Excellence of numerical differentiation method in calculating the coefficients of high temperature series expansion of the free energy and convergence problem of the expansion

Abstract

In this paper, it is shown that the numerical differentiation method in performing the coupling parameter series expansion [S. Zhou, J. Chem. Phys. 125, 144518 (2006); AIP Adv. 1, 040703 (2011)] excels at calculating the coefficients ai of hard sphere high temperature series expansion (HS-HTSE) of the free energy. Both canonical ensemble and isothermal-isobaric ensemble Monte Carlo simulations for fluid interacting through a hard sphere attractive Yukawa (HSAY) potential with extremely short ranges and at very low temperatures are performed, and the resulting two sets of data of thermodynamic properties are in excellent agreement with each other, and well qualified to be used for assessing convergence of the HS-HTSE for the HSAY fluid. Results of valuation are that (i) by referring to the results of a hard sphere square well fluid [S. Zhou, J. Chem. Phys. 139, 124111 (2013)], it is found that existence of partial sum limit of the high temperature series expansion series and consistency between the limit value and the true solution depend on both the potential shapes and temperatures considered. (ii) For the extremely short range HSAY potential, the HS-HTSE coefficients ai falls rapidly with the order i, and the HS-HTSE converges from fourth order; however, it does not converge exactly to the true solution at reduced temperatures lower than 0.5, wherein difference between the partial sum limit of the HS-HTSE series and the simulation result tends to become more evident. Something worth mentioning is that before the convergence order is reached, the preceding truncation is always improved by the succeeding one, and the fourth- and higher-order truncations give the most dependable and qualitatively always correct thermodynamic results for the HSAY fluid even at low reduced temperatures to 0.25.

Approximate Analytic Expression of Surface Charge Density/Surface Potential Relationship for a Spherical Colloidal Particle Immersed in a General Electrolyte Solution

“…To be specific, the variational iteration solution only applies to small x 0 value and y 0 value in the case of the PBE, and this fully demonstrates the variational iteration method keeps the property of a wide range of perturbation methods. Fortunately, in the case of the PBE the ranges of valid application of the variational iteration solution are complementary with those of previous approximate solutions, [15][16][17][18][19] and their combination consequently makes possible approximate solution of the PBE effectively workable across the entire x 0 range. On the other hand, we propose to approximate the highly nonlinear term of the PBE by a polynomial of proper order, and this makes analytically possible application of the variational iteration method to any complex differential equations; specific to the PBE of the present interest, advantage of the polynomial approximation of the nonlinear term resides in its universality, and this makes possible dealing with all kinds of electrolyte types in a unified way, and consequently surpasses well beyond the previously proposed approximate solutions of the PBE.…”

“…In the last 100 years, although a great deal of research had been done in analytically and approximately solving the PBE, the issue is not yet dealt with satisfactorily, and the acquired approximate solutions are exclusively valid only for high value of reduced colloid radius ja (Debye-Hü ckel screening parameter j being defined below, and a representing radius of the spherical particle), a relevant parameter of the PBE. [15][16][17][18][19] Moreover, many of these investigations are only concerned with specific electrolyte types, [15,[20][21][22][23][24] for example, symmetric electrolyte, single electrolyte, or salt-free water medium. It is noted that the non-linear term of the PBE is determined by the electrolyte types considered.…”