Elastic deformation of alloy AuSi with FCC structure under pressure

1. Introduction There are many theoretical and experimental works on thermodynamic and elastic properties of metals and interstitial alloys [1-27], for examples the theory of interstitial alloys [1, 2], the calculations from first principles, the many-body potentials, the molecular dynamics for defects in metals, alloys and solid solutions [3-5], the thermodynamic and elastic properties of ideal ternary and binary interstitial alloys [6-13] and the thermodynamic and elastic properties of metals [16-27]. In this paper, we build the theory of elastic deformation for binary interstitial AB with facecentered cubic (FCC) structure under pressure by the statistical moment method (SMM) [14, 15].

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74 HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0031 Natural Sciences 2018, Volume 63, Issue 6, pp. 74-83 This paper is available online at ELASTIC DEFORMATION OF ALLOY AuSi WITH FCC STRUCTURE UNDER PRESSURE Nguyen Quang Hoc 1 , Nguyen Duc Hien 2 and Dang Quoc Thang 1 1 Faculty of Physics, Hanoi National University of Education 2 Mac Dinh Chi High School, Gia Lai Province Abstract: The theory of elastic deformation for binary interstitial alloy with FCC structure under pressure is builded by the statistical moment method. The elastic deformation of main metal is special case of elastic deformation for binary interstitial alloy. The theoretical results are applied to alloy AuSi. The numerical results for alloy AuSi are compared with the numerical results for main metal Au and experiments. Keywords: Interstitial alloy, elastic deformation, Young modulus, bulk modulus, rigidity modulus, elastic constant, Poisson ratio. 1. Introduction There are many theoretical and experimental works on thermodynamic and elastic properties of metals and interstitial alloys [1-27], for examples the theory of interstitial alloys [1, 2], the calculations from first principles, the many-body potentials, the molecular dynamics for defects in metals, alloys and solid solutions [3-5], the thermodynamic and elastic properties of ideal ternary and binary interstitial alloys [6-13] and the thermodynamic and elastic properties of metals [16-27]. In this paper, we build the theory of elastic deformation for binary interstitial AB with face- centered cubic (FCC) structure under pressure by the statistical moment method (SMM) [14, 15]. 2. Content 2.1. Analytic results In interstitial alloy AB with FCC structure, the cohesive energy and the alloy parameter for the atom B (in body center of cubic unit cell), the atom A1 (the atom A stays in the face centers of cubic unit cell) and the atom A2 (the atom A stays in the peaks of cubic unit cell) in the approximation of three coordination are determined by [6-13]      0 1 1 14 3 12 5 ,3B AB AB ABr r ru      (2.1) 11 01 0 ( ),B B Ar r y T  (2.2) Received March 28, 2018. Revised June 29, 2018. Accepted July 9, 2018. Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn Elastic deformation of alloy AuSi with FCC structure under pressure 75        (2) (1) (2) (1)1 1 1 1 1 1 2 4 8 3 3 3 3 9 B AB B AB B AB B AB B B B k r r r r r r            (2) (1)1 1 1 8 5 4 5 5 , 5 AB B AB B B r r r    (2.3) (4) (2) (1) (4) (3) 1 1 1 1 1 12 3 1 1 1 1 1 1 1 2 3 ( ) ( ) ( ) ( 3) ( 3) 24 4 4 54 27 B AB B AB B AB B AB B AB B B B B r r r r r r r r            (2) (1) (4) (3) 1 1 1 12 3 1 1 1 2 2 3 17 8 5 ( 3) ( 3) ( 5) ( 5) 27 81 150 125 AB B AB B AB B AB B B B B r r r r r r r         (2) (1) 1 12 3 1 1 1 5 ( 5) ( 5), 25 125 AB B AB B B B r r r r    (2.4) (3) (2) (1) (3) 2 1 1 12 3 1 1 1 1 1 (4) 1 1 (2) (1) (4) (3) 1 1 1 12 3 1 1 1 3 3 2 ( ) ( ) ( ) 2 ( 2) 2 4 4 8 7 2 5 2 2 5 5 1 ( ) 4 7 4 26 ( ) ( ) ( ) ( ) 8 16 25 125 B AB B AB B AB B AB B B B B B AB B B AB B AB B AB B AB B B B 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 AB B AB B B B r r r r    (2.5)  1 24 ,B B B    (2.6)   1 10 0 1 ,A A AB Au u r (2.7) ),(0011 11 Tyrr BAA  (2.8)  1 1 (2) 1 ,A A AB Ak k r  (2.9) 1 1 (4) 1 11 1 ( ), 24 ABA AA r   (2.10) 1 1 1 12 1 1 1 (3) (2) (1) 2 21 1 13 1 1 1 1 1 1 ( ) ( ) ( ), 4 2 2 AA AB AB ABA A A A A A r r r r r r        (2.11)   1 1 11 2 4 ,A A A    (2.12)   2 20 0 1 ,A A AB Au u r  (2.13) ),( 222 0011 Tyrr AAA  (2.14)     2 2 2 2 (2) (1) 1 1 1 , 1 23 6 6 A A AB A AB A A k k r r r     (2.15) 2 2 2 2 2 2 2 2 1 1 (4) (3) (2) (1) 1 1 1 12 3 1 1 1 1 2 2 2 ( ) ( ) ( ) ( ), 54 9 9 9 A A AB A AB A AB A AB A A A A r r r r r r r           (2.16) Nguyen Quang Hoc, Nguyen Duc Hien and Dang Quoc Thang 76 2 2 2 2 2 2 2 2 (4) 2 1 2 1 1 (3) (2) (1) 2 1 1 13 1 1 4 14 ( ) 81 27 27 14 ( ) ( ) ( ), 27 A AB A A A A AB A AB A AB A A r r r r r r r         (2.17)   2 2 21 2 4 ,A A A    (2.18) where AB is the interaction potential between the atom A and the atom B, 1 1Br r is the nearest neighbor distance between the interstitial atom B and the metallic atom A at temperature T, 01Br 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 0Bu , 10 ( )Ay T is the displacement of the atom A1 (the atom A stays in the face centers of cubic unit cell) from equilibrium position at temperature T, ( ) ( ) ( ) / (m 1,2,3,4), , , , ,m m mAB i AB i ir r r x y z           and iu  is the displacement of the ith atom in the direction , 11 1A B r r is the nearest neighbor distance between the atom A1 and atoms in crystalline lattice, 201A r is the nearest neighbor distance between the atom A2 and atoms in crystalline lattice at 0K and is determined from the minimum condition of the cohesive energy 20 0 , ( )A Bu y T is the displacement of the atom C at temperature T, 0 1 2, , ,A A A Au k   are the coressponding quantities in clean metal A in the approximation of two coordination sphere [14, 15] The equation of state for interstitial alloy AB with FCC structure at temperature T and pressure P is written in the form 0 1 1 1 1 1 cth . 6 2 u k Pv r x x r k r            (2.19) At 0 K and pressure P, this equation has the form 0 0 1 1 1 . 4 u k Pv r r k r           (2.20) If we know the form of interaction potential 0 ,i eq. (2.6), we will determine the nearest neighbor distance   1 1 2,0 , , ,Xr P X B A A A at 0 K and pressure P. After we know  1 ,0Xr P , we can determine alloy parametrs 1 2( ,0), ( ,0), ( ,0), ( ,0)X X X Xk P P P P   at 0K and pressure P. After that, we can calculate the displacements [14, 15] 2 0 3 2 ( ,0) ( , ) ( , ) 3 ( ,0) .X X X X P y P T A P T k P    (2.21) From that, we derive the nearest neighbor distance  1 ,Xr P T at temperature T and pressure P       11 1 , ,0 , ,B B Ar P T r P y P T       1 1, ,0 ,A A Ar P T r P y P T        2 2 21 1 , ,0 , ,A A Ar P T r P y P T        1 11 01 , ,T ,A A Br P T r P y P T  (2.22) Then, we calculate the mean nearest neighbor distance in interstitial alloy AB by the expressions as follows [6-13]      1 1, ,0 , ,A Ar P T r P y P T  Elastic deformation of alloy AuSi with FCC structure under pressure 77  1 1 1 1 1( ,0) 1 ( ,0) ( ,0), ( ,0) 2 ( ,0),A B A B A A Br P c r P c r P r P r P                 1 2 , 1 15 , , 6 , 8 , ,B A B B B A B Ay P T c y P T c y P T c y P T c y P T     (2.23) where 1 ( , )Ar P T is the mean nearest neighbor distance between atoms A in interstitial alloy AB at pressure P and temperature T, 1 ( ,0)Ar P is the mean nearest neighbor distance between atoms A in interstitial alloy AB at pressure P and 0K, 1 ( ,0)Ar P is the nearest neighbor distance between atoms A in clean metal A at pressure P and 0K, 1 ( ,0)Ar P is the nearest neighbor distance between atoms A in the zone containing the interstitial atom B at pressure P and 0K and cB is the concentration of interstitial atoms B. The free energy of alloy AB with FCC structure and the condition B Ac c has the form   1 2 1 15 6 8 ,AB B A B B B A B A cc c c c TS          2 2 1 0 0 22 2 3 1 3 2 X X X X X X X X X U N X k                         3 2 2 2 1 1 24 2 4 1 2 2 1 1 , 3 2 2 X X X X X X X X X X X X X k                           2 0 3 ln(1 ) , coth , Xx X X X X XN x e X x x        (2.24) where X is the free energy of atom X, AB is the free energy of interstitial alloy AB, cS is the configuration entropy of interstitial alloy AB. The Young modulus of alloy AB with FCC structure at temperature T and pressure P is determined by 1 2 2 22 2 2 2 2 2 6 8 1 15 , A AB AB A B B A E E c c                         1 1 1 , . A A A E r A    2 2 1 4 21 1 1 1 1 , , 2 2 A A A A A A A A A A A x cthx x cthx x k k                  2 22 2 20 012 2 2 1 1 1 1 3 1 4 2 4 2 XX X X X X X X X X X U k k r r k r k r                           0 01 1 1 1 3 1 2 , , , 2 2 2 2 X X X X X X X X X X X X U k k cthx r x r k r m                (2.25) where  is the relative deformation. The bulk modulus KAB, the rigidity modulus GAB and the elastic constants C11AB, C12AB, C44AB of alloy AB with FCC structure at temperature T and pressure P has the form Nguyen Quang Hoc, Nguyen Duc Hien and Dang Quoc Thang 78 , 3(1 2 ) AB AB A E K    (2.26)   , 2 1 AB AB A E G    (2.27)     11 1 , 1 1 2 AB A AB A A E C        (2.28)    12 , 1 1 2 AB A AB A A E C       (2.29)   44 . 2 1 AB AB A E C    (2.30 The Poisson ratio of alloy AB with FCC structure has the form ,AB A A B B Ac c      (2.31) where A and B respectively are the Poisson ratioes of materials A and B and are determined from the experimental data. When the concentration of interstitial atom B is equal to zero, the obtained results for alloy AB become the coresponding results for main metal A. 2.2. Numerical results for alloy AuSi For alloy AuSi, we use the n-m pair potential 0 0( ) , n m r rD r m n n m r r                    (2.32) where the potential parameters are given in Table 1 [16]. Table 1. Potential parameters 0, , ,m n D r of materials Material m n 1610 ergD    10 0 10 mr    Au 5.5 10.5 6462.540 2.8751 Si 6.0 12.0 45128.24 2.2950 Approximately,  Au-Si Au-Au Si-Si 1 . 2    (2.33) According to our numerical results as shown in figures from Figure 1 to Figure 18 for AuSi at the same temperature and concentration of interstitial atoms when pressure increases, E, G, K and C11, C12, C44 increase. For example for AuSi at T = 300K, Si 5%c  when P increases from 0 to 70GPa, E increases from 0.5784. 10 11 Pato1,2431.10 11 Pa, G increases from 0.2019.10 11 Pa to 0.4340.10 11 Pa, K increases from 1.4176.10 11 Pa to 3.0467.10 11 Pa, C11 increases from 1.6869. 10 11 Pa to 3.6254.10 11 Pa, C12 increases from 1.2830.10 11 Pa to 2.7574.10 11 Pa and C44 increases from 0.2019.10 11 Pa to 0.4340.10 11 Pa. When the concentration of Si is equal to zero, Elastic deformation of alloy AuSi with FCC structure under pressure 79 the dependence of elastic moduli and elastic constants of alloy AuSi on pressure becomes the dependence of elastic moduli and elastic constants of main metal Au. 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 80 90 E ,G ,K ( 1 0 1 0 P a ) p (GPa) E G K 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 80 90 100 C 1 1 , C 1 2 , C 4 4 ( 1 0 1 0 P a ) p (GPa) C11 C12 C44 Figure 1. E, G, K (10 10 Pa)(P) for Au-1%Si at T = 300 K Figure 2. C11, C12, C44(10 10 Pa)(P) for Au-1%Si at T = 300 K 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 E ,G ,K ( 1 0 1 0 P a ) p (GPa) E G K 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 C 1 1 , C 1 2 , C 4 4 ( 1 0 1 0 P a ) p (GPa) C11 C12 C44 Figure 3. E, G, K (10 10 Pa) (P) for Au-3%Si at T = 300 K Figure 4. C11, C12, C44 (10 10 Pa)(P) for Au-3%Si at T = 300 K 0 10 20 30 40 50 60 70 0 5 10 15 20 25 30 E ,G ,K ( 1 0 1 0 P a ) p (GPa) E G K 0 10 20 30 40 50 60 70 0 5 10 15 20 25 30 35 40 C 1 1 , C 1 2 , C 4 4 ( 1 0 1 0 P a ) p (GPa) C11 C12 C44 Figure 5. E, G, K (10 10 Pa)(P) for Au-5%Si at T = 300 K Figure 6. C11, C12, C44 (10 10 Pa)(P) for Au-5%Si at T = 300 K Nguyen Quang Hoc, Nguyen Duc Hien and Dang Quoc Thang 80 0 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 16 18 20 E , G , K ( 1 0 1 0 P a ) T (K) E G K 0 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 16 18 20 22 24 C 1 1 , C 1 2 , C 4 4 ( 1 0 1 0 P a ) T (K) C11 C12 C44 Figure 7. E, G, K (10 10 Pa)(T) for Au-5%Si at P = 0 Figure 8. C11, C12, C44(10 10 Pa)(T) for Au-5%Si at P = 0 0 100 200 300 400 500 600 700 800 900 1000 0 5 10 15 20 25 30 E ,G ,K ( 1 0 1 0 P a ) T (K) E G K 0 100 200 300 400 500 600 700 800 900 1000 0 5 10 15 20 25 30 35 C 1 1 , C 1 2 , C 4 4 ( 1 0 1 0 P a ) T (K) C11 C12 C44 Figure 9. E, G, K (10 10 Pa) (T) for Au-5%Si at P = 30 GPa Figure 10. C11, C12, C44 (10 10 Pa) (T) for Au-5%Si at P = 30 Gpa 0 100 200 300 400 500 600 700 800 900 1000 0 5 10 15 20 25 30 35 40 E ,G ,K ( 1 0 1 0 P a ) T (K) E G K 0 100 200 300 400 500 600 700 800 900 1000 0 5 10 15 20 25 30 35 40 45 C 1 1 , C 1 2 , C 4 4 ( 1 0 1 0 P a ) T (K) C11 C12 C44 Figure 11. E, G, K (10 10 Pa) (T) for Au-5%Si at P = 70 GPa Figure 12. C11, C12, C44 (10 10 Pa) (T) for Au-5%Si at P = 70 GPa Elastic deformation of alloy AuSi with FCC structure under pressure 81 0 1 2 3 4 5 0 5 10 15 20 25 30 E ,G ,K ( 1 0 1 0 P a ) %C(Si) E G K 0 1 2 3 4 5 0 5 10 15 20 25 30 35 C 1 1 , C 1 2 , C 4 4 ( 1 0 1 0 P a ) %C(Si) C11 C12 C44 Figure 13. E, G, K (10 11 Pa)(cSi) for Au-xSi at P = 0 and T = 300 K Figure 14. C11, C12, C44(10 11 Pa) )(cSi) for Au-xSi at P = 0 and T = 300 K 0 1 2 3 4 5 0 5 10 15 20 25 30 E ,G ,K ( 1 0 1 0 P a ) %C(Si) E G K 0 1 2 3 4 5 0 5 10 15 20 25 30 35 C 1 1 , C 1 2 , C 4 4 ( 1 0 1 0 P a ) %C(Si) C11 C12 C44 Figure 15. E, G, K (10 11 Pa) (cSi) for Au-xSi at P = 0 and T = 1000 K Figure 16. C11, C12, C44 (10 11 Pa) (cSi) for Au-xSi at P = 0 and T =1000 K 0 1 2 3 4 5 0 20 40 60 80 100 E ,G ,K ( 1 0 1 0 P a ) %C(Si) E G K 0 1 2 3 4 5 0 10 20 30 40 50 60 70 80 90 100 110 C 1 1 , C 1 2 , C 4 4 ( 1 0 1 0 P a ) %C(Si) C11 C12 C44 Figure 17. E, G, K (10 11 Pa) (cSi) for Au-xSi at P = 70 GPa and T = 300 K Figure18. C11, C12, C44 (10 11 Pa) (cSi) for Au-xSi at P = 70 GPa and T = 300 K For AuSi at the same pressure and concentration of interstitial atoms when temperature increases, E, G, K and C11, C12, C44 descrease. For example for AuSi at P = 70 GPa, Si 5%c  when T increases from 50 K to 1000 K, E descreases from 1.3963.10 11 Pa to 1.2480.10 11 Pa, G descreases from 0.4875.10 11 Pa to 0.4358.10 11 Pa, K descreases from Nguyen Quang Hoc, Nguyen Duc Hien and Dang Quoc Thang 82 3.4323.10 11 Pa to 3.0588.10 11 Pa, C11 descreases from 4.0723.10 11 Pa to 3.6399.10 11 Pa, C12 descreases from 3.0973.10 11 Pa to 2.7684.10 11 Pa and C44 descreases from 0.4875.10 11 Pa to 0.4358.10 11 Pa. The dependence of elastic moduli and elastic constants of alloy AuSi on temperature is the same as the dependence of elastic moduli and elastic constants of main metal Au. For AuSi at the same pressure and temperature when the concentration of Si increases, E, G, K and C11, C12, C44 descrease. For example for AuSi at P = 70 GPa, T = 1000 K when the concentration of Si increases from 0 to 5%, E descreases from 1.3963.10 11 Pato 1,2480.10 11 Pa, G descreases from 0.4875.10 11 Pa to 0,4358.10 11 Pa, K descreases from 3.4223.10 11 Pa to 3,0588.10 11 Pa, C11 descreases from 4.0723.10 11 Pa to 3,6400.10 11 Pa, C12 descreases from 3,0973.10 11 Pa to 2,7684.10 11 Pa and C44 descreases from 0.4875.10 11 Pa to 0.4358.10 11 Pa. Table 2 gives the nearest neighbor distance and the elastic moduli of Au at T = 300 K, P = 0 according to the SMM and the experimental data [17, 18]. Table 2. Nearest neighbor distance and elastic moduli E, K, G of Au at P = 0, T = 300 K according to SMM and EXPT [17, 18] Method o Aa        1010 PaE  1010 PaK  1010 PaG SMM 2.8454 8.96 14.94 3.20 EXPT [17,18] 2.8838 8.91 16.70 3.10 Table 6. Elastic constants C11, C12, C44 of Au at T = 300K and P = 0 calculated by the SMM, other calculations[18-27] and from EXPT [18] SMM EXPT [18] Other calculations [19] [20] [21] [22] [23] [24] [25] [26] [27] C11 [10 11 Pa] 1.92 1.92 1.92 1.83 1.79 2.09 1.36 1.50 1.97 1.84 2.00 C12 [10 11 Pa] 1.28 1.65 1.66 1.54 1.47 1.75 0.91 1.29 1.84 1.54 1.73 C44 [10 11 Pa] 0.32 0.42 0.39 0.45 0.42 0.31 0.49 0.70 0.52 0.43 0.33 3. Conclusion The analytic expressions of the free energy, the mean nearest neighbor distance between two atoms, the elastic moduli such as the Young modulus, the bulk modulus, the rigidity modulus and the elastic constants depending on temperature, concentration of interstitial atoms for interstitial alloy AB with FCC structure under pressure are derived by the SMM. The numerical results for alloy AuSi are in good agreement with the numerical results for main metal Au. 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