Third-order thermodynamic perturbation theory for effective potentials that model complex fluids

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

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.

Convergence and low temperature adaptability analysis of the high temperature series expansion of the free energy

By appealing to the coupling parameter series expansion to calculate the first seven perturbation coefficients of the high temperature series expansion (HTSE) of the free energy, analysis of convergence and low temperature adaptability of the HTSE in calculating fluid thermodynamic properties is performed for the first time; the fluid thermodynamic properties considered include critical parameters, vapor-liquid coexistence curve, thermodynamic characteristic functions, chemical potential, pressure, and constant volume excess heat capacity. To proceed with the analysis, a well known square well model is used as sample; the well widths considered range over a wide interval, and the relevant temperatures amenable to simulation calculations (used as "exact" results to analyze the HTSE) can be both very high and very low. The main discoveries reached are summarized as follows: (1) The HTSE usually converges at the 4th-order truncation, but with decrease of the temperature considered, the lowest truncation order, which makes the HTSE to converge, tends to rise. As a conservative estimate, it is considered that the HTSE always converges for reduced temperature T* higher than 0.25, whereas for T* < 0.25 there appear signs indicating that the HTSE may diverge from the 7th-order truncation. (2) Within the temperature interval with T* ≥ 0.5, the HTSE converges approximately to the correct solution, and the HTSE can be reliably used to calculate the fluid thermodynamic properties, and within this temperature interval, the 4th-order truncation is enough; whereas for T* < 0.5, such as within the temperature interval with 0.275 ≤ T* ≤ 0.355, although the HTSE does converge, it does not converge to the correct solution, and the deviations between the HTSE calculations and MC simulations become an ever-prominent issue with the rising of the density, and the slopes of the thermodynamic properties over density are not satisfactorily represented. As a result, the HTSE is not suited for calculations for temperature interval T* < 0.5.

An improved first-order mean spherical approximation theory for the square-shoulder fluid

The theory, which utilizes an exponential enhancement of the first-order mean spherical approximation (FMSA) for the radial distribution functions of the hard-core plus square-well fluid, is adopted to study the properties of the simplest model of the core-softened fluids, i.e., the hard spheres with a square-shoulder interaction. The results for structure and thermodynamic properties are reported and compared against both the Monte Carlo simulation data as well as with those obtained within the conventional FMSA theory. We found that in the region of low densities and low temperatures, where the conventional FMSA theory fails, the exponential-based FMSA theory besides being qualitatively correct also provides with a notable quantitative improvement of the theoretical description.

Perturbation theory of liquids for short-ranged hard-core potentials: structure and thermodynamics of short-ranged square-well fluids

A new method is devised to derive a thermodynamic perturbation theory that is suitable for short-ranged potentials. We have calculated structure and thermodynamic of short-ranged square-well potential with good agreement with computer simulation data even at low temperatures. We have also compared the results with other perturbation theories. It is found that the proposed theory gives better results when used to compute the structure and thermodynamic of short-ranged hard-core potentials.