The analytic expressions of the Helmholtz free energy, the Gibbs thermodynamic potential the mean nearest neighbor distance between two atoms, the crystal parameters for bcc, fcc and hcp phases of defective and perfect substitutional alloys AB with interstitial atoms C and structural phase transition temperatures of these alloys at zero pressure and under pressure are derived by the statistical moment method.
Trang 1Natural Sciences, 2019, Volume 64, Issue 6, pp 57-67
This paper is available online at http://stdb.hnue.edu.vn
STUDY ON STRUCTURAL PHASE TRANSITIONS IN DEFECTIVE
AND PERFECT SUBSTITUTIONAL ALLOYS AB WITH INTERSTITIAL ATOMS C UNDER PRESSURE
Nguyen Quang Hoc1, Dinh Quang Vinh1, Le Hong Viet2,
Ta Dinh Van1 and Pham Thanh Phong1 1
Faculty of Physics, Hanoi National University of Education
2
Tran Quoc Tuan University, Co Dong, Son Tay, Hanoi
Abstract The analytic expressions of the Helmholtz free energy, the Gibbs
thermodynamic potential the mean nearest neighbor distance between two atoms, the crystal parameters for bcc, fcc and hcp phases of defective and perfect substitutional alloys AB with interstitial atoms C and structural phase transition temperatures of these alloys at zero pressure and under pressure are derived by the statistical moment method The structural phase transition temperatures of the main metal A, the substitutional alloy AB and the interstitial alloy AC are special cases
of ones of the substitutional alloy AB with interstitial atoms C
Keywords: Statistical moment method, Helmholtz free energy, Gibbs
thermodynamic potential, structural phase transition temperature
1 Introduction
Structural phase transitions of crystals in general and metals and interstitial alloys
in particular are specially interested by many theoretical and experimental researchers [1-7] In [8], the body centered cubic (bcc) - face centered cubic (fcc) phase transition temperature determined in solid nitrogen and carbon monoxide on the basis of the self-consistent field approximation In [9], this phase transition temperature in solid nitrogen
is calculated by the statistical moment method (SMM) The ( , bcc, fcc, hexagonal close packed (hcp)) phase transition temperature for rare-earth metals and substitutional alloys is also derived from the SMM [10]
In this paper, we build the theory of , bcc, fcc, hcp) structural phase transition for defective and perfect substitutional alloys AB with interstitial atoms C at zero pressure and under pressure by the SMM [11-13]
Received April 30, 2019 Revised June 15, 2019 Accepted July 22, 2019
Trang 258
2 Content
In the case of perfect interstitial alloy AC with bcc structure (where the main atom
A1 stays in body center, the main atom A2 stays in peaks and the interstitial atom C stays in face centers of cubic unit cell), the cohesive energy and the alloy’s parameters for atoms C, A1 and A2 in the approximation of three coordination spheres are determined by [11-13]
2 4 5, 2
) ( )
( 2
1
1 1
1 1
0
bcc AC bcc
AC bcc
AC n
i
i AC bcc
u
i
(1)
5 5
16 2
2 2
1
2 1 1
) 1 (
1 1
) 1 (
1 1 ) 2 ( 2
2
bcc C bcc C bcc
C bcc AC bcc bcc
AC bcc bcc
i
AC eq i
AC bcc
r
r r
r u
8
1 24
1 48
1
1 ) 2 ( 2 1 1
) 4 ( 4
4 1
bcc AC bcc
bcc AC i
eq i
ACF bcc
r
r
125
5 4 2 150
1 2 16
2
1 ) 3 ( 1 1
) 4 ( 1
) 1 ( 3 1
bcc AC bcc
bcc AC bcc
AC
r r
r r
AC bcc
bcc AC bcc
bcc AC i
bcc eq
i i
AC bcc
r
r r
r r
u
) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 ( 1 2
2 4 2
8
5 4
1 4
1 48
25
2 2 8
1 2
8
1 2
8
2
1 ) 4 ( 1
) 1 ( 3 1 1
) 2 ( 2 1 1
)
3
(
1
bcc AC bcc
AC bcc
bcc AC bcc
bcc AC
r
r r
r
5 25
3 5
25
2 5
5 25
3
1 ) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 ( 1
bcc AC bcc
bcc ACF bcc
bcc AC
r
r r
r
, 4 ,
1 1 1
1
0
bcc A bcc A bcc
A bcc A AC bcc A bcc
2
5 2
1
1 1
1 1
) 1 ( 1 1
) 2 ( 2
2
bcc A AC bcc A
bcc A AB bcc A i
r r eq i
AC bcc
A bcc
r r
k u
k k
bcc A
,
8
1 8
1 24
1 48
1
1 1
1 1
1 1
) 1 ( 3 1 1
) 2 ( 2 1 1
) 4 ( 1
4 4
1
1
bcc A AC bcc A
bcc A AC bcc A
bcc A AC bcc
A i
r r eq i
AC bcc
A
bcc
r
r r
r u
bcc A
,
4
3 4
3 2
1 48
6
1 1
1 1
1 1 1
) 1 ( 3 1 1 ) 2 ( 2 1 1 ) 3 ( 1 2 2
2 4 2
2
bcc A AC bcc A
bcc A AC bcc A
bcc A AC bcc A
bcc A i
r eq i i
AC bcc
A
bcc
r
r r
r r u
u
bcc A
(3)
2 , 2 4 2 2,
0
bcc A bcc A bcc
A bcc A AC bcc A bcc
4 , 2
2
1
2 2
2 2
) 1 ( 1 1 ) 2 ( 2
2
bcc A AC bcc A
bcc A AC bcc
A i
r r eq i
AC bcc
A bcc
r r k
u k
k
bcc A
Trang 3
bcc A AC bcc A
bcc A AC bcc
A i
r r eq i
AC bcc
A bcc
r
r u
bcc A
2 2
2 2
) 3 ( 1 1
) 4 ( 1
4
4 1
1
4
1 24
1 48
,
8
1 8
1
2 2
2 2
1 ) 1 ( 3 1 1
) 2 ( 2 1
A AC bcc A
bcc A AC bcc A
r r
r r
bcc A AC bcc A
bcc A AC bcc
A i
r r eq i i
AC bcc
A bcc
r
r u
u
bcc A
2 2
2 2
1
) 3 ( 1 1
) 4 ( 2
2 2
4 2
2
4
1 8
1 48
,
8
3 8
3
2 2
2 2
1 ) 1 ( 3 1 1
) 2 ( 2 1
bcc A AC bcc A
bcc A AC bcc A
r r
r r
where AC is the interaction potential between the atom A and the atom C, n is the i number of atoms on the ith coordination sphere with the radius r i i( 1, 2,3),
) (
1
0 01 1
r bcc bcc C bcc C bcc A is the nearest neighbor distance between the interstitial atom
C and the metallic atom A at temperature T, r01bcc C 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 u0bcc C , ( )
1
y bcc A is the displacement of the atom A1 (the atom A stays in the bcc unit cell) from equilibrium position
at temperature T
(AB m) m AC(r i)/r i m(m1,2,3,4, , x,y.z, and u i is the displacement
of the ith atom in the direction bcc
C bcc
1 is the nearest neighbor distance between the atom A1 and atoms in crystalline lattice r1bcc A2 r01bcc A2 y bcc0C (T),r01bcc A2 is the nearest neighbor distance between atom A2 and atoms in crystalline lattice at 0K and is determined from the minimum condition of the cohesive energy 0 , 0 ( )
2 y T
u bcc A bcc C is the
displacement of the atom C at temperature T In Eqs (3) and (4), u0bcc A,k bcc A ,1bcc A ,2bcc A are the coressponding quantities in clean bcc metal A in the approximation of two coordination sphere [11-13]
The equation of state for bcc interstitial alloy AC at temperature T and pressure P is
written in the form
3 3
4 ,
2
1 6
1 1
0 1
bcc bcc
bcc bcc bcc bcc bcc bcc
bcc bcc
v r
k k cthx x r
u r
At 0K and pressure P, this equation has the form
4
6
1
1 0 1
0
bcc bcc
bcc bcc
bcc
r
k k r
u r
(6)
Trang 460
If we know the interaction potential i0, Equation (6) permits us to determine the nearest neighbour distance r1bcc X (P,0)(XA,A1,A2,C)at pressure P and temperature
) 0 , ( ), 0 , ( ), 0 , ( ),
0
,
the displacement y0bcc X(P,T) of atom X from the equilibrium position at temperature T and pressure P is calculated as in [11] From that, we can calculate the nearest
neighbour distance r1bcc X (P,T)at temperature T and pressure P as follows:
), , ( )
0 , ( )
, ( ), , ( )
0 , ( )
,
)
, ( )
0 , ( )
, ( ), , ( )
, (
2 2
The mean nearest neighbour distance between two atoms in bcc interstitial alloy
AC has the form
, ) , ( )
0 , ( )
,
r bcc A bcc A bcc
1 ( ,0) ( ,0), ( ,0) 3 ( ,0), )
0 ,
r bcc A C bcc A C A bcc A bcc bcc C
1 7 ( , ) ( , ) 2 ( , ) 4 ( , ), )
,
(
2
1 P T c y P T y
c T P y c T P y c T
P
y bcc C bcc A C B bcc C bcc A C bcc A (8) where r1bcc A (P,T)is the mean nearest neighbor distance between atoms A in the
interstitial alloy AC at pressure P and temperature T, r1bcc A (P,0) is the mean nearest
neighbor distance between atoms A in the interstitial alloy AC at pressure P and
temperature 0K, r1bcc A (P,0) is the nearest neighbor distance between atoms A in the
deformed clean metal A at pressure P and temperature 0K, r1A bcc(P,0)is the nearest neighbor distance between atoms A in the zone containing the interstitial atom C at
pressure P and temperature 0K and c C is the concentration of interstitial atoms C
In the case of fcc interstitial alloy AC (where the main atom A1 stay in face centers, the main atom A2 stay in peaks and the interstitial atom C stays in body center of cubic unit cell), the corresponding formulas are as follows [11-13]:
3 12 5, 4
) ( 3 ) ( 2
1
1 1
1 1
0
fcc AC fcc
AC fcc
AC n
i
i AC fcc
u
i
(9)
9
3 8 3 3
4 2
2
1
1 ) 2 ( 1 1
) 2 ( 1
) 1 ( 1 1 ) 2 ( 2
2
fcc AC fcc fcc
AC fcc
AC i
fcc fcc AC eq i
AC fcc
r r
r r
r u
5
5 8 5
1 1
) 2
C fcc C fcc
C fcc
AC fcc fcc
r
54
1 4
1 4
1 24
1 48
1
1 ) 4 ( 1
) 1 ( 3 1 1
) 2 ( 2 1 1
) 4 ( 4
4
1
fcc AC fcc
AC bcc
fcc AC bcc
fcc AC i
eq i
AB fcc
r
r r
r
Trang 5
150
17 3 81
3 2 3 27
2 3
27
3
2
1 ) 4 ( 1
) 1 ( 3 1 1
) 2 ( 2 1 1
)
3
(
1
fcc AC fcc
AC fcc
fcc AC fcc
fcc AC
r
r r
r
125
5 5
25
1 5
125
5 8
1 ) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 ( 1
fcc AC fcc
fcc AC fcc
fcc AC
r
r r
r
AC fcc
fcc AC fcc
fcc AC i
fcc eq
i i
AC fcc
r
r r
r r
u
) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 ( 1 2
2 4 2
4
3 4
3 2
1 48
16
2 7 2 8
7 2
8
2 2
4
1
1 ) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 ( 1 1
)
4
AC fcc
fcc AC fcc
fcc AC fcc
fcc
r
r r
r r
125
5 3 5 25
3 5
125
5 26 5 25
4
1 ) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 (
1 1
)
4
AC bcc
bcc AC bcc
fcc AC fcc fcc
r
r r
r r
, 4 ,
1 1 1
1
0
fcc A fcc A fcc
A fcc A AC fcc A fcc
, 2
1
1 1
) 2 ( 2
2
fcc A AC fcc A i
r r eq i
AC fcc
A fcc
u k
k
fcc A
, 24
1 48
1
1 1
) 4 ( 1
4 4 1
1
fcc A AC fcc
A i
r r eq i
AC fcc
A fcc
u
fcc A
2
1 2
1 4
1 48
6
1 1
1 1
1 1 1
) 1 ( 3 1 1 ) 2 ( 2 1 1 ) 3 ( 1 2 2
2 4 2
2
fcc A AC fcc A
fcc A AC fcc A
fcc A AC fcc A
fcc A i
r eq i i AC fcc
A
fcc
r
r r
r r u
u
fcc A
(11)
2 , 2 4 2 2,
0
fcc A fcc A fcc
A fcc A AC fcc A fcc
6
23 6
1 2
1
2 2
2 2
) 1 ( 1 1
) 2 ( 2
2
fcc A AC fcc A
fcc A AC fcc
A i
r r eq i
AC fcc
A fcc
r r
k u
k k
fcc A
fcc A AC fcc A
fcc A AC fcc
A i
r r eq i
AC fcc
A fcc
r
r u
fcc A
2 2
2 2
1
) 3 ( 1 1
) 4 ( 1
4 4 1
1
9
2 54
1 48
,
9
2 9
2
2 2
2 2
1 ) 1 ( 3 1 1
) 2 ( 2 1
A AC fcc A
fcc A AC fcc A
r r
r r
Trang 662
fcc A AC fcc A
fcc A AC fcc
A i
r r eq i i
AC fcc
A
fcc
r
r u
u
fcc A
2 2
2 2
) 3 ( 1 1
) 4 ( 2
2 2 4 2
2
27
4 81
1 48
,
27
14 27
14
2 2
2 2
1 ) 1 ( 3 1 1
) 2 ( 2 1
fcc A AC fcc A
fcc A AC fcc A
r r
r r
, 2
2 ,
2
1 6
1 1
0 1
fcc fcc
fcc fcc fcc fcc fcc bcc
fcc fcc
v r
k k cthx x r
u r
, 4
6
1
1 0 1
0
fcc fcc
fcc fcc
fcc
r
k k r
u r
(14)
), , ( )
0 , ( )
, ( ), , ( )
0 , ( )
,
)
, ( )
0 , ( )
, ( ), , ( )
, (
2 2
, ) , ( )
0 , ( )
,
r fcc A fcc A fcc
1 ( ,0) ( ,0), ( ,0) 2 ( ,0), )
0 ,
r cc A C A fcc C A fcc A fcc C fcc
) ,
(
2
y c T P y c T P y c T
P
A C fcc
A C fcc
B C fcc
A C
The mean nearest neighbor distance between two atoms A in the perfect bcc
substitutional alloy AB with interstitial atom C at pressure P and temperature T is
, ,
bcc T
bcc TB bcc B B bccF T
bcc TAC bcc AC AC bcc
B
B a c B
B a c
), , ( ,
) , ( ,
) ,
r
, 3
3
1 4
3 3 2 1 ,
3
3
1 4
3 3 2 1
3
0
2 2
3
0
2 2
bcc B
bcc B
T
bcc B
bcc B bcc
B bcc
TB
bcc TB
bccF AC
bcc AC
T
bcc AC
bcc AC bccF
AC bcc
TAC
bcc
TAC
a a
a N a P B
a a
a N a
P B
2 2
2 2
2 2
2 2
2
2 2 1
1
T
bcc A
bcc A C
T
bcc A
bcc A C
T
bcc C
bcc C C
T bcc A
bcc A C
T
bcc
AC
bcc
AC
a
c a
c a
c a
c
C B, , A , A A, X , 2
1 4
6
1 3
1
2 1 2
2 2 2
0 2 2
2
bcc X
bcc X bcc X bcc
X
bcc X bcc X
bcc X bcc
X
bcc X T
bcc
X
bcc
X
a
k k a
k k
a
u a
N
(17)
The Helmholtz free energy of perfect bcc substitutional alloy AB with interstitial atom C before deformation with the condition c C c B c A has the form
c bccAC c bcc A bcc B B bcc AC bcc
2 1
bccAC c bcc A C bcc A C bcc C C bcc A C bcc
Trang 7
2
1 3
2
2 2 2 0
0
bcc X bcc
X bcc
X bcc X bcc X
bcc X bcc X bcc
X
Y Y
k N
2 1 2
2 2
1 3
4 2
2 1 2 1 2
4 3
X
bcc X bcc
X bcc X bcc
X
bcc X bcc
X bcc X bcc
X
Y Y
Y Y
k
ln1 , coth ,
0
bcc X bcc
X bcc X x bcc
X bcc
where bcc X is the Helmholtz free energy, Y X bcc is an atom X in clean metals A, B or interstitial alloy AC, bccAC
c
S is the configuration entropy of bcc interstitial alloy AC and
bccABC
c
S is the configuration entropy of bcc alloy ABC
In the case of fcc interstitial alloy AC, the corresponding formulas are as follows:
, ,
fcc T
fcc TB fcc B B fcc T
fcc TAC fcc AC AC fcc
B
B a c B
B a c
), , ( ,
) , ( ,
) ,
r
a ABC fcc A fccABC AC fcc fccAC A B fcc B fcc
, 3
3
1 2 2
1 ,
3
3
1 2 2
1
3
0
2 2
3
0
2 2
fcc B
fcc B
T fcc B
fcc B fccF
B fcc
TB
fcc TB
fcc AC
fcc AC
T fcc AC
fcc AC fccF
AC fccF
TAC
fcc
TAC
a a
a N a
P B
a a
a N a
P B
2 2
2 2
2 2
2 2
2
2 2 1
1
T
fcc A
fcc A C
T
fcc A
fcc A C
T
fcc C
fcc C C T fcc A
fcc A C
T
fcc
AC
fcc
AC
a
c a
c a
c a
c
C, B, , A , A A, X , 2
1 4
6
1 3
1
2 1 2
2 2 2
0 2 2
2
fcc X
fcc X fcc X fcc
X
fcc X fcc X
fcc X fcc
X
fcc X T
fcc
X
fcc
X
a
k k a
k k
a
u a
N
(19)
B fcc A fcc c fccAC c fccABC,
B fcc AC fcc
2 1
fccAC c fcc A B fcc A B fcc B B fcc A B fcc
2
1 3
2
2 2 2 0
0
fcc X fcc
X fcc
X fcc X fcc X
fcc X fcc X fcc X
Y Y
k N
2 1 2
2 2
1 3
4 2
2 1 2 1 2
4 3
X
fcc X fcc X fcc X fcc
X
fcc X fcc X fcc X fcc
X
Y Y
Y Y
k
ln1 , coth
0
fcc X fcc
X fcc X x fcc
X fcc
For perfect hcp interstitial alloy AC and perfect hcp substitutional alloy AB with interstitial atoms C, we have the same formulas as for perfect fcc interstitial alloy AC and perfect fcc substitutional alloy AB with interstitial atoms C The numerical
Trang 864
calculations of the cohesive energy u and the alloy parameters 0 k, , 1 2, of fcc alloy and ones of the hcp alloys are different
When the phase equilibrium happens between the phase and the phase of perfect substitutional alloy AB with interstitial atoms C at zero pressure,
,
ABC ABC T ABC T ABC T ABC (21) where we call bcc fcc
ABC
T as the - phase transition temperature of substitutional alloy AB with interstitial atoms C According to the thermodynamic relation, ETS
Therefore, the - phase transition temperature of substitutional alloy AB with interstitial atoms Cat zero pressure can be determined by the following formular
)
0
ABC ABC
ABC ABC
ABC
ABC ABC
S S
E E
S
E P
T
(22)
For example, when bcc, fcc,
2 1
bcc A bcc B B bcc A C bcc A C bcc C C bcc A C bcc
3
2 2 1
2 2
2
2 0
0
X bcc X bcc X bcc
X
bcc X bcc
X bcc X bcc
X
bcc X bcc X bcc
k
N E
U
, sinh ,
3
X
bcc X bcc
X bcc X bcc
X
x
x Z
Y N
2 1
bcc A bcc B B bcc A C bcc A C bcc B C bcc A C bcc
3
2 2 1
2 0
X bcc X bcc X bcc
X bcc X
bcc X bcc
X
Bo bcc
X bcc
k
Nk S
ln2sinh , 3
0
bcc X bcc
X Bo bcc
2 1
fcc A fcc B B fcc A C fcc A C fcc B C fcc A C fcc
3
2 2 1
2 2
2
2 0
0
X fcc X fcc X fcc
X
fcc X fcc
X fcc X fcc
X
fcc X bcc X fcc
k
N E
U
, sinh ,
3
X
fcc X fcc
X fcc X fcc
X
x
x Z
Y N
2 1
fcc A fcc B B fcc A C fcc A C fcc B C fcc A C fcc
3
2 2 1
2 0
X fcc X fcc X fcc
X fcc X
fcc X fcc
X
Bo fcc
X fcc
k
Nk S
ln2sinh , 3
0
ccc X fcc
X Bo fcc
where k Bo is the Boltzmann constant
When the phase equilibrium happens between the phase and the phase of
perfect substitutional alloy AB with interstitial atoms C at preesure P,
Trang 9, ,
G
G ABC ABC ABC ABC ABC ABC ABC
(24)
where G is the Gibbs thermodynamic potential According to the thermodynamic
relation, G PV UTSPV.Therefore, the - phase transition temperature of
substitutional alloy AB with interstitial atoms C at pressure P can be determined by the
following formular:
AB AB
AB AB AB
AB ABC
ABC ABC
ABC
S S
V V P E E S
V P E
T
(25)
For example, when bcc, fcc, we also have Eq (23) and
2
2 ,
3 3
ABC fcc
ABC
bcc ABC bcc
ABC bcc
ABC
a N Nv
V
a N Nv
The Helmholtz free energy of defective (or real) substitutional alloy AB with interstitial atoms C has the form
*
)
X
X X
XX X
XX XX f
where ABC is the Helmholtz free energy of perfect substitutional alloy AB with interstitial atoms C, g v f ( ABC) is the Gibbs thermodynamic potential change of substitutional alloy AB with interstitial atoms C when one vacancy is formulated,
)
( X
g v f isthe Gibbs thermodynamic potential change of an atom X when one vacancy is formulated, S c ABC* is the configurational entropy of alloy atoms and vacancies, N is the total numeber of atoms in alloy, n1 is the number of atoms on the first coordination sphere, XX(1) is the Helmholtz free energy of an atom X on the first coordination sphere with vacancy as centre,
The concentration of equilibrium atom is determined from the minimum condition
of the Helmholtz free energy
, ) ( exp
) (
T k
C g c T
k
B g c n
n
Bo
f v B Bo
f v B A
v v
A v
Bo
c g A c g A c g A n
k T
Approximately, the mean nearest neighbor distance between two atoms in defective alloy is equal to one in perfect alloy
The - phase transition temperature of defective substitutional alloy AB with
interstitial atoms C at zero pressure and at pressure P is determined by
Trang 1066
(29)
1 1
) 1 ( 1
Rbcc
ABC n n n B c c E n n c c E n n n B c E n n c E
E
1 1
) 1 ( 1
1
2 n v n n v B A c C E A n v n c C E A n v n n v B A c C E A n v n c C E A
1n v n1n vB C 1 c C E C n v n1c C E C(1),
1 1
) 1 ( 1
Rfcc
ABC n n n B c c E n n c c E n n n B c E n n c E
E
1 1
) 1 ( 1
1
6 n v n n v B A c C E A n v n c C E A n v n n v B A c C E A n v n c C E A
1n v n1n vB C 1 c C E C n v n1c C E C(1),
1 1
) 1 ( 1
Rbcc
ABC n n n B c c S n n c c S n n n B c S n n c S
S
1 1
) 1 ( 1
1
2 n v n n v B A c C S A n v n c C S A n v n n v B A c C S A n v n c C S A
1
1 1
) 1 ( 1
Rfcc
ABC n n n B c c S n n c c S n n n B c S n n c S
S
1 1
) 1 ( 1
1
6 n v n n v B A c C S A n v n c C S A n v n n v B A c C S A n v n c C S A
1n v n1n vB C 1 c C S C n v n1c C S C(1), (30)
R ABC R
ABC
ABC ABC R
ABC R
ABC R
ABC ABC R
ABC R
ABC
S S
V V P E
E S
V P E
T
(31)
When the concentration of interstitial atoms C is equal to zero, the theory of
structural phase transition of substitutional alloy AB with interstitial atoms C becomes
that of substitutional alloy AB When the concentration of substitutional atoms B is
equal to zero, the theory of structural phase transition of substitutional alloy AB with
interstitial atoms C becomes that of interstitial alloy AC When the concentrations of
substitutional and interstitial atoms are equal to zero, the theory of structural phase
transition of substitutional alloy AB with interstitial atoms C becomes that of main metal A
3 Conclusions
The analytic expressions of the alloy parameters, the mean nearest neighbour
distance between two atoms, the Helmholtz free energy, the Gibbs thermodynamic
potential, the energy and entropy for bcc, fcc and hcp phases of substitutional alloy AB
with interstitial atoms C and the structural phase transition temperatures of these alloys
at zero pressure and under pressure are derived by the statistical moment method The
structural phase transition temperature of substitutional alloy AB, interstitial alloy AC
and main metal A are special cases of that of substitutional alloy AB with interstitial
atoms C In next paper, we will carry out numertical calculations for some real
ternary ABC