1. Introduction There are many theoretical and experimental works on thermodynamic and elastic properties of metals and interstitial alloys [1-27], for examples the theory of interstitial alloys [1, 2], the calculations from first principles, the many-body potentials, the molecular dynamics for defects in metals, alloys and solid solutions [3-5], the thermodynamic and elastic properties of ideal ternary and binary interstitial alloys [6-13] and the thermodynamic and elastic properties of metals [16-27]. In this paper, we build the theory of elastic deformation for binary interstitial AB with facecentered cubic (FCC) structure under pressure by the statistical moment method (SMM) [14, 15].
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74
HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0031
Natural Sciences 2018, Volume 63, Issue 6, pp. 74-83
This paper is available online at
ELASTIC DEFORMATION OF ALLOY AuSi WITH FCC STRUCTURE
UNDER PRESSURE
Nguyen Quang Hoc
1
, Nguyen Duc Hien
2
and Dang Quoc Thang
1
1
Faculty of Physics, Hanoi National University of Education
2
Mac Dinh Chi High School, Gia Lai Province
Abstract: The theory of elastic deformation for binary interstitial alloy with FCC structure
under pressure is builded by the statistical moment method. The elastic deformation of main
metal is special case of elastic deformation for binary interstitial alloy. The theoretical results
are applied to alloy AuSi. The numerical results for alloy AuSi are compared with the
numerical results for main metal Au and experiments.
Keywords: Interstitial alloy, elastic deformation, Young modulus, bulk modulus, rigidity
modulus, elastic constant, Poisson ratio.
1. Introduction
There are many theoretical and experimental works on thermodynamic and elastic properties
of metals and interstitial alloys [1-27], for examples the theory of interstitial alloys [1, 2],
the calculations from first principles, the many-body potentials, the molecular dynamics for
defects in metals, alloys and solid solutions [3-5], the thermodynamic and elastic properties of
ideal ternary and binary interstitial alloys [6-13] and the thermodynamic and elastic properties of
metals [16-27].
In this paper, we build the theory of elastic deformation for binary interstitial AB with face-
centered cubic (FCC) structure under pressure by the statistical moment method (SMM) [14, 15].
2. Content
2.1. Analytic results
In interstitial alloy AB with FCC structure, the cohesive energy and the alloy parameter for
the atom B (in body center of cubic unit cell), the atom A1 (the atom A stays in the face centers of
cubic unit cell) and the atom A2 (the atom A stays in the peaks of cubic unit cell) in the
approximation of three coordination are determined by [6-13]
0 1 1 14 3 12 5 ,3B AB AB ABr r ru (2.1)
11 01 0
( ),B B Ar r y T
(2.2)
Received March 28, 2018. Revised June 29, 2018. Accepted July 9, 2018.
Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn
Elastic deformation of alloy AuSi with FCC structure under pressure
75
(2) (1) (2) (1)1 1 1 1
1 1
2 4 8 3
3 3
3 9
B
AB B AB B AB B AB B
B B
k r r r r
r r
(2) (1)1 1
1
8 5
4 5 5 ,
5
AB B AB B
B
r r
r
(2.3)
(4) (2) (1) (4) (3)
1 1 1 1 1 12 3
1 1 1
1 1 1 1 2 3
( ) ( ) ( ) ( 3) ( 3)
24 4 4 54 27
B
AB B AB B AB B AB B AB B
B B B
r r r r r
r r r
(2) (1) (4) (3)
1 1 1 12 3
1 1 1
2 2 3 17 8 5
( 3) ( 3) ( 5) ( 5)
27 81 150 125
AB B AB B AB B AB B
B B B
r r r r
r r r
(2) (1)
1 12 3
1 1
1 5
( 5) ( 5),
25 125
AB B AB B
B B
r r
r r
(2.4)
(3) (2) (1) (3)
2 1 1 12 3
1 1 1 1
1
(4)
1 1
(2) (1) (4) (3)
1 1 1 12 3
1 1
1 3 3 2
( ) ( ) ( ) 2 ( 2)
2 4 4 8
7 2 5
2 2 5 5
1
( )
4
7 4 26
( ) ( ) ( ) ( )
8 16 25 125
B
AB B AB B AB B AB
B B B B
B
AB B B
AB B AB B AB B AB B
B B
r r r
r r r r
r r
r r r r
r r r
(2) (1)
1 12 3
1 1
5
5 5
3 3
( ) ( ),
25 125
AB B AB B
B B
r r
r r
(2.5)
1 24 ,B B B (2.6)
1 10 0 1
,A A AB Au u r
(2.7)
),(0011 11 Tyrr BAA
(2.8)
1 1
(2)
1 ,A A AB Ak k r
(2.9)
1 1
(4)
1 11
1
( ),
24
ABA AA r
(2.10)
1 1 1 12
1 1 1
(3) (2) (1)
2 21 1 13
1 1 1
1 1 1
( ) ( ) ( ),
4 2 2
AA AB AB ABA A A
A A A
r r r
r r r
(2.11)
1 1 11 2
4 ,A A A
(2.12)
2 20 0 1
,A A AB Au u r
(2.13)
),(
222 0011
Tyrr AAA
(2.14)
2 2 2
2
(2) (1)
1 1
1
,
1 23
6 6
A A AB A AB A
A
k k r r
r
(2.15)
2 2 2 2 2
2 2 2
1 1
(4) (3) (2) (1)
1 1 1 12 3
1 1 1
1 2 2 2
( ) ( ) ( ) ( ),
54 9 9 9
A A AB A AB A AB A AB A
A A A
r r r r
r r r
(2.16)
Nguyen Quang Hoc, Nguyen Duc Hien and Dang Quoc Thang
76
2 2 2 2 2
2 2 2
(4)
2 1 2
1 1
(3) (2) (1)
2 1 1 13
1
1 4 14
( )
81 27 27
14
( ) ( ) ( ),
27
A AB A
A A
A AB A AB A AB A
A
r
r r
r r r
r
(2.17)
2 2 21 2
4 ,A A A (2.18)
where AB is the interaction potential between the atom A and the atom B, 1 1Br r is the nearest
neighbor distance between the interstitial atom B and the metallic atom A at temperature T, 01Br is
the nearest neighbor distance between the interstitial atom C and the metallic atom A at 0K and is
determined from the minimum condition of the cohesive energy 0Bu , 10 ( )Ay T is the displacement
of the atom A1 (the atom A stays in the face centers of cubic unit cell) from equilibrium position at
temperature T, ( ) ( ) ( ) / (m 1,2,3,4), , , , ,m m mAB i AB i ir r r x y z
and
iu is the
displacement of the ith atom in the direction ,
11 1A B
r r is the nearest neighbor distance
between the atom A1 and atoms in crystalline lattice,
201A
r is the nearest neighbor distance
between the atom A2 and atoms in crystalline lattice at 0K and is determined from the
minimum condition of the cohesive energy
20 0
, ( )A Bu y T is the displacement of the atom C at
temperature T, 0 1 2, , ,A A A Au k are the coressponding quantities in clean metal A in the
approximation of two coordination sphere [14, 15]
The equation of state for interstitial alloy AB with FCC structure at temperature T and
pressure P is written in the form
0
1
1 1
1 1
cth .
6 2
u k
Pv r x x
r k r
(2.19)
At 0 K and pressure P, this equation has the form
0 0
1
1 1
.
4
u k
Pv r
r k r
(2.20)
If we know the form of interaction potential 0 ,i eq. (2.6), we will determine the nearest neighbor
distance 1 1 2,0 , , ,Xr P X B A A A at 0 K and pressure P. After we know 1 ,0Xr P , we can
determine alloy parametrs 1 2( ,0), ( ,0), ( ,0), ( ,0)X X X Xk P P P P at 0K and pressure P. After
that, we can calculate the displacements [14, 15]
2
0 3
2 ( ,0)
( , ) ( , )
3 ( ,0)
.X
X X
X
P
y P T A P T
k P
(2.21)
From that, we derive the nearest neighbor distance 1 ,Xr P T at temperature T and pressure P
11 1
, ,0 , ,B B Ar P T r P y P T
1 1, ,0 ,A A Ar P T r P y P T
2 2 21 1
, ,0 , ,A A Ar P T r P y P T
1 11 01
, ,T ,A A Br P T r P y P T
(2.22)
Then, we calculate the mean nearest neighbor distance in interstitial alloy AB by the expressions
as follows [6-13]
1 1, ,0 , ,A Ar P T r P y P T
Elastic deformation of alloy AuSi with FCC structure under pressure
77
1 1 1 1 1( ,0) 1 ( ,0) ( ,0), ( ,0) 2 ( ,0),A B A B A A Br P c r P c r P r P r P
1 2
, 1 15 , , 6 , 8 , ,B A B B B A B Ay P T c y P T c y P T c y P T c y P T
(2.23)
where 1 ( , )Ar P T is the mean nearest neighbor distance between atoms A in interstitial alloy AB at
pressure P and temperature T, 1 ( ,0)Ar P is the mean nearest neighbor distance between atoms A
in interstitial alloy AB at pressure P and 0K, 1 ( ,0)Ar P is the nearest neighbor distance between
atoms A in clean metal A at pressure P and 0K, 1 ( ,0)Ar P is the nearest neighbor distance between
atoms A in the zone containing the interstitial atom B at pressure P and 0K and cB is the
concentration of interstitial atoms B.
The free energy of alloy AB with FCC structure and the condition B Ac c has the form
1 2
1 15 6 8 ,AB B A B B B A B 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
(2.24)
where X is the free energy of atom X, AB is the free energy of interstitial alloy AB, cS is the
configuration entropy of interstitial alloy AB.
The Young modulus of alloy AB with FCC structure at temperature T and pressure P is
determined by
1 2
2 22
2 2 2
2
2
6 8
1 15 ,
A AB
AB A B B
A
E E c c
1 1
1
,
.
A
A A
E
r A
2 2
1 4
21 1
1 1 1 , ,
2 2
A A
A A A A A A
A A
A x cthx x cthx x
k k
2
22 2
20
012 2 2
1 1 1
1 3 1
4
2 4 2
XX X X X
X
X X X X X
U k k
r
r k r k r
0
01
1 1
1 3 1
2 , , ,
2 2 2 2
X X X X
X X X X X
X X X
U k k
cthx r x
r k r m
(2.25)
where is the relative deformation.
The bulk modulus KAB, the rigidity modulus GAB and the elastic constants C11AB, C12AB,
C44AB of alloy AB with FCC structure at temperature T and pressure P has the form
Nguyen Quang Hoc, Nguyen Duc Hien and Dang Quoc Thang
78
,
3(1 2 )
AB
AB
A
E
K
(2.26)
,
2 1
AB
AB
A
E
G
(2.27)
11
1
,
1 1 2
AB A
AB
A A
E
C
(2.28)
12 ,
1 1 2
AB A
AB
A A
E
C
(2.29)
44 .
2 1
AB
AB
A
E
C
(2.30
The Poisson ratio of alloy AB with FCC structure has the form
,AB A A B B Ac c (2.31)
where A and B respectively are the Poisson ratioes of materials A and B and are determined
from the experimental data.
When the concentration of interstitial atom B is equal to zero, the obtained results for alloy
AB become the coresponding results for main metal A.
2.2. Numerical results for alloy AuSi
For alloy AuSi, we use the n-m pair potential
0 0( ) ,
n m
r rD
r m n
n m r r
(2.32)
where the potential parameters are given in Table 1 [16].
Table 1. Potential parameters 0, , ,m n D r of materials
Material m n 1610 ergD
10
0 10 mr
Au 5.5 10.5 6462.540 2.8751
Si 6.0 12.0 45128.24 2.2950
Approximately,
Au-Si Au-Au Si-Si
1
.
2
(2.33)
According to our numerical results as shown in figures from Figure 1 to Figure 18 for AuSi
at the same temperature and concentration of interstitial atoms when pressure increases, E, G, K
and C11, C12, C44 increase. For example for AuSi at T = 300K, Si 5%c
when P increases from 0
to 70GPa, E increases from 0.5784. 10
11
Pato1,2431.10
11
Pa, G increases from 0.2019.10
11
Pa to
0.4340.10
11
Pa, K increases from 1.4176.10
11
Pa to 3.0467.10
11
Pa, C11 increases from 1.6869.
10
11
Pa to 3.6254.10
11
Pa, C12 increases from 1.2830.10
11
Pa to 2.7574.10
11
Pa and C44
increases from 0.2019.10
11
Pa to 0.4340.10
11
Pa. When the concentration of Si is equal to zero,
Elastic deformation of alloy AuSi with FCC structure under pressure
79
the dependence of elastic moduli and elastic constants of alloy AuSi on pressure becomes the
dependence of elastic moduli and elastic constants of main metal Au.
0 10 20 30 40 50 60 70
0
10
20
30
40
50
60
70
80
90
E
,G
,K
(
1
0
1
0
P
a
)
p (GPa)
E
G
K
0 10 20 30 40 50 60 70
0
10
20
30
40
50
60
70
80
90
100
C
1
1
,
C
1
2
,
C
4
4
(
1
0
1
0
P
a
)
p (GPa)
C11
C12
C44
Figure 1. E, G, K (10
10
Pa)(P)
for Au-1%Si at T = 300 K
Figure 2. C11, C12, C44(10
10
Pa)(P)
for Au-1%Si at T = 300 K
0 10 20 30 40 50 60 70
0
10
20
30
40
50
60
E
,G
,K
(
1
0
1
0
P
a
)
p (GPa)
E
G
K
0 10 20 30 40 50 60 70
0
10
20
30
40
50
60
70
C
1
1
,
C
1
2
,
C
4
4
(
1
0
1
0
P
a
)
p (GPa)
C11
C12
C44
Figure 3. E, G, K (10
10
Pa) (P)
for Au-3%Si at T = 300 K
Figure 4. C11, C12, C44 (10
10
Pa)(P)
for Au-3%Si at T = 300 K
0 10 20 30 40 50 60 70
0
5
10
15
20
25
30
E
,G
,K
(
1
0
1
0
P
a
)
p (GPa)
E
G
K
0 10 20 30 40 50 60 70
0
5
10
15
20
25
30
35
40
C
1
1
,
C
1
2
,
C
4
4
(
1
0
1
0
P
a
)
p (GPa)
C11
C12
C44
Figure 5. E, G, K (10
10
Pa)(P)
for Au-5%Si at T = 300 K
Figure 6. C11, C12, C44 (10
10
Pa)(P)
for Au-5%Si at T = 300 K
Nguyen Quang Hoc, Nguyen Duc Hien and Dang Quoc Thang
80
0 100 200 300 400 500 600 700 800 900 1000
0
2
4
6
8
10
12
14
16
18
20
E
,
G
,
K
(
1
0
1
0
P
a
)
T (K)
E
G
K
0 100 200 300 400 500 600 700 800 900 1000
0
2
4
6
8
10
12
14
16
18
20
22
24
C
1
1
,
C
1
2
,
C
4
4
(
1
0
1
0
P
a
)
T (K)
C11
C12
C44
Figure 7. E, G, K (10
10
Pa)(T)
for Au-5%Si at P = 0
Figure 8. C11, C12, C44(10
10
Pa)(T)
for Au-5%Si at P = 0
0 100 200 300 400 500 600 700 800 900 1000
0
5
10
15
20
25
30
E
,G
,K
(
1
0
1
0
P
a
)
T (K)
E
G
K
0 100 200 300 400 500 600 700 800 900 1000
0
5
10
15
20
25
30
35
C
1
1
,
C
1
2
,
C
4
4
(
1
0
1
0
P
a
)
T (K)
C11
C12
C44
Figure 9. E, G, K (10
10
Pa) (T)
for Au-5%Si at P = 30 GPa
Figure 10. C11, C12, C44 (10
10
Pa) (T)
for Au-5%Si at P = 30 Gpa
0 100 200 300 400 500 600 700 800 900 1000
0
5
10
15
20
25
30
35
40
E
,G
,K
(
1
0
1
0
P
a
)
T (K)
E
G
K
0 100 200 300 400 500 600 700 800 900 1000
0
5
10
15
20
25
30
35
40
45
C
1
1
,
C
1
2
,
C
4
4
(
1
0
1
0
P
a
)
T (K)
C11
C12
C44
Figure 11. E, G, K (10
10
Pa) (T)
for Au-5%Si at P = 70 GPa
Figure 12. C11, C12, C44 (10
10
Pa) (T)
for Au-5%Si at P = 70 GPa
Elastic deformation of alloy AuSi with FCC structure under pressure
81
0 1 2 3 4 5
0
5
10
15
20
25
30
E
,G
,K
(
1
0
1
0
P
a
)
%C(Si)
E
G
K
0 1 2 3 4 5
0
5
10
15
20
25
30
35
C
1
1
,
C
1
2
,
C
4
4
(
1
0
1
0
P
a
)
%C(Si)
C11
C12
C44
Figure 13. E, G, K (10
11
Pa)(cSi)
for Au-xSi at P = 0 and T = 300 K
Figure 14. C11, C12, C44(10
11
Pa) )(cSi)
for Au-xSi at P = 0 and T = 300 K
0 1 2 3 4 5
0
5
10
15
20
25
30
E
,G
,K
(
1
0
1
0
P
a
)
%C(Si)
E
G
K
0 1 2 3 4 5
0
5
10
15
20
25
30
35
C
1
1
,
C
1
2
,
C
4
4
(
1
0
1
0
P
a
)
%C(Si)
C11
C12
C44
Figure 15. E, G, K (10
11
Pa) (cSi)
for Au-xSi at P = 0 and T = 1000 K
Figure 16. C11, C12, C44 (10
11
Pa) (cSi)
for Au-xSi at P = 0 and T =1000 K
0 1 2 3 4 5
0
20
40
60
80
100
E
,G
,K
(
1
0
1
0
P
a
)
%C(Si)
E
G
K
0 1 2 3 4 5
0
10
20
30
40
50
60
70
80
90
100
110
C
1
1
,
C
1
2
,
C
4
4
(
1
0
1
0
P
a
)
%C(Si)
C11
C12
C44
Figure 17. E, G, K (10
11
Pa) (cSi)
for Au-xSi at P = 70 GPa and T = 300 K
Figure18. C11, C12, C44 (10
11
Pa) (cSi)
for Au-xSi at P = 70 GPa and T = 300 K
For AuSi at the same pressure and concentration of interstitial atoms when temperature
increases, E, G, K and C11, C12, C44 descrease. For example for AuSi at P = 70 GPa,
Si 5%c when T increases from 50 K to 1000 K, E descreases from 1.3963.10
11
Pa to
1.2480.10
11
Pa, G descreases from 0.4875.10
11
Pa to 0.4358.10
11
Pa, K descreases from
Nguyen Quang Hoc, Nguyen Duc Hien and Dang Quoc Thang
82
3.4323.10
11
Pa to 3.0588.10
11
Pa, C11 descreases from 4.0723.10
11
Pa to 3.6399.10
11
Pa, C12
descreases from 3.0973.10
11
Pa to 2.7684.10
11
Pa and C44 descreases from 0.4875.10
11
Pa to
0.4358.10
11
Pa. The dependence of elastic moduli and elastic constants of alloy AuSi on
temperature is the same as the dependence of elastic moduli and elastic constants of main metal Au.
For AuSi at the same pressure and temperature when the concentration of Si increases, E, G,
K and C11, C12, C44 descrease. For example for AuSi at P = 70 GPa, T = 1000 K when the
concentration of Si increases from 0 to 5%, E descreases from 1.3963.10
11
Pato 1,2480.10
11
Pa,
G descreases from 0.4875.10
11
Pa to 0,4358.10
11
Pa, K descreases from 3.4223.10
11
Pa to
3,0588.10
11
Pa, C11 descreases from 4.0723.10
11
Pa to 3,6400.10
11
Pa, C12 descreases from
3,0973.10
11
Pa to 2,7684.10
11
Pa and C44 descreases from 0.4875.10
11
Pa to 0.4358.10
11
Pa.
Table 2 gives the nearest neighbor distance and the elastic moduli of Au at T = 300 K, P = 0
according to the SMM and the experimental data [17, 18].
Table 2. Nearest neighbor distance and elastic moduli E, K, G of Au at P = 0, T = 300 K
according to SMM and EXPT [17, 18]
Method o
Aa
1010 PaE 1010 PaK
1010 PaG
SMM 2.8454 8.96 14.94 3.20
EXPT [17,18] 2.8838 8.91 16.70 3.10
Table 6. Elastic constants C11, C12, C44 of Au at T = 300K and P = 0 calculated by the SMM,
other calculations[18-27] and from EXPT [18]
SMM EXPT
[18]
Other calculations
[19] [20] [21] [22] [23] [24] [25] [26] [27]
C11
[10
11
Pa]
1.92 1.92 1.92 1.83 1.79 2.09 1.36 1.50 1.97 1.84 2.00
C12
[10
11
Pa]
1.28 1.65 1.66 1.54 1.47 1.75 0.91 1.29 1.84 1.54 1.73
C44
[10
11
Pa]
0.32 0.42 0.39 0.45 0.42 0.31 0.49 0.70 0.52 0.43 0.33
3. Conclusion
The analytic expressions of the free energy, the mean nearest neighbor distance between two
atoms, the elastic moduli such as the Young modulus, the bulk modulus, the rigidity modulus and
the elastic constants depending on temperature, concentration of interstitial atoms for interstitial
alloy AB with FCC structure under pressure are derived by the SMM. The numerical results for
alloy AuSi are in good agreement with the numerical results for main metal Au. Temperature
changes from 5 K to 1000 K, pressure changes from 0 to 70 GPa, and the concentration of
interstitial atoms Si changes from 0 to 5%.
Elastic deformation of alloy AuSi with FCC structure under pressure
83
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[2] A. A. Smirnov, 1979. Theory of Interstitial Alloys. Nauka, Moscow (in Russian).
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