885.52 1878.76
30 T(K) 2962.25 2884.36 2860.03 2836.44
50
3790.97 3661.14 3620.88 3581.99
65
4356.58 4188.98 4137.17 4087.20
Nguyen Quang Hoc and Dinh Quang Vinh
68
Figure 3. Dependence of melting temperature on pressure and concentration
of interstitial atoms for alloy Au-0.06Cu-ySi
Figure 4. Dependence of melting temperature on pressure and concentration
of substitution atoms for alloy Au-xCu-0.05Si
Study on the melting of substitution alloy AB with interstitial atom C and FCC structure under pressure
69
Figure 5. Dependence of melting temperature on pressure and concentration
of interstitial atoms for alloy Ag-0.06Cu-ySi
Figure 6. Dependence of melting temperature on pressure and concentration
of substitution atoms for alloy Ag-xCu-0.05Si
Nguyen Quang Hoc and Dinh Quang Vinh
70
According to our numerical results for alloys AuCuSi and AgCuSi at the same concentration
of substitution atoms and the same concentration of interstitial atoms when pressure increases, the
melting temperature increases. For example
Cu Si6%, 5%c c when P increases from 5GPa to
65 GPa,
meltingT of alloy AuCuSi increases from 1844.11 K to 3712.51 K.
At the same pressure and concentration of substitution atoms when the concentration of
interstitial atoms increases, the melting temperature decreases. For example at P = 65 GPa,
Cu 6%c when Sic increases from 0 to 5%, mT of alloy AuCuSi decreases from 3842.18 K to