Calculation of the radial distribution function of hard-sphere mixtures in the Percus-Yevick approximation

A formulation is given which permits the rapid mechanical computation of the three radial distribution functions g_ij (r) of a binary hard-sphere mixture to any distance r, in the Percus-Yevick (P-Y) approximation. The consistency of the P-Y equation of state obtained by various methods is discussed.     

The mean spherical model for charged hard spheres

The reduced mean electrostatic potential v(r) and the radial distribution functions gij(r) for a system of charged hard spheres of equal diameter are calculated from the solution of the mean spherical model equation given by Waisman and Lebowitz. An analytical solution is given for v(r) and the gij(r) are shown to be the sum of the Percus-Yevick uncharged hard-sphere distribution function and an electrostatic term. The correct qualitative behaviour of the mean potential is predicted at high concentrations but the radial distribution functions are only accurate for low valency electrolytes at high concentrations.     

Analytical representations for the radial distribution functions of mixtures of adhesive spheres

A study has been made of the pole topology of the Laplace transforms of the pair distribution functions (PDFs) of a binary mixture of adhesive hard spheres (AHS) both for the Percus-Yevick equation and the mean spherical model (MSM). Expressions are given that describe how the distribution of the poles in the left half of the complex plane varies with the system parameters for the special case of the MSM for symmetric binary AHS mixtures. The locations of the poles closest to the imaginary axis are known to determine the asymptotic form of the PDFs, i.e. either exponentially monotonic or exponentially oscillatory decaying. As a byproduct of this inquiry analytical r space representations of the PDFs are derived that allow their accurate and efficient determination over the entire r range.     

New accurate binary hard sphere mixture radial distribution functions at contact and a new equation of state

New and very accurate formulae for additive binary hard sphere (HS) mixture radial distribution functions (RDFs) at contact are proposed in a simple analytical form. Using the virial theorem, the formulae also provide a new HS mixture equation of state (EOS). The new RDF formulae are the most accurate currently available. The new EOS is of comparable accuracy with that of Malijevsky, A., and Veverka, J. (1999, Phys. Chem. chem. Phys., 1, 4267), which is the most accurate HS mixture EOS currently available. However, the new EOS proposed here is of much simpler analytical form.     

Non-polar solute-water pair correlation functions

The microscopic theory of the solvation of non-polar solutes in water proposed by Pratt and Chandler has been generalized in order to obtain separate solute-oxygen and solute-hydrogen radial distribution functions, g(r). The g(r)s predicted by this method for a spherical solute have been tested by comparison with the corresponding functions from two computer simulation studies. The water-water interactions were described by the configuration intereaction potential (CI) in both cases. The agreement between theoretical and simulation results is good for the solute-oxygen g(r), less so for the solute-hydrogen function. Moreover, the influence of the model of water on the calculated solute-solvent g(r)s has been examined by comparing results obtained with partial structure functions derived from the CI model and from recent neutron diffraction measurements. It is found that CI model and real water yield remarkably different solute-water radial distribution functions. Finally, the solvation of a model two-site solute has been studied for various bond lengths and the results confirm that when the space between the sites is sufficient to host a water molecule, the solvation of each site is the same as that of an isolated site, with respect to oxygen as well as hydrogen.     

Universal cubic equation of state and contact values of the radial distribution functions for multi-component additive hard-sphere mixtures

A universal cubic equation of state (UC EOS) is proposed based on a modification of the virial Percus-Yevick (PY) integral equation EOS for hard-sphere fluid. The UC EOS is extended to multi-component hard-sphere mixtures based on a modification of Lebowitz solution of PY equation for hard-sphere mixtures. And expressions of the radial distribution functions at contact (RDFC) are improved with the form as simple as the original one. The numerical results for the compressibility factor and RDFC are in good agreement with the simulation results. The average errors of the compressibility factor relative to MC data are 3.40%, 1.84% and 0.92% for CP3P, BMCSL equations and UC EOS, respectively. The UC EOS is a unique cubic one with satisfactory precision among many EOSs in the literature both for pure and mixture fluids of hard spheres.     

A simultaneous electronic transition of the oxygen-naphthalene complex

Thorne's extension of the Enskog theory to thermal conductivity and viscosity of binary hard sphere fluid mixtures is examined. It is shown that for molecules of the same size and different mass the expression for the thermal conductivity of such a mixture obtained by Longuet-Higgins, Pople and Valleau is identical to the collisional term in Thorne's equation arising from the locally maxwellian velocity distribution. Thorne's equations are evaluated by using the Percus-Yevick approximation for the contact radial distribution functions obtained by Lebowitz. The variations of the transport coefficients are examined as functions of pressure, composition and ratios of diameters and masses of the two species. Some comparison is made with experimental results.     

Analytic solution of the Percus-Yevick equation for sticky hard sphere potential

It is shown that within the Percus-Yevick approximation the radial distribution function for sticky (i.e. with a surface adhesion) hard spheres satisfies a linear differential equation with retarded right-hand side. Using the theory of distributions and the Green's function technique the analytic solution of this equation is found and explicit formulas are given enabling one to evaluate the radial distribution function both for sticky and non-attractive hard spheres for any distance and any density.     

Percus-Yevick theory and the Chandler-Weeks-Andersen reference fluid

Chandler, Weeks and Andersen have recently developed a successful perturbation theory of liquids. In their theory, the radial distribution function of the reference fluid is calculated from that of the hard-sphere fluid. In their published work, the Percus-Yevick theory is used to calculate the hard-sphere radial distribution function. In this paper, the Percus-Yevick theory is used to calculate directly the thermodynamic properties and radial distribution function of the reference fluid. If the Carnahan and Starling averaging procedure is used, the Percus-Yevick thermodynamic properties are excellent. However, the radial distribution function shows the same discrepancies as that of Chandler, Weeks and Andersen. Finally, recent calculations of Chandler, Weeks and Andersen, using the Monte Carlo estimates of the hardsphere radial distribution function are shown to give good results for the reference fluid distribution function. This indicates that the Percus-Yevick theory, rather than fundamental errors in the Chandler, Weeks and Andersen theory, is responsible for the discrepancies.     

Perturbative solution to order βɛ of the Percus-Yevick equation for triangular well potential forn=2

The radial distribution function for a fluid in which the molecules interactvia a triangular well potential is considered. Expanding the radial distribution function in pwoers of βɛ, where ɛ is the depth of the potential andβ=1/k _BT the first-order terms are calculated analytically using the Percus-Yevick theory in the Baxter’s formulation. The first-order terms in the direct correlation functionc(r) are also calculated. The first- and second-order terms in the free energy obtained from the energy equation of state are calculated and compared with other calculations. An erratum to this article is available at .     

Analytic solution of Percus-Yevick equation for fluid of hard spheres

A new method of analytic solution of the Percus-Yevick equation for the radial distribution functiong(r) of hard-sphere fluid is proposed. The original non-linear integral equation is reduced to non-homogeneous linear integral equation of Volterra's type of the second order. The kernel of this new equation has a polynomial form which allows to find analytic expression forg(r) itself without using the Laplace transformation. In addition, the first three moments of the total correlation function can be found.     

Perturbation theory for the square-well dimer fluid

An analytical equation of state is presented for the square-well dimer fluid of variable well width (1 ≤ λ ≥ 2) based on Barker-Henderson perturbation theory using the recently developed analytical expression for radial distribution function of hard dimers. The integral in the first- and the second-order perturbation terms utilizes the Tang, Y and Lu, B. C.-Y., 1994, J. chem. Phys., 100, 6665 formula for the Hilbert transform. To test the equation of state, NVT and Gibbs ensemble Monte Carlo simulations for square-well dimer fluids are performed for three different well widths (λ = 1.3, 1.5 and 1.8). The prediction of the perturbation theory is also compared with that of thermodynamic perturbation theory in which the equation of state for the square-well dimer is written in terms of that of square-well monomers and the contact value of the radial distribution function.     

A comparative study of aqueous DMSO mixtures by computer simulations and integral equation theories

ABSTRACT

Several computer simulation studies of aqueous-dimethylsulfoxide with different force field models, and conducted by different authors, point out to an anomalous depressing of second and third neighbour correlations of the water–water radial distribution functions. This seemingly universal feature can be interpreted as the formation of linear water clusters. We test here the ability of liquid state integral equation theories to reproduce this feature. It is found that the incorporation of the water bridge diagram function is required to reproduce this feature. These theories are generally unable to properly reproduce atom–atom distribution functions. However, the near-ideal Kirkwood–Buff integrals are relatively well reproduced. We compute the X-ray scattering function and compare with available experimental results, with the particular focus to explain why these data do not reproduce the cluster pre-peak observed in the water–water structure factor.     

Second-order Percus–Yevick theory for the radial distribution functions of a mixture of hard spheres in the limit of zero concentration of the large spheres

The radial distribution functions of a mixture of hard spheres are quite interesting when the ratio of diameters is large and the concentration of the large spheres is very small. In this regime, the radial distrbution functions change rapidly with concentration. The usual PercusYevick theory, which is adequate over most of the concentration range, fails at low concentrations of the large spheres. Values are reported of the radial distribution functions for zero concentration of the large spheres using the most accurate theory presently available, secondorder Percus-Yevick theory. Agreement with recent formulae for the contact values of these functions is very good except for the contact value for a pair of large spheres, where the agreement is fairly good. It is possible that the radial distribution function for a pair of large spheres may be a little larger than the already large values given by this recent formula.     

Calculation of distribution functions for two-component rigid-sphere molecules of a fluid

Analytical expressions are derived and a computational algorithm and program for calculating distribution functions of rigid spheres g _ij (r) necessary for calculations of the thermodynamic parameters of a binary fluid mixture are developed in the context of perturbation theory using the procedure based on the inversion of the Laplace transform for functions rg _ij (r) obtained from the Percus–Yevick equation.     

Analytical representation of the Percus-Yevick hard-sphere radial distribution function

Explicit analytical expressions, written in terms of complex variables and suitable for rapid computer evaluation, are presented for the Percus-Yevick hard-sphere radial distribution function, g(R), for R ˇ- 5σ. Some effects of truncating g(R) to unity past R = 5 σ are discussed.     

Is it possible to infer the equation of state of a mixture of hard discs from that of the one-component system?

Based on exact asymptotic properties of the composition-independent virial coefficients of a binary mixture of hard discs in the limits α = σ₂/σ₁ → 0, α → 1 and α → ∞, R. J. Wheatley (1998, Molec. Phys., 93, 965) has recently proposed an approximate interpolation equation for these coefficients. In this note, the equation of state equivalent to this interpolation is obtained, expressing the compressibility factor of the mixture in terms of that of the pure system. An extension to an arbitrary number of components is also given. The equation of state derived here is compared with another one recently proposed by following a different route (Santos, A., Yuste, S. B., and López de Haro, M., 1999, Molec. Phys., 96, 1) and with Monte Carlo simulation results. It is shown that the latter equation is more accurate than the former one, at least for not too disparate mixtures (0.7 < α < 1).     

Density expansion of the radial distribution and bridge functions of the hard sphere fluid

The bridge function and the background correlation function (and consequently the radial distribution function) of the pure hard sphere fluid are expanded up to the sixth power in density. The calculations are based on the Ree–Hoover representation of the diagrams and Monte Carlo integration. The coefficients as functions of the particle–particle separation are fitted to splines taking into account discontinuities in higher derivatives up to the term of the order of (r- const)⁵.     

Microwave propagation and scattering in a dense distribution of non-tenuous spheres: experiment and theory

Abstract

Controlled laboratory experimental results of coherent microwave propagation through a random medium are reported. The medium consisted of layers of styrofoam with spherical glass beads embedded at predetermined random positions generated by computer. The magnitude and phase of the transmitted field was measured over the frequency range 18-20.4 GHz for media with volume fractional densities ranging from 0.5% to 11%. The results are compared with independent scattering, Foldy's approximation, and the quasicrystalline approximation (QCA) using the solution of the Percus-Yevick (PY) equation for the pair distribution function. The effects of a size distribution are included. Experimental results indicate that at low densities, the measured extinction rate increases linearly with concentration in agreement with independent scattering. As concentration further increases, the extinction curve turns convex and is lower than independent scattering. However, it is higher than that predicted by QCA-PY. Using the known particle positions the authors have also computed the pair correlation function and good agreement is obtained with the Percus-Yevick approximation.