Thermodynamic properties of binary interstitial alloy with fcc structure: Dependence on temperature and concentration of interstitial atoms

72 JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2017-0033 Mathematical and Physical Sci. 2017, Vol. 62, Iss. 8, pp. 72-81 This paper is available online at THERMODYNAMIC PROPERTIES OF BINARY INTERSTITIAL ALLOY WITH FCC STRUCTURE: DEPENDENCE ON TEMPERATURE AND CONCENTRATION OF INTERSTITIAL ATOMS Nguyen Quang Hoc and Dinh Quang Vinh Faculty of Physics, Hanoi National University of Education Abstract. The analytic expressions of the thermodynamic quantities such as the mean nearest neighbor distance, free energy, isothermal and adiabatic compressibilities, the isothermal and adiabatic elastic modulus, the thermal expansion coefficient, heat capacities at constant volume and at constant pressure, and the entropy of binary interstitial alloy with face-centered cubic (FCC) structure when the concentration of interstitial atoms is very small are derived by the statistical moment method. The obtained expressions of these quantities depend on temperature and concentration of interstitial atoms. The theoretical results are applied to interstitial alloy AuSi. In the case when the concentration of interstitial atoms is equal to zero, we obtain the thermodynamic quantities of the main metal and the numerical results for alloy AuSi give the numerical results for Au. The calculated results of the thermal expansion coefficient and the heat capacity at constant pressure in the temperature interval of 100 to 700 K for Au are in good agreement with experiments. Keywords: Binary interstitial alloy, interstitial atom, statistical moment method. 1. Introduction Thermodynamic and elastic properties of interstitial alloys are of special interest to many theoretical and experimental researchers [1-7]. In this paper, we build the thermodynamic theory for binary interstitial alloy with face- centered cubic (FCC) structure using the statistical moment method (SMM) [8] and apply the obtained theoretical results to the alloy AuSi. 2. Content 2.1. Thermodynamic quantities of binary interstitial alloy with FCC structure Consider the interstitial alloy AC. The cohesive energy of the atom C (in body center of cubic unit cell) with the atoms A (in face centers and peaks of cubic unit cell) in the approximation of three coordination spheres with the center C and the radii 1 1 1, 3, 5r r r is determined by Received February 26, 2017. Accepted September 6, 2017. Contact Nguyen Quang Hoc, email: hocnq@hnue.edu.vn Thermodynamic properties of binary interstitial alloy with FCC structure: dependence on temperature 73        0 1 1 1 1 1 1 6 8 3 24 5 2 2 i C AC AC AC n AC i i r r ru r                 1 1 14 3 12 5 ,3 AC AC ACr r r    (2.1) where AC is the interaction potential between the atom A and the atom C, in is the number of atoms on the ith coordination sphere with the radius ( 1,2,3),ir i  11 1 01 0 ( )C C Ar r r y T   is the nearest neighbor distance between the interstitial atom C and the metallic atom A at temperature T, 01Cr 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 0Cu , 10 ( )Ay T is the displacement of the atom A1 (the atom A stays in the face center of cubic unit cell) from equilibrium position at temperature T. The alloy’s parameters for atom C in the approximation of three coordination spheres have the following form         2 2 (2) (1) (2) (1) 1 1 1 1 1 1 1 2 2 4 8 3 3 3 3 9 AC i i eq C AC AC AC ACk u r r r r r r                         (2) (1)1 1 1 2 1 4 8 5 4 5 5 , , 5 AC AC C C Cr r r        4 4 (4) (2) (1) (4) (3) 1 1 1 1 1 12 3 1 1 1 (2) (1) (4) (3) 1 1 1 12 3 1 1 1 1 48 1 1 1 1 2 3 ( ) ( ) ( ) ( 3) ( 3) 24 4 4 54 27 2 2 3 17 8 5 ( 3) ( 3) ( 5) ( 5) 27 81 150 125 AC i i eq C AC AC AC AC AC AC AC AC AC u r r r r r r r r r r r r r r r                                 (2) (1) 1 12 3 1 1 1 5 ( 5) ( 5), 25 125 AC ACr r r r    4 2 2 (3) (2) (1) 2 1 1 12 3 1 1 1 (3) 1 1 (4) 1 (2) (1) (4) (3) 1 1 1 1 12 3 1 1 6 48 1 3 3 ( ) ( ) ( ) 2 2 4 4 2 7 2 5 ( 2) 2 2 5 5 8 1 ( ) 4 7 4 26 ( ) ( ) ( ) ( ) 8 16 25 125 AC i i i eq C AC AC AC AC AC AC C AC C AC AC u u r r r r r r r r r r r r r r r r                                   (2) (1) 1 12 3 1 1 5 5 5 3 3 ( ) ( ), 25 125 AC ACr r r r    (2.2) where  ( ) 1 1 1 1( ) ( ) / 1,2,3,4, , 2, 3, 5i i iAC i ACr r r i r r r r r        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 Nguyen Quang Hoc and Dinh Quang Vinh 74   1 10 0 1 ,A A AC Au u r     1 1 1 1 1 1 2 1 22 (2) 1 1 2 , 4 , A i AC A A A A A A i eq r r AC Ak k u k r                      1 1 1 4 1 1 14 (4) 1 1 48 1 ( ), 24 A i AC A A A i eq r r AC A u r                     1 1 1 1 1 1 1 1 4 2 2 22 2 (3) (2) (1) 1 1 12 3 1 1 1 6 48 1 1 1 ( ) ( ) ( ) 4 2 2 A i AC A A A i i eq r r AC A AC A AC A A A Au u r r r r r r                         (2.3) where 11 1 2A Cr r is the nearest neighbor distance between atom A1 and other atoms in a crystalline lattice. The cohesive energy of atom A2 (which contains the interstitial atom C on the first coordination sphere) with the atoms in the crystalline lattice and the corresponding alloy’s parameters in the approximation of three coordination spheres with center A2 is determined by   2 20 0 1 ,A A AC Au u r        2 2 2 2 2 2 2 2 2 1 22 (2) (1) 1 1 1 1 2 , 4 , 1 23 6 6 A i AC A A A A A A i eq r r AC A AC A A k k u k r r r                       2 1 2 2 2 2 2 2 2 4 1 1 14 (4) (3) 1 1 1 (2) (1) 1 12 3 1 1 1 48 1 2 ( ) ( ) 54 9 2 2 ( ) ( ), 9 9 A i AC A A A i eq r r AC A AC A A AC A AC A A A u r r r r r r r                             2 2 2 4 (4) 2 2 12 2 2 6 48 1 ( ) 81 A i AC A A AC A i i eq r r A r u u                      2 2 2 2 2 2 2 1 1 (3) (2) (1) 1 1 13 1 4 14 27 27 14 ( ) ( ) ( ), 27A A AC A AC A AC A Ar r r r r r      (2.4) where 2 2 21 01 0 01 ( ),A A C Ar r y T r  is the nearest neighbor distance between the atom A2 and other atoms in crystalline lattice at 0K and is determined from the minimum condition of the cohesive energy     2 20 0 0 ,A C Au y T y T is the displacement of the atom C at temperature T. Thermodynamic properties of binary interstitial alloy with FCC structure: dependence on temperature 75 In Eqs. (2.3) and (2.4), 0 1 2, , ,A A A Au k   are the coresponding quantities in clean metal A in the approximation of two coordination sphere [8]. The nearest neighbor distances 1 1 2(0, )( , , , )Xr T X A A A C in interstitial alloy at pressure P = 0 and temperature T are derived from 1 1 1 1(0, ), (0, ),(0, ) (0,0) (0, ) (0,0)A CA A C CT Tr T r y r T r y    1 2 21 1 1 1 (0, )(0, ) 2 (0, ), (0, ) (0,0) CA C A A Tr T r T r T r y   (2.5) 1 1 2 (0,0)( , , , ) X r X A A A C is determined from the equation of state or the minimum condition of cohesive energy.  1 0,0Xr is determined using Maple software and from that we can determine the parameters      0,0 , 0,0 , 0,0 .X X Xk   After that, we can calculate the displacement [8] 2 0 3 2 (0,0) (0, ) (0, ) 3 (0,0) ,X X X X y T A T k    5 2 1 12 2 2 3 2 3 4 2 3 2 3 4 5 4 , 1 2 , , , , 2 13 47 23 1 25 121 50 16 1 , 3 6 6 2 3 6 3 3 2 43 93 169 83 22 1 , 3 2 3 3 4 2 i X X X X X iX X X X X i X X X X X X X X X X X X X X X X Y A a a k m x a k a Y Y Y a Y Y Y Y a Y Y Y Y Y                                       2 3 4 5 6 5 103 749 363 733 148 53 1 , 3 6 3 3 3 6 2 X X X X X X Xa Y Y Y Y Y Y              2 3 4 5 6 7 6 , 561 1489 927 733 145 31 1 65 coth . 2 3 2 3 2 3 2 X XX X X X X X X X Xa Y Y Y Y Y Y Y Y x x         (2.6) Then, we calculate the mean nearest neighbor distance in interstitial alloy AC by the expressions as follows  1 1 1 1 1(0, ) (0,0) (0, ), (0,0) 1 (0,0) (0,0),A A A C A C Ar T r y T r c r c r       1 21 1 (0,0) 2 (0,0), (0, ) 1 15 (0, ) (0, ) 6 (0, ) 8 (0, ),A C C A C C C A C Ar r y T c y T c y T c y T c y T       (2.7) where 1 (0, )Ar T is the mean nearest neighbor distance between atoms A in interstitial alloy AC at P = 0 and temperature T, 1 (0,0)Ar is the mean nearest neighbor distance between atoms A in interstitial alloy AC at P = 0 and 0K, 1 (0,0)Ar is the nearest neighbor distance between atoms A in clean metal at P = 0 and 0K, 1 (0,0)Ar is the nearest neighbor distance between atoms A in the zone containing the interstitial atom C at P = 0 and 0K and cC is the concentration of interstitial atoms C. The free energy per mole of interstitial alloy AC is determined by Nguyen Quang Hoc and Dinh Quang Vinh 76   1 2 ,1 15 6 8 cAC C A C C C A C Ac c c c TS            2 2 2 1 0 0 22 3 2 2 2 1 1 24 2 1 3 coth 1 coth 3 2 2 4 1 1 coth 1 coth 2 2 1 coth 1 coth , 3 2 2 X X X X X X X X X X X X X X X X X X X X X X X U N x x x x k x x x x x x x x k                                                0 3 ln 1 .XxX XN x e       (2.8) where X is the free energy for an atom 1 2( , , , )X X A A A C and Sc is the configurational entropy of alloy The isothermal compressibility of interstitial alloy AC has the form [8] 3 0 2 2 23 , 3 2 3 2 TAC AC AC AC AC ACAC T a a a aa                   1 2 1 2 2 22 2 22 22 2 2 2 2 1 1 15 6 8 , (0, ) A AAC AC CA C C C C AC A C A ATT TA T TT c c c c a a a a ar T                                                       2 22 2 0 2 2 2 , 31 1 2 4 2 XX X X X X X X X X XT u k k a a k a k a                             (2.9) where 1 0 1(0, ) and (0,0).AC A AC Aa r T a r  The thermal expansion coefficient of interstitial alloy AC has the form [8] 2 2 0 0 , 3 3 AC B TAC AC AC ACB TAC AC AC AC AC da k a ak d a v a                    1 2 1 2 2 22 22 2 2 32 1 2 2 2 1 15 6 8 , coth3 6 2 2 sinh 3 sinh A AAC CA C C C C AC A C A A X X X X X X X X X X X X X X X c c c c a a a a a k x k x x a k a x k k a x                                              2 3 1 2 2 2 2 2 coth1 4 coth . 6 sinh sinh X X X X X X X X X X X X X X X x k x x x x a x k a a x                          (2.10) The energy of interstitial alloy AC is determined by [8]   1 2 1 15 6 8 ,AC C A C C C A C AE c E c E c E c E     Thermodynamic properties of binary interstitial alloy with FCC structure: dependence on temperature 77 2 32 2 1 0 0 2 22 2 2 coth3 coth 2 2 , 3 sinh sinh X X X X X X X X X X X X X X x x xN E U E x x k x x                  0 3 coth .X X XE N x x (2.11) The entropy of interstitial alloy AC is determined by [8]   1 2 1 15 6 8 ,AC C A C C C A C AS c S c S c S c S     2 3 1 0 22 2 2 3 coth 4 coth 2 , 3 sinh sinh B X X X X X X X X X X X X Nk x x x S S x x k x x                  0 3 coth ln 2sinh .X B X X XS Nk x x x    (2.12) The heat capacity at constant volume of interstitial alloy AC is determined by [8]   1 2 1 15 6 8 ,VAC C VA C VC C VA C VAC c C c C c C c C     2 3 4 4 2 1 1 2 22 2 2 4 2 coth 2 2 coth2 3 2 , sinh 3 sinh 3 sinh sinh X X X X X X X X VX B X X X X X X x x x x x x C Nk x k x x x                      (2.13) The heat capacity at constant pressure of interstitial alloy AC is determined by [8] 2 . 9 AC TAC PAC VAC TAC TV C C     (2.14) The adiabatic compressibility of interstitial alloy AC has the form [8] .VACSAC TAC PAC C C   (2.15) 2.2. Numerical results for interstitial alloy AuSi In numerical calculations for alloys of AuSi, we use the n-m pair potential 0 0( ) , n m r rD r m n n m r r                    (2.16) where potential parameters are given in Table 1 [ 9] Table 1. The parameters 0, , ,m n D r of materials Au, Si Material m n D(10 -16 erg) ro(10 -10 m) Au 5.5 10.5 4683 2.8751 Si 6 12 4518.24 2.295 Our numerical results are described by figures from Figure 1 to Figure 14. When the concentration Si 0,c  we obtain thermodynamic quantities of Au. Our calculated results in Table 2 and Table 3 are in good agreement with experimental data. Nguyen Quang Hoc and Dinh Quang Vinh 78 Table 2. Dependence of thermal expansion coefficient  5 110 KT   on temperature for Au from SMM and EXPT[10] T(K) 100 200 300 500 700 1000 T -SMM 1.19 1.37 1.44 1.55 1.66 1.91 T -EXPT 1.15 1.34 1.41 1.50 1.59 1.95 Table 3. Dependence of heat capacity at constant pressure  J/mol.KPC on temperature for Au from SMM and EXPT[10] T(K) 100 200 300 500 700 PC -SMM 21.7695 24.9163 25.9581 27.3465 28.6754 PC -EXPT 21.4364 24.3253 25.1208 25.9581 26.7955 At the same temperature, when the concentration of interstitial atoms increases, thermodynamic quantities such as the thermal expansion coefficient, isothermal and adiabatic compressibilities, entropy of alloy decrease and thermodynamic quantities such as the mean nearest neighbor distance, the heat capacities at constant volume and at constant pressure of alloy increase. In the same concentration of interstitial atoms when temperature increases, all thermodynamic quantities of alloy increase. This temperature dependence of thermodynamic quantities for interstitial alloy and metal is similar. 0 200 400 600 800 1000 2.84 2.85 2.86 2.87 2.88 2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 r 1 1 0 -1 0 m T (K) 0% 0.3% 1% 3% 5% 0 2 4 6 8 10 0 2 4 6 8 10 Figure 1. 1( )r T at Sic  0 - 5%, P = 0 0 1 2 3 4 5 2.84 2.85 2.86 2.87 2.88 2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 r 1 1 0 -1 0 m C Si (%) 100K 300K 500K 700K 1000K 0 2 4 6 8 10 0 2 4 6 8 10 Figure 2. 1 Si( )r c at T = 100 - 1000K, P = 0 Thermodynamic properties of binary interstitial alloy with FCC structure: dependence on temperature 79 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 10 11 12 X T 1 0 -1 2 P a -1 C Si (%) 100K 300K 1000K 0 2 4 6 8 10 0 2 4 6 8 10 Figure 3.  SiT c at T = 100 - 1000 K, P = 0 0 200 400 600 800 1000 0 2 4 6 8 10 12 X T 1 0 -1 2 P a -1 T (K) 0% 1% 5% 0 2 4 6 8 10 0 2 4 6 8 10 Figure 4.  T T at Sic  0 - 5%, P = 0 0 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 a n p h a (1 0 -5 K -1 ) T (K) 0% 1% 5% 0 2 4 6 8 10 0 2 4 6 8 10 Figure 5.  T T at Sic  0 - 5%, P = 0 0 1 2 3 4 5 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 a n p h a ( 1 0 -5 K -1 ) C Si (%) 100K 300K 1000K 0 2 4 6 8 10 0 2 4 6 8 10 Figure 6. Si( )T c at T =100 – 1000 K, P = 0 0 200 400 600 800 1000 8 10 12 14 16 18 20 22 24 26 28 30 32 34 C v (J /m o l. K ) T(%) 0% 1% 5% 0 2 4 6 8 10 0 2 4 6 8 10 Figure 7.  VC T at Sic  0 - 5%, P = 0 0 1 2 3 4 5 25.5 26.0 26.5 27.0 27.5 28.0 28.5 29.0 29.5 30.0 30.5 31.0 31.5 32.0 32.5 C V (J /m o l. K ) C Si (%) 300K 500K 1000K 0 2 4 6 8 10 0 2 4 6 8 10 Figure 8.  SiVC c at T = 100 - 1000 K, P = 0 Nguyen Quang Hoc and Dinh Quang Vinh 80 1 2 3 4 5 25.5 26.0 26.5 27.0 27.5 28.0 28.5 29.0 29.5 30.0 30.5 31.0 31.5 32.0 32.5 C p (J /m o l. K )) C Si (%) 300K 500K 1000K 0 2 4 6 8 10 0 2 4 6 8 10 Figure 9.  SiPC c at T =100 – 1000 K, P = 0 0 200 400 600 800 1000 8 10 12 14 16 18 20 22 24 26 28 30 32 C v (J /m o l. K ) T (%) 0% 1% 5% 0 2 4 6 8 10 0 2 4 6 8 10 Figure 10.  PC T at Sic  0 - 5%, P = 0 0 200 400 600 800 1000 0 2 4 6 8 10 X S (1 0 -1 2 P a -1 ) T (K) 0% 1% 5% 0 2 4 6 8 10 0 2 4 6 8 10 Figure 11.  S T at Sic  0 - 5%, P = 0 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 10 11 X S (1 0 -1 2 P a -1 ) C Si (%) 100K 300K 1000K 0 2 4 6 8 10 0 2 4 6 8 10 Figure 12.  SiS c at T = 100 - 1000K, P = 0 0 1 2 3 4 5 15 20 25 30 35 40 45 50 55 60 65 70 75 80 S (J /K ) C Si (%) 100K 300K 1000K 0 2 4 6 8 10 0 2 4 6 8 10 Figure 13.  SiS c at T = 100 – 1000 K, P = 0 0 200 400 600 800 1000 0 10 20 30 40 50 60 70 80 90 S (J /K ) T (K) 0% 3% 5% 0 2 4 6 8 10 0 2 4 6 8 10 Figure 14.  S T at Sic  0 - 5%, P = 0 Thermodynamic properties of binary interstitial alloy with FCC structure: dependence on temperature 81 3. Conclusion From the SMM, the minimum condition of cohesive energies and the method of three coordination spheres, we find that the mean nearest neighbor distance, the free energy, the isothermal and adiabatic compressibilities, the isothermal and adiabatic elastic modulus, the thermal expansion coefficient, the heat capacities at constant volume and at constant pressure, the entropy of binary interstitial alloy with FCC structure with the concentration of interstitial atoms is very small (from 0 to 5%). The obtained expressions of these quantities depend on temperature and concentration of interstitial atoms. At zero concentration of interstitial atoms, thermodynamic quantities of interstitial alloy become ones of main metal in an alloy. The theoretical results are applied to interstitial alloy AuSi. 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