WCA Theory Research Articles

Effect of cut-off distance used in molecular dynamics simulations on fluid properties

The effect of cut-off distance used in molecular dynamics (MD) simulations on fluid properties was studied systematically in both canonical (NVT) and isothermal–isobaric (NPT) ensembles. Results show that the cut-off distance in the NVT ensemble plays little role in determining the equilibrium structure of fluid if the ensemble has a high density. However, pressures calculated in the same NVT ensembles strongly depend on the cut-off distance used. In the NPT ensemble, cut-off distance plays a key role in determining fluid equilibrium structure, density and self-diffusion coefficient. The characteristic of the radial distribution function of fluid in NPT ensembles depending on the cut-off distance used in MD simulations means that the WCA theory (a perturbation theory developed by Weeks, Chandler and Andersen) is not suitable for NPT ensembles because the assumption (the effect of the attractive force in determining the liquid structure is negligible) used in the WCA theory is not valid. The dependence of fluid properties on the cut-off distance also indicates that using the WCA potential (the repulsive part of the intermolecular potential proposed in the WCA theory) to calculate fluid transport in heterogeneous systems could lead to significant errors or incorrect results.

Radial distribution function of a hard-sphere fluid for the nearest neighbor surroundings

An expression for RDF as a function of two variables, the distance r and the packing factor η, was obtained by approximating the results of Monte Carlo simulation of a hard-sphere fluid. The mean square accuracy of the expressions presented is about ±0.0002 (1 ≤ r ≤ 1.5, 0 ≤ η ≤ 0.5). A continuous extension of RDF to the region of r < 1 is proposed, which provides the continuity of the first and second derivatives of RDF at the point r = 1. Analysis of the problem of determining the hard sphere diameter in WCA theory of simple liquids shows that the proposed expression makes it possible to directly calculate the hard sphere diameter without any simplifying approximations.

Perturbation theory for liquids with a hard sphere plus square well potential

Numerical data have been obtained for a series of perturbation theory in the form of Barker-Henderson discrete representation. A hard sphere liquid (864 particles) was modeled by the Monte Carlo method for 106 values of occupancy from the range η = 0.005–0.530 (step 0.005); the well width was varied from 0 to 1.5 hard sphere diameters. For each occupancy, averaging was done over 60·106 ensembles. Approximating analytical expressions are given as functions of well density and width for the first three terms of the free energy decomposition. For reduced temperature T* = 1.0, the standard error in the free energy calculation (three orders of perturbation theory if the above expressions are used) is of the order of ±0.001. Analysis of four-particle correlations revealed peaks at distances of 1.35 A and 1.65 A (η = 0.53) in the high-density region of hard sphere liquids, which are absent in crystals and dense liquids. It is analyzed whether the results are useful for realization of WCA theory of simple liquids in second order perturbation theory.

Reformulation of Weeks−Chandler−Andersen Perturbation Theory Directly in Terms of a Hard-Sphere Reference System

A new variant of fluid thermodynamic perturbation theory is constructed by building on approximations developed by Andersen, Chandler, and Weeks (WCA), directly using a hard-sphere (HS) reference system rather than indirectly via a soft-repulsive reference system. The resulting HS-WCA theory is more forgiving with respect to the choice of a reference HS diameter than the original WCA theory. For example, a simple analytical Boltzmann factor criteria, which relates the reference HS diameter to the distance at which the repulsive pair-potential is equal to kT, is shown to produce global Lennard-Jones fluid thermodynamic predictions that are essentially identical to those of the original WCA theory (or its self-consistent refinement by Lado).

Intermolecular forces and the structure of uniform and nonuniform fluids

We discuss the ramifications of Widom's idea that attractive intermolecular forces essentially cancel in dense uniform liquids. This idea was used directly in the WCA theory of uniform liquids, where the structure of the liquid is approximated by that of a simpler reference fluid with purely repulsive intermolecular forces. To take account of the unbalanced attractive forces found in nonuniform fluids, Weeks, Selinger, and Broughton (WSB) developed a new method where the structure is related to that of a nonuniform reference fluid in an external field chosen to yield a self-consistent description of correlations induced by the repulsive forces and a mean field treatment of the attractive forces. Using simulations, we provide a quantitative test of the accuracy of both methods for the uniform fluid at different points in the phase diagram by relating correlation functions in the uniform fluid to those in a nonuniform fluid with a particle fixed at the origin. We find that at high densities the WSB approach can correct most of the small errors in the structure of the WCA reference fluid. At lower densities the WSB method provides a considerable improvement over the WCA theory. A simplified version of the WSB method is presented that is of comparable accuracy.

An extensive study of the Helmholtz free energy of Lennard-Jones fluids using WCA theory

On the basis of results from molecular dynamics and using Weeks-Chandler-Andersen theory, a simple and exact analytical expression for the Helmholtz free energy of a system of particles interacting through the Lennard-Jones potential is obtained for a wide range of densities and temperatures. The values obtained from this expression and its derivative, the potential energy, are compared with existing published values, and are in reasonable agreement. The combination of accuracy with simplicity—only five parameters are used — could be very useful in theoretical and chemical engineering applications, where straightforward mathematical handling of thermodynamic properties is an important requirement. The high temperature approximation for the radial distribution function and for the Helmholtz free energy are compared with results from exact calculation. The results show this approximation to work well for the radial distribution function at high densities only, but to be good even at low densities for the Helmholtz free energy.

Concentration fluctuations at the liquid-vapour coexistence in NeAr mixture

A new analytical formula is prepared for the concentration fluctuations, in the long-wavelength limits, S cc(0), for Lennard-Jones fluids based on the WCA theory and RPA approximation. This has made it possible to investigate the dependence of S cc(0) on concentration, pressure and temperature throughout the liquid and vapour phases, as well as across the liquid-vapour coexistence regions of the phase diagram of NeAr mixture.

Computer simulation and perturbation theory of fluids modelled using three- and six-site Lennard-Jones potentials

Systematic NVT molecular dynamics simulations have been performed for pure fluids of nonspherical molecules modelled using a triangular three-site and a hexagonal six-site Lennard-Jones (12:6) potential in order to obtain thermodynamic properties of these fluids, particularly the residual Helmholtz free energy. Molecular geometries, shown earlier to work well for propane and benzene, have been utilized in this study. Free energy values obtained from these simulations have been compared with predictions from the reference fluid simulations in a Weeks-Chandler-Andersen type site-frame perturbation expansion to the first order. Comparisons clearly show this site-frame expansion to work well at high densities and no significant effect of molecular anisotropy has been found on the deviations of these predictions, from the exact simulation values, vis-a-vis the WCA theory for the Lennard-Jones (12 : 6) fluid.

Interatomic Core Forces Deduced From Observed Liquid Structure Factors

Abstract A simplified version of the WCA theory allows us to start from an observed structure factor and derive a core potential and gradient at an effective collision diameter. The quality of fit and internal consistency checks indicate that the method works for liquids of inert gases, certain simple metals and the noble metals. It is less accurate, although still useful, for transition metals and lanthanides. In the latter case, there is a peaking in core diameter at Eu(4f76s2) as the series is traversed. This has no counterpart in the transition metal series which shows a steady decrease (although with an anomalously small value for Cr(3d54s1)).

Soft Sphere Characterisation from Liquid Structure Factors; Application to Lead

Abstract We adapt the WCA theory so as to provide a convenient formalism for gauging core softness from the evidence of a measured liquid structure factor. The method is illustrated for lead.

A simplified WCA theory of liquid structure

A linearized version of the WCA theory allows simple calculation of the structure factor of simple liquids. The degree of accuracy attained is comparable with that of the full method. The results are used to suggest why different potentials can be found to satisfy the experimental data for a given system beyond the principal peak.

Equilibrium Theory of Simple Liquids

The perturbation theory of liquids developed recently by Weeks, Chandler, and Andersen (WCA) is examined in detail: Each assumption introduced by these authors is tested by comparison with "exact" computer results. It is shown that the basic assumptions of WCA are justified. An improved expression for the radial distribution function of the hard-sphere gas enables us to correct for some further inconsistent assumptions of the WCA theory. We then succeed in giving simple analytical expressions for the thermodynamic functions of the Lennard-Jones fluid shown to be quite good at high density. We show that the remainder of the perturbation series, which converges slowly at lower density, can be evaluated with the help of the Percus-Yevick equation. We therefore possess a unified theory of liquids which is especially simple at high density. Finally we reexpress the original WCA theory in an analytical form.