Nguyen Quang Hoc

57 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 Hoc

¹

, Dinh Quang Vinh

¹

, Le Hong Viet

²

, Ta Dinh Van

¹

and Pham Thanh Phong

¹

1Faculty of Physics, Hanoi National University of Education

2Tran 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.

Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn

58

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

C r r r r

u

i      







(1)

    

⁵



^{, 4}

 

^,

5 5 2 16 2

2 1

2 1 1

) 1 ( 1 1

) 1 ( 1 1 ) 2 ( 2

2

bcc C bcc

C bcc

C bcc bcc AC

bcc bcc AC

bcc i

AC i eq

bcc AC

C r

r r r

u r

k       











 





















 

^

   

^

 



















 ²

8 1 24

1 48

1

1 ) 2 ( 2 1 1

) 4 ( 4

4 1

bcc bcc AC

bcc AC

i i eq

ACF bcc

C r

r

u  r 

 



      

⁵



^,

125 5 2 4

150 2 1 16

2

1 ) 3 ( 1 1

) 4 ( 1

) 1 ( 3 1

bcc bcc AC

bcc AC bcc

bcc AC r

r r r

r





  



 

^

   

^

   

^

 





















 ^{AC bcc} _bcc ^{AC bcc} _bcc ^{AC bcc}

i

bcc i eq

i AC bcc

C r

r r

r r r

u

u ¹

) 1 ( 3 1 1

) 2 ( 2 1 1

) 3 ( 1 2

2 4 2

8 5 4

1 4

1 48

6    







 

^

   

^

   

^

 

^

 5

25 2 2 8

2 1 8

2 1 8

2

1 ) 4 ( 1

) 1 ( 3 1 1

) 2 ( 2 1 1

) 3 ( 1

bcc AC bcc

bcc AC bcc

bcc AC bcc

bcc AC r r

r r

r

r  r   

        

⁵



^,

5 25 5 3 25

5 2 5

25 3

1 ) 1 ( 3 1 1

) 2 ( 2 1 1

) 3 ( 1

bcc bcc AC

bcc bcc ACF

bcc

bcc AC r

r r

r r

r     

 (2)

 

₁ ^, ₁ ⁴



_{1 1}



^,

1 0 1 1 2

0

bcc A bcc

A bcc

A bcc

A AC bcc

A bcc

A u r

u      

   

^,

2 5 2

1

1 1

1

11

1 1

) 1 ( 1 1

) 2 ( 2

2

bcc A bcc AC

A bcc

A AB bcc A i

r eq r i

AC bcc

A bcc

A r

r r u k

k k

bcc A



 







 



























 







         

^,

8 1 8

1 24

1 48

1

1 1

1 1

1

11

1 1

) 1 ( 3 1 1

) 2 ( 2 1 1

) 4 ( 4 1

4 1

1

bcc A bcc AC

A bcc

A bcc AC

A bcc

A AC bcc

A i

r eq r i bcc AC

A bcc

A r

r r

r u r

bcc A







 













 



























 







         

^,

4 3 4

3 2

1 48

6

1 1

1 1

1 1 11

1 1

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

2 4 2

2

bcc A bcc AC

A bcc A bcc AC

A bcc

A bcc AC A bcc

A i

r eq r i i

AC bcc

A bcc

A r

r r r

r r u

u

bcc A







 















 





























 







(3)

 

₂ , ₂ 4



_{2 2}



,

2 0 1 1 2

0

bcc A bcc

A bcc

A bcc

A AC bcc

A bcc

A u r

u      

 

⁴

 

^,

2 2 1

2 2

2

12

1 1

) 1 ( 1 1 ) 2 ( 2

2

bcc A bcc AC

A bcc

A AC bcc

A i

r eq r i bcc AC

A bcc

A r

r r u k

k k

bcc A



 







 



























 







59

 

^

 

^



 



























 







bcc A bcc AC

A bcc

A AC bcc

A i

r eq r i bcc AC

A bcc

A r

r r u

bcc A

2 2

2

12

2 1

) 3 ( 1 1

) 4 ( 4 1

4 1

1 4

1 24

1 48

1    







       

^,

8 1 8

1

2 2

2 2

1 ) 1 ( 3 1 1

) 2 ( 2 1





 _AC ^bcc_A

bcc A bcc

A bcc AC

A

r r

r r





 

^

 

^



 





























 







bcc A bcc AC

A bcc

A AC bcc

A i

r eq r i i bcc AC

A bcc

A r

r r u

u

bcc A

2 2

2

1 2

2 1

) 3 ( 1 1

) 4 ( 2 2

2 4 2

2 4

1 8

1 48

6    









       

^,

8 3 8

3

2 2

2 2

1 ) 1 ( 3 1 1

) 2 ( 2 1

bcc A bcc AC

A bcc

A bcc AC

A

r r

r r



 

 (4) where _AC is the interaction potential between the atom A and the atom C, n_i is the number of atoms on the ith coordination sphere with the radius r i_i( 1, 2,3),

)

1(

0 01 1

1 r r y T

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, r₀₁^bcc_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 u₀^bcc_C , ( )

0 1 T

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(m1,2,3,4, ,  x,y.z,  and u_i is the displacement of the ith atom in the direction .r₁^bcc_A r₁^bcc_C

1  is the nearest neighbor distance between the atom A₁ and atoms in crystalline lattice. r₁^bcc_A2 r₀₁^bcc_A2  y^bcc_0C (T),r₀₁^bcc_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), u₀^bcc_A,k^bcc_A ,₁^bcc_A ,₂^bcc_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 ³

1 1

0 1

bcc bcc

bcc bcc bcc bcc bcc bcc

bcc bcc

bcc r

r v k cthx k

r x r u

Pv  



 







 



 

  (5)

At 0K and pressure P, this equation has the form 4 . 6

1

1 0 1

0

1 



 







 



 

 _{bcc bcc}^bcc

bcc bcc

bcc bcc

bcc

r k k r

r u

Pv 

(6)

60

If we know the interaction potential _i0, Equation (6) permits us to determine the nearest neighbour distance r₁^bcc_X (P,0)(XA,A₁,A₂,C)at pressure P and temperature 0K. When we know r₁^bcc_X (P,0) we can determine the parameters

) 0 , ( ), 0 , ( ), 0 , ( ), 0 ,

(P ₁ P ₂ P P

k_X^bcc ^bcc_X ^bcc_X ^bcc_X at pressure P and 0K for each case of X. Then, the displacement y₀^bcc_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 r₁^bcc_X (P,T)at temperature T and pressure P as follows:

), , ( )

0 , ( )

, ( ), , ( )

0 , ( )

,

( _{1 1 1}

1 PT r P y ₁ PT r PT r P y PT

r^bcc_B  ^bcc_B  ^bcc_A ^bcc_A  ^bcc_A  ^bcc_A ).

, ( )

0 , ( )

, ( ), , ( )

,

( 2 2

1 1 1 1

1 PT r PT r PT r P y PT

r^bcc_A  ^bcc_B ^bcc_A  ^bcc_A  _B^bcc (7) The mean nearest neighbour distance between two atoms in bcc interstitial alloy AC has the form

, ) , ( )

0 , ( )

,

( ₁

1 PT r P y PT

r^bcc_A  ^bcc_A  ^bcc



1



( ,0) ( ,0), ( ,0) 3 ( ,0), )

0 ,

( _{1 1 1 1}

1 P c r P c r P r P r P

r^bcc_A   _C ^bcc_A  _C _A^bcc _A^bcc  ^bcc_C



1 7



( , ) ( , ) 2 ( , ) 4 ( , ),

) ,

(PT c y PT c y PT c y 1 PT c y 2 PT

y^bcc   _C ^bcc_A  _{C B}^bcc  _C ^bcc_A  _C ^bcc_A (8) where r₁^bcc_A (P,T)is the mean nearest neighbor distance between atoms A in the interstitial alloy AC at pressure P and temperature T, r₁^bcc_A (P,0) is the mean nearest neighbor distance between atoms A in the interstitial alloy AC at pressure P and temperature 0K, r₁^bcc_A (P,0) is the nearest neighbor distance between atoms A in the deformed clean metal A at pressure P and temperature 0K, r₁_A^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 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]:



³



¹²



⁵



^,

4 ) ( 3 ) 2 (

1

1 1

1 1

0

fcc AC fcc

AC fcc

AC n

i

i AC fcc

C r r r r

u

i       







(9)

 

^

 

^

 

^

 

^

 















 ₂¹



 ² ₃^{4 3} ₉^{8 3 (2) 1 3}

1 1

) 2 ( 1

) 1 ( 1 1 ) 2 ( 2

2

fcc fcc AC

fcc AC fcc

AC i

fcc fcc AC i eq

fcc AC

C r

r r r r

u r

k     



  

⁵



^{, 4}

 

^,

5 5 5 8

4 ⁽¹⁾ _{1 1 2}

1 1

) 2

( fcc

C fcc C fcc

C fcc

fcc AC fcc

AC r

r r    

   



 

^

   

^

   

^

 

^

 



















 ₅₄^{1 3}

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

bcc AC fcc

bcc AC fcc

AC

i i eq

fcc AB

C r r

r r

r

u  r   





61

 

^

   

^

   

^

 

^

 5

150 3 17 81

3 3 2

27 3 2 27

3 2

1 ) 4 ( 1

) 1 ( 3 1 1

) 2 ( 2 1 1

) 3 ( 1

fcc AC fcc

fcc AC fcc

fcc AC fcc

fcc AC r r

r r

r

r  r   

        

⁵



^,

125 5 5 25

5 1 125

5 8

1 ) 1 ( 3 1 1

) 2 ( 2 1 1

) 3 ( 1

fcc fcc AC

fcc fcc AC

fcc

fcc AC r

r r

r

r  r    



 

^

   

^

   

^

 





















 ^{AC fcc} _fcc ^{AC fcc} _fcc ^{AC fcc}

i

fcc i eq

i fcc AC

C r

r r

r r r

u

u ¹

) 1 ( 3 1 1

) 2 ( 2 1 1

) 3 ( 1 2

2 4

2 4

3 4

3 2

1 48

6    







 

^

 

^

   

^

   

^

 2

16 2 2 7

8 2 7 8

2 2 4

1

1 ) 1 ( 3 1 1

) 2 ( 2 1 1

) 3 ( 1 1

) 4

( fcc

fcc AC fcc

fcc AC fcc

fcc AC fcc

AC r

r r

r r r

r   



          

⁵



^,

125 5 5 3

25 5 3 125

5 5 26

25 4

1 ) 1 ( 3 1 1

) 2 ( 2 1 1

) 3 ( 1 1

) 4

( bcc

bcc AC bcc

bcc AC fcc

fcc AC fcc

AC r

r r

r r r

r   

   

 (10)

 

₁ ^, ₁ ⁴



_{1 1}



^,

1 0 1 1 2

0

fcc A fcc A fcc

A fcc A AC fcc

A fcc

A u r

u      

 

^,

2 1

1

11

1 1

) 2 ( 2

2

fcc A AC fcc A i

r eq r i fcc AC

A fcc

A k r

k u k

fcc A

 





 



























 







 

^,

24 1 48

1

1

11

1 1

) 4 ( 4 1

4 1

1

fcc A AC fcc

A i

r eq r i

AC fcc

A fcc

A r

u

fcc A



 









 



























 







         

^,

2 1 2

1 4

1 48

6

1 1

1 1

1 1 11

1 1

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

2 4 2

2

fcc A fcc AC

A fcc A fcc AC

A fcc A fcc AC A fcc

A i

r eq r i i fcc AC

A fcc

A r

r r r

r r u

u

fcc A







 















 





























 

 



(11)

 

₂ ^, ₂ ⁴



_{2 2}



^,

2 0 1 1 2

0

fcc A fcc A fcc

A fcc A AC fcc

A fcc

A u r

u      

   

^,

6 23 6

1 2

1

2 2

2

12

1 1

) 1 ( 1 1

) 2 ( 2

2

fcc A fcc AC

A fcc

A AC fcc

A i

r eq r i fcc AC

A fcc

A r

r r u k

k k

fcc A



 







 



























 







 

^

 

^



 



























 







fcc A fcc AC

A fcc

A AC fcc

A i

r eq r i fcc AC

A fcc

A r

r r

u fcc

A

2 2

2

1 2

2 1

) 3 ( 1 1

) 4 ( 4 1

4 1

1 9

2 54

1 48

1    







       

^,

9 2 9

2

2 2

2 2

1 ) 1 ( 3 1 1

) 2 ( 2 1





 _AC ^fcc_A

fcc A fcc

A fcc AC

A

r r

r r





62

 

^

 

^



 





























 







fcc A fcc AC

A fcc

A AC fcc

A i

r eq r i i fcc AC

A fcc

A r

r r u

u fcc

A

2 2

2

12

2 1

) 3 ( 1 1

) 4 ( 2 2

2 4 2

2 27

4 81

1 48

6    









       

^,

27 14 27

14

2 2

2 2

1 ) 1 ( 3 1 1

) 2 ( 2 1

fcc A fcc AC

A fcc

A fcc AC

A

r r

r r



 

 (12)

 

, 2 , 2

2 1 6

1 ₁ ³

1 1

0 1

fcc fcc

fcc fcc fcc fcc fcc bcc

fcc fcc

fcc r

r v k cthx k

r x r u

Pv  



 







 



 

  (13)

4 , 6

1

1 0 1

0

1 



 







 



 

 _{fcc fcc}^fcc

fcc fcc

fcc fcc

fcc

r k k r

r u

Pv 

(14) ),

, ( )

0 , ( )

, ( ), , ( )

0 , ( )

,

( _{1 1 1}

1 PT r P y ₁ PT r PT r P y PT

r_B^fcc  _B^fcc  _A^{fcc fcc}_A  _A^fcc  _A^fcc ).

, ( )

0 , ( )

, ( ), , ( )

,

( 2 2

1 1 1 1

1 PT r PT r PT r P y PT

r^fcc_A  _B^{fcc fcc}_A  ^fcc_A  _B^fcc (15) ,

) , ( )

0 , ( )

,

( ₁

1 PT r P y PT

r^fcc_A  ^fcc_A  ^fcc



1



( ,0) ( ,0), ( ,0) 2 ( ,0), )

0 ,

( _{1 1 1 1}

1 P c r P c r P r P r P

r^cc_A   _{C A}^fcc  _C _A^fcc _A^fcc  _C^fcc



1 15



( , ) ( , ) 6 ( , ) 8 ( , ), )

,

(PT c y PT c y PT c y 1 PT c y 2 PT

y^fcc   _{C A}^fcc  _{C B}^fcc  _{C A}^fcc  _{C A}^fcc (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

, ,

, _T^bcc _{AC TAC}^bcc _{B TB}^bcc _{AC A C}

bcc T

bcc bcc TB B bccF B

T bcc bcc TAC AC AC bcc

ABC B c B c B c c c

B a B c B

a B c

a      

), , ( ,

) , ( ,

) ,

( _{1 1}

1 PT a r PT a r PT

r

a^bcc_ABC  ^bccABC_A ^bcc_AC  ^bccAC_A ^bcc_B  ^bcc_B

, 3

3 1 4

3 2 3

, 1

3 3

1 4

3 2 3

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

N a P a

B a

a N a P a

B











1 7



2 4 ₂ ,

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

c a c a

c a c a

a 













 















 



 







 



 







 

 



 







     

. C B, , A , A A, X 2 ,

1 6 4

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 N a



 

(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^bccABC,

bccAC c bcc A bcc B B bcc AC bcc

ABC  c   TS TS





1 7



2 4 ,

2 1

bccAC c bcc A C bcc A C bcc C C bcc A C bcc

AC   c  c   c   c  TS



63

   

^















 



 

 



 



 1 2

3

3 _{2 2} ² 2 ¹

2 0

0

bcc X bcc

bcc X X bcc bcc X

X bcc

X bcc

X bcc X

Y Y k

N

U    



      

¹



^,

1 2 2

2 2 3 1

4 2

2 1 2 1 4 2

3

















  

 



 







 



 

 ^bcc_{X X}^{bcc Xbcc bcc}_X ^bcc_X ^bcc_X ^Xbcc _X^bcc

bcc X