Elastic deformation of binary and ternary interstitial alloys with FCC structure at zero pressure: Dependence on temperature, concentration of substitution atoms and concentration of interstitial atoms

Abstract. 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 for interstitial alloy AC and interstitial alloy ABC (substitution alloy AB with interstitial atom C) with FCC structure at zero pressure are derived using the statistical moment method. We apply the theoretical results to the interstitial alloys AuLi and AuCuLi at zero pressure and at different temperatures, concentrations of substitution atoms and interstitial atoms. Calculated results for the main metal in these alloys are compared with the experimental data and other calculations.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0031 Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 47-57 This paper is available online at ELASTIC DEFORMATION OF BINARY AND TERNARY INTERSTITIAL ALLOYS WITH FCC STRUCTURE AT ZERO PRESSURE: DEPENDENCE ON TEMPERATURE, CONCENTRATION OF SUBSTITUTION ATOMS AND CONCENTRATION OF INTERSTITIAL ATOMS Nguyen Quang Hoc1, Bui Duc Tinh1, Le Dinh Tuan2 and Nguyen Duc Hien3 1Faculty of Physics, Hanoi National University of Education 2Faculty of Natural Sciences, Hong Duc University 3Mac Dinh Chi High School, Chu Pah District, Gia Lai Province Abstract. 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 for interstitial alloy AC and interstitial alloy ABC (substitution alloy AB with interstitial atom C) with FCC structure at zero pressure are derived using the statistical moment method. We apply the theoretical results to the interstitial alloys AuLi and AuCuLi at zero pressure and at different temperatures, concentrations of substitution atoms and interstitial atoms. Calculated results for the main metal in these alloys are compared with the experimental data and other calculations. Keywords: Binary and ternary interstitial alloy, statistical moment method, Young modulus, bulk modulus, rigidity modulus, elastic constant. 1. Introduction The elastic deformation of metals and alloys has attracted the attention of many researchers [1-16]. In this paper the authors have used the statistical moment method (SMM) [1, 2, 16] to derive the analytic expressions of the free energy, the mean nearest neighbor distance between two atoms, the elastic moduli such as the Young modulus E, the bulk modulus G and the rigidity modulus K, and the elastic constants C11, C12, C44 depending on temperature, concentration of substitution atoms and concentration of interstitial atoms for interstitial alloy AC and interstitial alloy ABC (substitution alloy AB with interstitial atom C) with an FCC structure at zero pressure. We applied the theoretical results to the interstitial alloys AuLi and AuCuLi at zero pressure and at different temperatures, concentration of substitution atoms and interstitial atoms. Some of the calculated results for the main metals in the alloys are compared with the experimental data and the other calculations. Received March 1, 2016. Accepted July 25, 2016. Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn 47 Nguyen Quang Hoc, Bui Duc Tinh, Le Dinh Tuan and Nguyen Duc Hien 2. Content 2.1. Elastic deformation of interstitial alloy AC with FCC structure The bound energy of atom C (in the body center of a cubic unit cell) with atoms A (in the face centers and peaks of the cubic unit cell) in the approximation of three coordination spheres with a center C and radii r1, r1 √ 3, r1 √ 5 is determined by [1, 2] u0C = ni∑ i=1 ϕAC (ri) = 6ϕAC (r1) + 8ϕAC ( r1 √ 3 ) + 24ϕAC ( r1 √ 5 ) , (2.1) where ϕAC is the interaction potential between atom A and atom C, r1 ≡ r1C = r01C + y0A1(T ) is the nearest neighbor distance between interstitial atom C and metallic atom A at temperature T, r01C is the nearest neighbor distance between interstitial atom C and metallic atom A at 0 K and is determined from the minimum condition of the bound energy u0C , y0A1(T ) is the displacement of atom A1 (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 form [1, 2] kC = ϕ (2) AC (r1) + 2 r1 ϕ (1) AC (r1) + 4 3 ϕ (2) AC ( r1 √ 3 ) + 8 √ 3 9r1 ϕ (1) AC ( r1 √ 3 ) +4ϕ (2) AC ( r1 √ 5 ) + 8 √ 5 5r1 ϕ (1) AC ( r1 √ 5 ) , γC = 4 (γ1C + γ2C) , γ1C = 1 24 ϕ (4) AC(r1) + 1 4r21 ϕ (2) AC(r1)− 1 4r31 ϕ (1) AC(r1) + 1 54 ϕ (4) AC(r1 √ 3) + 2 √ 3 27r1 ϕ (3) AC(r1 √ 3)− 2 27r21 ϕ (2) AC(r1 √ 3) + 2 √ 3 81r31 ϕ (1) AC(r1 √ 3) + 17 150 ϕ (4) AC(r1 √ 5) + 8 √ 5 125r1 ϕ (3) AC(r1 √ 5) + 1 25r21 ϕ (2) AC(r1 √ 5)− √ 5 125r31 ϕ (1) AC(r1 √ 5), γ2C = 1 2r1 ϕ (3) AC(r1)− 3 4r21 ϕ (2) AC(r1) + 3 4r31 ϕ (1) AC(r1) + 1 4 ϕ (4) AC(r1 √ 2) + √ 2 8r1 ϕ (3) AC(r1 √ 2) + 7 8r21 ϕ (2) AC(r1 √ 2) − 7 √ 2 16r31 ϕ (1) AC(r1C √ 2) + 4 25 ϕ (4) AC(r1 √ 5) + 26 √ 5 125r1 ϕ (3) AC(r1 √ 5) − 3 25r21 ϕ (2) AC(r1 √ 5) + 3 √ 5 125r31 ϕ (1) AC(r1 √ 5), (2.2) 48 Elastic deformation of binary and ternary interstitial alloys with fcc structure at zero pressure... where ϕ(k)AC(ri) ≡ ∂ kϕAC ∂rk ≤i (k = 1, 2, 3, 4). The bound energy of atom A1 (which contains interstitial atom C on the first coordination sphere) with the atoms in a crystalline lattice and the corresponding alloy’s parameters in the approximation of three coordination spheres with the center A1 is determined by [1, 2] u0A1 = u0A + ϕAC (r1A1) , kA1 = kA + ϕ (2) AC (r1A1) , γA1 = 4 (γ1A1 + γ2A1) , γ1A1 = γ1A + 1 24 ϕ (4) AC(r1A1), γ2A1 = γ2A + 1 4r1A1 ϕ (3) AC(r1A1)− 1 2r21A1 ϕ (2) AC(r1A1) + 1 2r31A1 ϕ (1) AC(r1A1), (2.3) where r1A1 ≈ r1C is the nearest neighbor distance between atom A1 and atoms in the crystalline lattice. The bound 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 the center A2 is determined by [1, 2] u0A2 == u0A + ϕAC (r1A2) , kA2 = kA + 1 6 ϕ (2) AC (r1A2) + 23 6r1A2 ϕ (1) AC (r1A2) , γA2 = 4 (γ1A2 + γ2A2) , γ1A2 = γ1A + 1 54 ϕ (4) AC(r1A2) + 2 9r1A2 ϕ (3) AC(r1A2)− 2 9r21A2 ϕ (2) AC(r1A2) + 2 9r31A2 ϕ (1) AC(r1A2), γ2A2 = γ2A + 1 81 ϕ (4) AC(r1A2) + 4 27r1A2 ϕ (3) AC(r1A2) + 14 27r21A2 ϕ (2) AC(r1A2)− 14 27r31A2 ϕ (1) AC(r1A2), (2.4) where r1A2 = r01A2 + y0C(T ), r01A2 is the nearest neighbor distance between atom A2 and atoms in the crystalline lattice at 0 K and is determined from the minimum condition of the bound energy u0A2 , y0C(T ) is the displacement of atom C at temperature T. In Eqs. (2.3) and (2.4), u0A, kA, γ1A, γ2A are the coressponding quantities in clean metal A in the approximation of two coordination sphere [1, 2]. The nearest neighbor distances r1X(0, T )(X = A,A1, A2, C) in the interstitial alloy at pressure P = 0 and temperature T are derived from [16] r1A(0, T ) = r1A(0, 0) + yA(0, T ), r1C (0, T ) = r1C(0, 0) + yC(0, T ), r1A1(0, T ) ≈ r1C(0, T ), r1A2(0, T ) = r1A2(0, 0) + yC(0, T ) (2.5) r1X(0, 0)(X = A,A1, A2, C) is determined from the equation of state or the minimum condition of bound energy. From the r1X(0, 0) obtained with the use of Maple software, we can determine the parameters kX(0, 0), γX (0, 0), ωX (0, 0) at 0 K. After that, we can calculate the displacements [1, 2] y0X(0, T ) = √ 2γX(0, 0)θ2 3k3X(0, 0) AX(0, T ),X = A,A1, A2, C, (2.6) 49 Nguyen Quang Hoc, Bui Duc Tinh, Le Dinh Tuan and Nguyen Duc Hien where θ = kBT, kB is the Boltzmann constant and has the form as in [1, 2]. Then, we calculate the mean nearest neighbor distance in interstitial alloy AC using the expressions as follows [16] r1A(0, T ) = r1A(0, 0) + y(0, T ), r1A(0, 0) = (1− cC) r1A(0, 0) + cCr′1A(0, 0), r′1A(0, 0) ≈ √ 3r1C(0, 0), y(0, T ) = (1− 15cC) yA(0, T ) + cCyC(0, T ) + 6cCyA1(0, T ) + 8cCyA2(0, T ), (2.7) The Young modulus of interstitial alloy AC is determined by EAC (cC , T ) = EA (cC , T )  1− 15cC + cC ∂ 2ψC ∂ε2 + 6 ∂2ψA1 ∂ε2 + 8 ∂2ψA2 ∂ε2 ∂2ψA ∂ε2   , EA (cC , T ) = 1 π.r1 (cC , T )A1 (cC , T ) , A1 (cC , T ) = 1 kA (cC , T ) [ 1 + 2γ2A (cC , T ) θ 2 k4A (cC , T ) ( 1 + 1 2 XA ) (1 +XA) ] , XA ≡ xA cothxA, xA = ~ωA (cC , T ) 2θ , ∂2ψX ∂ε2 = { 1 2 ∂2U0X ∂r21 + 3 4 ~ωX kX [ ∂2kX ∂r21 − 1 2kX ( ∂kX ∂r1 )2]} 4r201+ + ( 1 2 ∂U0X ∂r1 + 3 2 ~ωXcthxX 1 2kX ∂kX ∂r1 ) 2r01, xX = ~ωX 2θ , ωX = √ kX m , (2.8) where ε is the relative deformation, cC is the concentration of interstitial atoms C, EA is the Young modulus of pure metal A, ψA is the free energy of atom A in pure metal A, and ψX(X = C,A1, A2) is the free energy of atom X in interstitial alloy and has the form in [1, 2], U0X = Nu0X . The bulk modulus and the rigidity modulus of interstitial alloy AC have the form KAC (cC , T ) = EAC (cC , T ) 3(1 − 2νA) , GAC (cC , T ) = E (cC , T ) 2 (1 + νA) . (2.9) The elastic constants of interstitial alloy AC have the form C11AC (cC,T ) = E (cC,T ) (1− νA) (1 + νA) (1− 2νA) , C12AC (cC,T ) = E (cC,T ) νA (1 + νA) (1− 2νA) , C44AC (cC , T ) = E (cC , T ) 2 (1 + νA) . (2.10) The Poisson ratio of interstitial alloy AC has the form νAC = cAνA + cCνC ≈ νA, (2.11) where νA and νC are the Poisson ratio of the materials A, C and cA is the concentration of main atoms A. 50 Elastic deformation of binary and ternary interstitial alloys with fcc structure at zero pressure... 2.2. Elastic deformation of ABC interstitial alloy with FCC structure The mean nearest neighbor distance between two atoms in the interstitial alloy ABC with FCC structure at temperature T is given by aABC = cACaAC BTAC BT +cBaB BTB BT , BT = cACBTAC+cBBTB, cAC = cA+cC , aAC = r1A(0, T ), BTAC = 1 χTAC , χTAC = 3 ( aAC a0AC )3 2a2 AC 3 √ 2a3 AC ( ∂2ψAC ∂a2 AC ) T , ( ∂2ψAC ∂a2AC ) T = [ ∂2ψAC ∂r1A2(0, T ) ] T ≈ (1− 15cC) ( ∂2ψA ∂a2A ) T +cC ( ∂2ψC ∂a2C ) T + 6cC ( ∂2ψA1 ∂a2A1 ) T + 8cC ( ∂2ψA2 ∂a2A2 ) T , ( ∂2ψX ∂a2X ) T = ( ∂2ψX ∂r21X(0, T ) ) T , 1 3N ( ∂2ΨX ∂a2X ) T = 1 6 ∂2u0X ∂a2X + ~ωX 4kX [ ∂2kX ∂a2X − 1 2kX ( ∂kX ∂aX )2] . (2.12) The Young modulus of interstitial alloy ABC is determined by EABC = EAB − (cA + cB)EA + EAC , EAB = cAEA + cBEB, EAC = EA  1− 15cC + cC ∂ 2ψC ∂ε2 + 6 ∂2ψA1 ∂ε2 + 8 ∂2ψA2 ∂ε2 ∂2ψA ∂ε2   , (2.13) where EAB is the Young modulus of substitution alloy AB and EAC is the Young modulus of interstitial alloy AC. The bulk modulus and rigidity modulus of interstitial alloy ABC have the form KABC = EABC 3(1− 2νABC) , GABC = EABC 2 (1 + νABC) . (2.14) The elastic constants of interstitial alloy ABC have the form C11ABC = EABC (1− νABC) (1 + νABC) (1− 2νABC) , C12ABC = EABCνABC (1 + νABC) (1− 2νABC) , C44ABC = EABC 2 (1 + νABC) . (2.15) The Poisson ratio of interstitial alloy ABC has the form νABC = cAνA + cBνB + cCνC ≈ cAνA + cBνB = νAB. (2.16) where cB is the concentration of substitution atoms B and νAB is the Poisson ratio of substitution alloy AB 51 Nguyen Quang Hoc, Bui Duc Tinh, Le Dinh Tuan and Nguyen Duc Hien 2.3. Numerical results and discussion for alloys AuLi and AuCuLi For metallic crystal, we can apply the n-m pair potential in the form as follows [1] ϕ (a) = D n−m [ m (r0 a )n − n(r0 a )m] , (2.17) where the potential parameters and the Poisson ratio for materials Au, Cu and Li are presented in Table 1 [3]. Table 1. The n-m potential parameters and the Poisson ratio for materials Au, Cu, Li [3] Material m n D (10−16 erg) r0 (10−10 m) ν Au 5.5 10.5 6462.54 2.8751 0.39 Cu 5.5 11 4693.518 2.5487 0.37 Li 1.66 3.39 6800.502 3.0077 For the interaction between the main atom A and the interstitial atom C in the interstitial alloy, ϕAC(a) ≈ 1 2 [ϕAA(a) + ϕCC(a)] . (2.18) According to Figures 1 and 2 at the same temperature when the concentration of interstitial atoms increases, the mean nearest neighbor distance in alloy AuLi increases. In the same concentration of interstitial atoms when the temperatures increases, the mean nearest neighbor distance in alloy AuLi increases. The increase in concentration of interstitial atoms is stronger than the increase with temperature (when the concentration of intersitial atoms increases from 0 to 5%, the mean nearest neighbor distance increases from 2.8402 to 2.9457 A˚ but when the temperature increases from 50 to 1000 K, the mean nearest neighbor distance increases from 2.8402 to 2.8848 A˚). At zero concentration of interstitial atoms, we obtain the nearest neighbor distance of pure metal Au in [1]. According to Figures 3 and 4 at the same temperature when the concentration of interstitial atoms increases, the elastic moduli E, K and the elastic contants of alloy AuLi decrease. In the same concentration of interstitial atoms, when the temperatures increases, the elastic moduli E, K of alloy AuLi decrease. At zero concentration of interstitial atoms, we obtain the elastic moduli E and K of pure Au in [1]. According to Figures 5 and 6 at the same temperature when the concentration of interstitial atoms increases, the elastic contants C11, C12, C44 of alloy AuLi decrease. At the same concentration of interstitial atoms when the temperatures increases, the elastic contants C11, C12, C44 of alloy AuLi decrease. At zero concentration of interstitial atoms, we obtain the elastic contants C11, C12 and C44 of pure Au in [1]. The calculated (CAL) results from the SMM and the other calculations and the experimental data (EXPT) for the nearest neighbor distance, the elastic moduli and the elastic constants of main metal Au in interstitial alloys AuLi and AuCuLi are shown in Tables 2 and 3. Some calculated results from the SMM are in rather good agreement with experimental data and better than that obtained using other calculation methods. 52 Elastic deformation of binary and ternary interstitial alloys with fcc structure at zero pressure... Table 2. The nearest neighbor distance and elastic moduli of Au at P = 0, T = 300 K from the SMM and the experimental data [5, 6] Method a(A˚) E(1010Pa) K(1010Pa) G(1010Pa) SMM 2.8454 8.96 14.94 3.20 EXPT[5,6] 2.8838 2.8838 16.70 3.10 53 Nguyen Quang Hoc, Bui Duc Tinh, Le Dinh Tuan and Nguyen Duc Hien Table 3. The elastic constants of Au at P = 0, T = 300 K from the SMM, other calculations [7-15] and experimental data [6] Elastic constant SMM EXPT Other calculations (1010Pa) [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] C11 1.92 1.92 1.92 1.83 1.79 2.09 1.36 1.50 1.97 1.84 2.00 C12 1.28 1.65 1.66 1.54 1.47 1.75 0.91 1.29 1.84 1.54 1.73 C44 0.32 0.42 0.39 0.45 0.42 0.31 0.49 0.70 0.52 0.43 0.33 According to Figures 7-10 for alloy AuCuLi at the same temperature and concentration of substitution atoms when the concentration of interstitial atoms increases, the mean nearest neighbor distance increases (for example at 50 K and cCu = 6%, a increases from 2.8069 đến 2.9162 A˚ when cLi increases from 0 to 5%). For alloy AuCuLi at the same concentration of substitution atoms and interstitial atoms when the temperature increases, the mean nearest neighbor distance increases (for example at cCu = 6%, cLi = 5%, a increases from 2.9162 54 Elastic deformation of binary and ternary interstitial alloys with fcc structure at zero pressure... to 3.6067 A˚ when T increases from 50 to 1000 K). For alloy AuCuLi at the same temperature and concentration of interstitial atoms when the concentration of substitution atoms increases, the mean nearest neighbor distance decreases (for example at 1000 K, cLi = 0.6%, a decreases from 2.9743 to 2.9521 A˚ when cCu increases from 0 to 6%). At zero concentration of substitution atoms, andinterstitial atoms, the mean nearest neighbor distance of interstitial alloy AuCuLi becomes the mean nearest neighbor distance of metal Au in [1]. The change of mean nearest neighbor distance with temperature for interstitial alloy AuCuLi is similar to that for interstitial alloy AuLi. The change of mean nearest neighbor distance with temperature and concentration of substitution atoms for interstitial alloy AuCuLi is similar to that for substitution alloy AuCu [4]. According to Figures 11-13 for alloy AuCuLi in the same temperature and concentration of substitution atoms when the concentration of interstitial atoms increases, the elastic moduli E and K decrease (for example at 300 K, E decreases from 11.4259.1010 to 6.3259.1010 Pa when cLi increases from 0 to 5%). For alloy AuCuLi in the same concentration of substitution atoms 55 Nguyen Quang Hoc, Bui Duc Tinh, Le Dinh Tuan and Nguyen Duc Hien and concentration of interstitial atoms when the temperature increases, the elastic moduli E and K decrease (for example cCu = 10%, cLi = 5%, at E decreases from 6.7681.1010 to 5.2362.1010Pa when T increases from 100 to 700 K). For alloy AuCuLi at the same temperature and concentration of interstitial atoms, when the concentration of substitution atoms increases, the elastic moduli E and K increase (for example at 300 K, cLi = 5%, E increases from 5.97.1010 to 6.8597.1010Pa when cCu increases from 0 to 25%). At zero concentrations of substitution atoms and interstitial atoms, the elastic moduli E and K of interstitial alloy AuCuLi become the elastic moduli E and K of metal Au in [1]. The change of elastic moduli E and K in temperature for interstitial alloy AuCuLi is similar to that for interstitial alloy AuLi. The change of elastic moduli E and K in temperature and concentration of substitution atoms for interstitial alloy AuCuLi is similar to that for substitution alloy AuCu [4]. According to Figures 14-16 for alloy AuCuLi at the same temperature and concentration of substitution atoms, when the concentration of interstitial atoms increases, the elastic constants C11, C12 decrease (for example at 300 K and cCu = 10%, C11 decreases from 22.4909.1010 to 12.4519.1010 Pa when cLi increases from 0 to 5%). For alloy AuCuLi with the same concentration of substitution atoms and concentration of interstitial atoms, when the temperature increases, the elastic constants C11, C12 decrease (for example at cCu = 10%, cLi = 5% , C11 decreases from 13.3223.1010 to 10.2952.1010Pa when T inceases from 100 to 700 K). For alloy AuCuLi at the same temperature and concentration of interstitial atoms, when the concentration of substitution atoms increases, the elastic constants C11, C12 increase (for example at 300 K, cLi = 5%, C11 increases from 11.9089.1010 to 13.2435.1010Pa when cCu increases from 0 to 25%). At zero concentration of substitution atoms and interstitial atoms, the elastic constants of interstitial alloy AuCuLi become the elastic constants C11, C12 of metal Au in [1]. The change of elastic constants C11, C12 with temperature for interstitial alloy AuCuLi is similar to that for interstitial alloy AuLi. The change of elastic constants C11, C12 with temperature and concentration of substitution atoms for interstitial alloy AuCuLi is similar to that for substitution alloy AuCu [4]. 56 Elastic deformation of binary and ternary interstitial alloys with fcc structure at zero pressure... 3. Conclusion Our results in using the elastic theory for interstitial alloys AC, ABC with FCC structure based on the SMM are applied to alloys AuLi and AuCuLi at zero pressure in the temperature intervals from 100 to 1000 K, in the range of concentration of substitution atoms from 0 to 25% and in the range of concentration of interstitial atoms from 0 to 5%. The calculated results for main metal Au in the interstitial alloys are in rather good agreement with experimental data and are compared with other calculated results. Acknowledgements. This work was carried out thanks to the financial support provided by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under Grant No. 103.01-2013.20. REFERENCES [1] V. V. Hung, 2009. Statistical moment method in studying thermodynamic and elastic property of crystal. HNUE Publishing House, pp.1-231. [2]