Ornstein-Zernike Integral Equation Research Articles

Theoretical Investigation of Uniform and Non-uniform Penetrable SphereFluid

A bridge function approximation is proposed for a single-componentfluid consisting of penetrable sphere interacting via a potential thatremains finite and constant for center-center distance smaller than theparticle diameter and is zero otherwise. The radial distribution function fromthe Ornstein–Zernike integral equation combined with the present bridgefunction approximation is in satisfactory agreement with the correspondingsimulation data for all of the investigated state points. The presentlycalculated excess Helmholtz free energy respectively based on virial routeand compressibility route is highly self-consistent, and is in very goodagreement with simulational results for the case of low temperatures.The present bridge function approximation, combined with the bridge densityfunctional approximation, can reproduce very accurately density profiles ofthe penetrable sphere fluid confined in a hard spherical cavity for all thecases where simulational results are available.

Thermodynamics of asphaltene structure and aggregation

We develop an analytical solution of the associative Ornstein-Zernike integral equation in the Percus-Yevick approximation to model asphaltene structural properties and phase separation in model oil dispersions as a function of pressure, temperature and composition. To account for association, asphaltenes are characterized as hard spheres with active association sites placed on its surface; these sites are capable of linking with other similar species. Solvent molecules are modeled as hard spheres flexible linear chains. There is a sticky attraction between asphaltene and solvent. Using a calculation scheme similar to branched polymerization, asphaltene precipitation and the structural properties of the mixture are studied at various degrees of aggregation. Results show how the occurrence of instability correlates with pair potential parameters, sizes, density and composition of the system. These results compare well with experimental evidences.

A comparative study of electrical resistivity of liquid alkali metals

The resistivities of liquid alkali metals are investigated using Ziman formula and the alternative expression based on Kubo formula suggested by Ünal and Alkan [B. Ünal, B. Alkan, J. Phys. Soc. Jpn. 62 (1993) 2425]. As the input pseudopotential, the individual version of the electron-ion potential proposed by Fioalhais and coworkers which was originally developed for solid state is used and static structure factors are derived from the solution of Ornstein–Zernike integral equation with Rogers–Young closure. Good agreement with experiment is found for liquid Na and K using both formulas. For Li, Rb and Cs the agreement between the calculated values using Ziman formula and experimental data is less satisfactory. From present investigations, it is shown that the formula based on Kubo formula suggested by Ünal and Alkan is the better chose for the resistivity calculations of liquid Li, Rb and Cs metals.

Pair-correlation entropy of hydrophobic hydration: Decomposition into translational and orientational contributions and analysis of solute-size effects

We develop an efficient method to evaluate the translational and orientational contributions to the solute-water pair-correlation entropy that is a major component of the hydration entropy. A water molecule is modeled as a hard sphere of diameter dS=0.28 nm in which a point dipole and a point quadrupole of tetrahedral symmetry are embedded. A hard sphere of diameter dM, a hydrophobic solute, is immersed at infinite dilution in the model water. The pair-correlation entropy is decomposed into the translational and orientational contributions in an analytical manner using the angle-dependent Ornstein-Zernike integral equation theory. The two contributions are calculated for solutes with a variety of sizes (0.6<or=dM/dS<or=30). The effects of the solute-water attractive interaction are also studied. As dM becomes larger, the percentage of the orientational contribution first increases, takes a maximum value at dM=DM (DM/dS depends on the strength of the solute-water attractive interaction and is in the range of 1.4-2), and then decreases toward a limiting value. The percentage of the orientational contribution reduces progressively as the solute-water attractive interaction becomes stronger. The physical origin of the maximum orientational restriction at dM=DM is discussed in detail.

Study on multi-Yukawa potential between charged colloid particles

The Yukawa potential has the function to express both the attractive dispersion interaction and the double-layer repulsive electrostatic interaction. In this paper, a comprehensive review of recent advances in the use of mean spherical approximation (MSA) to solve the Ornstein–Zernike integral equation to Yukawa potential is introduced. The explicit analytical Yukawa EOSs used for neutral fluids and for like-charged colloid solutions are also described. The one-component multi-Yukawa EOS is deduced and tested by MC data. The pair correlation functions and the electrostatic potentials of mean force between like-charged colloid particles are determined by hypernetted-chain (HNC) approximation. The electrostatic attractive forces are existing between like-charged bovine serum albumin molecules in aqueous NaCl solutions under certain conditions. So the multi-Yukawa potentials are needed.

Excess volume of homonuclear diatomic mixtures

A new fast and robust numerical method of solving the Ornstein–Zernike integral equation is described and tested on the binary mixture of hard diatomic molecules. The method combines direct iterations and the Newton-GMRES algorithm. Excess volumes of a mixture of hard homonuclear dumbbells with reduced elongations L* = 0.3 and 0.6 are calculated and compared to computer simulations and several equations of state including the virial expansion up to B 5.

Integral equation theory for hard spheres confined on a cylindrical surface: Anisotropic packing entropically driven

The structure of two-dimensional (2D) hard-sphere fluids on a cylindrical surface is investigated by means of the Ornstein-Zernike integral equation with the Percus-Yevick and the hypernetted-chain approximation. The 2D cylindrical coordinate breaks the spherical symmetry. Hence, the pair-correlation function is reformulated as a two-variable function to account for the packing along and around the cylinder. Detailed pair-correlation function calculations based on the two integral equation theories are compared with Monte Carlo simulations. In general, the Percus-Yevick theory is more accurate than the hypernetted-chain theory, but exceptions are observed for smaller cylinders. Moreover, analysis of the angular-dependent contact values shows that particles are preferentially packed anisotropically. The origin of such an anisotropic packing is driven by the entropic effect because the energy of all the possible system configurations of a dense hard-sphere fluid is the same. In addition, the anisotropic packing observed in our model studies serves as a basis for linking the close packing with the morphology of an ordered structure for particles adsorbed onto a cylindrical nanotube.

Effective interactions in the colloidal suspensions from hypernetted-chain theory

The hypernetted-chain (HNC) Ornstein-Zernike integral equations are used to determine the properties of simple models of colloidal solutions where the colloids and ions are immersed in a solvent considered as a dielectric continuum and have a size ratio equal to 80 and a charge ratio varying between 1 and 4000. At an infinite dilution of colloids, the effective interactions between colloids and ions are determined for ionic concentrations ranging from 0.001 to 0.1 mol/l and compared to those derived from the Poisson-Boltzmann theory. At finite concentrations, we discuss on the basis of the HNC results the possibility of an unambiguous definition of the effective interactions between the colloidal molecules.

Chemical potentials and phase equilibria of Lennard-Jones mixtures: A self-consistent integral equation approach

We explore the vapor-liquid phase behavior of binary mixtures of Lennard-Jones-type molecules where one component is supercritical, given the system temperature. We apply the self-consistency approach to the Ornstein-Zernike integral equations to obtain the correlation functions. The consistency checks include not only thermodynamic consistencies (pressure consistency and Gibbs-Duhem consistency), but also pointwise consistencies, such as the zero-separation theorems on the cavity functions. The consistencies are enforced via the bridge functions in the closure which contain adjustable parameters. The full solution requires the values of not only the monomer chemical potentials, but also the dimer chemical potentials present in the zero-separation theorems. These are evaluated by the direct chemical-potential formula [L. L. Lee, J. Chem. Phys. 97, 8606 (1992)] that does not require temperature nor density integration. In order to assess the integral equation accuracy, molecular-dynamics simulations are carried out alongside the states studied. The integral equation results compare well with simulation data. In phase calculations, it is important to have pressure consistency and valid chemical potentials, since the matching of phase boundaries requires the equality of the pressures and chemical potentials of both the liquid and vapor phases. The mixtures studied are methane-type and pentane-type molecules, both characterized by effective Lennard-Jones potentials. Calculations on one isotherm show that the integral equation approach yields valid answers as compared with the experimental data of Sage and Lacey. To study vapor-liquid phase behavior, it is necessary to use consistent theories; any inconsistencies, especially in pressure, will vitiate the phase boundary calculations.

Potential of mean force in confined colloids: Integral equations with fundamental measure bridge functions

The potential of mean force for uncharged macroparticles suspended in a fluid confined by a wall or a narrow pore is computed for solvent-wall and solvent-macroparticle interactions with attractive forces. Bridge functions taken from Rosenfeld's density-functional theory are used in the reference hypernetted chain closure of the Ornstein-Zernike integral equations. The quality of this closure is assessed by comparison with simulation. As an illustration, the role of solvation forces is investigated. When the "residual" attractive tails are given a range appropriate to "hard sphere-like" colloids, the unexpected role of solvation forces previously observed in bulk colloids is confirmed in the confinement situation.

A simple analysis of thermodynamic properties for classical plasmas: II. Validation

The generalized hole corrected Debye–Hückel theory (Penfold et al J. Stat. Mech. (2005) P06009)is implemented. Predictions of thermodynamic functions and simple structural propertiescompare favourably with results from closure of the Ornstein–Zernike integral equation inthe mean spherical approximation, and with Monte Carlo simulation of the charged hardsphere primitive model. A strictly nonelectroneutral system was simulated using aconventional electrolyte program and the properties subsequently corrected for theconfiguration independent background terms. No convergence difficulties were encounteredover the concentration range studied. With the new theory, activity coefficients of goodaccuracy can be obtained in a simple analytical form that is suitable for use with anapproximate free energy density functional describing ion–ion correlations in screeningatmospheres.

Quantitative Description of Potential of Mean Force BetweenMacroparticles in Fluid with Attactive Forces

A statistical mechanics method is proposed for calculation ofpotential of mean force (PMF). In the case of solvophobic or solvophilicmacroparticles immersed in solvent bath of soft sphere or Lennard-Jonesparticles, prediction accuracy for the PMF and MF from the simplestimplementation of the proposed method, where hypernetted chain approximationis adopted for correlation of the macroparticle-macroparticle at infinitelydilute limit, is comparable to that of a recent more sophisticated approachbased on mixture Ornstein–Zernike integral equation / bridge function fromfundamental measure functional. Adaptation of the present method for generalcomplex fluids is discussed, and method for improving the accuracy issuggested. Differences and relative merits of the present recipe comparedwith that based on potential distribution theory is discussed.

Local Solvent Density Augmentation around a Solute in Supercritical Solvent Bath: 1. A Mechanism Explanation and a New Phenomenon

A recently proposed partitioned density functional (DF) approximation (Phys. Rev. E 2003, 68, 061201) and an adjustable parameter-free version of a Lagrangian theorem-based DF approximation (LTDFA: Phys. Lett. A 2003, 319, 279) are combined to propose a DF approximation for nonuniform Lennard-Jones (LJ) fluid. Predictions of the present DF approximation for local LJ solvent density inhomogeneity around a large LJ solute particle or hard core Yukawa particle are in good agreement with existing simulation data. An extensive investigation about the effect of solvent bath temperature, solvent-solute interaction range, solvent-solute interaction magnitude, and solute size on the local solvent density inhomogeneity is carried out with the present DF approximation. It is found that a plateau of solvent accumulation number as a function of solvent bath bulk density is due to a coupling between the solvent-solute interaction and solvent correlation whose mathematical expression is a convolution integral appearing in the density profile equation of the DF theory formalism. The coupling becomes stronger as the increasing of the whole solvent-solute interaction strength, solute size relative to solvent size, and the closeness to the critical density and temperature of the solvent bath. When the attractive solvent-solute interaction becomes large enough and the bulk state moves close enough to the critical temperature of the solvent bath, the maximum solvent accumulation number as a function of solvent bath bulk density appears near the solvent bath critical density; the appearance of this maximum is in contrast with a conclusion drawn by a previous investigation based on an inhomogeneous version of Ornstein-Zernike integral equation carried out only for a smaller parameter space than that in the present paper. Advantage of the DFT approach over the integral equation is discussed.

A simplified expression for the hard-sphere dimer fluid radial distribution function

Recently, in our laboratory a closed form expression for the correlation function of the hard-sphere dimer fluid obtained from Wertheims multidensity Ornstein-Zernike integral equation theory with Percus-Yevick approximation was presented by Kim et al. [2001]. However, it is difficult to apply its expression to perturbation theory and vapor-liquid equilibria calculations, since it is of very complex form. In this work, we present a simplified expression for the first shell of the radial distribution function (RDF) of the hard-sphere dimer fluid using a series expansion of the analytical expression. The expansion is carried out in terms of both the packing fraction and the radial distance. Expressions are also obtained for the coordination number and its first and second derivatives as functions of radial distance and packing fraction. These expressions, which are useful in perturbation theory, are simpler to use than those obtained from the starting equation, while giving good agreement with the original expression results. Then we present an simplified equation of state for the square-well dimer fluid of variable well width (λ) based on Barker-Henderson perturbation theory using its expression for the radial distribution function of the hard-sphere dimer fluid, and test its expression with NVT and Gibbs ensemble Monte Carlo simulation data [Kim et al., 2001].

Solid–liquid transition of charge-stabilized colloidal dispersions: a single-component structure-function approach

We have extended the Raveché–Mountain–Streett one-phasecriterion that governs the freezing of Lennard-Jones systems to a hard-core repulsive Yukawa-model (HCRYM) system. We find in the framework of the Rogers–Young (RY) approximation for an Ornstein–Zernike integral equation that an HCRYM fluid freezes when the ratio α = g(rmin)/g(rmax), where rmax is the distance corresponding to the maximum in the radial distribution function g(r) and rmin is the distance corresponding to the subsequent minimum in g(r), is approximately 0.215. To describe the freezing of charge-stabilized colloidal dispersions in electrolytes, which consist of colloidal macroions,electrolyte small ions, and solvent molecules, we employ the single-component model in which the colloidal particles interact through the effective screened Coulomb potential of Belloni. Whenthe macroion surface effective charge number is taken as an adjustable parameter, the theoretical freezing line predicted by the RY g(rmin)/g(rmax) = 0.215 Raveché–Mountain–Streett one-phase criterion is in very good agreement with the corresponding experimental data.PACS Nos.: 61.25.Em, 61.20.Gy

Polyion monolayers and halos around large weakly-charged colloids

In this paper we present a theory for the structure of a suspension of highly charged spherical polyions near the surface of a much larger but more weakly charged spherical colloid. Starting from a level of description involving effective screened Coulomb potentials between the particles of both species, we calculate the radial distribution function of the polyions around isolated big colloids. Our theory is based on the use of the Ornstein–Zernike integral equation, complemented with the hyper-netted chain approximate closure, whose numerical solutions are selectively contrasted with the results of Monte Carlo simulations generated for this purpose. Our results indicate that, under certain conditions, our model suspension of polyions is predicted to adsorb onto the surface of the larger colloid, even if both species carry charges of the same sign. Under extreme conditions, the structure of the collection of adsorbed particles corresponds strictly to a monolayer, strongly bound to the surface of the larger particle by electrostatic forces. The complex formed by one large particle plus its adsorbed polyions then bears a large effective charge, which may enhance the electrostatic stability of the former. At some threshold condition, however, this monolayer structure becomes loose enough to be better described as a halo. One might thus associate this threshold condition with the threshold for the stability of the colloidal species formed by the larger particles. We discuss the possible connection of our theoretical results with the recent experimental observations that motivated this work.

Analytic solutions for Baxter's model of sticky hard sphere fluids within closures different from the Percus-Yevick approximation.

We discuss structural and thermodynamical properties of Baxter's adhesive hard sphere model within a class of closures which includes the Percus-Yevick (PY) one. The common feature of all these closures is to have a direct correlation function vanishing beyond a certain range, each closure being identified by a different approximation within the original square-well region. This allows a common analytical solution of the Ornstein-Zernike integral equation, with the cavity function playing a privileged role. A careful analytical treatment of the equation of state is reported. Numerical comparison with Monte Carlo simulations shows that the PY approximation lies between simpler closures, which may yield less accurate predictions but are easily extensible to multicomponent fluids, and more sophisticate closures which give more precise predictions but can hardly be extended to mixtures. In regimes typical for colloidal and protein solutions, however, it is found that the perturbative closures, even when limited to first order, produce satisfactory results.

Ion pairing in model electrolytes: A study via three-particle correlation functions

A novel integral equations approach is applied for studying ion pairing in the restricted primitive model electrolyte, i.e., the three-point extension (TPE) to the Ornstein–Zernike integral equations. In the TPE approach, the three-particle correlation functions g[3](r1,r2,r3) are obtained. The TPE results are compared to molecular dynamics (MD) simulations and other theories. Good agreement between TPE and MD is observed for a wide range of parameters, particularly where standard integral equations theories fail, i.e., low salt concentration and high ionic valence. Our results support the formation of ion pairs and aligned ion complexes.

Study of dipolar fluid inclusions in charged random matrices

Structural, thermodynamic, and dielectric properties of a dipolar fluid confined in a charged random matrix are studied by means of grand canonical Monte Carlo simulation and replica Ornstein–Zernike integral equations in the hypernetted chain approximation. The fluid is modeled by a system of dipolar hard spheres. Two matrix topologies are considered: a frozen restricted primitive model matrix and a frozen hard sphere fluid with randomly distributed negative and positive charges. Both models lead to similar results in most cases, with significant deviations from the behavior of the corresponding equilibrated mixtures. The dielectric behavior is particularly interesting, since the effect of partial quenching on the equilibrated mixture recovers the electrostatics of the pure dipolar fluid but with the presence of Coulomb tails in the dipole–dipole total correlations. Differences between the two matrix models arise more vividly in the low density regime, in which the matrix with randomly distributed charges tends to enhance dipole association around the matrix particles. The integral equation results are in relatively good agreement with the computer simulation estimates.

Computer simulation and replica Ornstein—Zernike integral equation studies of a hard-sphere dipolar fluid adsorbed into disordered porous media

By means of extensive grand canonical Monte Carlo simulations and replica Ornstein-Zernike integral equation calculations, we explore the thermodynamics and dielectric behaviour of a dipolar fluid confined in disordered matrices. Different matrix topologies are modelled using, on the one hand, quenched hard-sphere configurations and, on the other, randomly positioned spheres. This illustrates the influence of the pore size and shape on the properties of the adsorbed fluid. For the same purpose, various sizes of the matrix particles have been considered. The integral equation calculations in the hypernetted chain approximation agree quantitatively with the simulation results. As in other studies on quenched disorder, one observes that the effect of confinement when considering hard-sphere matrices is rather limited, exhibiting however a tendency to facilitate the transition to ferroelectric states, as well as favouring the gas-liquid transition. Additionally, one observes that the sensitivity of the fluid properties to changes in the size of the matrix particles is considerably larger when these are randomly positioned.