Thermodynamic properties of some rare earth alloys

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0039 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 104-114 This paper is available online at THERMODYNAMIC PROPERTIES OF SOME RARE EARTH ALLOYS Dang Thanh Hai1, Vu Van Hung1 and Pham Thi Minh Hanh2 1Vietnam Education Publishing House 2Hanoi Pedagogical University No. 2 Abstract. Thermodynamic properties of some rare-earth alloys have been studied using the statistical moment method (SMM) and analytic expressions of the Helmholtz free energy and thermodynamic properties of some rare-earth alloys have been obtained. Present statistical moment method results of nearest neighbor distance, linear thermal expansion coefficient and specific heats, and pressure dependence of molecular volume for AlCe3, Th57Ce43 and AgCe3 are compared with experimental and other calculation results. The influence of Ce concentration on thermodynamic properties of Al1−xCex, Ag1−xCex and Th1−xCex alloys have been studied. Keywords: Thermodynamic properties, rare-earth alloys, statistical moment method. 1. Introduction Study of rare earths and rare earth alloys have been attractive in recent years because of their applications in many fields of technology. The theories about a transition state of metal such as a theory of pseudo-potential [1-3] were used to the study rare earths of the actinide series La, Yb, Ce and Th [4-6]. A. Rosengren, I. Ebbsjo and B. Johansson [7] studied crystalline lattice dynamics of thori by using pseudo-potential of Krasko and Gurskii [8]. N. Singh and S. P. Singh [9] calculated phonon scattering of La, Yb, Ce and Th. Using the calculated phonon scattering curve, phonon density of states of thori were presented by T. C. Panelya, P. R. Vyas, C. V. Pandya and V. B. Gohel [10]. In [11], J.K. Baria and A.R. Jani also calculated the phonon scattering curve, phonon density of states, Debye temperature, the Gru¨neisen parameters and elastic dynamic constants of La, Yb, Ce and Th. Tanju Gurel and Resul Eryigit have performed an ab initio study of structural, elastic, lattice-dynamic and thermodynamic properties of the rare-earth hexaborides, LaB6 and CeB6 [12]. The calculations have been carried out within the density functional theory and linear response formalism using pseudopotentials and a plane-wave basis. Thermodynamic properties of LaB6 and CeB6 obtained from quasiharmonic approximation are in a good agreement with the available experimental data. They also present complete phonon-dispersion curves, phonon density of states and mode-Gru¨neisen parameters, and they compare well with experimental measurements. A sizable difference between the vibrational contribution to entropy of LaB6 and CeB6 is found. The thermal electronic contribution to entropy and specific heat is found to be important for CeB6. Received March 10, 2015. Accepted August 17, 2015. Contact Dang Thanh Hai, e-mail address: dthai@nxbgd.vn 104 Thermodynamic properties of some rare earth alloys A comprehensive first principles study of structural, elastic, electronic and phonon properties of rare-earth CeBi and LaBi is reported within the density functional theory scheme [13]. Ground state properties such as lattice constant, elastic constants, bulk modulus and finally the phase transition and lattice dynamic properties of rare-earths CeBi and LaBi of rock salt and CsCl structures are determined. The electron band structures for the two phases of both rare-earth crystals are presented. The influence of rare earth additions on the microstructure and some properties of gold and gold alloys have been studied [14]. Rare earth additions can refine the grain size of gold alloys but they show a tendency to segregation, both dendritic segregation in cast alloys and grain boundary segregation in annealed alloys. For gold alloys, rare earth additions are generally used in trace amounts or dilute concentrations in order to avoid a large segregation of rare earth and the potential embrittlement of gold alloys. The experimental results demonstrate that rare earth additions can inhibit recovery softening, increase the recrystallization temperature and enhance the strength of gold alloys. The strengthening mechanisms of rare earth additions in gold alloys are discussed. Some gold alloys with rare earth additions have been developed and their applications are illustrated briefly. M.G. Shelyapina et al. [15] used the ab initiomethod combined with X-ray analysis to study the influence of magnetic properties and V concentration in NbFV alloy on their thermodynamic properties. The phase diagrams of Ce, Th, and Pu metals have been studied by means of the density functional theory [16]. In addition to these metals, the phase stability of Ce-Th and Pu-Am alloys has been also investigated from first-principles calculations. Equations of state for Ce, Th, and the Ce-Th alloys have been calculated up to 1 Mbar pressure with good comparison to experimental data. This work focuses on studying the thermodynamic properties of AgCe3, AlCe3 and Th57Ce43 alloys using the statistical moment method. 2. Content 2.1. Free energy of substitutional alloy A-B with FCC and BCC structures Considering the free energy of the substitutional alloy A-Bmodel with fcc and bcc structure in which there are two types of lattice point a and b corresponding to the atomic A and B types. In the N atomics collection, NA atoms belong to the atomic A type and NB atoms belong to the atomic B type. The concentration of A and B is determined as follows: cA = NA N ; cB = NB N , (2.1) where, cA and cB are the concentration of A and B respectively. According to the definition of free energy in statistical physics ψAB = −kBT lnZAB , (2.2) where ZAB is overall statistics in A-B alloy; ZAB is described in the form ZAB = ∑ n exp ( − En kBT ) , (2.3) 105 Dang Thanh Hai, Vu Van Hung and Pham Thi Minh Hanh where En is energy of system corresponding to the principle quantum number n. On the other hand, En is expressed through configurative energy Et and vibration energy Em by the expression En = Et + Em, (2.4) here, vibration energy Em is assumed approximately, not to depend on configuration. Substituting Eq. (2.4) to Eq. (2.3) and changing the sum depending on n to the sum depending on i andm, we have ZAB = ∑ i,m exp ( −Ei + Em kBT ) = ∑ m exp ( − Em kBT )∑ i exp ( − Ei kBT ) . (2.5) In approximate calculation without correlation effect, the configuration energy Et is the same for all the configurations and equal to E. Replace Et by E in equation (2.5) and move exp[−E/(kBT )] to the sum of i, and the results are ZAB = ∑ m exp ( −E + Em kBT ) W = z1W, (2.6) z1 = ∑ m exp ( −E ′ m kBT ) , (2.7) here E′m = E + Em is the energy of alloy which was determined by state quantum number m of the system and the average energy configuration Eβα of the effective system (α, β) in the form E = ∑ α, β υαP β αE β α . (2.8) If Em is assumed not to depend on configuration or effective systems, replacing Eq. (2.8) with Eq. (2.7) we obtain z1 = ∑ m exp − ∑ α,β υαP β α (E β α)′m kBT  . (2.9) From (2.2), (2.6) and (2.9) we have ψAB = ∑ m υαP β αψ β α − kBT lnW. (2.10) Notice that the second term in (2.10) is calculated by kBT lnW = TSc,where Sc is entropy of alloy configuration Sc = −kBN ∑ α,β υαP β α lnP β α . (2.11) Here, Hemlholtz free energy ψAB of an alloy is in the form ψAB = ∑ α,β υβP β αψ β α − TSc, (2.12) 106 Thermodynamic properties of some rare earth alloys where P βα is the probability for atomic α (α = A,B) located on a lattice point (β = a, b) satisfying the relations cA = υaP a A + υbP b A; cB = υaP a B + υbP b B ;P a A + P b B = P b A + P a B = 1;N = NA +NB, (2.13) where υβ is concentration of lattice point β. In Eq. (2.12), ψ β α is the free energy of the effective system (α, β) in the form ψβα = 3N { uβ0α 6 + θ [ xβα + ln(1− e−2x β α) ]} , (2.14) uβ0α = ∑ i φβαi (|ai|) . (2.15) xβα = ~ωβα 2θ = ~ 2θ √ kβα/m∗; m∗ = cAmA + cBmB (2.16) where uβ0α is potential interactions between atomics in the effective system (α, β);mA,mB are the mass of atomic A or B in an A-B alloy and the second differential coefficient of (kβα) is in the form kβα = 1 2 ∑ i ( ∂2φβαi ∂u2αl ) eq ≡ m∗ ( ωβα )2 , (l = x, y, z). (2.17) In case cB << cA, we can ignore the terms that are smaller than u0α and kα. This means that uβ0A ≈ u0A, kβA ≈ kA. Therefore, the free energy of the substitutional alloys A-B which disorders when cB << cA has distributions as a function of atomic concentration as follows: ψAB = cAψA + cBψB − TSc. (2.18) Expression (2.18) shows the relationship between free energy ψAB of the disordered substitutional alloy A-B (cB << cA) and the free energy of component metals ψA and ψB as follows: ψα = 3N {u0α 6 + θ [ xα + ln(1− e−2xα) ]} . (2.19) 2.2. Lattice parameters of alloys To calculate the lattice parameters of alloys with fcc and bcc structures, we use the thermodynamic equilibrium condition of system for calculating the lattice parameters of substitutional alloy A-B. As we know, free energy ψAB Eq. (2.12) is a function of the nearest neighbor distance in the substitutional alloy A-B so that is approximately equal to aaA, a b A, a a B , a b B . Therefore, we can expand ψAB by (a − aβα) up to the second order term approximation as follows: ψβα (a) = ψ β α ( aβα ) + 1 2 ( ∂2ψβα ∂a2 ) T ( a− aβα )2 . (2.20) 107 Dang Thanh Hai, Vu Van Hung and Pham Thi Minh Hanh Using the definition of bulk modulus, we have BABT = 1 χABT = −V0 ( ∂2ψAB ∂V 2AB ) . (2.21) According to the expressions (2.12), (2.20) and thermodynamic equilibrium condition(( ∂ψ ∂a ) T, P,N = 0 ) , the nearest neighbor distance has the form aAB ≈ ( υaP a Aa a A + υbP b Aa b A ) BAT + ( υaP a Ba a B + υbP b Ba b B ) BBT cABAT + cBB B T , (2.22) B¯T = cAB A T + cBB B T , (2.23) where B¯T is the average value of the bulk modulus. We have aAB ≈ ( υaP a Aa a A + υbP b Aa b A ) BT,A B¯T + ( υaP a Ba a B + υbP b Ba b B ) BT,B B¯T . (2.24) For the completely disordered alloy, using the approximated condition aaA ≈ abA = aA, aaB ≈ abB = aB , we are able to easily obtain the equation to calculate a0AB and aAB as follows: a0AB = cAa0A B0T,A B¯0T + cBa0B B0T,B B¯0T at 0K (2.25) aAB = cAaA BT,A B¯T + cBaB BT,B B¯T at T (K) (2.26) where a0α and aα(α = A,B) are the nearest neighbor distance of α metal at 0K and T (K); B0T, α and BT, α are bulk modulus of α metal at 0K and T (K); B¯0T is the average value of the bulk modulus of the substitutional alloys A-B at 0K which is described similarly to the Eq. (2.23). 2.3. Thermodynamic quantity of substitutional alloy A-B According to [17], isothermal compressibility is obtained in the form χABT = 3 ( aAB a0AB )3 2P + a2 3VAB ( ∂2ψAB ∂a2AB ) T , (2.27) where P is pressure, VAB is the volume of crystal including N particles: VAB = N.vAB and vAB is the atomic volume of alloy at T temperature. Here vAB is determined by equations vAB = √ 2 2 a3AB applied for FCC structure (2.28) vAB = 4 3 √ 3 a3AB applied for BCC structure (2.29) 108 Thermodynamic properties of some rare earth alloys and ( ∂2ψAB ∂a2AB ) T ≈ cA∂ 2ψA ∂a2A + cB ∂2ψB ∂a2B . (2.30) If we know the isothermal compressibility, we can determine the bulk moduli as follows: BABT = 1 χABT . (2.31) The linear thermal expansion coefficient is defined as follows: αABT = kB a0AB · daAB dθ . (2.32) Using the mathematical transformation, we determine the linear thermal expansion coefficient by equation αABT = a0A a0AB cAα A T BTA B¯T + a0B a0AB cBα B T BTB B¯T . (2.33) Appling the results of thermodynamics, we can write it in the form αABT = − kBχ AB T 3 · ( a0AB aAB )2 · aAB 3VAB · ( ∂2ψAB ∂θ∂aAB ) T , (2.34) where ( ∂2ψAB ∂θ∂aAB ) T ≈ cA ∂ 2ψA ∂θ∂aA + cB ∂2ψB ∂θ∂aB . (2.35) This result allows us to calculate αT if we know χT . Similar to the case of metals, the energy of an alloy is written in the form EAB = ψAB − θ · ∂ψAB ∂θ = cA ( ψA − θ∂ψA ∂T ) + cB ( ψB − θ∂ψB ∂T ) . (2.36) The specific heat at constant volume has the form CABV = ∂EAB ∂T = −cAT ∂ 2ψA ∂T 2 − cBT ∂ 2ψB ∂T 2 . (2.37) According to the definition of specific heat at constant volume for metal, we have a simpler equation to determine CABV = cAC A V + cBC B V . (2.38) The specific heat at constant pressure is determined from the well established thermodynamic relations CABP = C AB V + 9TVAB(α AB T ) 2 χABT . (2.39) 109 Dang Thanh Hai, Vu Van Hung and Pham Thi Minh Hanh 2.4. Results and discussions Using the moment method in statistical dynamics, we calculated the thermodynamic properties of rare earth alloys AlCe3, Th57Ce43 and AgCe3. For simplicity, we take the effective pair interaction energy in rare earth metals as the power law, similar to the Lennard - Jones ϕ(r) = D (n−m) [ m (r0 r )n − n(r0 r )m] , (2.40) where D, r0 are determined to fit the experimental data [18]. Table 1. Temperature dependence of the thermodynamic quantities of AgCe3 alloy at P = 10GPa T (K) 300 400 500 600 700 800 900 1000 a(A˚) 3.181 3.187 3.193 3.199 3.205 3.212 3.218 3.225 α(10−6K−1) 9.16 9.35 9.48 9.60 9.71 9.81 9.92 10.02 BT (10 10Pa) 4.422 4.314 4.208 4.104 4.000 3.897 3.795 3.693 χT (10 −12/Pa) 22.61 23.17 23.75 24.36 24.99 25.65 26.34 27.07 CV (J/mol.K) 25.30 25.85 26.28 26.68 27.06 27.45 27.84 28.24 CP (J/mol.K) 25.44 25.99 26.42 26.82 27.21 27.59 27.98 28.38 Table 2. Temperature dependence of the thermodynamic quantities of Al2Ce3 alloy at P = 10GPa T (K) 300 400 500 600 700 800 900 1000 a(A˚) 3.064 3.071 3.078 3.085 3.092 3.099 3.107 3.115 α(10−6K−1) 9.59 10.07 10.37 10.59 10.77 10.94 11.10 11.25 BT (10 10Pa) 4.639 4.492 4.354 4.221 4.091 3.962 3.834 3.708 χT (10 −12/Pa) 21.55 22.26 22.96 23.68 24.44 25.23 26.07 26.96 CV (J/mol.K) 24.45 25.49 26.20 26.78 27.32 27.84 28.36 28.89 CP (J/mol.K) 24.59 25.64 26.35 26.94 27.48 28.00 28.52 29.05 Table 3. Temperature dependence of the thermodynamic quantities of Th57Ce43 alloy at P = 10GPa T (K) 300 400 500 600 700 800 900 1000 a(A˚) 3.391 3.394 3.397 3.400 3.403 3.406 3.409 3.412 α(10−6K−1) 6.978 7.057 7.114 7.163 7.207 7.250 7.292 7.333 BT (10 10Pa) 3.970 3.918 3.866 3.814 3.763 3.712 3.661 3.611 χT (10 −12/Pa) 25.18 25.52 25.86 26.21 26.57 26.93 27.30 27.69 CV (J/mol.K) 25.15 25.50 25.77 26.01 26.24 26.47 26.69 26.92 CP (J/mol.K) 25.24 25.59 25.86 26.10 26.33 26.56 26.78 27.01 110 Thermodynamic properties of some rare earth alloys Tables 1, 2 and 3 present the temperature dependence of the thermodynamic quantities of AgCe3, Al2Ce3 and Th57Ce43 alloys at pressure 10GPa. We can see that the nearest neighbor distance, linear thermal expansion coefficient, specific heat at constant volume and specific heat at constant pressure increases when the temperature increases. The nearest neighbor distance increased 10% and the linear thermal expansion coefficient increased 10.9% for AgCe3, 11.7% for Th57Ce43 and 10.5% for Al2Ce3. The bulk modulusBT decreases when the temperature increases, reduction of AgCe3 is 11.97%, Th57Ce43 is 12.5% and Al2Ce3 is 11%. The results obtained in the above table is in accordance with the laws of experiment. Table 4. Pressure dependence of thermodynamic quantities of Al2Ce3 alloy at T = 300K P (GPa) 10 20 30 40 50 60 70 80 a(A˚) 3.064 3.015 2.976 2.943 2.915 2.875 2.855 2.836 α(10−6K−1) 9.594 7.118 5.711 4.788 4.131 3.636 3.248 2.934 BT (10 10Pa) 4.639 6.430 8.176 9.895 11.59 13.27 14.95 16.61 χT (10 −12/Pa) 21.55 15.55 12.22 10.10 8.625 7.531 6.688 6.018 CV (J/mol.K) 24.45 23.86 23.38 22.97 22.61 22.27 21.96 21.67 CP (J/mol.K) 24.59 23.96 23.47 23.04 22.66 22.32 22.00 21.70 Table 5. Pressure dependence of thermodynamic quantities of AgCe3 alloy at T = 300K P (GPa) 10 20 30 40 50 60 70 80 a(A˚) 3.181 3.124 3.079 3.042 3.010 2.983 2.959 2.938 α(10−6K−1) 9.318 6.960 5.642 4.785 4.176 3.719 3.360 3.071 BT (10 10Pa) 4.213 5.899 7.533 9.135 10.71 12.27 13.82 15.35 χT (10 −12/Pa) 23.73 16.95 13.27 10.94 9.333 8.146 7.235 6.511 CV (J/mol.K) 25.30 24.97 24.72 24.51 24.33 24.17 24.02 23.87 CP (J/mol.K) 25.44 25.07 24.80 24.58 24.39 24.22 24.06 23.92 Table 6. Pressure dependence of thermodynamic quantities of Th57Ce43 alloy at T = 300K P (GPa) 10 20 30 40 50 60 70 80 a(A˚) 3.391 3.314 3.256 3.209 3.171 3.137 3.108 3.082 α(10−6K−1) 6.978 5.334 4.393 3.772 3.326 2.988 2.722 2.505 BT (10 10Pa) 3.970 5.486 6.944 8.367 9.762 11.13 12.49 13.84 χT (10 −12/Pa) 25.18 18.22 14.39 11.95 10.24 8.978 8.001 7.223 CV (J/mol.K) 25.15 24.95 24.79 24.66 24.55 24.45 24.36 24.27 CP (J/mol.K) 25.24 25.01 24.84 24.71 24.59 24.48 24.39 24.30 Tables 4, 5 and 6 present the pressure dependence of the thermodynamic quantities of AgCe3, Al2Ce3 and Th57Ce43 alloys at T = 300K . The thermodynamic quantities such as the linear thermal expansion coeffic ient, specific heat at constant volume and specific heat at constant pressure decreased when the pressure increased. When pressure P increases from 10GPa 111 Dang Thanh Hai, Vu Van Hung and Pham Thi Minh Hanh to 80GPa, the isothermal compression ratio of Al2Ce3 reduction is 72.1%, AgCe3 reduction is 72.6% and Th57Ce43 decreased 71.32%. The linear thermal expansion coefficient of Al2Ce3 is a 43% reduction, and there was a 72% reduction of AgCe3. However, the bulk modulusBT increases when the pressure increases. When the pressure increases from 10GPa to 80GPa, the bulk modulus BT of Al2Ce3 increases 3.5 times. Table 7. Ce concentration dependence of thermodynamic quantities of Ag1−xCex at P = 0 and T = 500 K cCe 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 a(A˚) 2.892 2.931 2.975 3.025 3.083 3.149 3.228 3.321 α(10−6K−1) 17.94 17.72 17.48 17.20 16.90 16.56 16.16 15.71 BT (10 10Pa) 3.618 3.401 3.186 2.973 2.762 2.554 2.348 2.147 χT (10 −12/Pa) 27.63 29.39 31.38 33.63 36.20 39.15 42.57 46.56 CV (J/mol.K) 28.05 27.90 27.76 27.61 27.47 27.32 27.17 27.03 CP (J/mol.K) 28.59 28.42 28.25 28.08 27.91 27.74 27.57 27.40 Table 8. Ce concentration dependence of thermodynamic quantities of Al1−xCex at P = 0 and T = 500K cCe 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 a(A˚) 2.875 2.916 2.961 3.013 3.073 3.141 3.222 3.317 α(10−6K−1) 18.39 18.14 17.87 17.56 17.21 16.82 16.37 15.86 BT (10 10Pa) 3.558 3.343 3.131 2.922 2.716 2.513 2.315 2.123 χT (10 −12/Pa) 28.10 29.90 31.92 34.21 36.81 39.77 43.18 47.09 CV (J/mol.K) 27.81 27.69 27.57 27.45 27.33 27.21 27.09 26.97 CP (J/mol.K) 28.36 28.21 28.07 27.92 27.78 27.64 27.49 27.35 Table 9. Ce concentration dependence of thermodynamic quantities of Th1−xCex at P = 0 and T = 500K cCe 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 a(A˚) 3.495 3.501 3.508 3.515 3.523 3.531 3.540 3.550 α(10−6K−1) 9.747 10.13 10.55 11.00 11.48 12.00 12.56 13.17 BT (10 10Pa) 2.532 2.450 2.366 2.282 2.197 2.111 2.025 1.938 χT (10 −12/Pa) 39.47 40.81 42.24 43.80 45.50 47.34 49.37 51.59 CV (J/mol.K) 25.97 26.05 26.14 26.22 26.31 26.40 26.48 26.57 CP (J/mol.K) 26.17 26.26 26.36 26.46 26.55 26.65 26.75 26.85 We found that the nearest neighbor distance, linear thermal expansion coefficient and isothermal compressibility increased while bulk modulus decreased when the concentration of Ce in the alloy increased. We also found that the concentration of Ce strongly influenced the thermodynamic properties of the alloy when concentration of Ce in the alloy increased. In Figure 1 we displayed the good agreement of the SMM calculations of the nearest neighbor distance of Al2Ce3, AgCe3 and Th57Ce43 alloys. The results calculated by the statistical moment metho