Percus-Yevick Approximation Research Articles

Simulation of binary hard-sphere systems with 1 : 5 and 1 : 10 size ratios

Hard-sphere molecular dynamics simulations were performed on asymmetric binary systems to obtain pair-correlation functions. Particle-diameter ratios of 1 : 5 and 1 : 10 were applied to mimic the colloid/solvent situation, where the large particles were the colloids and the small ones represented the solvent molecules. The packing fractions were chosen systematically between 0.1–0.5 for the large spheres and 0.0–0.5 for the small ones. We focused on two questions: a) To what extent can one approximate the inter-particle structure of the large spheres in binary systems with one-component hard-sphere data. b) How reliable are the theoretical pair-correlation functions of the Percus-Yevick and Rational Function Approximation models. Overlap integrals of pair-correlation function pairs were calculated to answer quantitatively. The results supported quantitatively the limits of the one-component hard-sphere approximation. On contrary, the selected theoretical methods reproduced the simulation results satisfactorily, if the packing fraction of the solvent was not more than 0.2. The theoretically important contact values of the pair-correlation functions were calculated, too.

On the radial distribution function of a hard-sphere fluid

Two related approaches, one fairly recent [A. Trokhymchuk et al., J. Chem. Phys.123, 024501 (2005)] and the other one introduced 15years ago [S. B. Yuste and A. Santos, Phys. Rev. A43, 5418 (1991)], for the derivation of analytical forms of the radial distribution function of a fluid of hard spheres are compared. While they share similar starting philosophy, the first one involves the determination of 11 parameters while the second is a simple extension of the solution of the Percus-Yevick equation. It is found that the second approach has a better global accuracy and the further asset of counting already with a successful generalization to mixtures of hard spheres and other related systems.

Effect of many-body interactions on the bulk and interfacial phase behavior of a model colloid-polymer mixture

We study a model suspension of sterically stabilized colloidal particles and nonadsorbing ideal polymer coils, both in bulk and adsorbed against a planar hard wall. By integrating out the degrees of freedom of the polymer coils, we derive a formal expression for the effective one-component Hamiltonian of the colloids. We employ an efficient Monte Carlo simulation scheme for this mixture based on the exact effective colloid Hamiltonian; i.e., it incorporates all many-body interactions. The many-body character of the polymer-mediated effective interactions between the colloids yields bulk phase behavior and adsorption phenomena that differ substantially from those found for pairwise simple fluids. We determine the phase behavior for size ratios q=sigma(p)/sigma(c)=1, 0.6, and 0.1, where sigma(c) and sigma(p) denote the diameters of the colloids and polymer coils, respectively. For q=1 and 0.6, we find both a fluid-solid and a stable colloidal gas-liquid transition with an anomalously large bulk liquid regime caused by the many-body interactions. We compare the phase diagrams obtained from simulations with the results of the free-volume approach and with direct simulations of the true binary mixture. Although we did not simulate the polymer coils explicitly, we are able to obtain the three partial structure factors and radial distribution functions. We compare our results with those obtained from density functional theory and the Percus-Yevick approximation. We find good agreement between all results for the structure. We also study the mixture in contact with a single hard wall for q=1. Upon approach of the gas-liquid binodal, we find far from the triple point, three layering transitions in the partial wetting regime.

Thermodynamically self-consistent integral equation theory for pair-correlation functions of molecular fluids-II

A closure for the pair-correlation functions of molecular fluids is described in which the hypernetted-chain and the Percus–Yevick approximations are “mixed” as a function of interparticle separation. An adjustable parameter α in the mixing function is used to enforce thermodynamic consistency, by which it is meant that identical results are obtained when the equations of state are calculated via the virial and compressibility routes, respectively. The mixed integral equation for the pair-correlation functions has been solved for two model fluids: (i) a fluid of the hard Gaussian overlap model, and (ii) a fluid the molecules of which interact via a modified Gay–Berne model potential. For the modified Gay–Berne fluid we have slightly modified the original Gay–Berne potential to study the effect of attraction on hard core systems. The pair-correlation functions of the isotropic phase which enter in the density-functional theory as input informations have been calculated from the integral equation theories for these model fluids. We have used two different versions of the density-functional theory known as the second order and modified weighted-density-functional theory to locate the isotropic-nematic (I–N) transitions and calculate the values of transition parameters for the hard Gaussian overlap and modified Gay–Berne model fluids. We have compared our results with those of computer simulations wherever they are available. We find that the density-functional theory is good to study the I–N transition in molecular fluids if the values of the pair-correlation functions in the isotropic phase are accurately known.

Molecular correlation functions for uniaxial ellipsoids in the isotropic state

We perform event-driven molecular dynamics simulations of a system composed by uniaxial hard ellipsoids for different values of the aspect ratio and packing fraction. We compare the molecular orientational-dependent structure factors previously calculated within the Percus-Yevick approximation with the numerical results. The agreement between theoretical and numerical results is rather satisfactory. We also show that, for specific orientational quantities, the molecular structure factors are sensitive to the particle shape and can be used to distinguish prolate from oblate ellipsoids. A first order theoretical expansion around the spherical shape and a geometrical analysis of the configurations confirms and explains such an observation.

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.

Equation of state of the hard-disk fluid on a sphere from Percus–Yevick equation

The Percus–Yevick equation is solved numerically for hard disks on a sphere. The effect of curvature of the hosting surface is calculated, and an empirical correction to the equation of state is proposed.

Structure of penetrable-rod fluids: Exact properties and comparison between Monte Carlo simulations and two analytic theories

Bounded potentials are good models to represent the effective two-body interaction in some colloidal systems, such as the dilute solutions of polymer chains in good solvents. The simplest bounded potential is that of penetrable spheres, which takes a positive finite value if the two spheres are overlapped, being 0 otherwise. Even in the one-dimensional case, the penetrable-rod model is far from trivial, since interactions are not restricted to nearest neighbors and so its exact solution is not known. In this paper the structural properties of one-dimensional penetrable rods are studied. We first derive the exact correlation functions of the penetrable-rod fluids to second order in density at any temperature, as well as in the high-temperature and zero-temperature limits at any density. It is seen that, in contrast to what is generally believed, the Percus-Yevick equation does not yield the exact cavity function in the hard-rod limit. Next, two simple analytic theories are constructed: a high-temperature approximation based on the exact asymptotic behavior in the limit T--> infinity and a low-temperature approximation inspired by the exact result in the opposite limit T--> 0. Finally, we perform Monte Carlo simulations for a wide range of temperatures and densities to assess the validity of both theories. It is found that they complement each other quite well, exhibiting a good agreement with the simulation data within their respective domains of applicability and becoming practically equivalent on the borderline of those domains. A comparison with numerical solutions of the Percus-Yevick and the hypernetted-chain approximations is also carried out. Finally, a perspective on the extension of our two heuristic theories to the more realistic three-dimensional case is provided.

Structural properties of liquid cesium metal by application of cohesive energy density

The structural property of liquid cesium is investigated in the temperature range 900–1900 K by application of the semiempirical effective Lennard–Jones (8.5–4) pair potential function and employing Gillan's algorithm to solve Percus–Yevick equation. The potential function has been derived accurately by application of cohesive energy density in a wide range of pressure–density–temperature ( PρT) data including data at the proximity of absolute zero temperature. The method is very much responsive to the liquid dynamics and leads to indication of three distinct ranges of metal, nonmetal, and metal–nonmetal transition states. The resulted pair correlation functions are well compared with the reported experimental and first-principle molecular dynamic. The calculated coordination numbers in the liquid range are in agreement with experiment particularly at low temperatures, though it is singular at about 1400 K. This observation is similar to the change from one liquid structure to another one and is verified by the heights of the first peak of experimental pair correlation function as a function of temperatures. At T = 973 K, the position of the first peak ( r = 5.27 Å ) is in agreement well with experimental ( r = 5.31 Å ) and with the first-principle DFT molecular dynamics ( r = 5.24 Å ).

Ionic interaction and conductivity of metallic hydrogen

We calculate the electroresistivity of metallic hydrogen within the framework of perturbation theory in electronproton interaction. To this end we employ the Kubo linear response theory while using the two-time retarded Green functions method to calculate the relaxation time. The expressions for the second and third order contributions are given. To describe the electron subsystem, the random phase approximation is used, allowing for the exchange interactions and correlations in a local field approximation. Thermodynamics of the proton subsystem is assumed to be given by the Percus-Yevick equation. At a given density and temperature the only parameter of the theory is the hard sphere diameter, which is calculated from effective pair ionic interaction. For a completely degenerated electron gas, the latter is determined by the density of the system. The dependence of the second and the third order contributions on the parameters of the theory is investigated. For all densities and temperatures examined here the third order contribution constitutes more than half of the second order term. The corresponding magnitude of resistivity is about 100 � 250µ cm.

Thermodynamic instabilities of a binary mixture of sticky hard spheres

The thermodynamic instabilities of a binary mixture of sticky hard spheres (SHS) in the modified mean spherical approximation (mMSA) and the Percus-Yevick (PY) approximation are investigated using an approach devised by Chen and Forstmann [corrected] [J. Chem. Phys. [corrected] 97, 3696 (1992)]. This scheme hinges on a diagonalization of the matrix of second functional derivatives of the grand canonical potential with respect to the particle density fluctuations. The zeroes of the smallest eigenvalue and the direction of the relative eigenvector characterize the instability uniquely. We explicitly compute three different classes of examples. For a symmetrical binary mixture, analytical calculations, both for mMSA and for PY, predict that when the strength of adhesiveness between like particles is smaller than the one between unlike particles, only a pure condensation spinodal exists; in the opposite regime, a pure demixing spinodal appears at high densities. We then compare the mMSA and PY results for a mixture where like particles interact as hard spheres (HS) and unlike particles as SHS, and for a mixture of HS in a SHS fluid. In these cases, even though the mMSA and PY spinodals are quantitatively and qualitatively very different from each other, we prove that they have the same kind of instabilities. Finally, we study the mMSA solution for five different mixtures obtained by setting the stickiness parameters equal to five different functions of the hard sphere diameters. We find that four of the five mixtures exhibit very different type of instabilities. Our results are expected to provide a further step toward a more thoughtful application of SHS models to colloidal fluids.

Hard-sphere radial distribution function again

A theoretically based closed-form analytical equation for the radial distribution function, g(r), of a fluid of hard spheres is presented and used to obtain an accurate analytic representation. The method makes use of an analytic expression for the short- and long-range behaviors of g(r), both obtained from the Percus-Yevick equation, in combination with the thermodynamic consistency constraint. Physical arguments then leave only three parameters in the equation of g(r) that are to be solved numerically, whereas all remaining ones are taken from the analytical solution of the Percus-Yevick equation.

Effect of shape anisotropy on the phase diagram of the Gay-Berne fluid

We have used the density functional theory to study the effect of molecular elongation on the isotropic-nematic, isotropic-smectic A and nematic-smectic A phase transitions of a fluid of molecules interacting via the Gay-Berne intermolecular potential. We have considered a range of length-to-width parameter 3.0 < or = x(0) < or = 4.0 in steps of 0.2 at different densities and temperatures. Pair correlation functions needed as input information in density functional theory are calculated using the Percus-Yevick integral equation theory. Within the small range of elongation, the phase diagram shows significant changes. The fluid at low temperature is found to freeze directly from isotropic to smectic A phase for all the values of x(0) considered by us on increasing the density while the nematic phase stabilizes in between isotropic and smectic A phases only at high temperatures and densities. Both isotropic-nematic and nematic-smectic A transition density and pressure are found to decrease as we increase x(0). The phase diagram obtained is compared with computer simulation result of the same model potential and is found to be in good qualitative agreement.

Structure of Nonuniform Hard Sphere Fluids from Shifted Linear Truncations of Functional Expansions

Percus showed that approximate theories for the structure of nonuniform hard sphere fluids can be generated by linear truncations of functional expansions of the nonuniform density rho(r) about that of an appropriately chosen uniform system. We consider the most general such truncation, which we refer to as the shifted linear response (SLR) equation, where the density response rho(r) to an external field phi(r) is expanded to linear order at each r about a different uniform system with a locally shifted chemical potential. Special cases include the Percus-Yevick (PY) approximation for nonuniform fluids, with no shift of the chemical potential, and the hydrostatic linear response (HLR) equation, where the chemical potential is shifted by the local value of phi(r). The HLR equation gives exact results for very slowly varying phi(r) and reduces to the PY approximation for hard core phi(r), where generally accurate results are found. We show that a truncated expansion about the bulk density (the PY approximation) also gives exact results for localized fields that are nonzero only in a "tiny" region whose volume V(phi) can accommodate at most one particle. The SLR equation can also exactly describe a limit where the fluid is confined by hard walls to a very narrow slit. This limit can be related to the localized field limit by a simple shift of the chemical potential, leading to an expansion about the ideal gas. We then try to develop a systematic way of choosing an optimal local shift in the SLR equation for general phi(r) by requiring that the predicted rho(r) is insensitive to small variations about the appropriate local shift, a property that the exact expansion to all orders would obey. The resulting insensitivity criterion (IC) gives a theory that reduces to the HLR equation for slowly varying phi(r) and is much more accurate than HLR both for very narrow slits, where the IC agrees with exact results, and for fields confined to "tiny" regions, where the IC gives very accurate (but not exact) results. However, the IC is significantly less accurate than the PY and HLR equations for single hard core fields. Only a small change in the predicted reference density is needed to correct this remaining limit.

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].

Effective interaction between two giant spheres suspended in a size polydisperse hard-sphere fluid

Analytic expressions for the Laplace transform of the interaction energy and force between two exceedingly large hard spheres at infinite dilution in a polydisperse hard-sphere suspending fluid are presented. The equations are based on the Percus–Yevick approximation for the many-component suspending fluid, supplemented by the hypernetted chain approximation for the correlation function of the suspended spheres. By applying the Derjaguin approximation, the energy and force results for two spheres are related to the energy per unit area and the disjoining pressure between two flat walls suspended in a polydisperse fluid. Numerical results for the representative Schultz distributions of the diameters of the species comprising the suspending fluid are presented and discussed.

Ordering of droplets and light scattering in polymer dispersed liquid crystal films

We study the effects of droplet ordering in initial optical transmittance through polymerdispersed liquid crystal (PDLC) films prepared in the presence of an electrical field. Theexperimental data are interpreted by using a theoretical approach to light scattering inPDLC films that explicitly relates optical transmittance and the order parameters,characterizing both the orientational structures inside bipolar droplets and orientationaldistribution of the droplets. The theory relies on the Rayleigh–Gans approximation anduses the Percus–Yevick approximation to take into account the effects due to dropletpositional correlations.

Structure factors and phonon dispersion in liquid Li0.61Na0.39 alloy

The phonon spectra for liquid Li and Na have been computed through the phenomenological model of Bhatia and Singh for disordered systems like liquids and glasses and the obtained results have been compared with the available data obtained by inelastic neutron scattering (INS) and inelastic X-ray scattering (IXS) experiments. The effective pair potentials and their space derivatives are important ingredients in the computation of the dispersion curves. The pair potentials are obtained using the pseudo-potential theory. The empty core model proposed by Ashcroft is widely used for pseudo-potential calculations for alkali metals. But, it is thought to be unsuitable for Li because of its simple 1s electronic structure. However, it can be used with an additional term known as Born-Mayer (BM) core term. The influence of the BM core term on the phonon dispersion is discussed. The same pseudo-potential formalism has been employed to obtain the dispersion relation in liquid Li0.61Na0.39 alloy. Apart from the phonon spectra, the Ashcroft-Langreth structure factors in the alloy are derived in the Percus-Yevick approximation.

Ornstein-Zernike equation and Percus-Yevick theory for molecular crystals.

We derive the Ornstein-Zernike equation for molecular crystals of axially symmetric particles and apply the Percus-Yevick approximation. The one-particle orientational distribution function rho((1)) (Omega) has a nontrivial dependence on the orientation Omega, in contrast to a liquid, and is needed as an input. Despite some differences, the Ornstein-Zernike equation for molecular crystals has a similar structure as for liquids. We solve both equations numerically for hard ellipsoids of revolution on a simple cubic lattice. Compared to molecular liquids, the orientational correlators in direct and reciprocal space exhibit less structure. However, depending on the lengths a and b of the rotation axis and the perpendicular axes of the ellipsoids, respectively, different behavior is found. For oblate and prolate ellipsoids with b greater, similar 0.35 (in units of the lattice constant), damped oscillations in distinct directions of direct space occur for some of the orientational correlators. They manifest themselves in some of the correlators in reciprocal space as a maximum at the Brillouin zone edge, accompanied by a maximum at the zone center for other correlators. The oscillations indicate alternating orientational fluctuations, while the maxima at the zone center originate from ferrorotational fluctuations. For a less, similar 2.5 and b less, similar 0.35, the oscillations are weaker, leading to no marked maxima at the Brillouin zone edge. For a greater, similar 3.0 and b less, similar 0.35, no oscillations occur any longer. For many of the orientational correlators in reciprocal space, an increase of a at fixed b or vice versa leads to a divergence at the zone center q=0, consistent with the formation of ferrorotational long-range fluctuations, and for some oblate and prolate systems with b less, similar 1.0 a simultaneous tendency to divergence of few other correlators at the zone edge is observed. Comparison of the orientational correlators with those from Monte Carlo simulations shows satisfactory agreement. From these simulations we also obtain a phase boundary in the a-b plane for order-disorder transitions.

Structure of Nanometric Silica Clusters in Polymeric Composite Materials

Composite films were prepared from nanometric silica suspension and poly(vinyl alcohol) (PVA) solution. Both were mixed at increasing silica contents and left until complete evaporation of the solvent. Small angle neutronscattering (SANS) and transmission electron microscopy (TEM) were used to characterize the local and mean particle distribution of silica particles in the film. The interaction between the polymer and the surface was characterized through an adsorption isotherm. The SANS results were processed within the Percus-Yevick approximation. At a given drying stage, the colloidal system undergoes a microphase separation and submicron size clusters are produced. Both SANS and TEM showed the presence of dense and well-ordered clusters in the dry film. The size of the clusters and the surface-to-surface distance between the silica primary particles within the clusters both depend on the initial ionic strength.