Percus-Yevick Approximation Research Articles

Effective medium theory of light scattering in polymer dispersed liquid crystal films

We present calculations of the optical properties of PDLC films subjected to normally incident unpolarized light, in the long-wavelength régime under which the Rayleigh-Gans approximation applies. The calculations use an effective medium theory to take account of dependent scattering effects due to close packing of droplets, and use the Percus-Yevick approximation for hard spheres to account for interference effects. The director configurations inside the droplets are found using the effective medium theory which we have developed in earlier papers. Typical results are presented for radial and bipolar droplets. Detailed quantitative results have been derived for the optical characteristics in the presence of an applied field for the technologically important case of PDLC films containing partially ordered bipolar droplets.

Replica Ornstein-Zernike equations and the structure of a simple fluid mixture in disordered porous media: application to a composite monomolecular adlayer

A two-dimensional (2D) model for a simple fluid mixture, consisting of spherical particles with different diameters that is adsorbed on a disordered porous medium, i.e., in a plane claiming hard discs in a frozen configuration, is studied by means of replica Ornstein–Zernike equations. The hypernetted chain and Percus–Yevick approximations are used. The structural and thermodynamic properties of the composite monomolecular adlayer are addressed, and the density and composition of the monolayer and their dependence on the presence of a disordered quenched 2D matrix in the plane of absorption at a fixed chemical potential is examined. It is shown that porous 2D media exhibit the effect of selectivity. The concentration of the smaller particles is larger on the porous media compared with the bulk phase. The model and methodology can be extended to study phase transitions in 2D partly quenched systems.

The mean spherical model for a Lorentz-Berthelot mixture of sticky hard spheres

We have analyzed the Percus-Yevick (PY) and the mean spherical model (MSM) equation for an N-component system of sticky hard spheres. The PY approximation leads to a set of N(N+1)/2 coupled quadratic equations for the unknown coefficients. While for this closure, the pair distribution functions have to be calculated numerically, we can proceed in the MSM one step further if we assume a Lorentz-Berthelot-type rule for the interactions: then the structure functions can be calculated analytically. We show that under these conditions in the limit N→∞ (stochastic limit) the analyticity of the solution is preserved. General expressions both for the discrete and continuous (polydisperse) case are presented.

Replica Ornstein-Zernike equations and the structure of a simple fluid mixture in disordered porous media: application to a composite monomolecular adlayer

A two-dimensional (2D) model for a simple fluid mixture, consisting of spherical particles with different diameters that is adsorbed on a disordered porous medium, i.e., in a plane claiming hard discs in a frozen configuration, is studied by means of replica Ornstein–Zernike equations. The hypernetted chain and Percus–Yevick approximations are used. The structural and thermodynamic properties of the composite monomolecular adlayer are addressed, and the density and composition of the monolayer and their dependence on the presence of a disordered quenched 2D matrix in the plane of absorption at a fixed chemical potential is examined. It is shown that porous 2D media exhibit the effect of selectivity. The concentration of the smaller particles is larger on the porous media compared with the bulk phase. The model and methodology can be extended to study phase transitions in 2D partly quenched systems.

A Pulsed Field Gradient NMR Technique for the Determination of the Structure of Suspensions of Non-Brownian Particles with Application to Packings of Spheres

The internal structure of systems of particles in a liquid is studied with a novel NMR technique based on the measurement of the squared modulus of the magnetization in presence of a pulsed field gradient. The formalism is analogous to the one used in classical scattering techniques (light, X-rays, neutrons); it allows similar information to be obtained about the structure (in particular, the structure factorS(q)). The main improvement is that the range of particles sizes is 10 μm to 1 mm, as compared with the range of the scattering techniques (<5 μm). The NMR technique was validated by studying packings of spherical particles of mean diameter 240 μm created by sedimentation. The profile of the experimental squared modulus of the magnetization versus the wave vector provides results for the mean size of particles and the compacity. The main feature is that it depends on the pair distribution function, and the present results are in good agreement with a model based on the Percus–Yevick approximation. This technique is then particularly adapted to systems such as non-Brownian suspensions, fluidized beds, porous media, and sediments.

A Theoretically Improved Perturbation Model for Activity Coefficients of Amino Acids and Peptides in Aqueous Solutions

A theoretically improved model, based on the perturbation theory, to correlate the activity coefficients of amino acids and peptides in aqueous solutions is presented. The radial distribution function for the reference system and for the perturbation term have been obtained from the solution of the Percus-Yevick equation for hard sphere systems. The pair interaction energies required by the model are those due to dispersion forces and dipole-dipole interactions. These interactions are represented by a Lennard-Jones (6-12) and a Keesom expression, respectively. The depth of potential well in the Lennard-Jones model and the size parameter of hard spheres were evaluated by the minimization of the Helmholtz free energy of the systems. The significance of the parameters is justified by comparison of the model with the Gibbs-Bogoliubov inequality. Independently evaluated dipole moments of the amino acids are used. The model can accurately correlate the activity coefficients of amino acids and peptides in aqueous solutions over a wide range of concentration.

Iterative solutions of integral equations and structural stability of fluids

We argue that the threshold density of structural stability, ${\ensuremath{\rho}}_{\mathrm{inst}},$ of a classical fluid can be determined from the Floquet matrix for the iterative form of the integral equation for the pair structure. A measure of the structural stability of the fluid is provided by the Lyapunov exponent related to the perturbed dynamics. The hypernetted-chain and Percus-Yevick equations yield, for hard spheres, a value of ${\ensuremath{\rho}}_{\mathrm{inst}}$ that is about $10%$ smaller than the freezing density.

Adsorption of Dimerizing and Dimer Fluids in Disordered Porous Media

The pair distribution functions and the adsorption isotherms for a hard-core dimerizing fluid in disordered hard-sphere matrices have been determined using the associative extension of replica Ornstein-Zernike equations. Some grand canonical Monte Carlo simulation results for tangent hard-sphere dumbbells adsorbed in hard-sphere matrices are presented. Theoretical predictions for the structure from the solutions of integral equations with the associative hypernetted chain (HNC) and Percus-Yevick (PY) approximation agree with those from the simulations. However, the HNC closure has been found to work better than the PY one.

Structure of multi-component hard-sphere mixtures

A method to obtain (approximate) analytical expressions for the radial distribution functions and structure factors in a multi-component mixture of additive hard spheres is introduced. In this method, only contact values of the radial distribution function and the isothermal compressibility are required and thermodynamic consistency is achieved. The approach is simpler than but yields equivalent results to the Generalized Mean Spherical Approximation. Calculations are presented for a binary and a ternary mixture at high density in which the Boublı́kMansoori-Carnahan-Starling-Leland equation of state is used. The results are compared with the Percus-Yevick approximation and the most recent simulation data.

Note on light attenuation by scattering: comparison of coherent and incoherent (diffusion) approximations

A simple procedure characterizing light attenuation by transmission (based on the diffusion approximation of the transfer theory, the asymmetry factor, and the effective refractive index of the scattering medium) is compared with the coherent approaches based on (1) published Monte Carlo simulations, (2) scattering from a system of closely packed spherical particles with interparticle correlation characterized by the Percus-Yevick approximation, and (3) experimental data. Use of the procedure in the analysis of phase separation in polymer blends is presented. It is shown that the results obtained by this procedure are in reasonable accord with theoretical and experimental data so far published for systems with a volume fraction of minor phase less than ~0.2 and without any specific interaction in the system.

Theoretical studies of the available volume for adsorption in a random quenched and depleted disordered medium

The use of a removable template during synthesis of a disordered porous material can result in a material of greatly differing pore structure. We present here a theoretical study of the influence of a template on the material’s available volume. We model the material as a medium formed by quenching and depleting an equilibrium fluid and develop a theoretical method by which the available volume for adsorption in this model material may be investigated. This method makes use of the known relationship between the direct correlation function and the available volume. An analysis of the diagrams contributing to the available volume shows that two types of field points are present: those corresponding to the density of the medium following depletion (ρ) and those corresponding to the density of the original quenched fluid (ρq). A perturbation term to the available volume is derived that accounts for the diagrams containing ρq field points. The resulting expression contains only the direct and total correlation function; these quantities are known exactly for a hard sphere system within the Percus–Yevick approximation. Using this approach, we show that for materials of a given density, those formed by depletion can possess available volumes enhanced by up to an order of magnitude.

Structure of Liquid Sulfur Through Polymer Percus-Yevick Theory

The static structure factor S(k) for liquid sulfur is calculated via an analytical solution of the Polymer Percus-Yevick (PPY) equation using the fused hard sphere polyatomic model. As the first step in this direction we have applied the ideal chain approximation in which formation of ring polymers is neglected. The chain molecule is represented to be formed by hard sphere beads with two independent attractive sites displaced inside the hard core. The beads adjucent within a given chain penetrate each other at a distance L between their centres. Theoretical results are comparted with experimental data. The theory gives the correct qualitative form for the small and large wavelength region of structure factor. The analysis of the S(k → 0) limit is presented also.

The effective interaction between colloidal hard sphere particles in a polymerizing solvent. Application of Wertheim's theory of association

The effective interaction between colloidal particles in a solvent of hard spheres, which can polymerize with the formation of chains and rings, is studied. Polymerization results in changes of the effective interaction between the colloid particles, compared with the effective interaction for unpolymerized solvents that has been extensively studied in previous publications. To describe the mixture, we use the associative Percus-Yevick approximation for Wertheim's Ornstein-Zernike integral equation. Both the infinite dilution and the nonvanishing concentration limits for the colloid species are considered. It is shown that, in the model of Wertheim for polymerization, the intercolloidal potential of mean force (PMF) depends primarily on the solvent density and to a lesser extent on the degree of polymerization. A depletion of the solvent density around the colloid particles is observed. If the solvent consists of longer chains, compared with hard spheres, the correlations between the colloid particles are of longer range. The oscillations of the PMF depend upon the average chain length but not so strongly as in the model with chains having a fixed number of beads; this is probably due to the flexibility of the chain conformations in Wertheim's model. We also study the effect of the size ratio of the colloid spheres and the solvent monomers. We observe that the polymer hypernetted chain closure is difficult to apply to the systems under study. This closure shares difficulties, especially near the critical point, that are well known in the study of the phase diagram of fluids with both repulsive and attractive interactions. For large colloidal particles, the mixed closure of Henderson is useful.

Influence of spatial correlations on crack bridging by frictional fibres

The effect of fibre interaction on matrix cracking in a unidirectional fibre-reinforced composite is analyzed. It is assumed that the matrix material contains a crack in a plane perpendicular to the fibres. Fibres, remaining intact, debond from the matrix and then act as bridging ligaments in the crack wake. The debonding process is accompanied by frictional sliding governed by a Coulomb friction law. Fibres are considered to be randomly located in the transverse plane. The fibre axial stress and longitudinal displacement are expressed in terms of the solution to a model problem for a single fibre in an ambient stress field due to all other fibres and applied load. The stress field produced by the other fibres is described using an ensemble averaging procedure. The radial distribution function g( r) that provides a quantitative measure of the correlations between the positions of different fibres is evaluated numerically from the Percus-Yevick equation for hard disks. The dependence of the fibre axial stress on the relative fibre-matrix displacement is examined for different values of the volume fraction of fibres. The resulting stress-displacement law is compared with results for other choices of the function g( r) and with a law given by a concentric cylinder model.

Cooperative adsorption of network forming fluids onto crystalline surfaces: Structure and connectivity of the interface

Density profiles and first layer adsorption isotherms of a network forming fluid near a crystalline surface are investigated using the Percus–Yevick approximation for the associative Henderson–Abraham–Barker (HAB) equation. The fluid molecule is represented as a hard sphere with four independent attractive sites. The surface is described as a hard wall with a lattice of adsorbing sites. It is shown that the association induces a new type of cooperativity related to a bridging between the adsorbing sites due to tree-like clusters adsorption. The density profiles calculated exhibit reasonably good agreement with the Monte Carlo computer simulation data. The connectivity properties of the interface are studied and an interfacial percolation threshold is predicted to exist. The mean thickness τ of the adsorbed layer and the connectedness wall-particle correlation functions are calculated via the connectivity analogue of the HAB equation. It is shown that the adlayer may become infinitely thick with the mean cluster size S in the bulk being finite.

Solution of the Percus–Yevick equation for the molecular pair correlation function of a linear solute molecule in a simple liquid

The short-range translational and orientational order around a rigid linear solute molecule immersed in a dense simple liquid is investigated within the Percus–Yevick theory. The orientational correlations induced by the anisotropic solute–solvent interaction are analyzed by a spherical harmonics expansion. The smearing of the local shell structure is studied for increasingly anisotropic model molecules and for different liquid densities.

Second-order Percus-Yevick theory for a confined hard-sphere fluid

A fluid of hard spheres confined between two hard walls and in equilibrium with a bulk hard-sphere fluid is studied using a second-order Percus-Yevick approximation. We refer to this approximation as second-order because the correlations that are calculated depend upon the position of two hard spheres in the confined fluid. However, because the correlation functions depend upon the positions of four particles (two hard spheres and two walls treated as giant hard spheres), this is the most demanding application of the second-order theory that has been attempted. When the two walls are far apart, this calculation reduces to our earlier second-order approximation calculations of the properties of hard spheres near a single hard wall. Our earlier calculations showed this approach to be accurate for the single-wall case. In this work we calculate the density profiles and the pressure of the hard-sphere fluid on the walls. We find, by comparison with grand canonical Monte Carlo results, that the second-order approximation is very accurate, even when the two walls have a small separation. We compare with a singlet approximation (in the sense that correlation functions that depend on the position of only one hard sphere are considered). The singlet approach is fairly satisfactory when the two walls are far apart but becomes unsatisfactory when the two walls have a small separation. We also examine a simple theory of the pressure of the confined hard spheres, based on the usual Percus-Yevick theory of hard-sphere mixtures. Given the simplicity of the latter approach the results of this simple (and explicit) theory are surprisingly good.

On the contact values of the derivatives of the hard-sphere radial distribution function

Approximate analytical expressions for the values of the derivatives of the radial distribution function at contact in a hard-sphere fluid are derived. These expressions are obtained within a rational function approximation (RFA) method recently developed. The results are compared with those based on the Percus-Yevick approximation, as well as with those obtained from expressions for the cavity function inside the core or through computer simulations. While the qualitative dependence on the packing fraction is similar for the values of the first two derivatives at contact in all cases, the one corresponding to the third derivative is definitely different in our approach from that previously reported. The integral moments of the total correlation function as given by the RFA method are also computed.

Adsorption of a Hard Sphere Fluid in a Disordered Polymerized Matrix: Application of the Replica Ornstein–Zernike Equations

A model of hard spheres adsorbed in disordered porous media is studied using the associative replica Ornstein–Zernike (ROZ) equations. Extending previous studies of adsorption in a hard sphere matrices, we investigate a polymerized matrix. We consider an associating fluid of hard spheres with two intracore attractive sites per particle; consequently chains consisting of overlapping hard spheres can be formed due to the chemical association. This is the generalization of the model with sites on the surface of Wertheim that has been studied in the bulk by Chang and Sandler. The matrix structure is obtained in the polymer Percus–Yevick approximation. We solve the ROZ equations in the associative hypernetted chain approximation. The pair distribution functions, the fluid compressibility, the equation of state and chemical potential of the adsorbed fluid are obtained and discussed. It is shown that the adsorption of a hard sphere fluid in a matrix at given density, but consisting of longer chains of overlapping hard spheres, is higher than the adsorption of this fluid in a hard sphere matrix.

Adsorption of fluids in confined disordered media from inhomogeneous replica Ornstein-Zernike equations

The density distribution and pair distribution function for a fluid adsorbed in a slitlike pore filled with a quenched hard-sphere fluid have been investigated using the inhomogeneous replica Ornstein-Zernike equations. The one-particle and two-particle functions are related via Born-Green-Yvon equation and the inhomogeneous Percus-Yevick approximation is used. The effect of the matrix is to lower the amount of adsorbed fluid in the entire pore at low chemical potentials. For high values of the fluid chemical potential, layering of adsorbed fluid is observed. The solution of the problem is important for gel-exclusion chromatography and in separation science.