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Study on structural phase transitions in defective and perfect substitutional alloys AB with interstitial atoms C under pressure

HNUE JOURNAL OF SCIENCE
DOI: 10.18173/2354-1059.2019-0031
Natural 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 selfconsistent 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.
Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn
57


Nguyen Quang Hoc, Dinh Quang Vinh, Le Hong Viet, Ta Dinh Van and Pham Thanh Phong

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].



k Cbcc 

1   2 AC

2 i  ui2





2
bcc
1

8r







( 3)
 AC
r1bcc 2 

3
bcc
1

25 5r

(1)



 



(1)
 AC
r1bcc 2 

 




1
( 4)
  1  AC
r1bcc 

8 r1bcc
 eq 24



2

bcc 3
1

  4 AC
6

48 i  ui2 ui2





 

16 r






2 (1) bcc
16
( 2)
(1)
bcc
   AC
r1bcc  bcc  AC
r1
2 
 AC
r1bcc 5 ,  Cbcc  4  1bcc
C   2C ,
bcc

r
5
5
r
1
1
 eq

  4 ACF
1


48 i  u i4

bcc
1C



 2bcc
C 



1 ni
 AC (ri )   AC (r1bcc )  2 AC r1bcc 2  4 AC r1bcc 5 ,

2 i 1

u 0bcc
C 



 

 

8r







( 2)
 AC
r1bcc 2 



( 3)
 AC
r1bcc 5 







( 2)
 AC
r1bcc 2 



 

bcc 2
1







1 ( 4) bcc
4 5 (3) bcc
 AC r1 2 
 AC r1 5 ,
150
125r1bcc


1
( 3)
  1  AC
r1bcc 
bcc

4 r1bcc
 eq 4r1

1

2



2

 

bcc 2
1

25 r

 

2



1

 

8r



( 2)
 ACF
r1bcc 5 

 

5

 

bcc 3
1

8r



(1)
 AC
r1bcc 2 

bcc 3
1



( 2)
r1bcc  
 AC



3

 

bcc 3
1

25 5 r

(1)
r1bcc  
 AC





2 ( 4) bcc
 AC r1 5 
25





(1)
 AC
r1bcc 5 ,

(2)



bcc
bcc
bcc
bcc
bcc
u0bcc
A1  u 0 A   AC r1 A1 ,  A1  4  1 A1   2 A1 ,

k Abcc
 k Abcc 
1

bcc
 1bcc
A   1A 
1

  4
1
  AC
48 i  u i4


 2bccA   2bccA 
1

1   2 AC

2 i  u i2


 
5
( 2)
(1)
 
k Abcc   AB
r1bcc

 AC
r1bcc
A
A1 ,
bcc
1
 
2
r
1 A1
 eq  r r bcc

 

1 A1

 
1 ( 4) bcc
1
 
 1bcc
 AC r1 A1 
A 
 
24
8 r1bcc
 eq  r r bcc
A1

  4
6
AC


48 i  u i2 u i2


 

 

 

2

( 2)
r1bccA  
 AC
1

1 A1

1

 

bcc 3
1 A1

8r

(1)
r1bccA ,
 AC

 
1
3
3
( 3)
( 2)
(1)
 
 2bcc
 AC
r1bcc
 AC
r1bcc
 AC
r1bcc
A 
A1 
A1 
A1 ,
bcc
 
bcc 2
bcc 3
2
r
4 r1 A1
4 r1 A1
1 A1
 eq  r r bcc

 

1 A1

 

 



 

 



bcc
bcc
bcc
bcc
bcc
u0bcc
A2  u 0 A   AC r1 A2 ,  A2  4  1 A2   2 A2 ,

k Abcc
 k Abcc 
1
58

1

1   2 AC

2 i  u i2


 
4 (1) bcc
( 2)
 
k Abcc  2 AC
r1bcc
 bcc  AC
r1 A2 ,
A
2
 
r
1
A
 eq  r r bcc
2

 

1 A2

 

 

(3)


Study on structural phase transitions in defective and perfect substitutional alloys AB

  4
1
  AC
48 i  u i4


bcc
 1bcc
A   1A 
2



1

 

bcc 2
1 A2

8r

 2bccA   2bccA 
2

 
1 ( 4) bcc
1
( 3)
 
 1bcc
  AC
r1 A2  bcc  AC
r1bcc
A
A2 
 
24
4
r
1
A
 eq  r r bcc
2

 

1 A2

( 2)
r1bccA  
 AC

  4
6
 2 AC2

48 i  u i u i



3

 

bcc 2
1 A2

8r

 

2

1

 

bcc 3
1 A2

8r

(1)
r1bccA ,
 AC
2

 
1 ( 4) bcc
1
( 3)
 
 2bcc
 AC
r1bcc
A   AC r1 A2 
A2 
bcc
 
8
4
r
1 A2
 eq  r r bcc

 

 

1 A2

( 2)
r1bccA  
 AC
2

3

 

bcc 3
1 A2

8r

(1)
r1bccA ,
 AC

(4)

2

where  AC is the interaction potential between the atom A and the atom C, ni is the
number of atoms on the ith coordination sphere with the radius ri (i  1, 2,3),
bcc
bcc
r1bcc  r1bcc
C  r01C  y 0 A1 (T ) is the nearest neighbor distance between the interstitial atom

C and the metallic atom A at temperature T, r01bccC is the nearest neighbor distance
between the interstitial atom C and the metallic atom A at 0K and is determined from
bcc
the minimum condition of the cohesive energy u 0bcc
C , y0 A1 (T ) is the displacement of the
atom A1 (the atom A stays in the bcc unit cell) from equilibrium
position
at temperature T.
( m)
 AB
  m AC (ri ) / ri m (m  1,2,3,4,  ,   x, y.z,    and ui is the displacement
bcc
of the ith atom in the direction  . r1bcc
A1  r1C is the nearest neighbor distance between
bcc
bcc
bcc
the atom A1 and atoms in crystalline lattice. r1bcc
A2  r01A2  y 0C (T ), r01A2 is the nearest
neighbor distance between atom A2 and atoms in crystalline lattice at 0K and is
bcc
determined from the minimum condition of the cohesive energy u 0bcc
A2 , y 0C (T ) is the
bcc
bcc
bcc
displacement of the atom C at temperature T. In Eqs. (3) and (4), u0bcc
A , k A ,  1 A ,  2 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

 

 1 u 0bcc
1 k bcc  bcc 4 r1bcc
bcc
bcc
, v 
Pv  r 


x
cthx
.
bcc
2k bcc r1bcc 
3 3
 6 r1
At 0K and pressure P, this equation has the form
3

bcc

bcc
1

 1 u 0bcc 0bcc k bcc 
.
Pv bcc  r1bcc 
 bcc
bcc
4k r1bcc 
 6 r1

(5)

(6)

59


Nguyen Quang Hoc, Dinh Quang Vinh, Le Hong Viet, Ta Dinh Van and Pham Thanh Phong

If we know the interaction potential i0 , Equation (6) permits us to determine the
nearest neighbour distance r1bcc
X ( P,0)(X  A, A1 , A 2 , C) at pressure P and temperature
we can determine the parameters
r1bcc
X ( P,0)
bcc
bcc
bcc
bcc
k X ( P,0),  1X ( P,0),  2 X ( P,0),  X ( P,0) at pressure P and 0K for each case of X. Then,
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:
0K.

When

we

know

bcc
bcc
bcc
bcc
bcc
r1bcc
B ( P, T )  r1B ( P,0)  y A1 ( P, T ), r1 A ( P, T )  r1 A ( P,0)  y A ( P, T ),
bcc
bcc
bcc
bcc
r1bcc
A1 ( P, T )  r1B ( P, T ), r1 A2 ( P, T )  r1 A2 ( P,0)  y B ( P, T ).

(7)

The mean nearest neighbour distance between two atoms in bcc interstitial alloy
AC has the form
bcc
bcc
r1bcc
( P, T ),
A ( P, T )  r1 A ( P,0)  y

bcc
bcc
bcc
r1bcc
3r1bcc
A ( P,0)  1  cC r1 A ( P,0)  cC r1 A ( P,0), r1 A ( P,0) 
C ( P,0),

bcc
bcc
bcc
y bcc ( P, T )  1  7cC y bcc
A ( P, T )  cC y B ( P, T )  2cC y A1 ( P, T )  4cC y A2 ( P, T ), (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, r1Abcc ( 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 cC 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]:

u 0fcc
C 
k

fcc
C



1   2 AC
 
2 i  u i2

 



60





  4 AB
1


48 i  ui4

(9)


2 (1) fcc 4 ( 2) fcc
8 3 ( 2) fcc
( 2)
   AC
r1 fcc  fcc  AC
r1   AC r1
3  fcc  AC
r1
3 

3
r1
9r1
 eq



( 2)
 4 AC
r1 fcc 5 

 1fcc
C 



1 ni
 AC (ri )  3 AC (r1 fcc )  4 AC r1 fcc 3  12 AC r1 fcc 5 ,
2 i 1















8 5 (1) fcc
fcc
 AC r1 5 ,  Cfcc  4  1fcc
C   2C ,
fcc
5r1


1
( 4)
  1  AC
r1 fcc 

4 r1bcc
 eq 24

 



 

 

2

( 2)
r1 fcc  
 AC

1

 

bcc 3
1

4r

(1)
r1 fcc  
 AC





1 ( 4) fcc
 AC r1 3 
54


Study on structural phase transitions in defective and perfect substitutional alloys AB







2 3 (3) fcc
2
 AC r1 3 
fcc
27r1
27 r1 fcc





fcc
2C

 



2





( 2)
 AC
r1 fcc 3 



8 5
1
( 3)
 AC
r1 fcc 5 
fcc
125r1
25 r1 fcc

  4 AC
6


48 i  ui2 ui2



 

2






3
( 3)
  1  AC
r1 fcc 
fcc

4 r1 fcc
 eq 2r1

 





 

81 r1

fcc 3











 

4 ( 4) fcc
26 5 (3) fcc
3
 AC r1 5 
 AC r1 5 
25
125r1 fcc
25 r1bcc

 

fcc 3

( 2)
r1 fcc  
 AC



2



 

2



 

4 r1

fcc 3



fcc 3

3 5

 

bcc 3
1

125 r



(1)
r1 fcc  
 AC









(1)
 AC
r1 fcc 2 

 

16 r1





3

7 2

( 2)
 AC
r1bcc 5 



(1)
 AC
r1 fcc 5 ,

 

125 r1



17 ( 4) fcc
 AC r1 5 
150



5

( 2)
 AC
r1 fcc 2 

 



2



(1)
 AC
r1 fcc 3 

( 2)
 AC
r1 fcc 5 

1 ( 4) fcc
2 (3) fcc
7
  AC
r1
2  fcc  AC
r1
2 
4
8r1
8 r1 fcc


2 3

(1)
 AC
r1bcc 5 , (10)

fcc
u0fccA1  u0fccA   AC r1Afcc1 ,  Afcc
 4  1fcc
A1   2 A1 ,
1

k Afcc
 k Afcc 
1

fcc
 1fcc
A   1A 
1

 2fccA   2fccA 
1

1   2 AC

2 i  u i2


  4
1
AC


4

48 i  u i


  4
6
AC


48 i  u i2 u i2


 
( 2)
 
k Afcc   AC
r1 Afcc1 ,
 
 eq  r r fcc

 

1 A1

 
1 ( 4) fcc
 
 1fcc
  AC
r1 A1 ,
A
 
24
 eq  r r fcc

 

1 A1

 
1
1
( 3)
 
 2fccA  fcc  AC
r1 Afcc1 
 
fcc
4
r
2
r
1 A1
 eq  r r fcc
1 A1

 

 

2

( 2)
r1Afcc  
 AC
1

1 A1

 



1

 

2 r1 Afcc1

3

(1)
r1Afcc ,
 AC

(11)

1



fcc
u0fccA2  u0fccA   AC r1Afcc2 ,  Afcc2  4  1fcc
A2   2 A2 ,

k

fcc
A1

k

fcc
A

fcc
 1fcc
A   1A 
2

1   2 AC
 
2 i  u i2


 
1 ( 2) fcc
23 (1) fcc
 
k Afcc   AC
r1 A2  fcc  AC
r1 A2 ,
 
6
6r1 A2
 eq  r r fcc

  4
1
  AC
48 i  u i4


 
1 ( 4) fcc
2
( 3)
 
 1fcc
  AC
r1 A2  fcc  AC
r1 Afcc2 
A
 
54
9
r
1 A2
 eq  r r fcc



2

 

fcc 2
1 A2

9r

 

 

1 A2

 

 

1 A2

( 2)
r1Afcc  
 AC
2

2

 

fcc 3
1 A2

9r

(1)
r1Afcc ,
 AC
2

61


Nguyen Quang Hoc, Dinh Quang Vinh, Le Hong Viet, Ta Dinh Van and Pham Thanh Phong

 2fccA   2fccA 
2

14



 

fcc 2
1 A2

27 r
Pv

fcc

  4
6
AC
 
48 i  u i2 u i2


 r1

fcc

( 2)
r1Afcc  
 AC
2

 
1 ( 4) fcc
4
( 3)
 
 2fccA   AC
r1 A2 
 AC
r1 Afcc2 
fcc
 
81
27
r
1 A2
 eq  r r fcc

 

 

1 A2

14

 

fcc 3
1 A2

27 r

(1)
r1Afcc ,
 AC

(12)

2

 1 u 0fcc
1 k fcc
fcc
fcc



x
cthx
 6 r bcc
2k fcc r1 fcc
1


 fcc
, v 



 1 u 0fcc 0fcc k fcc
Pv fcc  r1 fcc 

fcc
6

r
4k fcc r1 fcc
1


 

2 r1 fcc
2

3

,

(13)


,



(14)

r1Bfcc ( P, T )  r1Bfcc ( P,0)  y Afcc
( P, T ), r1Afcc ( P, T )  r1Afcc ( P,0)  y Afcc ( P, T ),
1
r1Afcc1 ( P, T )  r1Bfcc ( P, T ), r1Afcc2 ( P, T )  r1Afcc2 ( P,0)  y Bfcc ( P, T ).

(15)

r1Afcc ( P, T )  r1Afcc ( P,0)  y fcc ( P, T ),

r1ccA ( P,0)  1  cC r1Afcc ( P,0)  cC r1Afcc ( P,0), r1Afcc ( P,0)  2r1Cfcc ( P,0),
y fcc ( P, T )  1  15cC y Afcc ( P, T )  cC y Bfcc ( P, T )  6cC y Afcc
( P, T )  8cC y Afcc2 ( P, T ),
1

(16)

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
a bcc
ABC  c AC a AC

bcc
BTAC
bccF
T

B

 c B a Bbcc

bcc
BTB
bcc
T

B

bcc
bcc
, BTbcc  c AC BTAC
 c B BTB
, c AC  c A  cC ,

bccABC
bccAC
a bcc
( P, T ), a bcc
( P, T ), a Bbcc  r1bcc
ABC  r1 A
AC  r1 A
B ( P, T ),

2P 
bcc
BTAC


1



bcc
TAC



bcc

3 3 1   2 AC


bccF
bcc
2
4a AC 3N  a AC  T

 a bcc
AC
3 bccF
 a0 AC






3

2P 
1

bcc
, BTB


bcc
 TB



3 3 1   2 Bbcc 


4a Bbcc 3N  a Bbcc2  T
 a bcc 
3 Bbcc 
 a0 B 

3

,

bcc
  2 bcc 
  2 bcc 
  2 AC

  2 bcc 
  2 bcc 
 bcc2   1  7cC  bccA2   cC  bccC 2   2cC  bccA12   4cC  bccA22  ,
 a

 a

 a A 
 a A 
 a A  T
 AC  T
 C T
1
2

T

T

 Xbcc
1   2 Xbcc 
1  2 u 0bcc
X
 bcc2  
 bcc
2
3N  a X  T 6 a bcc
4k X
X

2
  2 k bcc
1  k bcc  
 bccX 2  bcc  Xbcc  , X  A, A1 , A 2 , B, C.
2k X  a X  
 a X


(17)

The Helmholtz free energy of perfect bcc substitutional alloy AB with interstitial
atom C before deformation with the condition cC  c B  c A has the form
bcc
bcc
 ABC
  AC
 c B  Bbcc  Abcc   TScbccAC  TScbccABC ,

bcc
 AC
 1  7cC  Abcc  cC Cbcc  2cC Abcc  4cC Abcc  TScbccAC ,
1

62

2


Study on structural phase transitions in defective and perfect substitutional alloys AB
2

 

bcc
 Xbcc  U 0bcc
X   0 X  3N 

 bcc bcc
 YX
2  2X


 

bcc

 kX







2 3  4 bcc bcc  YXbcc 
  2  1bcc
  2 X YX 1 
X
bcc 4 3
2 

kX


 





2 x
bcc
 0bcc
X  3N x X  ln 1  e

bcc
X

,Y



2

 YXbcc 
2 1bcc
X
1 
 
3 
2 





YXbcc 
bcc 

 1  YXbcc
 2 1bcc

1

X 2X 
2 




2





,
 


bcc
 x bcc
X coth x X ,

bcc
X

(18)

where  Xbcc is the Helmholtz free energy, YXbcc is an atom X in clean metals A, B or
interstitial alloy AC, S cbccAC is the configuration entropy of bcc interstitial alloy AC and
S cbccABC is the configuration entropy of bcc alloy ABC.

In the case of fcc interstitial alloy AC, the corresponding formulas are as follows:
fcc
fcc
fcc
fcc BTAC
fcc BTB
fcc
a ABC  c AC a AC
 cB aB
, BTbcc  c AC BTAC
 c B BTBfcc , c AC  c A  cC ,
fcc
fcc
BT
BT
fcc
fcc
a ABC
 r1AfccABC ( P, T ), a AC
 r1AfccAC ( P, T ), a Bfcc  r1Bfcc ( P, T ),

2P 
fcc
BTAC


fcc
  2 AC

 a fcc 2
 AC

1
fccF
 TAC

2
fccF
a AC

fcc
1   2 AC

fcc 2
3N  a AC

fcc
 a AC

3 fcc
 a0 AC






2P 
, BTBfcc 

3


1  2 u 0fccX  Xfcc
 

fcc 2
4k Xfcc
 T 6 a X

  2 k fcc
1
 fccX 2  fcc
2k X
 a X

1

 TBfcc



2
a BfccF

 k Xfcc
 fcc
 a X





2

1   2 Bfcc

3N  a Bfcc 2



T

3

,


  2 Afcc2
  8cC 

 a Afcc 2
2
T



 ,

T

 a Bfcc
3 fcc
 a0 B

  2 Afcc

1
  6cC 

 a Afcc 2
T
1


  2 Cfcc

  cC 
fcc 2
T
 aC


  2 Afcc
  1  15cC 
fcc 2

 a A
T

1   2 Xfcc

3N  a Xfcc 2




T







, X  A, A1 , A 2 , B, C,


(19)

fcc
fcc
 ABC
  AC
 cB  Bfcc  Afcc   TScfccAC  TScfccABC ,

fcc
 AB
 1  15c B  Afcc  c B Bfcc  6c B Afcc  8c B Afcc  TScfccAC ,
1

2

 

 Xfcc  U 0fccX   0fccX  3N 
2 3  4 fcc fcc  YXfcc
 2 X YX 1 
4 
2
k Xfcc  3


 







 


k



 fcc fcc
 YX
2  2X


fcc
X




  2  1fcc
X


 0fccX  3N x Xfcc  ln 1  e

2 x Xfcc

,Y



2

2



2



2 1fcc
X
3

 YXfcc
1 
2




YXfcc
fcc 

 2 1fcc

1

X 2X 
2


fcc
X

 x Xfcc coth x Xfcc .


 



 1  YXfcc






,
 


(20)

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
63


Nguyen Quang Hoc, Dinh Quang Vinh, Le Hong Viet, Ta Dinh Van and Pham Thanh Phong

calculations of the cohesive energy u0 and the alloy parameters 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,



 
(21)
 ABC   ABC
, TABC
 TABC
 TABC
,
bcc fcc
where we call TABC
as the - phase transition temperature of substitutional alloy AB
with interstitial atoms C. According to the thermodynamic relation,   E  TS.
Therefore, the - phase transition temperature of substitutional alloy AB with
interstitial atoms Cat zero pressure can be determined by the following formular

 
TABC
( P  0) 


E ABC
E   E ABC
 ABC
.

S ABC
S ABC  S ABC

(22)

For example, when   bcc,   fcc,





bcc
E ABC
 1  7cC E Abcc  cC ECbcc  2cC E Abcc
 4cC E Abcc
 c B EBbcc  E Abcc ,
1
2

bcc
E Xbcc  U 0bcc
X  E0 X 

3N 2  bcc bcc
 YX
2  2X
k Xbcc 

 

 

2



 1bcc
X
3

bcc
bcc
E0bcc
X  3 NY X , Z X 

2  Z   2
bcc 2
X

x bcc
X
,
sinh x bcc
X

bcc bcc
2X X

Y

Z  ,
bcc 2
X







bcc
S ABC
 1  7cC S Abcc  cC S Bbcc  2cC S Abcc
 4cC S Abcc
 cB S Bbcc  S Abcc ,
1
2

S

bcc
X

S

bcc
0X



3Nk Bo  1bcc
X

4  YXbcc  Z Xbcc

bcc 2
kX
 3

 
S

bcc
0X





 3Nk Bo Y

bcc
X

  2
2



 ln 2 sinh x

bcc
X

bcc bcc
2X X

Y

,

Z  ,
bcc 2
X







fcc
E ABC
 1  15cC E Afcc  cC EBfcc  6cC E Afcc
 8cC E Afcc
 c B EBfcc  E Afcc ,
1
2
fcc
E Xfcc  U 0bcc
X  E0 X 

3N 2  fcc fcc
 YX
2  2X
k Xfcc 

 

 

2



 1fcc
X
3

E0fccX  3NYXfcc , Z Xfcc 

2  Z   2
fcc 2
X

x Xfcc
,
sinh x Xfcc

fcc
2X

2
YXfcc Z Xfcc ,










fcc
S ABC
 1  15cC S Afcc  cC S Bfcc  6cC S Afcc
 8cC S Afcc
 c B S Bfcc  S Afcc ,
1
2

S Xfcc  S 0fcc
X 



3Nk Bo   1fcc
X
4  YXfcc  Z Xfcc

fcc 2
kX
 3

 









  2

fcc
S 0fcc
 ln 2 sinh x Xccc ,
X  3Nk Bo YX

2

fcc
2X

2
YXfcc Z Xfcc ,






(23)

where kBo 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,
64


Study on structural phase transitions in defective and perfect substitutional alloys AB



 


G ABC  G ABC
, TABC
 TABC
 TABC
, PABC
 PABC
 P,

(24)

where G is the Gibbs thermodynamic potential. According to the thermodynamic
relation, G    PV  U  TS  PV . Therefore, the - phase transition temperature of
substitutional alloy AB with interstitial atoms C at pressure P can be determined by the
following formular:
 
TABC









E ABC  PV ABC
E   E AB
 P V AB
 V AB
 AB
.

S ABC
S AB  S AB

(25)

For example, when   bcc,   fcc, we also have Eq. (23) and



bcc
ABC

4 a bcc
ABC





3



3

fcc
2 a ABC
(26)
V
 Nv
N
,V
 Nv
N
.
2
3 3
The Helmholtz free energy of defective (or real) substitutional alloy AB with
interstitial atoms C has the form
R
 ABC
  ABC  ngvf ( ABC )  TScABC* ,
bcc
ABC

fcc
ABC

fcc
ABC

gvf ( ABC )  cA gvf ( A)  cC gvf (C )  cA1 gvf  A1   cA2 gvf  A2   cB gvf ( B),
(1)
g vf ( X )  n1  XX
 XX   B X  1 XX , B X  1 

U0X

X

(1)
, N XX
  X(1) , N XX   X , (27)

where  ABC is the Helmholtz free energy of perfect substitutional alloy AB with
interstitial atoms C, g vf (ABC ) is the Gibbs thermodynamic potential change of
substitutional alloy AB with interstitial atoms C when one vacancy is formulated,
g vf (X ) isthe Gibbs thermodynamic potential change of an atom X when one vacancy is
formulated, ScABC * 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
(1)
sphere,  XX
is the Helmholtz free energy of an atom X on the first coordination sphere
with vacancy as centre, cA  1  cB  7cC , cA1  2cC , cA2  4cC for bcc alloy and
cA  1  cB  15cC , cA1  6cC , cA2  8cC for fcc alloy.

The concentration of equilibrium atom is determined from the minimum condition
of the Helmholtz free energy

 c g f ( B) 
 c B g vf (C ) 
nv  nvA exp  B v
exp


,
k Bo T 
k Bo T 


 cA gvf ( A)  cA1 gvf ( A1 )  c A2 gvf ( A2 ) 
n  exp  
.
kBoT


A
v

(28)

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
65


Nguyen Quang Hoc, Dinh Quang Vinh, Le Hong Viet, Ta Dinh Van and Pham Thanh Phong
R  
ABC

T

R
R
R
EABC
E ABC
 E ABC
( P  0) 
 R
,
R
R
S ABC
S ABC  S ABC

(29)

Rbcc
E ABC
 1  nv n1  nv BA  11  c B  7cC E A  nv n1 1  cB  7cC E A(1)  1  nv n1  nv BB  1cB EB  nv n1cB EB(1) 













 2 1  nv n1  nv BA1  1 cC E A1  2nv n1cC E A(11)  4 1  nv n1  nv BA2  1 cC E A2  4nv n1cC E A(12) 

 1  nv n1  nv BC  1cC EC  nv n1cC EC(1) ,
Rfcc
E ABC
 1  nv n1  nv BA  11  cB  15cC E A  nv n1 1  cB  15cC E A(1)  1  nv n1  nv BB  1cB EB  nv n1cB EB(1) 













 6 1  nv n1  nv BA1  1 cC E A1  6nv n1cC E A(11)  8 1  nv n1  nv BA2  1 cC E A2  8nv n1cC E A(12) 

 1  nv n1  nv BC  1cC EC  nv n1cC EC(1) ,
Rbcc
S ABC
 1  nv n1  nv BA  11  cB  7cC S A  nv n1 1  cB  7cC S A(1)  1  nv n1  nv BB  1cB S B  nv n1cB S B(1) 













 2 1  nv n1  nv BA1  1 cC S A1  2nv n1cC S A(11)  4 1  nv n1  nv BA2  1 cC S A2  4nv n1cC S A(12) 

 1  nv n1  nv BC  1cC S C  nv n1cC S C(1) ,
Rfcc
S ABC
 1  nv n1  nv BA  11  c B  15cC S A  nv n1 1  c B  15cC S A(1)  1  nv n1  nv BB  1c B S B  nv n1c B S B(1) 













 6 1  nv n1  nv BA1  1 cC S A1  6nv n1cC S A(11)  8 1  nv n1  nv BA2  1 cC S A2  8nv n1cC S A(12) 

 1  nv n1  nv BC  1cC S C  nv n1cC S C(1) ,
R  
TABC




(30)



R
R
R


E ABC
 PV ABC
E ABC
 E ABC
 P V ABC
 V ABC

.
R
R
R
S ABC
S ABC
 S ABC

(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.
66


Study on structural phase transitions in defective and perfect substitutional alloys AB

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