Build theory of nonlinear deformation for BCC and FCC substitutional alloys AB with interstitial atom C under pressure

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.

HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2019-0030 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 Hoc1, Nguyen Thi Hoa2 and Nguyen Duc Hien3 Faculty of Physics, Hanoi National University of Education University of Transport and Communications Mac 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 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 45 Nguyen Quang Hoc, Nguyen Thi Hoa and Nguyen Duc Hien 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] 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 r1bcc , r1bcc , r1bcc are determined by [3, 15] u 0bcc C     2 AC  i  ui2 k Cbcc    4 ACF  48 i  u i4   bcc 16 r bcc 8r    ( 3)  AC r1bcc  bcc 25 5r      ( 4)    AC r1bcc   r1bcc  eq 24    (1)  AC r1bcc    4 AC  48 i  ui2 ui2        (1)  (1) bcc 16 ( 2) (1) bcc    AC r1bcc  bcc  AC r1   AC r1bcc ,  Cbcc   1bcc C   2C , bcc  r1 5r1  eq  1bcc C   2bcc C   ni  AC (ri )   AC (r1bcc )  2 AC r1bcc  4 AC r1bcc ,  i 1      8r  ( 3)  AC r1bcc        ( 2)  AC r1bcc    bcc 25 r   ( 2)  AC r1bcc     bcc   ( 4) bcc ( 3)  AC r1   AC r1bcc , 150 125r1bcc  ( 3)    AC r1bcc  bcc  r1bcc  eq 4r1    bcc 8r   ( 2) r1bcc    AC    bcc 8r  (1)  AC r1bcc  ( 2)  ACF r1bcc    bcc 25 r (1) r1bcc    AC   ( 4) bcc  AC r1  25   (1)  AC r1bcc , (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 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, 46 Build theory of nonlinear deformation for bcc and fcc substitutional alloys AB with interstitial… ( m)  AB   m AC (ri ) / ri m (m  1,2,3,4,  ,   x, y.z,    and ui 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]     bcc bcc bcc bcc bcc u0bcc A1  u A   AC r1 A1 ,  A1   A1   A1 , k bcc A1 bcc  1bcc A   1A   2bccA   2bccA  k bcc A   2 AC   i  u i2    4   AC 48 i  u i4    4 AC   48 i  u i2 u i2    ( 2) (1)   k Abcc   AB r1bcc  AC r1bcc A1  A1 , bcc   2r1 A1  eq  r r bcc     A1   ( 4) bcc    1bcc   AC r1 A1  A   24 r1bcc  eq  r r bcc A1   ( 2) r1bccA    AC   1   bcc A1 8r A1   ( 3)    2bcc  AC r1bcc A  A1  bcc   r r1bcc A1  eq  r r bcc A1     ( 2) r1bccA    AC 1 A1   bcc A1 4r (1) r1bccA ,  AC (1) r1bccA ,  AC (3) bcc where r1bcc is the nearest neighbor distance between atom A1 and atoms in A1  r1C 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]     bcc bcc bcc bcc bcc u0bcc A2  u A   AC r1 A2 ,  A2   A2   A2 , k bcc A1 k bcc A bcc  1bcc A   1A    2 AC   i  u i2    4 AC    48 i  u i     bcc A2 8r  bcc A2  bcc 2A   bcc A2 8r     A2   ( 4) bcc ( 3)    1bcc   AC r1 A2  bcc  AC r1bcc A A2    24 r A  eq  r r bcc     A2 ( 2) r1bccA    AC   4    AC2 48 i  u i u i     (1) bcc ( 2)   k Abcc  2 AC r1bcc A2  bcc  AC r1 A2 ,   r1 A2  eq  r r bcc   bcc A2 8r (1) r1bccA ,  AC   ( 4) bcc ( 3)    2bcc  AC r1bcc A   AC r1 A2  A2  bcc   4r1 A2  eq  r r bcc     A2 ( 2) r1bccA    AC   bcc A2 8r (1) r1bccA ,  AC (4) 47 Nguyen Quang Hoc, Nguyen Thi Hoa and Nguyen Duc Hien bcc bcc bcc where r1bcc between the atom A2  r01A2  y 0C (T ), r01A2 is the nearest neighbor distance A2and atoms in crystalline lattice at 0K and is determined from the minimum condition bcc of the cohesive energy u 0bcc A2 , y 0C (T ) is the displacement of the atom C at temperature T bcc bcc bcc In Eqs (3) and (4), u0bcc A , k A ,  A ,  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 r1bccF X (X  A, A1 , A , C) at temperature T has the form bcc bcc bcc bcc r1bccF  r1bcc X X  r01X   r01X  1     r1 X  r01X  2   , where   E  (5) bcc (  is the stress and E is the Young modulus), r1bcc X  r1 X ( P, T ) is the nearest neighbour distance in bcc alloy before deformation When the alloy is deformed, the mean nearest neighbour distance r01bccF X at 0K has the form bcc r01bccF X  r01X 1    (6) The equation of state for bcc interstitial alloy AC at temperature T and pressure P is written in the form [3]    u 0bcc k bcc  bcc r1bcc bcc bcc , v  Pv  r    x cthx bcc 2k bcc r1bcc  3  r1 At 0K and pressure P, this equation has the form bcc bcc (7)  u 0bcc 0bcc k bcc  Pv bcc  r1bcc   bcc bcc  bcc 4k r1   r1 (8) If we know the interaction potential i0 , the equation (8) permits us to determine the nearest neighbour distance r1bcc X ( P,0)(X  A, A1 , A , C) at pressure P and temperature 0K.After finding r1bcc X ( P,0), we can determine  bcc r1bccF X P,0  r1 X P,0  2    (9) ( P,0),  ( P,0),  ( P,0) at and then determine the parameters k ( P,0),  pressure Pand 0K for each case of X when alloy is deformed Then, the displacement y0bccF X P, T  of atom X from the equilibrium position at temperature T and pressure P is calculated a in [3, 15] bccF X bccF 1X bccF 2X bccF X When alloy is deformed, the nearest neighbour distance r1bccF X ( P, T ) is determined by [3] bccF bccF bccF r1bccF ( P, T )  r1bccF ( P,0)  y bccF ( P, T ), C C A1 ( P, T ), r1 A ( P, T )  r1 A ( P,0)  y A bccF bccF bccF r1bccF ( P, T ), r1bccF ( P, T ) A1 ( P, T )  r1C A2 ( P, T )  r1 A2 ( P,0)  yC 48 (10) Build theory of nonlinear deformation for bcc and fcc substitutional alloys AB with interstitial… When alloy is deformed, the mean nearest neighbour distance r1bccACF ( P, T ) has the A form [3] r1bccACF ( P, T )  r1bccACF ( P,0)  y bccACF ( P, T ), A A    r1bccACF ( P,0)  1  cC r1bccF ( P,0)  cC r1AbccACF ( P,0)  2   , r1AbccF ( P,0)  3r1bccF ( P,0), A A C bccF y bccACF ( P, T )  1  7cC y bccF ( P, T )  cC yCbccF ( P, T )  2cC y bccF A A1 ( P, T )  4cC y A2 ( P, T ), (11) where r1bccACF ( P, T ) is the mean nearest neighbor distance between two atoms A in the A deformed bcc interstitial alloy AC at pressure P and temperature T, r1bccACF ( P,0) is the A mean nearest neighbor distance between two atoms A in the deformed bcc interstitial alloy AC at pressure P and temperature 0K, r1bccF 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AbccACF ( 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] u k Cfcc  fcc 0C    2 AC  i  u i2       4 AB   48 i  ui4    2fcc C           4 AC  48 i  ui2 ui2    (12)      81 r1 4r fcc     bcc      ( 4) fcc (3) fcc   AC r1  fcc  AC r1  8r1 r1 fcc      (1) r1 fcc    AC   (1)  AC r1 fcc  ( 2)  AC r1 fcc      ( 3)    AC r1 fcc  fcc  r1 fcc  eq 2r1    ( 2) r1 fcc    AC ( 2)  AC r1 fcc  ( 3)  AC r1 fcc  fcc 125r1 25 r1 fcc   (1) fcc fcc  AC r1 ,  Cfcc   1fcc C   2C , 5r1 fcc        ( 4)    AC r1 fcc   r1bcc  eq 24 (3) fcc  AC r1  fcc 27r1 27 r1 fcc    (1) fcc ( 2) fcc ( 2) fcc ( 2)    AC r1 fcc  fcc  AC r1   AC r1  fcc  AC r1   r r 1  eq ( 2)  4 AC r1 fcc   1fcc C   ni   AC (ri )  3 AC (r1 fcc )  4 AC r1 fcc  12 AC r1 fcc , i 1   125 r1 fcc ( 2) r1 fcc    AC  ( 2)  AC r1 fcc  ( 4) fcc  AC r1  54   17 ( 4) fcc  AC r1  150   (1)  AC r1 fcc ,   r1 fcc   16 r1 fcc (1) r1 fcc    AC   (1)  AC r1 fcc  49 Nguyen Quang Hoc, Nguyen Thi Hoa and Nguyen Duc Hien      ( 4) fcc 26 (3) fcc  AC r1   AC r1  fcc 25 125r1 25 r1bcc     ( 2)  AC r1bcc        125 r   (1)  AC r1bcc , (13) bcc fcc u0fccA1  u0fccA   AC r1Afcc1 ,  Afcc   1fcc A1   A1 , k Afcc  k Afcc  fcc  1fcc A   1A   2fccA   2fccA    4 AC   48 i  u i2 u i2    2 AC  i  u i2    4 AC    48 i  u i    ( 2)   k Afcc   AC r1 Afcc1 ,    eq  r r fcc   A1   ( 4) fcc    1fcc  AC r1 A1 , A    24  eq  r r fcc   A1   1 ( 3)    2fccA  fcc  AC r1 Afcc1    4r1 A1 r1 Afcc1  eq  r r fcc     ( 2) r1Afcc    AC 1 A1      fcc A1 2r (1) r1Afcc , (14)  AC  fcc u0fccA2  u0fccA   AC r1Afcc2 ,  Afcc2   1fcc A2   A2 , k Afcc  k Afcc  fcc  1fcc A   1A    2 AC  i  u i2    4   AC 48 i  u i4     fcc A2 9r  2fccA   2fccA     fcc A2 27 r   A2     A2 ( 2) r1Afcc    AC (1) r1Afcc ,  AC   fcc A2 9r   ( 4) fcc ( 3)    2fccA   AC r1 A2   AC r1 Afcc2  fcc   81 27r1 A2  eq  r r fcc ( 2) r1Afcc    AC     ( 4) fcc ( 3)    1fcc  AC r1 A2  fcc  AC r1 Afcc2  A    54 r A2  eq  r r fcc   4  AC2  48 i  u i u i  14   ( 2) fcc 23 (1) fcc   k Afcc   AC r1 A2  fcc  AC r1 A2 ,   6 r A2  eq  r r fcc     A2 14   fcc A2 27 r (1) r1Afcc ,  AC (15) r1XfccF  r1Xfcc  r01fccX   r01fccX  1     r1Xfcc  r01fccX  2   , fcc r01fccF X  r01X 1   ,  u 0fcc k fcc fcc fcc Pv fcc  r1 fcc    x cthx bcc 2k fcc r1 fcc  r1 (17)  fcc , v     u 0fcc 0fcc k fcc Pv fcc  r1 fcc   fcc  r 4k fcc r1 fcc  50 (16)  ,     r1 fcc , (18) (19) Build theory of nonlinear deformation for bcc and fcc substitutional alloys AB with interstitial…   r1XfccF P,0  r1Xfcc P,0  2   , fccF 1C r ( P, T )  r fccF 1C ( P,0)  y bccF A1 fccF 1A ( P, T ), r (20) ( P, T )  r fccF 1A ( P,0)  y fccF A ( P, T ), r1 AfccF ( P, T )  2r1CfccF ( P, T ), r1 AfccF ( P, T )  r1 AfccF ( P,0)  yCfccF ( P, T ), 2 (21) r1AfccACF ( P, T )  r1AfccACF ( P,0)  y fccACF ( P, T ),    r1 AfccACF ( P,0)  1  cC r1AfccF ( P,0)  cC r1AfccACF ( P,0)  2   , r1AfccF ( P,0)  2r1CfccF ( P,0), y fccACF ( P, T )  1  15cC y AfccF ( P, T )  cC yCfccF ( P, T )  6cC y AfccF ( P, T )  8cC y AfccF ( P, T ) (22) The mean nearest neighbor distance between two atoms A in the deformed bcc substitutional alloy AB with interstitial atom C at pressure P and temperature T is determined by [3, 15] bccF a bccF ABC  c AC a AC bccF BTAC bccF T B  c B a BbccF bccF BTB bccF T B bccF bccF , BTbccF  c AC BTAC  c B BTB , c AC  c A  cC , bccABCF a bccF ( P, T ), a bccF  r1bccACF ( P, T ), a BbccF  r1bccF ( P, T ), ABC  r1 A AC A B 2P  bccF BTAC   bccF   2 AC  bccF  a  AC bccF TAC  bccF 3   2 AC  3N  a bccF 4a bccF AC AC  a bccF AC 3 bccF  a0 AC    2 AbccF   1  7cC  bccF   a A T        T 2P  bccF , BTB    2 CbccF    cC  bccF T  aC  bccF TB   a bccF 3 BbccF  a0 B   2 AbccF    2cC  bccF1   a A T        T    2 AbccF   4cC    a bccF T  A2   ,  T  , X  A, A1 , A2 , B, C  (23) The mean nearest neighbor distance between atoms A in the deformed bcc substitutional alloy AB with interstitial atom C at pressure P and temperature T = 0K is determined by bccF bccF bccF B0TAC bccF B0TB bccF a0bccF  c a  c a , B0bccF  c AC B0bccF ABC AC AC B 0B T TAC  c B B0TB , bccF bccF B0T B0T   2 XbccF  3N  a bccF X   XbccF  u 0bccF X    bccF 4k XbccF  T a X   k bccF X  bccF  bccF 2k X  a X 3   2 BbccF  4a BbccF 3N  a BbccF  k XbccF  bccF  a X    bccABCF bccACF a0bccF ( P,0), a0bccF ( P,0), a0bccF  r1bcc ABC  r1 A AC  r1 A B B ( P,0) (24) The Helmholtz free energy of bcc substitutional alloy AB with interstitial atom C before deformation with the condition cC  c B  c A has the form [3] bcc bcc  ABC   AC  c B  Bbcc  Abcc   TScbccAC  TScbccABC , 51 , Nguyen Quang Hoc, Nguyen Thi Hoa and Nguyen Duc Hien bcc  AC  1  7cC  Abcc  cC Cbcc  2cC Abcc  4cC Abcc  TScbccAC , 2     bcc bcc  X YX bcc  k  X  bcc  Xbcc  U 0bcc X   X  3N       2  bcc bcc  YXbcc     1bcc   X YX 1  X bcc kX        bcc  0bcc X  3N x X  ln  e 2 x bcc X ,Y     YXbcc  2 1bcc X 1        , YXbcc  bcc     YXbcc  2 1bcc   X 2X    bcc X     bcc  x bcc X coth x X , (25) where  Xbcc is the Helmholtz free energy is an atom X in clean metals A, B or interstitial alloy AC before deformation, S cbccAC is the configuration entropy of bcc interstitial alloy AC before deformation and S cbccABC is the configuration entropy of bcc alloy ABC before deformation In the case of fcc interstitial alloy AC, the corresponding formulas are as follows [3, 15]: fccF fccF a ABC  c AC a AC fccF BTAC fccF T B  c B a BfccF BTBfccF fccF T B fccF , BTbccF  c AC BTAC  c B BTBfccF , c AC  c A  cC , fccF fccF a ABC  r1 AfccABCF ( P, T ), a AC  r1AfccACF ( P, T ), a BfccF  r1BfccF ( P, T ), 2P  fccF BTAC   fccF  TAC fccF   2 AC   a fccF  AC a fccF AC fccF   2 AC  fccF 3N  a AC  a fccF 3 AC fccF  a0 AC        2 AfccF   1  15cC  fccF   a A T   2 XfccF  3N  a XfccF a  c AC a fccF AC B0fccF TAC fccF 0T B 2P  , BTBfccF    2 CfccF    cC  fccF T  aC   XfccF  u 0fccF X    fccF 4k XfccF  T a X fccF ABC    T  TBfccF  cB a B0fccF TB fccF 0T B a fccF B   2 BfccF  3N  a BfccF  a fccF 3 BfccF  a0 B   2 AfccF    6cC    a AfccF T    k fccF X  fccF  fccF 2k X  a X fccF 0B   k XfccF  fccF  a X       2 AfccF   8cC    a AfccF 2 T  ,   ,  T  , X  A, A1 , A2 , B, C  (26) fccF , B0fccF  c AC B0fccF T TAC  c B B0TB , fccABCF fccACF a0fccF ( P,0), a0fccF ( P,0), a0fccF  r1BfccF ( P,0) ABC  r1 A AC  r1 A B fcc fcc  ABC   AC  cB  Bfcc  Afcc   TScfccAC  TScfccABC , fcc  AB  1  15cB  Afcc  c B Bfcc  6c B Afcc  8cB Afcc  TScfccAC , 52       T (27) Build theory of nonlinear deformation for bcc and fcc substitutional alloys AB with interstitial…     fcc fcc  X YX fcc  k  X   Xfcc  U 0fccX   0fccX  3N   2  fcc fcc  YXfcc  X YX 1   k Xfcc                1fcc X   0fccX  3N x Xfcc  ln  e 2 x fcc X ,Y  fcc X  2 1fcc X  YXfcc 1    YXfcc fcc    2 1fcc   X 2X          YXfcc    ,     x Xfcc coth x Xfcc (28) When the process of nonlinear deformation in both fcc and bcc alloy happens, the relationship between the stress and the strain is decribed by σ1 ABC ε Fα ABC  σ oABC  εF (29) Here, oABC and ABC are constant depending on every interstitial alloy We can find the strain F corresponding to the maximum value of the real stress through the density of deformation energy In order to determine the stress - strain dependence according to the above formula, it is necesary to determine two constants oABC and ABC for every intestitial alloy Therefore, we can calculate the density of deformation energy of substitutional alloy AB with interstitial atom C in the form F Ψ ABC Ψ ABC f ABC (ε)  F   VABC N VABC  Ψ AF1 Ψ A1  c A1  F   v ABC v ABC  F  Ψ ABC  F  v ABC  Ψ ABC v ABC  ΨF Ψ   c A  A2  A2 F   v ABC v ABC       N   Ψ AF c A  F   v ABC  ΨA v ABC   F   cB  Ψ B  Ψ B  vF   ABC v ABC    Ψ AF   cC  F   v ABC  ΨA v ABC  ΨF Ψ   cB  F A  A  v   ABC v ABC         cA   cB  7cC , cA1  2cC , cA2  4cC for bcc alloy, cA   cB  15cC , cA1  6cC , cA2  8cC for fcc alloy (30) Since  is very small (