Thermodynamic properties of binary interstitial alloys with a BCC structure: Dependence on temperature and concentration of interstitial atoms

Abstract. Thermodynamic quantities such as mean nearest neighbor distance, free energy, isothermal and adiabatic compressibilities, isothermal and adiabatic elastic modulus, thermal expansion coefficient, heat capacities at constant volume and constant pressure, and entropy of binary interstitial alloys with a body-centered cubic (BCC) structure with a very small concentration of interstitial atoms are derived using the statistical moment method. The obtained expressions of these quantities depend on temperature and concentration of interstitial atoms. The theoretical results are applied to the interstitial alloy FeSi. In the case where the concentration of interstitial atoms of Si is equal to zero, we have the thermodynamic quantities of the main metal and the numerical results for the alloy FeSi give the numerical results for Fe. The calculated results of the thermal expansion coefficient and heat capacity at constant pressure in the interval of temperature from 100 to 700 K for Fe are in good agreement with the experimental data.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0044 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 146-155 This paper is available online at THERMODYNAMIC PROPERTIES OF BINARY INTERSTITIAL ALLOYS WITH A BCC STRUCTURE: DEPENDENCE ON TEMPERATURE AND CONCENTRATION OF INTERSTITIAL ATOMS Nguyen Quang Hoc, Dinh Quang Vinh, Bui Duc Tinh, Tran Thi Cam Loan, Ngo Lien Phuong, Tang Thi Hue and Dinh Thi Thanh Thuy Faculty of Physics, Hanoi National University of Education Abstract. Thermodynamic quantities such as mean nearest neighbor distance, free energy, isothermal and adiabatic compressibilities, isothermal and adiabatic elastic modulus, thermal expansion coefficient, heat capacities at constant volume and constant pressure, and entropy of binary interstitial alloys with a body-centered cubic (BCC) structure with a very small concentration of interstitial atoms are derived using the statistical moment method. The obtained expressions of these quantities depend on temperature and concentration of interstitial atoms. The theoretical results are applied to the interstitial alloy FeSi. In the case where the concentration of interstitial atoms of Si is equal to zero, we have the thermodynamic quantities of the main metal and the numerical results for the alloy FeSi give the numerical results for Fe. The calculated results of the thermal expansion coefficient and heat capacity at constant pressure in the interval of temperature from 100 to 700 K for Fe are in good agreement with the experimental data. Keywords: Binary interstitial alloy, statistical moment method, coordination sphere. 1. Introduction Thermodynamic and elastic properties of interstitial alloys are of special interest to many theoretical and experimental researchers, for example [1-3]. In [4, 5] the equilibrium vacancy concentration in BCC substitution and interstitial alloys is calculated taking into account thermal redistribution of the interstitial component in different types of interstices. The conditions where this effect gives rise to a decrease or increase in vacancy concentration are formulated. Coatings based on interstitial alloys of transition metals have a wide range of applicability. However, interest in synthesizing coatings from new materials with requisite service properties is limited due to the scarceness of data on their melting temperature. In [6], the authors considered calculating the melting temperature for interstitial alloys of transition metals based on the characteristics of intermolecular interaction. In [7], the authors attempt to present a survey of the order-disorder transformations of the interstitial alloys of transition metals with hydrogen, deuterium and oxygen. Special attention Received August 19, 2015. Accepted October 26, 2015. Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn 146 Thermodynamic properties of binary interstitial alloys with a BCC structure... is given to the formation of interstitial superstructures, the stepwise process of disordering and property changes attributed to order-disorder. Four groups of interstitial alloys are considered: (1) TO, ZrO, HfO, (2) VO, (3) VH, VD and (4) TaH, TaD. Characteristic features of the phase transformations in each group and each system are presented and discussed in comparison with others. In this paper, we build the thermodynamic theory for binary interstitial alloy with a BCC structure using the statistical moment method (SMM) [8] and applying the obtained theoretical results to the alloy FeSi. 2. Content 2.1. Thermodynamic quantities of binary interstitial alloys with a BCC structure The cohesive energy of atom C (in face centers of cubic unit cell) with atoms A (in body centers and peaks of cubic unit cell) in the approximation of three coordination spheres with center C and radii r1, r1 √ 2, r1 √ 5 is determined by u0C = 1 2 ni∑ i=1 ϕAC (ri) = 1 2 [ 2ϕAC (r1) + 4ϕAC ( r1 √ 2 ) + 8ϕAC ( r1 √ 5 )] = ϕAC (r1) + 2ϕAC ( r1 √ 2 ) + 4ϕAC ( r1 √ 5 ) , (2.1) where ϕAC is the interaction potential between atom A and atom C, ni is the number of atoms on the ith coordination sphere with radius ri(i = 1, 2, 3), 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 cohesive energy u0C , y0A1(T ) is the displacement of atom A1 (atom A stays in the body center of the cubic unit cell) from the equilibrium position at temperature T. The alloy’s parameters for atoms C in the approximation of three coordination spheres have the following form kC = 1 2 ∑ i ( ∂2ϕAC ∂u2iβ ) eq = ϕ (2) AC (r1) + √ 2 r1 ϕ (1) AC ( r1 √ 2 ) + 4 5r21 ϕ (2) AC ( r1 √ 5 ) + 16 5 √ 5r1 ϕ (1) AC ( r1 √ 5 ) , γC = 4 (γ1C + γ2C) , γ1C = 1 48 ∑ i ( ∂4ϕAC ∂u4iβ ) eq = 1 24 ϕ (4) AC(r1) + 1 8r21 ϕ (2) AC ( r1 √ 2 ) − √ 2 16r31 ϕ (1) AC ( r1 √ 2 ) + 1 150 ϕ (4) AC(r1 √ 5) + 4 √ 5 125r1 ϕ (3) AC(r1 √ 5), γ2C = 6 48 ∑ i ( ∂4ϕAC ∂u2iα∂u 2 iβ ) eq = 1 4r1 ϕ (3) AC(r1)− 1 2r21 ϕ (2) AC(r1) + 5 8r31 ϕ (1) AC(r1) 147 N. Q. Hoc, D. Q. Vinh, B. D. Tinh, T. T. C. Loan, N. L. Phuong, T. T. Hue and D. T. T. Thuy + √ 2 8r1 ϕ (3) AC(r1 √ 2)− 1 8r21 ϕ (2) AC(r1 √ 2) + 1 8r31 ϕ (1) AC(r1 √ 2) + 2 25 ϕ (4) AC(r1 √ 5) + 3 25 √ 5r1 ϕ (3) AC(r1 √ 5) + 2 25r21 ϕ (2) AC(r1 √ 5)− 2 25 √ 5r31 ϕ (1) AC(r1 √ 5), (2.2) where ϕ(i)AC(ri) = ∂ 2ϕAC(ri)/∂r 2 i (i = 1, 2, 3, 4), α, β = x, y, z, α 6= β and uiβ is the displacement of the ith atom in direction β When atom A1, which contains interstitial atom C on the first coordination sphere, is taken as the coordinate origin, the cohesive energy of atom A1 with the atoms in crystalline lattice and the corresponding alloy’s parameters in the approximation of three coordination spheres mentioned above is determined by u0A1 = u0A + ϕAC (r1A1) ; kA1 = kA + 1 2 ∑ i (∂2ϕAC ∂u2iβ ) eq  r=rA1 = kA + ϕ (2) AC (r1A1) + 5 2r1A1 ϕ (1) AC (r1A1) , γA1 = 4 (γ1A1 + γ2A1) , γ1A1 = γ1A + 1 48 ∑ i (∂4ϕAC ∂u4iβ ) eq  r=rA1 = γ1A + 1 24 ϕ (4) AC(r1A1) + 1 8r21A1 ϕ (2) AC(r1A1)− 1 8r31A1 ϕ (1) AC(r1A1), γ2A1 = γ2A + 6 48 ∑ i ( ∂4ϕAC ∂u2iα∂u 2 iβ ) eq  r=rA1 = γ2A + 1 2r1A1 ϕ (3) AC(r1A1)− 3 4r21A1 ϕ (2) AC(r1A1) + 3 4r31A1 ϕ (1) AC(r1A1), (2.3) where u0A, kA, γ1A, γ2A are the coressponding quantities in clean metal A in the approximation of two coordination spheres [8] and r1A1 ≈ r1C is the nearest neighbor distance between atom A1, atom A stays in body of cubic unit cell and atoms in the crystalline lattic. When atom A2 (atom A stays in peaks of cubic unit cell) which contain interstitial atom C on the first coordination sphere is taken as the coordinate origin, the cohesive energy of atom A2 with the atoms in crystalline lattice and the corresponding alloy’s parameters in the approximation of three coordination spheres mentioned above is determined by u0A2 = u0A + ϕAC (r1A2) , kA2 = kA + 1 2 ∑ i (∂2ϕAC ∂u2iβ ) eq  r=rA2 = kA + 2ϕ (2) AC (r1A2) + 4 r1A2 ϕ (1) AC (r1A2) , γA2 = 4 (γ1A2 + γ2A2) , 148 Thermodynamic properties of binary interstitial alloys with a BCC structure... γ1A2 = γ1A + 1 48 ∑ i (∂4ϕAC ∂u4iβ ) eq  r=rA1 = γ1A + 1 24 ϕ (4) AC(r1A2) + 1 4r1A2 ϕ (3) AC(r1A2)− 1 8r21A2 ϕ (2) AC(r1A2) + 1 8r31A2 ϕ (1) AC(r1A2), γ2A2 = γ2A + 6 48 ∑ i ( ∂4ϕAC ∂u2iα∂u 2 iβ ) eq  r=rA2 = γ2A + 1 8 ϕ (4) AC(r1A2) + 1 4r1A2 ϕ (3) AC(r1A2) + 3 8r21A2 ϕ (2) A2C (r1A2)− 3 8r31A2 ϕ (1) A2C (r1A2), (2.4) where r1A2 = r01A2 + y0C(T ), r01A2 is the nearest neighbor distance between atom A2 and atoms in crystalline lattice at 0 K and is determined from the minimum condition of the cohesive energy u0A2 , y0C(T ) is the displacement of atom C at temperature T. 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 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) where r1X(0, 0)(X = A,A1, A2, C) is determined from the equation of state or the minimum condition of cohesive energy. From the obtained r1X(0, 0) using 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 [8] y0X(0, T ) = √ 2γX(0, 0)θ2 3k3X(0, 0) AX(0, T ).X = A,A1, A2, C, AX = a1X + 5∑ i=2 ( γXθ k2X )i aiX , kX = mω 2 X , xX = ~ωX 2θ , a1X = 1 + XX 2 , a2X = 13 3 + 47 6 XX + 23 6 X2X + 1 2 X3X , a3X = − ( 25 3 + 121 6 XX + 50 3 X2X + 16 3 X3X + 1 2 X4X ) , a4X = 43 3 + 93 2 XX + 169 3 X2X + 83 3 X3X + 22 4 X4X + 1 2 X5X , a5X = − ( 103 3 + 749 6 XX + 363 3 X2X + 733 3 X3X + 148 3 X4X + 53 6 X5X + 1 2 X6X ) , a6X = 65 + 561 2 XX + 1489 3 X2X + 927 2 X3X + 733 3 X4X + 145 2 X5X + 31 3 X6X + 1 2 X7X , XX ≡ xX cothxX . (2.6) Then, we calculate the mean nearest neighbor distance in interstitial alloy AC by the expressions as follows: 149 N. Q. Hoc, D. Q. Vinh, B. D. Tinh, T. T. C. Loan, N. L. Phuong, T. T. Hue and D. T. T. Thuy 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− 7cC) yA(0, T ) + cCyC(0, T ) + 2cCyA1(0, T ) + 4cCyA2(0, T ), (2.7) where r1A(0, T ) is the mean nearest neighbor distance between atoms A in interstitial alloy AC at P = 0 and temperature T, r1A(0, 0) is the mean nearest neighbor distance between atoms A in interstitial alloy AC at P = 0 and 0 K, r1A(0, 0) is the nearest neighbor distance between atoms A in clean metal A at P = 0 and 0 K, r′1A(0, 0) is the nearest neighbor distance between atoms A in the zone containing the interstitial atom C at P = 0 and 0 K and cC is the concentration of interstitial atoms C. The free energy per mole of interstitial alloy AC is determined by ψAC = (1− 7cC)ψA + cCψC + 2cCψA1 + 4cCψA2 − TSc, ψX = U0X + ψ0X + 3N { θ2 k2X [ γ2XX 2 X − 2γ1X 3 ( 1 + 1 2 XX )] + 2θ3 k4X [ 4 3 γ22XXX ( 1 + 1 2 XX ) − 2γ21X + 2γ1Xγ2X ( 1 + 1 2 XX ) (1 +XX) ]} , ψ0X = 3Nθ [ xX + ln ( 1− e−xX)] . (2.8) where SC is the configuration entropy. The isothermal compressibility of interstitial alloy AC has the form χTAC = 3 ( aAC a0AC )3 a2 AC 3VAC ( ∂2ΨAC ∂a2 AC ) T = 3 ( aAC a0AC )3 2a2 AC 3 √ 2a3 AC ( ∂2ψAC ∂a2 AC ) T = ( aAC a0AC )3 √ 2 3aAC 1 3N ( ∂2ΨAC ∂a2 AC ) T , ( ∂2ψAC ∂a2AC ) T = [ ∂2ψAC ∂r1A2(0, T ) ] T = (1− 7cC) ( ∂2ψA ∂a2A ) T + cC ( ∂2ψC ∂a2C ) T +2cC ( ∂2ψA1 ∂a2A1 ) T + 4cC ( ∂2ψA2 ∂a2A2 ) T , 1 3N ( ∂2ΨX ∂a2X ) T = 1 6 ∂2u0X ∂a2X + ~ωX 4kX [ ∂2kX ∂a2X − 1 2kX ( ∂kX ∂aX )2] , ( ∂2ψX ∂a2X ) T = 1 2 ∂2u0X ∂a2X + 3~ωX 4kX [ ∂2kX ∂a2X − 1 2kX ( ∂kX ∂aX )2] , (2.9) where ΨAC = NψAC , aAC = r1A(0, T )anda0AC = r1A(0, 0). 150 Thermodynamic properties of binary interstitial alloys with a BCC structure... The thermal expansion coefficient of interstitial alloy AC has the form αTAC = kB α0AC daAC dθ = −kBχTAC 3 ( a0AC aAC )2 aAC 3VAC ∂2ΨAC ∂θ∂aAC = −kBχTAC 3 ( a0AC aAC )2 aAC 3vAC ∂2ψAC ∂θ∂aAC , ∂2ψAC ∂θ∂aAC = (1− 7cC) ∂ 2ψA ∂θ∂aA + cC ∂2ψC ∂θ∂aC + 2cC ∂2ψA1 ∂θ∂aA1 + 4cC ∂2ψA2 ∂θ∂aA2 , ∂2ψX ∂θ∂aX = 3 2kX ∂kX ∂aX Y 2X + 6θ2 k2X [ γ1X 3kX ∂kX ∂aX ( 2 +XXY 2 X ) −1 6 ∂γ1X ∂aX ( 4 +XX + Y 2 X )− (2γ2X kX ∂kX ∂aX − ∂γ2X ∂aX ) XXY 2 X ] , YX ≡ xX sinhxX . (2.10) The energy of interstitial alloy AC is determined by EAC = (1− 7cC)EA + cCEC + 2cCEA1 + 4cCEA2 , EX = U0X + E0X + 3Nθ2 k2X [ γ2Xx 2 X coth xX + γ1X 3 ( 2 + x2X sinh2xX ) − 2γ2X x 3 X coth xX sinh2xX ] , E0X = 3NθxX coth xX . (2.11) The entropy of interstitial alloy AC is determined by SAC = (1− 7cC)SA + cCSC + 2cCSA1 + 4cCSA2 , SX = S0X + 3NkBθ k2X [γ1X 3 ( 4 +XX + Y 2 X )− 2γ2XXXY 2X] , S0X = 3NkB [XX − ln (2 sinhxX)] . (2.12) The heat capacity at constant volume of interstitial alloy AC is determined CV AC = (1− 7cC)CV A + cCCV C + 2cCCV A1 + 4cCCV A2 , CV X = 3NkB { Y 2X + 2θ k2X [( 2γ2 + γ1X 3 ) XXY 2 X + 2γ1X 3 − γ2X ( Y 4X + 2X 2 XY 2 X )]} . (2.13) The heat capacity at constant pressure of interstitial alloy AC is determined by CPAC = CV AC + 9TVACα 2 TAC χTAC . (2.14) The adiabatic compressibility of interstitial alloy AC has the form χSAC = CV AC CPAC χTAC . (2.15) 151 N. Q. Hoc, D. Q. Vinh, B. D. Tinh, T. T. C. Loan, N. L. Phuong, T. T. Hue and D. T. T. Thuy 2.2. Numerical results for interstitial alloy FeSi In numerical calculations for alloy FeSi, we use the n-m pair potential ϕ(r) = D n−m [ m (r0 r )n − n(r0 r )m] , (2.16) where potential parameters are given in Table 1 [9]. Our numerical results are described by figures in Figures 1-14. When the concentration cSi → 0, we obtain thermodynamic quantities of Fe. Our calculated results in Table 2 are in rather good agreement with the experimental data. At the same temperature, when the concentration of interstitial atoms increases, the thermodynamic quantities of alloy decrease. When the concentration of interstitial atoms remains the same and temperature increases, the thermodynamic quantities of alloy increase. Figure 1. r1FeSi(T ) at P = 0, cSi = 0 - 5% Figure 2. r1FeSi(cSi) at P = 0, T =100 - 1000 K Figure 3. χTFeSi (cSi) at P = 0, T =100 - 1000 K Figure 4. χTFeSi (T ) at P = 0, cSi = 0 - 5% 152 Thermodynamic properties of binary interstitial alloys with a BCC structure... Table 1. Parameters m, n, D, r0 of materials Fe, Si Material m n D(10−16 erg) r0(10−10 m) Fe 7 11.5 6416.448 2.4775 Si 6 12 45128.340 2.2950 Table 2. Dependence of the thermal expansion coefficient on temperature for Fe T (K) 100 200 300 500 700 αT ( 10−6K−1 ) This paper 5.69 10.90 12.74 14.82 16.12 Expt [10] 5.6 10.0 11.7 14.3 16.3 Figure 5. αTFeSi (T ) at P = 0, cSi = 0 - 5% Figure 6. αTFeSi (cSi) at P = 0, T =100 - 1000 K Figure 7. CV FeSi (T ) at P = 0, cSi = 0 - 5% Figure 8. CV FeSi (cSi) at P = 0, T = 100 - 1000 153 N. Q. Hoc, D. Q. Vinh, B. D. Tinh, T. T. C. Loan, N. L. Phuong, T. T. Hue and D. T. T. Thuy Figure 9. CPFeSi (cSi) at P = 0, T =100 - 1000 K Figure 10. CPFeSi (T ) at P = 0, cSi = 0 - 5% Figure 11. χSFeSi(T ) at P = 0, cSi = 0 - 5% Figure 12. χSFeSi(cSi) at P = 0, T =100 - 1000 K Figure 13. SFeSi(cSi) at P = 0, T =100 - 1000 K Figure 14. SFeSi(T ) at P = 0, cSi = 0 - 5% 154 Thermodynamic properties of binary interstitial alloys with a BCC structure... 3. Conclusion From the SMM, the minimum condition of cohesive energies and the method of three coordination spheres, we find 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 and the entropy of binary interstitial alloy with BCC structure with very small concentration of interstitial atoms. The obtained expressions of these quantities depend on the temperature and concentration of interstitial atoms. At zero concentration of interstitial atoms Si, thermodynamic quantities of interstitial alloy FeSi become ones of main metal Fe. At zero concentration of interstitial atoms Si, our calculated results for the thermal expansion coefficient and the heat capacity at constant pressure of interstitial alloy are in rather good agreement with experimental data. We have only considered the interstitial alloy FeSi in the interval of temperature of 100 to 1000 K where the anharmonicity of lattice vibrations has a considerable influence. Acknowledgements. This work was carried out with financial support from the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under Grant No. 103.01-2013.20. REFERENCES [1] K. E. Mironov, 1967. Interstitial alloy. Plenum Press, New York. [2] A. A. Smirnov, 1979. Theory of Interstitial Alloys. Nauka, Moscow (in Russian). [3] A. G. Morachevskii and I. V. Sladkov, 1993. Thermodynamic Calculations in Metallurgy. Metallurgiya, Moscow (in Russian). [4] V. V. Heychenko, A. A. Smirnov, 1974. Reine und angewandte Metallkunde in Einzeldarstellungen 24, 80. [5] V. A. Volkov, G. S. Masharov and S. I. Masharov, 2006. Rus. Phys. J., No. 10, 1084. [6] S. E. Andryushechkin, M. G. Karpman, 1999. Metal Science and Heat Treatment, 41, 2, 80. [7] M. Hirabayashi, S. Yamaguchi, H. Asano, K. 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