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

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

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. JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-004

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  1. 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 http://stdb.hnue.edu.vn 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
  2. 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 1h  √   √ i ni 1X u0C = ϕAC (ri ) = 2ϕAC (r1 ) + 4ϕAC r1 2 + 8ϕAC r1 5 2 2 i=1  √   √  = 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 ! √ 1 X ∂ 2 ϕAC (2) 2 (1)  √  4 (2)  √  kC = = ϕ AC (r1 ) + ϕ r1 2 + ϕ r1 5 2 ∂u2iβ r1 AC 5r12 AC i eq 16  √  (1) + √ ϕAC r1 5 , γC = 4 (γ1C + γ2C ) , 5 5r1 ! √ 1 X ∂ 4 ϕAC 1 (4) 1 (2)  √  2 (1)  √  γ1C = = ϕ (r 1 ) + ϕ r1 2 − ϕ r1 2 48 ∂u4iβ 24 AC 8r12 AC 16r13 AC i eq √ 1 (4) √ 4 5 (3) √ + ϕAC (r1 5) + ϕAC (r1 5), 150 125r1 ! 6 X ∂ 4 ϕAC 1 (3) 1 (2) 5 (1) γ2C = 2 2 = ϕAC (r1 ) − 2 ϕAC (r1 ) + 3 ϕAC (r1 ) 48 ∂uiα ∂uiβ 4r1 2r1 8r1 i eq 147
  3. 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 (3) √ 1 (2) √ 1 (1) √ 2 (4) √ + ϕAC (r1 2) − 2 ϕAC (r1 2) + 3 ϕAC (r1 2) + ϕAC (r1 5) 8r1 8r1 8r1 25 3 (3) √ 2 (2) √ 2 (1) √ + √ ϕAC (r1 5) + 2 ϕAC (r1 5) − √ 3 ϕAC (r1 5), (2.2) 25 5r1 25r1 25 5r1 (i) where ϕAC (ri ) = ∂ 2 ϕAC (ri )/∂ri2 (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  !  1X  ∂ 2 ϕAC  u0A1 = u0A + ϕAC (r1A1 ) ; kA1 = kA + 2 i ∂u2iβ eq r=rA1 (2) 5 (1) = kA + ϕAC (r1A1 ) + ϕ (r1A1 ) , γA1 = 4 (γ1A1 + γ2A1 ) , 2r1A1 AC  !  1 X  ∂ 4 ϕAC  γ1A1 = γ1A + 4 48 ∂u iβ i eq r=rA1 1 (4) 1 (2) 1 (1) = γ1A + ϕAC (r1A1 ) + 2 ϕAC (r1A1 ) − 3 ϕAC (r1A1 ), 24 8r1A1 8r1A1  !  6 X  ∂ 4 ϕAC  γ2A1 = γ2A + 48 i ∂u2iα ∂u2iβ eq r=rA1 1 (3) 3 (2) 3 (1) = γ2A + ϕAC (r1A1 ) − 2 ϕAC (r1A1 ) + 3 ϕAC (r1A1 ), (2.3) 2r1A1 4r1A1 4r1A1 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  !  1 X  ∂ 2 ϕAC  u0A2 = u0A + ϕAC (r1A2 ) , kA2 = kA + 2 i ∂u2iβ eq r=rA2 (2) 4 (1) = kA + 2ϕAC (r1A2 ) + ϕAC (r1A2 ) , γA2 = 4 (γ1A2 + γ2A2 ) , r1A2 148
  4. Thermodynamic properties of binary interstitial alloys with a BCC structure...  !  1 X 4 γ1A2 = γ1A +  ∂ ϕAC  48 ∂u4 i iβ eq r=rA1 1 (4) 1 (3) 1 (2) 1 (1) = γ1A + ϕAC (r1A2 ) + ϕAC (r1A2 ) − 2 ϕAC (r1A2 ) + 3 ϕAC (r1A2 ), 24 4r1A2 8r1A2 8r1A2  !  6 X 4 γ2A2 = γ2A +  ∂ ϕAC  48 ∂u 2 ∂u2 i iα iβ eq r=rA2 1 (4) 1 (3) 3 (2) 3 (1) = γ2A + ϕAC (r1A2 ) + ϕAC (r1A2 ) + 2 ϕA2 C (r1A2 ) − 3 ϕA2 C (r1A2 ), (2.4) 8 4r1A2 8r1A2 8r1A2 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] s 2γX (0, 0)θ 2 y0X (0, T ) = 3 (0, 0) AX (0, T ).X = A, A1 , A2 , C, 3kX 5   X γX θ i 2 ~ωX XX AX = a1X + 2 aiX , kX = mωX , xX = , a1X = 1 + , kX 2θ 2 i=2   13 47 23 2 1 3 25 121 50 2 16 3 1 4 a2X = + XX + XX + XX , a3X = − + XX + XX + XX + XX , 3 6 6 2 3 6 3 3 2 43 93 169 2 83 3 22 4 1 5 a4X = + XX + XX + XX + XX + XX , 3 2 3 3 4 2   103 749 363 2 733 3 148 4 53 5 1 6 a5X =− + XX + X + X + X + XX + XX , 3 6 3 X 3 X 3 X 6 2 561 1489 2 927 3 733 4 145 5 31 6 1 7 a6X = 65 + XX + XX + XX + XX + XX + XX + XX , 2 3 2 3 2 3 2 XX ≡ xX coth xX . (2.6) Then, we calculate the mean nearest neighbor distance in interstitial alloy AC by the expressions as follows: 149
  5. 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) + cC r1A (0, 0), ′ √ r1A (0, 0) ≈ 3r1C (0, 0), y(0, T ) = (1 − 7cC ) yA (0, T ) + cC yC (0, T ) + 2cC yA1 (0, T ) + 4cC yA2 (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, r1A ′ (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 + cC ψC + 2cC ψA1 + 4cC ψA2 − T Sc ,  2    θ 2 2γ1X 1 ψX = U0X + ψ0X + 3N 2 γ2X XX − 1 + XX kX 3 2       2θ 3 4 2 1 2 1 + 4 γ X X 1 + XX − 2γ + 2γ γ 1X 2X 1 + X X (1 + XX ) , kX 3 2X 2 1X 2   ψ0X = 3N θ xX + ln 1 − e−xX . (2.8) where SC is the configuration entropy. The isothermal compressibility of interstitial alloy AC has the form  3  3  3 3 aa0AC AC 3 aa0AC AC aAC a0AC χT AC = a2  2  =  2  = √  2  , AC ∂ ΨAC 2a2 ∂ ψAC 2 1 ∂ ΨAC √ AC 2 3VAC ∂a2AC T 3 2a3AC ∂a2AC T 3aAC 3N ∂aAC T        2  ∂ 2 ψAC ∂ 2 ψAC ∂ 2 ψA ∂ ψC = = (1 − 7cC ) + cC ∂a2AC T ∂r1A 2 (0, T ) T 2 ∂aA T ∂a2C T ! ! ∂ 2 ψA1 ∂ 2 ψA2 +2cC + 4cC , ∂a2A1 ∂a2A2 T T  2  "   # 1 ∂ ΨX 1 ∂ 2 u0X ~ωX ∂ 2 kX 1 ∂kX 2 = + − , 3N ∂a2X T 6 ∂a2X 4kX ∂a2X 2kX ∂aX  2  "   # ∂ ψX 1 ∂ 2 u0X 3~ωX ∂ 2 kX 1 ∂kX 2 = + − , (2.9) ∂a2X T 2 ∂a2X 4kX ∂a2X 2kX ∂aX where ΨAC = N ψAC , aAC = r1A (0, T )anda0AC = r1A (0, 0). 150
  6. Thermodynamic properties of binary interstitial alloys with a BCC structure... The thermal expansion coefficient of interstitial alloy AC has the form   kB daAC kB χT AC a0AC 2 aAC ∂ 2 ΨAC αT AC = =− α0AC dθ 3 aAC 3VAC ∂θ∂aAC  2 kB χT AC a0AC aAC ∂ 2 ψAC =− , 3 aAC 3vAC ∂θ∂aAC ∂ 2 ψAC ∂ 2 ψA ∂ 2 ψC ∂ 2 ψA1 ∂ 2 ψA2 = (1 − 7cC ) + cC + 2cC + 4cC , ∂θ∂aAC ∂θ∂aA ∂θ∂aC ∂θ∂aA1 ∂θ∂aA2  ∂ 2 ψX 3 ∂kX 2 6θ 2 γ1X ∂kX  = YX + 2 2 + XX YX2 ∂θ∂aX 2kX ∂aX kX 3kX ∂aX   1 ∂γ1X 2  2γ2X ∂kX ∂γ2X  − 4 + XX + YX − − XX YX2 , 6 ∂aX kX ∂aX ∂aX xX YX ≡ . (2.10) sinh xX The energy of interstitial alloy AC is determined by EAC = (1 − 7cC ) EA + cC EC + 2cC EA1 + 4cC EA2 ,     3N θ 2 γ1X x2X x3X coth xX EX = U0X + E0X + 2 γ2X x2X coth xX + 2+ − 2γ2X , kX 3 sinh2 xX sinh2 xX E0X = 3N θxX coth xX . (2.11) The entropy of interstitial alloy AC is determined by SAC = (1 − 7cC ) SA + cC SC + 2cC SA1 + 4cC SA2 , 3N kB θ h γ1X 2  2 i SX = S0X + 2 4 + X X + Y X − 2γ 2X XX Y X , kX 3 S0X = 3N kB [XX − ln (2 sinh xX )] . (2.12) The heat capacity at constant volume of interstitial alloy AC is determined CV AC = (1 − 7cC ) CV A + cC CV C + 2cC CV A1 + 4cC CV A2 ,  2θ h γ1X  CV X = 3N kB YX2 + 2 2γ2 + XX YX2 kX 3 2γ1X  + − γ2X YX4 + 2XX 2 2 YX . (2.13) 3 The heat capacity at constant pressure of interstitial alloy AC is determined by 9T VAC α2T AC CP AC = CV AC + . (2.14) χT AC The adiabatic compressibility of interstitial alloy AC has the form CV AC χSAC = χT AC . (2.15) CP AC 151
  7. 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 D h  r0 n  r m i 0 ϕ(r) = m −n , (2.16) n−m r r 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. χT FeSi (cSi ) at P = 0, T =100 - 1000 K Figure 4. χT FeSi (T ) at P = 0, cSi = 0 - 5% 152
  8. 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 10−6 K−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. αT FeSi (T ) at P = 0, cSi = 0 - 5% Figure 6. αT FeSi (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
  9. 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. CP FeSi (cSi ) at P = 0, Figure 10. CP FeSi (T ) at P = 0, T =100 - 1000 K cSi = 0 - 5% Figure 11. χSFeSi (T ) at P = 0, Figure 12. χSFeSi (cSi ) at P = 0, cSi = 0 - 5% 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
  10. 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. Hiraga, 1974. Reine und angewandte Metallkunde in Einzeldarstellungen, 24, 266. [8] N. Tang, V. V. Hung, 1988, 1990. Phys. Stat. Sol. (b) 149, 511; 161, 165; 162, 371; 162, 379. [9] M. N. Magomendov, 1987. Phys. Chem. J., 61, 1003 (in Russian). [10] American Institute of Physics Handbook, 1961. 3rd Ed., McGraw-Hill book company, New York, Toronto, London. 155
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