Estimation of spinodal pressure by extrapolation of Modified general Lennard-Jones equation of state (MGL-J EOS)

An expression for spinodal pressure has been formulated by extrapolating variation of bulk modulus with pressure which were evaluated by using modified general Lennard-Jones equation of state (MGL-J EOS). INTRODUCTION: The equation of state (EOS) can be extrapolated into the pressure range in which the material is metastable. This extrapolation gives an estimate of high pressure results which couldn’t be obtained experimentally. Moreover, such extrapolation can gives an estimation of the spinodal pressure (i.e.) The pressure at which the bulk modulus go to zero (Maris, 2009). MGL-J EOS can describe fluid state and is one of important tools to calculate the (PVT) relation and vapor-liquid equilibrium (Yutaka et al., 2007) and have all the merits of ideal universal (EOS) as it is energy analytic, pressure and volume analytic, beside it satisfies the following spinodal condition (Jiuxun, 2005): 2 / 1 sp P P B with 0 sp P P B (1) Estimation of spinodal pressure by extrapolation of Modified general ... 40 And it is based on the generalized Lennard-Jones (GL-J) potential which is a mathematically simple model that describes the interaction between a pair of neutral atoms or molecules. The most common expression of the GL-J potential is (Lennard-Jones, 1924): 6 12 / / 4 r r V 6 12 / 2 / r r r r m m (2) Where: depth of potential well. distance at which the inter particle potential is zero. r distance between the particles. m r distance at which the potential reaches its minimum, at m r the potential function has the value . The distances are related as 6 / 1 2 m r these parameters can be fitted to reproduce experimental data or accurate quantum chemistry calculations. The 12 r term describe the Pauli repulsion at short ranges due to overlapping electron orbitals and the 6 r term describes attraction at long ranges (Vander Waals force or dispersion force). (Jiuxun, 2005) adopt an all neighbor model to replace the nearest neighbor to obtain a modified lennard-jones equation of state (MGL-J EOS) in the form: 1 / / / n n V Vo V Vo n Bo P (3) Where o B n 3 / 1 (4) Bo , o B are the bulk modulus and its first derivative at zero pressure Vo is the volume at zero applied pressure; V is the volume at pressure P . Equation (3) is energy analytic; pressure and volume analytic, beside it satisfy the spinodal condition of equation (1). THEORITICAL DETAILS AND RESULTS: VARIATION OF BULK MODULUS WITH PRESSURE: The bulk modulus of material determines how much it will compress under a given amount of external pressure: dV Vdp B / (5) or means the reciprocal of compressibility. Thus, it is a measure of resistance to compressibility. (Gaudoin and Foulkes, 2002). From equation (3): n n o n n o o n n o n n o o V V V V n B V V V V n B P 2 2 1 1 1 2 2 2 / / n n n n V n Vo V n Vo n Bo dV dP 1 1 2 2 2 / V V nVo V V nVo n Bo n n n n V V Vo V V Vo Bo n n / 1 / / 1 / 2 2 A. M. AL-Sheikh & H. B. Mohammed 41 1 / 2 / / n n V Vo V Vo V Bo 1 / 2 / / n n V Vo V Vo V Bo (6) From equation (5) and (6) T T dV dp V B / 1 / 2 / n n V Vo V Vo Bo (7) EVALUATION OF SPINODAL pressure: By relating equation(3) and equation(7) and using values of Bo and o B shown in table(1) with n evaluated from equation(4) we obtain variation of T B with V Vo / and consequently with pressure as shown in figure(1) and figure(2) for C and α-Sn respectively. Table (1): Experimental values of Bo and o B (Tripathi et al., 2006) material Bo (GPa) o B C 442.0 4.69 α-Sn 53.0 4.22 Then by definition of spinodal pressure, as the pressure at which the bulk modulus go to zero, and extrapolating results shown in figure (1) and figure (2), we obtain an estimation for spinodal pressure in the form: dB BodP Psp / (8) Which differ from the estimation given by (jiuxun, 2005): n Bo Psp 4 / , where 3 / 1 n , then o B Bo Psp / 75 . 0 (9) Figure (3) and (4) shows the values of sp P for C and α-Sn, respectively, comparison between these values and values obtained by using equation (9) [jiuxun, 2005] shown in table (2): Table (2): A comparison for values of sp P obtained in present work and other reference.