Thermodynamic properties of a ternary interstitial alloy with BCC structure: Dependence on temperature, concentration of substitution atoms and concentration of interstitial atoms

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0033 Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 65-74 This paper is available online at THERMODYNAMIC PROPERTIES OF A TERNARY INTERSTITIAL ALLOY WITH BCC STRUCTURE: DEPENDENCE ON TEMPERATURE, CONCENTRATION OF SUBSTITUTION ATOMS AND CONCENTRATION OF INTERSTITIAL ATOMS Nguyen Quang Hoc1, Dinh Quang Vinh1, Nguyen Thi Hang, Nguyen Thi Nguyet, Luong Xuan Phuong, Nguyen Nhu Hoa, Nguyen Thi Phuc and Trinh Thi Hien2 1Faculty of Physics, Hanoi National University of Education 2Faculty of Natural Sciences, Hong Duc University, Thanh Hoa City Abstract. The analytic expressions of the thermodynamic quantities such as mean nearest neighbor distance, free energy, free isothermal and adiabatic compressibilities, free isothermal and adiabatic elastic moduli, thermal expansion coefficient, the heat capacities at constant volume and at constant pressure and entropy of a ternary interstitial alloy with a body-centered cubic (BCC) structure with a very small concentration of interstitial atoms are all derived using the statistical moment method. The obtained expressions of these quantities depend on temperature, concentration of substitution atoms and concentration of interstitial atoms. In this study, the theoretical results are applied to the interstitial alloy FeCrSi. The thermodynamic properties of main metal Fe, substitution alloy FeCr and interstitial alloy FeSi are special cases for the thermodynamic properties of interstitial alloy FeCrSi. The calculated results of the thermal expansion coefficient in the temperature interval 100 to 1000 K and heat capacity at constant pressure in the temperature interval from 100 to 700 K for main metal Fe are in good agreement with the experimental data. Keywords: Ternary interstitial alloy, statistical moment method. 1. Introduction Thermodynamic and elastic properties of interstitial alloys are of special interest to many theoretical and experimental researchers [1-5]. In this paper, we build a thermodynamic theory for a ternary interstitial alloy with a body-centered cubic (BCC) structure using the statistical moment method (SMM) [6] and we apply the obtained theoretical results to the alloy FeCrSi. Received March 4, 2016. Accepted July 24, 2016. Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn 65 N. Q. Hoc, D. Q. Vinh, N. T. Hang, N. T. Nguyet, L. X. Phuong, N. N. Hoa, N. T. Phuc and T. T. Hien 2. Content 2.1. Thermodynamic quantities of interstitial alloy ABC with BCC structure The interstitial alloy ABC (the substitution alloy AB with the interstitial atom C) with BCC structure has N atoms (NA main atoms A, NB substitution atoms B and NC interstitial atoms C). The concentration of alloy component are cA = NA N , cB = NB N and cC = NC N . The alloy ABC satisfies the condition cC << cB << cA. In order to have the interstitial alloy ABC with BCC structure we take the interstitial alloy AC with BCC structure (interstitial atom C stays in face centers of cubic unit cell) and then the atom B substitutes the atom A in body center of cubic unit cell. The free energy of interstitial alloy ABC has the form ψABC = ψAC + cB (ψB − ψA) + TSACc − TSABCc , ψAC = (1− 7cC)ψA + cCψC + 2cCψA1 + 4cCψA2 − TSACc , ψX = U0X + ψ0X + 3N { θ2 k2X [ γ2Xx 2 Xcoth 2xX − 2γ1X 3 ( 1 + 1 2 xX coth xX )] + 2θ3 k4X [ 4 3 γ22XxX coth xX ( 1 + 1 2 xX coth xX ) −2 (γ21X + 2γ1Xγ2X) ( 1 + 1 2 xX coth xX ) (1 + xX coth xX) ]} , ψ0X = 3Nθ [ xX + ln ( 1− e−2xX)] ,X = A,C,A1, A2, (2.1) where ψAC is the free energy of interstitial alloy AC, ψA is the free energy of atom A in pure metal A, ψC is the free energy of atom C in alloy ABC, ψA1 is the free energy of atom A1 (atom A in body center of cubic unit cell), ψA2 is the free energy of atom A2 (atom A in peaks of cubic unit cell), SACc is the configuration entropy of interstitial alloy AC, S ABC c is the configuration entropy of interstitial alloy ABC, the quantities U0X , kX , γ1X , γ2X , xX are defined as corresponding quantities in [6], θ = kBT and kB is the Boltzmann constant. The mean nearest neighbor distance between two atoms in the interstitial alloy ABC with BCC 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 = 4 √ 3aAC ( aAC a0AC )3 ( ∂2ψAC ∂a2 AC ) T , ( ∂2ψAC ∂a2AC ) T = [ ∂2ψAC ∂r1A2(0, T ) ] T ≈ (1− 7cC) ( ∂2ψA ∂a2A ) T + cC ( ∂2ψC ∂a2C ) T 66 Thermodynamic properties of a ternary interstitial alloy with BCC structure... +2cC ( ∂2ψA1 ∂a2A1 ) T + 4cC ( ∂2ψA2 ∂a2A2 ) T , ( ∂2ψX ∂a2X ) T = ( ∂2ψA ∂r21X(0, T ) ) T , 1 3N ( ∂2ΨX ∂a2X ) T = 1 6 ∂2u0X ∂a2X + ~ωX 4kX [ ∂2kX ∂a2X − 1 2kX ( ∂kX ∂aX )2] . (2.2) The mean nearest neighbor distance between two atoms in the interstitial alloy ABC with BCC structure at 0 K is given by a0ABC = cACa0AC B0TAC B0T + cBa0B B0TB B0T . (2.3) The quantities in (2.3) is the same as in (2.2) but are determined at 0 K. The isothermal compressibility and the isothermal elastic modulus of interstitial alloy ABC has the form χTABC = 3 ( aABC a0ABC )3 a2 ABC 3VABC ( ∂2ΨABC ∂a2 ABC ) T = 3 ( aABC a0ABC )3 a2 ABC 3vABC ( ∂2ψABC ∂a2 ABC ) T = ( aABC a0ABC )3 √ 3 4aABC 1 3N ( ∂2ΨABC ∂a2 ABC ) T , BTABC = 1 χTABC , ( ∂2ψABC ∂a2ABC ) T ≈ ( ∂2ψAC ∂a2A ) T + cB [( ∂2ψB ∂a2B ) T − ( ∂2ψA ∂a2A ) T ] . (2.4) The thermal expansion coefficient of interstitial alloy ABC has the form αTABC = kB α0ABC daABC dθ = −kBχTABC 3 ( a0ABC aABC )2 aABC 3VABC ∂2ΨABC ∂θ∂aABC = −kBχTABC 3 ( a0ABC aABC )2 aABC 3vABC ∂2ψABC ∂θ∂aABC , ∂2ψABC ∂θ∂aABC ≈ ∂ 2ψAC ∂θ∂aAC + cB ( ∂2ψB ∂θ∂aB − ∂ 2ψA ∂θ∂aA ) , ∂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 x2X sinh2xX + 6θ2 k2X [ γ1X 3kX ∂kX ∂aX ( 2 + x3X cothxX sinh2xX ) −1 6 ∂γ1X ∂aX ( 4 + xX coth xX + x2X sinh2xX ) − ( 2γ2X kX ∂kX ∂aX − ∂γ2X ∂aX ) x3X cothxX sinh2xX ] . (2.5) The energy of interstitial alloy ABC is determined by EABC ≈ EAC + cB (EB − EA) , EAC = (1− 7cC)EA + cCEC + 2cCEA1 + 4cCEA2 , 67 N. Q. Hoc, D. Q. Vinh, N. T. Hang, N. T. Nguyet, L. X. Phuong, N. N. Hoa, N. T. Phuc and T. T. Hien 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 . (2.6) The entropy of interstitial alloy ABC is determined by SABC ≈ SAC + cB (SB − SA) , SAC = (1− 7cC)SA + cCSC + 2cCSA1 + 4cCSA2 , SX = S0X + 3NkBθ k2X [ γ1X 3 ( 4 + xX cothxX + x2X sinh2xX ) − 2γ2X x 3 X coth xX sinh2xX ] , S0X = 3NkB [xX coth xX − ln (2 sinhxX)] . (2.7) The heat capacity at constant volume of interstitial alloy ABC is determined by CV ABC ≈ CV AC+cB (CV B −CV A) , CV AC = (1− 7cC)CV A+cCCV C+2cCCV A1+4cCCV A2 , CV X = 3NkB { x2X sinh2xX + 2θ k2X [( 2γ2 + γ1X 3 ) x3X coth xX sinh2xX + 2γ1X 3 − γ2X ( x4X sinh4xX + 2x4Xcoth 2xX sinh2xX )]} , (2.8) The heat capacity at constant pressure of interstitial alloy ABC is determined by CPABC = CV ABC + 9TVABCα 2 TABC χTABC . (2.9) The adiabatic compressibility and elastic modulus of interstitial alloy ABC has the form χSABC = CV ABC CPABC χTABC , BSABC = 1 χSABC . (2.10) 2.2. Numerical results for interstitial alloy FeCrSi In numerical calculations for alloy FeCrSi, we use the n-m pair potential ϕ(r) = D n−m [ m (r0 r )n − n (r0 r )m] , (2.11) where potential parameters are given in Table 1 [7]. Table 1. The parameters m,n,D, r0 of materials Fe, Cr and Si Material m n D ( 10−16erg ) r0 ( 10−10m ) Fe 7 11.5 6416.448 2.4775 Cr 6 15.5 6612.960 2.495 Si 6 12 45128.34 2.295 68 Thermodynamic properties of a ternary interstitial alloy with BCC structure... Our numerical results are described by figures from Figure 1 to Figure 24. When the concentration cSi → 0 and the concentration cCr → 0, we obtain thermodynamic quantities of Fe. Our calculated results in Table 2 and Table 3 are in good agreement with experiments. Table 2. Dependence of thermal expansion coefficient αT ( 10−6K−1 ) on temperature for Fe from SMM and EXPT [8] T(K) 100 200 300 500 700 1000 αT -SMM 5.69 10.9 12.74 14.62 16.12 18.61 αT -EXPT 5.60 10.0 11.7 14.30 16.3 19.2 Table 3. Dependence of heat capacity at constant pressure CP (J/mol.K) on temperature for Fe from SMM and EXPT[8] T(K) 100 200 300 500 700 CP -SMM 11.38 21.79 25.53 29.43 32.65 CP –EXPT 12.067 21.503 25.131 29.639 34.618 According to figures from Figure 1 to Figure 24, for alloy FeCrSi in the same temperature and concentration of substitution atoms Cr when the concentration of interstitial atoms Si increases, the mean nearest neighbor distance increases (for example at 10 K and cCr = 5% when cSi increases from 0 to 5%, a increases from 2.42 to 2.492 o A ) and other quantities decrease (for example at 10 K and cCr = 5%when cSiincreases from 0 to 5%, χT decreases from 3.046.10−12 to 1.821.10−12 Pa−1, αT decreases from 4.537.10−6 to 1.426.10−6 K−1, CV decreases from 5.605 to 5.404 J/mol.K, CP decreases from 5.606 to 5.404 J/mol.K and χS decreases from 3.046.10−12 to 1,821.10−12 Pa−1). For alloy FeCrSi in the same concentration of substitution atoms Cr and concentration of interstitial atoms Si when the temperature increases, all quantities increase (for example at cCr = 5% and cSi = 5% when T increases from 50 to 1000 K, a increases from 2.494 to 2.528 o A , χT increases from 1.836. 10−12 to 2.252. 10−12 Pa−1, αT increases from 1.426.10−6 at 10 K to 14.522. 10−6 K−1, CV increases from 5.404 to 32.417 J/mol.K, CP increases from 5.404 to 31.11 J/mol.K and χS increases from 1.837.10−12 to 2.251.10−12 Pa−1). For alloy FeCrSi in the same temperature and concentration of interstitial atoms when the concentration of substitution atoms increases, some quantities such as a, χT increase (for example at 1000 K and cSi = 1% when cCr increases from 0 to 10%, a increases from 2.471 to 2.473 o A , χT increases from 3.771. 10−12 to 3,775. 10−12 Pa−1) and some quantities such as αT , CV , CP and χS decrease (for example at 1000 K and cSi = 1% when cCr increases from 0 to 10%, αT decreases from 13.81.10−6 to 13.805.10−6 K−1, CV decreases from 33.549 to 33.545 J/mol.K, CP decreases from 36.477 to 36.473 J/mol.K and χS decreases from 3.792.10−12 to 3.774. 10−12 Pa−1). At zero concentration of substitution atoms and zero concentration of interstitial atoms, the thermodynamic quantities of interstitial alloy FeCrSi becomes the thermodynamic quantities of metal Fe in [6]. The change of thermodynamic quantities with temperature for interstitial alloy FeCrSi is similar to that for interstitial alloy FeSi [9]. The change of thermodynamic quantities with temperature and concentration of substitution atoms for interstitial alloy FeCrSi is similar to that for substitution alloy FeCr [10]. 69 N. Q. Hoc, D. Q. Vinh, N. T. Hang, N. T. Nguyet, L. X. Phuong, N. N. Hoa, N. T. Phuc and T. T. Hien 70 Thermodynamic properties of a ternary interstitial alloy with BCC structure... 71 N. Q. Hoc, D. Q. Vinh, N. T. Hang, N. T. Nguyet, L. X. Phuong, N. N. Hoa, N. T. Phuc and T. T. Hien 72 Thermodynamic properties of a ternary interstitial alloy with BCC structure... 73 N. Q. Hoc, D. Q. Vinh, N. T. Hang, N. T. Nguyet, L. X. Phuong, N. N. Hoa, N. T. Phuc and T. T. Hien 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 moduli, the thermal expansion coefficient, the heat capacities at constant volume and constant pressure and the entropy of the ternary interstitial alloy with BCC structure with very small concentration of interstitial atoms (from 0 to 5%). The obtained expressions of these quantities depend on temperature, concentration of substitution atoms and concentration of interstitial atoms. At zero concentration of substitution atoms, the thermodynamic quantities of the interstitial alloy ABC become that of the interstitial alloy AC. At zero concentration of interstitial atoms, the thermodynamic quantities of the interstitial alloy ABC become that of the substitution alloy AB. 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