ENTHALPY, ENTROPY AND HELMHOLTZ FREE ENERGY OF TRANSITION AND RARE EARTH LIQUID METALS

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1 Digest Journal of Nanomaterials and iostructures Vol. 5, No, April 010, p ENTHALPY, ENTROPY AND HELMHOLTZ FREE ENERGY OF TRANSITION AND RARE EARTH LIQUID METALS J. K. ARIA a, A. R. JANI b a V. P. & R. P. T. P. Science College, Vallabh Vidyanagar , Gujarat, India b Department of Physics, Sardar Patel University, Vallabh Vidyanagar , Gujarat, India Recently proposed model potential of aria and Jani is used to calculate Enthalpy, Entropy and Helmholtz free energy of liquid transition and rare earth metals with the variational approach. The parameter of the potential is determined with the standard zero pressure condition along with well established Sarkar et al screening function for exchange and correlation effect. Here we strongly emphasize that the parameter of the potential is independent of any fitting procedure either with any experimental data or with any theoretical values of any physical properties. The structure factor derived by Percus- Yevick solution for hard sphere fluids, which is characterized by hard sphere diameter, is used. A good agreement between theoretical investigations and experimental findings has confirmed the ability of the model potential to the liquid d and f shell metals. (Received April 5, 010; accepted April 1, 010) Keywords: Pseudopotential, Entropy, Enthalpy, Helmholtz free energy. In the last two decades, the considerable efforts have been made to the understanding of structure and thermodynamic properties of several liquid noble, transition and rare-earth metals [1-13]. A large number of experimental measurements on various properties of these metals are now available in the literature [14-15]. The interatomic potentials of simple metals [16-18] have been fully investigated and their thermodynamic properties could be derived with sufficient accuracy, but in the case of transition metals the hybridization of d tron with s tron makes the things complex. Despite the success of the theory in the solid state, results for the structure factor of liquid 3d transition metals using molecular dynamics and other complicated liquid state theories have not been that reliable. [3, 19]. Wills and Harrison [0] have derived the interatomic potentials for transition metals and obtained thermodynamic properties reasonably good. The potentials of Wills and Harrison have also been used for the determination of the surface properties of noble metals [1] and thermodynamic properties of 3d transition metals in the liquid state [3, 11]. The variational technique with a hard sphere fluid as a reference system have been used by retonnet and Deroache [11] for the calculation of Helmholtz free energy of a series of liquid transition metals. The potentials of Wills and Harrison [0] and retonnet and Deroache [11] are also successfully exploited for the computation of several equilibrium properties. A number of theoretical calculations based on the atomistic model and the use of interatomic potentials have been used for the study of liquid metals [-4]. Computer experiments are always intended to propose a plausible interpretation of experimental results in some cases to give the solution to an experimentally inaccessible problem. The reliability of the predicted values, however, is entirely depends on the validity of a given interatomic potential and the model used. Recently aria and Jani [5-7] have proposed a model potential, using this potential they have reported large number of physical properties of d and f-shell metals successfully [5-7].

2 318 For the investigation of thermodynamic properties of liquid metals, the Helmholtz free energy, F, lies at the heart of the pseudopotential perturbation scheme. The standard thermodynamic relation for the free energy is [8, 9] F = E T S (1) where, E is the internal energy i.e. Enthalpy and S is the entropy of the system at a temperature T. Under the usual perturbation theory the internal energy E can be expressed as [8, 9] E = E + E + E. () ion ion Here the first term E ion is composed of kinetic energy {(3/) k T} of ions plus the contribution due to ion-ion interactions, usually known as Madelung contribution. Thus, we write [8, 9], E ion 3 Z π = k T + { a( q) 1}dq (3) where Z is the valency and a(q) is the structure factor. For the structure dependent contribution, the structure factor, a(q), for liquid metals is calculated from the Percus-Yevick solution for HS fluids which is characterized by the HS diameter (σ) or, equivalently, by the packing fraction η = (πσ 3 /6Ω). The term E in equation () is the energy of the homogeneous tron gas, which is the sum of kinetic energy of trons, exchange energy, correlation energy, and the low temperature specific heat contribution for the tron gas. Hence the expression for E becomes [8, 9] π k E = NZ k F k F ln k F T (4) 10 4π k F with k F is the Fermi wave vector and N is the total number of atoms. In the framework of pseudopotential second ordered perturbation theory E -ion has been obtained by [8, 9], 8π Z Z E ion = lim W ( q) + q 0 + q Ω (5) 1 16 π ε (q) 1+ [ε (q) 1][1 G (q) ] H 4 { W ( q) } a ( q) q dq { } Where, the first term on right hand side represents the first order energy and the second is the band structure energy. Here, W ( q) is bare ion pseudopotential, of aria and Jani [5-7], its q-space form (in Ryd. Unit) is given by, 4 ( cos( q r ) ) ( ) ( ) [ ] c 1 4 sin sin q r 3 c 8 π Z (1+ ) (6) W (q) = ( ) Ω0 q cos e ( 3) + [ 11q r + + ] + ( ) 3 c 4 cos q r 3 c (1+ ) (1+ ) Where r c is the parameter of the potential and e is the base of natural logarithm. ε H (q) is the modified Hartree ditric function and G (q) is the exchange and correlation function due to Sarkar et al [30]. Using zero pressure condition we have evaluated the parameter of the potential r c which is shown in Table 1. The necessary parameters used in the calculations are also displayed in table 1. H

3 The second most essential part to investigate the free energy is the entropy, S, of the HS fluids. Using the information of HS diameter (σ) or packing fraction η, one can evaluate the entropy, S, as [8, 9]. 319 with S η S = S + S + S (6) gas η 1 5 k T S gas = k + k ln Ω (7) 4π = k η 3η 4 1 η (8) ( )( ) π k T S = (9) k F We have calculated the various contributions to the internal energy i. e., enthalpy, entropy and Helmholtz free energy of some d and f-shell metals at melting temperature. In expressions (3) and (5), the integration has been carried out up to 40k F to avoid any artificial cutoff in the calculation and to achieve proper convergence. The input parameters used in present calculation are shown in Table 1. The presently calculated values of internal energy i. e., enthalpy, entropy and Helmholtz free energy are displayed in table to 4 and were ever possible we have either compared it with experimental data or with others such theoretical findings. Thus it gets confirmed from Tables 3 and 4 that the presently calculated values of entropy are in good agreement with experimental data and are comparable to the other reported data [9, 10, 31]. This confirms the applicability of our model potential in the investigation of thermodynamic properties of liquid d and f-shell metals. So, present model potential is equally useful in the calculation of various properties of liquid metals without any modification. Table 1. Input parameters used in the present calculations. Temperature in (K) r c (au) Z η Ω 0 (au) Cu Ag Au Ni Pd Pt Rh Ir La Yb Ce Th

4 30 Table. Enthalpy of liquid transition and rare earth metals. Various contributions to the Enthalpy x 10-3 (au/atom) H H ion H -ion Total Enthalpy (H) x 10-3 (au/atom) Cu Ag Au Ni Pd Pt Rh Ir La Yb Ce Th Table 3. Entropy and helmholtz free energy of liquid transition and rare earth metals. Various contributions to the Entropy S gas / k T -Sη / k T S / k T Total Entropy S / k T Helmholtz free energy (F) x 10-3 (au/atom) Cu Ag Au Ni Pd Pt Rh Ir La Yb Ce Th

5 31 Table 4. Entropy of the liquid transition and rare earth metals. Entropy in unit of s/k T Others Present Expt. [31] [31] [1] Cu Ag Au Ni Pd Pt Rh Ir La Yb Ce Th Acknowledgement One of the authors (JK) is thankful to the University Grants Commission (UGC-WRO, Pune) for providing financial support under minor research project grant (F. No.: /08). References [1] C. Hausleitner, J. Hafner, J. Phys. F18, 105 (1988). [] R. N. Joarder, R. V. Gopal Rao, Phys. Stat. Sol. (b) 109, 137 (198). [3] C. Regnault, Z. Phys. 76, 179 (1989). [4] S. Natio, T. Arai, I. Yokoyama, Y. Waseda, High. Temp. Mater. And Proc. (G) 8, 15 (1989). [5] O. Akinlade, A. M. Umar, L. A. Hussain, PRAMANA, J. of Phys. 47, 71 (1996). [6] I. Yokoyama, T. Arai, Mater. Trans. JIM 37, 57 (1996). [7] H. Sasajima, T. Arai, I. Yokoyama, Mater. Trans. JIM 39, 551 (1998). [8] I. Yokoyama, I. Ohkoshi, T. Satoh, J. Phys. F.: Met Phys. 13, 79 (1983). [9] J. L. retonnet, G. M. huiyan, M. Silbert, J. Phys. : Condens. Matte. 4, 5359 (199). [10] N. Jakse, J. F. Wax, J. L. retonnet, A. Pasturel, J. Non-Cryst. Solids 05, 434 (1996). [11] J. L. retonnet, A. Derouiche, Phys. Rev. 43, 894 (1991). [1] G. M. huiyan, M. Silbert, M. J. Stott, Phys. Rev. 53, 636 (1996). [13] A. ari, T. Das, R. N. Joarder, J. Non-Cryst. Solids 136, 173 (1991). [14] R. Hultgren, Sted Values of Thermodynamic properties of elements, Amer. Soc. For s, Ohio [15] Y. Waseda, The Structure of Non-Crystalline Materials, McGraw-Hill, Publ. Co. New York [16] N. W. Aschroft, D. Stroud, Solid State Physics, Academic Press, New York [17] M. Simoji, Liquid s, Academic Press London [18] J. Hafner, Amorphous Solids and Liquid states, Plenum Press, New York [19] C. Hausleitner, G. Kahi, J. Hafner, J. Phys. : Condens. Matt. 3, 1589 (1991). [0] J. M. Wills, W. A. Harrison, Phys. Rev. 8, 4363 (1983). [1] V. Russier, C. Regnaut, J. P. adiali, Z. Phys. Chem. 156, 489 (1988). [] K. C. Jain, N. Gupta, N. S. Saxena, Phys. State. Sol. (b) 160, 433 (1990). [3] Arun Pratap, Meeta Rani, K. C. Jain, N. S. Saxena, Ind. J. of Pure & Appl. Phys.

6 3 8, 657 (1990). [4] Meeta Rani, Arun Pratap, N. S. Saxena, Ind. J. of Pure & Appl. Phys. 7, 69 (1989). [5] J. K. aria, A. R. Jani, Physica 38, 317 (003). [6] J. K. aria, A. R. Jani, Pramana J. Phys. 60, 135 (003). [7] J. K. aria, Physica 371, 80 (006). [8] J. Hafner, Phys. Rev. A 16, 351 (1977). [9] S. Singh, R. N. Singh, Phys. Chem. Liq. 4, 03 (1994). [30] A. Sarkar, D. Sen, S. Haldar, and D. Roy,Modern Physics Letters 1, 639 (1998). [31] M. M. G. Alemany, M. Calleja, C. Rey, L. J. Gallego, J. Casa and L. E. Gonzalez, J. of Non- Cryst. Solids 50, 53 (1999).

Digest Journal of Nanomaterials and Biostructures Vol. 5, No 1, March 2010, p

Digest Journal of Nanomaterials and Biostructures Vol. 5, No 1, March 2010, p. 23-27 NEW CRITERIA FOR THE SELECTION OF EXCHANGE AND CORRELATION FUNCTION J. K. BARIA *, A. R. JANI a V. P. & R. P. T. P.