Nguyen Quang Hoc

45 Natural Sciences, 2019, Volume 64, Issue 6, pp. 45-56

This paper is available online at http://stdb.hnue.edu.vn

BUILD THEORY OF NONLINEAR DEFORMATION FOR BCC AND FCC SUBSTITUTIONAL ALLOYS AB

WITH INTERSTITIAL ATOM C UNDER PRESSURE

Nguyen Quang Hoc

¹

, Nguyen Thi Hoa

²

and Nguyen Duc Hien

³

1Faculty of Physics, Hanoi National University of Education

2University of Transport and Communications

3Mac Dinh Chi High School, Chu Pah, Gia Lai

Abstract. Analytic expressions of characteristic nonlinear deformation quantities such as the density of deformation energy, the maximum real stress and the limit of elastic deformation for bcc and fcc substitutional alloys AB with interstitial atom C under pressure are derived by the statistical moment method. The nonlinear deformations of the main metal A, the substitutional alloy AB and the interstitial alloy AC are special cases for nonlinear deformation of substitutional alloy AB with interstitial atom C and the same structure.

Keywords: Interstitial and substitutional alloy, binary and ternary alloys, nonlinear deformation, density of deformation energy, maximum real stress, limit of elastic deformation, statistical moment method.

1. Introduction

Thermodynamic and elastic properties of metals and interstitial alloys are specially interested by many theoretical and experimental researchers [1-14]. For example in [1], strengthening effects interstitial carbon solute atoms in (i.e., ferritic of bcc) Fe-C alloys are understood, owning chiefly to the interaction of C with crystalline defects (e.g.

dislocations and grain boundaries) to resist plastic deformation via dislocation glide.

High-strength steels developed in current energy and infrastructure applications include alloys where in the bcc Fe matrix is thermodynamically supersaturated in carbon. In [2], structural, elastic and thermal properties of cementite (Fe₃C) were studied using a Modified Embedded Atom Method (MEAM) potential for iron-carbon (Fe-C) alloys.

The predictions of this potential are in good agreement with first-principle calculations and experiments. In [3], the thermodynamic properties of binary interstitial alloys with bcc structure are considered by the statistical moment method (SMM).

Received April 28, 2019. Revised June 22, 2019. Accepted June 29, 2019.

Contact Nguyen Quang Hoc, email address: hocnq@hnue.edu.vn

46

The analytic expressions of the elastic moduli for anharmonic fcc and bcc crystals are also obtained by the SMM and the numerical calculation results are carried out for metals Al, Ag, Fe, W and Nb in [4].

In this paper, we build the theory of nonlinear deformation for bcc and fcc substitutional alloys AB with interstitial atom C by the SMM [3, 4, 15, 16].

2. Content

In the case of interstitial alloy AC with bcc structure (where the main atoms A stay in body center and peaks, the interstitial atom C stays in face centers of cubic unit cell), the cohesive energy of the atom C(in face centers of cubic unit cell) with the atoms A (in body center and peaks of cubic unit cell) and the alloy’s parameters in the approximation of three coordination spheres with the center C and the radii

5 , 2

, _{1 1}

1

bcc bcc

bcc r r

r are determined by [3, 15]



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

AC bcc

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

bcc ACF

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) 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_Cis 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 energyu₀^bcc_C , y₀^bcc_A1(T) is the displacement of the atom A1(the atom A stays in the bcc unit cell) from equilibrium position at temperature T,

47











⁽_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 .

The cohesive energy of the atom A1 (which contains the interstitial atom C on the first coordination sphere) with the atoms in crystalline lattice and the corresponding alloy’s parameters in the approximation of three coordination spheres with the center A1

is determined by [3, 15].

 

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 bcc AC

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 bcc AC

A bcc

A r

r r

r r r

u u

bcc A







 















 





























 







(3)

where r₁^bcc_A r₁^bcc_C

1  is the nearest neighbor distance between atom A1 and atoms in crystalline lattice.

The cohesive energy of the atom A₂ (which contains the interstitial atom C on the first coordination sphere) with the atoms in crystalline lattice and the corresponding alloy’s parameters in the approximation of three coordination spheres with the center A2

is determined by [3, 15]

 

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

AC bcc

A bcc

A r

r r u k

k k

bcc A



 







 



























 







 

^

 

^



 



























 







bcc A bcc AC

A bcc

A AC bcc

A i

r eq r i

AC bcc

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)

48

wherer^bcc_A r^bcc_A y^bcc_C T r^bcc_A

2 2

2 01 0 01

1   ( ), is the nearest neighbor distance between the atom A2and 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 [3, 15, 16].

In the action of rather large external force F, the alloy transfers to the process of nonlinear deformation. When the bcc interstititial alloy AC is deformed, the nearest neighbour distance r₁^bccF_X (XA,A₁,A₂,C)at temperature T has the form



1



_{1 01}



2



,

01 01

1

1^bccF_X r^bcc_X r^bcc_X r^bcc_X  r^bcc_X r^bcc_X 

r (5)

where

 ^E( is the stress and E is the Young modulus),r₁^bcc_X r₁^bcc_X (P,T)is the nearest neighbour distance in bcc alloy before deformation. When the alloy is deformed, the mean nearest neighbour distance r₀₁^bccF_X at 0K has the form



¹



^.

01 01^bccF_X r^bcc_X 

r (6) The equation of state for bcc interstitial alloy AC at temperature T and pressure P is written in the form [3]

 

. 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  

 







 



 

  (7)

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 

(8) If we know the interaction potential _i0, the equation (8) permits us to determine the nearest neighbour distance r₁^bcc_X (P,0)(XA,A₁,A₂,C)at pressure P and temperature 0K.After findingr₁^bcc_X (P,0),we can determine

 

1

  

²



1 P,0 r P,0 12 

r^bccF_X ^bcc_X (9) and then determine the parametersk^bccF_X (P,0),₁^bccF_X (P,0),₂^bccF_X (P,0),^bccF_X (P,0)at pressure Pand 0K for each case of X when alloy is deformed. Then, the displacement



^{P T}



y^bccF_0X , of atom X from the equilibrium position at temperature T and pressure P is calculated a in [3, 15].

When alloy is deformed, the nearest neighbour distance r₁^bccF_X (P,T)is determined by [3]

), , ( )

0 , ( )

, ( ), , ( )

0 , ( )

,

( _{1 1 1}

1 PT r P y 1 PT r PT r P y PT

r^bccF_C  ^bccF_C  ^bccF_A ^bccF_A  ^bccF_A  ^bccF_A ).

, ( )

0 , ( )

, ( ), , ( )

,

( 2 2

1 1 1 1

1 PT r PT r PT r P y PT

r^bccF_A  ^bccF_C ^bccF_A  ^bccF_A  _C^bccF (10)

49 When alloy is deformed, the mean nearest neighbour distance r₁^bccACF_A (P,T)has the form [3]

, ) , ( )

0 , ( )

,

( ₁

1 PT r P y PT

r^bccACF_A  ^bccACF_A  ^bccACF

 



^{1 ( ,0) ( ,0)}

 

^{1 2}



^{, ( ,0) 3 ( ,0),}

) 0 ,

( _{1 1} ² _{1 1}

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

r^bccACF_A   _C ^bccF_A  _C _A^bccACF   _A^bccF  ^bccF_C



1 7



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

,

(PT c y P T c y PT c y 1 P T c y 2 PT

y^bccACF   _C ^bccF_A  _{C C}^bccF  _C ^bccF_A  _C ^bccF_A (11)

wherer₁^bccACF_A (P,T)is the mean nearest neighbor distance between two atoms A in the deformed bcc interstitial alloy AC at pressure P and temperature T,r₁^bccACF_A (P,0) is the mean nearest neighbor distance between two atoms A in the deformed bcc interstitial alloy AC at pressure P and temperature 0K, r₁^bccF_A (P,0)is the nearest neighbor distance between two atoms A in the deformed bcc clean metal A at pressure P and temperature 0K, r₁_A^bccACF(P,0) is the nearest neighbor distance between two atoms A in the zone containing the interstitial atom C when the bcc alloy AC is deformed at pressure P and temperature 0K and cC is the concentration of interstitial atomsC.

In the case of fcc interstitial alloy AC (where the main atom A1 stay in face centers, themain atom A2 stay in peaks and the interstitial atom C stays in body center of cubic unit cell), the corresponding formulas are as follows [3, 15]



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       







(12)

 

^

 

^

 

^

 

^

 















 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

AC fcc

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    

   



 

^

   

^

   

^

 

^

 



















 ³

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

bcc AC fcc

bcc AC fcc

AC

i i eq

fcc AB

C r r

r r

r

u  r   

 



 

^

_{ }  

^

_{ }  

^

 

^

 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   



50

    _{ }   _{ } 

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   

   

 (13)

 

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 fcc AC

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







 















 





























 







(14)

 

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

AC fcc

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

12

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





 

^

 

^



 





























 







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



 



(15)



¹



1 01



²



^,

01 01

1

1^fccF_X r^fcc_X r^fcc_X r^fcc_X  r^fcc_X r^fcc_X 

r (16)



1



,

01 01^fccF_X r^fcc_X 

r (17)

 

, 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  

 







 



 

  (18)

4 , 6

1

1 0 1

0

1 



 







 



 

 ^fcc _fcc^fcc _fcc^fcc _fcc^fcc

fcc

r k k r

r u

Pv 

(19)