Percus-Yevick Approximation Research Articles

Replica Ornstein-Zernike self-consistent theory for mixtures in random pores.

We present a self-consistent integral equation theory for a binary liquid in equilibrium with a disordered medium, based on the formalism of the replica Ornstein-Zernike (ROZ) equations. Specifically, we derive direct formulas for the chemical potentials and the zero-separation theorems (the latter provide a connection between the chemical potentials and the fluid cavity distribution functions). Next we solve a modified-Verlet closure to ROZ equations, which has built-in parameters that can be adjusted to satisfy the zero-separation theorems. The degree of thermodynamic consistency of the theory is also kept under control. We model the binary fluid in random pores as a symmetrical binary mixture of nonadditive hard spheres in a disordered hard-sphere matrix and consider two different values of the nonadditivity parameter and of the quenched matrix packing fraction, at different mixture concentrations. We compare the theoretical structural properties as obtained through the present approach with Percus-Yevick and Martinov-Sarkisov integral equation theories, and assess both structural and thermodynamic properties by performing canonical standard and biased grand canonical Monte Carlo simulations. Our theory appears superior to the other integral equation schemes here examined and provides reliable estimates of the chemical potentials. This feature should be useful in studying the fluid phase behavior of model adsorbates in random pores in general.

Effective Depletion Potential of Colloidal Spheres

A new semianalytical method, which is a combination of thedensity functional theory with Rosenfeld density functional andthe Ornstein–Zernike equation, is proposed for the calculationof the effective depletion potentials between a pair of bigspheres immersed in a small hard sphere fluid. The calculatedresults are almost identical to the integral equation method withthe Percus–Yevick approximation, and are also in agreement well with theMonte Carlo simulation results.

The structure factor and equation of state of hard-sphere fluids

In a paper by Verberg et al (2002 Dynamics: Models and Kinetic Methods for Non-Equilibrium ManyBody Systems ed J Karkheck (Dordrecht: Kluwer)) it was shown that the static structure factorS(k) of a hard-sphere fluid plays an important role in the computation of the transportproperties of colloidal suspensions. To perform actual calculations, those authors used asemiempirical form due to Grundke and Henderson which stems out of the Percus–Yevickapproximation. In this work we present calculations of the static structure factor of ahard-sphere fluid within the rational function approximation method (Bravo Yuste et al 1996 Phys. Rev. E 53 4820) which is wholly compatible with the equation ofstate used for the fluid. The Carnahan–Starling equation of state (Carnahanand Starling 1969 J. Chem. Phys. 51 635), the Padé (4, 3) of van Rensburg (1993J. Phys. A: Math. Gen. 26 4805) and Sánchez (1994 J. Chem. Phys. 101 7003), the equation ofstate of Malijevský and Veverka (1999 Phys. Chem. Chem. Phys. 1 4267) and a recent proposalby us (Robles and López de Haro 2003 Europhys. Lett. 62 56) will be explicitlyexamined.

Analytic Solution to Integral Equations of Liquid State Theories for Potentials with a Hard Core at Low Densities

We present in this paper a general analytical solution to the integral equations of liquid state theories (Born–Green–Yvon, hyper-netted-chain, and Percus–Yevick Equations) at low-density limit for potentials with a hard core. For the specific case of the Lennard-Jones potential with a hard core, we have derived an analytical function for the radial distribution function at high temperature and low density. We have noted that this function has two humps which is the characteristic feature of the radial distribution function at low densities. In addition, this function has been used to calculate the third virial coefficient for such a fluid exactly. We see that for the especial case of Lennard-Jones fluid with a hard core, which its radial distribution function has explicitly been calculated at high temperatures, the correct behavior of the third virial coefficient with temperature is obtained. The magnitude of hard-core diameter has significant effect on the thermodynamic properties of fluid: for instance...

Equation of state of a seven-dimensional hard-sphere fluid. Percus-Yevick theory and molecular-dynamics simulations.

Following the work of Leutheusser [Physica A 127, 667 (1984)], the solution to the Percus-Yevick equation for a seven-dimensional hard-sphere fluid is explicitly found. This allows the derivation of the equation of state for the fluid taking both the virial and the compressibility routes. An analysis of the virial coefficients and the determination of the radius of convergence of the virial series are carried out. Molecular-dynamics simulations of the same system are also performed and a comparison between the simulation results for the compressibility factor and theoretical expressions for the same quantity is presented.

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.

Direct correlation functions of the Widom–Rowlinson model

We calculate, through Monte Carlo (MC) numerical simulations, the partial total and direct correlation functions of the three dimensional symmetric Widom–Rowlinson mixture. We find that the differences between the partial direct correlation functions from simulation and from the Percus–Yevick approximation (calculated analytically by Ahn and Lebowitz) are well fitted by Gaussians. We provide an analytical expression for the fit parameters as function of the density. We also present MC simulation data for the direct correlation functions of a couple of non-additive hard sphere systems to discuss the modification induced by finite like diameters.

Phase equilibria of bulk and confined colloid–polymer systems

The phase equilibria of aqueous colloid–polymer systems are studied theoretically and experimentally in three and two dimensions. The colloidal interactions are modeled by the depletion model, and the thermodynamic properties are evaluated from the colloid pair correlation functions, obtained in the Percus–Yevick approximation. The phases are identified by direct observation in three dimensions, and by enhanced video microscopy in two dimensions. Comparison is made between these theoretical predictions and experimental results.

Hard-Like Equation of State

In a recent article, we have presented a hard-like equation of state, HLEOS, based on the hard-like behavior of fluids at high densities. It has been shown that the hard-like behavior of any real fluid starts at the common bulk modulus point for each isotherm. To confirm our claim, we have shown that the structure factor of real fluids for such conditions may be predicted just using the Percus–Yevick equation for hard fluids. We have introduced and presented an analytical expression for effective pair correlation function at contact, g eff (σ), for real fluids that it shows a new corresponding states. The HLEOS predicts thermodynamic properties of all kind of fluids, including non-polar, polar, and hydrogen-bonded systems such as Ar, Xe, CH 4 , DMB, N 2 , C 2 H 4 , CO, C 2 H 5 OH, and H 2 O successfully. The hard-like EOS has two temperature-dependent parameters, which are related to the soft and hardness of the repulsive branch of the potential. This EOS is also able to predict some known regularities fo...

Heat capacity of square-well fluids of variable width

We have obtained by Monte Carlo NVT simulations the constant-volume excess heat capacity of square-well fluids for several temperatures, densities and potential widths. Heat capacity is a thermodynamic property much more sensitive to the accuracy of a theory than other thermodynamic quantities, such as the compressibility factor. This is illustrated by comparing the reported simulation data for the heat capacity with the theoretical predictions given by the Barker-Henderson perturbation theory as well as with those given by a non-perturbative theoretical model based on Baxter's solution of the Percus-Yevick integral equation for sticky hard spheres. Both theories give accurate predictions for the equation of state. By contrast, it is found that the Barker-Henderson theory strongly underestimates the excess heat capacity for low to moderate temperatures, whereas a much better agreement between theory and simulation is achieved with the non-perturbative theoretical model, particularly for small well widths, although the accuracy of the latter worsens for high densities and low temperatures, as the well width increases.

Solution of the Percus-Yevick equation for square well spherocylinders.

The Percus-Yevick equation for square-well spherocylinders has been numerically solved for some selected orientations following a methodology proposed previously for different fluids of elongated molecules. The equation is solved for particles of aspect ratios ranging from L/sigma=0.3 up to L/sigma=5.0, attractive range lambda/sigma=1.5, and packing fractions within eta=0.1-0.3. The resulting pair correlation functions are checked against isothermal-isobaric Monte Carlo simulations and good agreement is found for the short-range structure, at intermolecular distances within one molecular diameter sigma to contact for each of the selected orientations. At larger distances, the integral equation tends to overestimate the pair correlations. The results confirm the prediction of reference-system average Mayer-function perturbation theory for short aspect ratios, reaching the Onsager limit for the greater aspect ratios. Some instabilities of the solution for the longest models and higher densities are tentatively discussed in terms of their possible relation to frustration phenomena found in some polymer and complex systems.

Employing functional counterpart of Lagrangian theorem to improve on density functional theory for density profile of non-uniform fluids

Abstract A recently proposed density functional theory [New J. Phys. 4 (2002) 36], which is based on functional counterpart of Lagrangian theorem of differential calculus, is extended to calculate density distribution of an adhesive hard sphere fluid confined between two hard walls. The needed input parameters, i.e., bulk second order direct correlation function and equation of state, are employed from the analytical Percus–Yevick approximation and the compressibility equation. The present predictions are in very good agreement with available computer simulation data. Then the present formalism is applied to investigate the affecting factor for the adhesive hard sphere fluid adsorption behaviors. The continuity and differentiability of the density functional C (1) ( r ;[ρ]) , which underlies the application of the functional counterpart of Lagrangian theorem of differential calculus, is proven.

Prediction of the viscosity of dense fluid mixtures

The Vesovic-Wakeham (VW) method of predicting the viscosity of dense fluid mixtures has been improved by implementing new mixing rules based on the rigid sphere formalism. The proposed mixing rules are based on both Lebowitz's solution of the Percus-Yevick equation and on the Carnahan-Starling equation. The predictions of the modified VW method have been compared with experimental viscosity data for a number of diverse fluid mixtures: natural gas, hexane + hheptane, hexane + octane, cyclopentane + toluene, and a ternary mixture of hydrofluorocarbons (R32 + R125 + R134a). The results indicate that the proposed improvements make possible the extension of the original VW method to liquid mixtures and to mixtures containing polar species, while retaining its original accuracy.

Structure of nonuniform three-component fluid mixtures: A density-functional approach

The structure of nonuniform three-component fluid mixtures is studied using a self-consistent density-functional approach as well as a simple weighted density approximation and also by computer simulation. The second-order direct correlation function and the bridge function of the corresponding uniform fluid mixture required as input in the self-consistent theory are obtained from integral equation theory using an accurate closure relation, whereas the available analytic expressions of second-order correlation function within the Percus–Yevick approximation is used in the simple weighted density approximation. This new approach is employed to predict the structure of ternary hard sphere fluid mixtures near a hard wall as well as the structure of ternary uniform hard sphere mixtures. The calculated density and concentration profiles of the nonuniform fluid mixtures as well as the radial distribution function profiles of the uniform fluid mixtures are shown to compare well with the computer simulation results performed in the present work.

Statistical mechanics in collective coordinates

We study the transformation of the statistical mechanics of N particles to the statistical mechanics of fields, that are the collective coordinates, describing the system. We give an explicit expression for the functional Fourier transform of the Jacobian of the transformation from particle to collective coordinate and derive the Fokker–Planck equation in terms of the collective coordinates. Simple approximations, leading to Debye–Huckel theory and to the hard sphere Percus–Yevick equation are discussed.

Contact values of the radial distribution functions of additive hard-sphere mixtures in d dimensions: A new proposal

The contact values gij(σij) of the radial distribution functions of a d-dimensional mixture of (additive) hard spheres are considered. A “universality” assumption is put forward, according to which gij(σij)=G(η,zij), where G is a common function for all the mixtures of the same dimensionality, regardless of the number of components, η is the packing fraction of the mixture, and zij=(σiσj/σij)〈σd−1〉/〈σd〉 is a dimensionless parameter, 〈σn〉 being the nth moment of the diameter distribution. For d=3, this universality assumption holds for the contact values of the Percus–Yevick approximation, the scaled particle theory, and, consequently, the Boublík–Grundke–Henderson–Lee–Levesque approximation. Known exact consistency conditions are used to express G(η,0), G(η,1), and G(η,2) in terms of the radial distribution at contact of the one-component system. Two specific proposals consistent with the above-mentioned conditions (a quadratic form and a rational form) are made for the z dependence of G(η,z). For one-dimensional systems, the proposals for the contact values reduce to the exact result. Good agreement between the predictions of the proposals and available numerical results is found for d=2, 3, 4, and 5.

Thermodynamic and structural properties of a sticky hard-sphere heteronuclear dimer fluid

The thermodynamic and structural properties of a sticky hard-sphere heteronuclear dimer fluid are studied using the multidensity Ornstein–Zernike integral equation theory. In this heteronuclear dimer fluid, each molecule consists of two sticky hard-sphere beads with identical diameters but interact with different strengths of stickiness. In this theoretical formalism, a monodisperse system of the heteronuclear sticky dimer is modeled as an equal molar binary mixture of associating sticky hard spheres (with different sticky strengths) in the complete association limit. A general analytical solution to the model was obtained within the Percus–Yevick and polymer Percus–Yevick approximations for any degree of association including the complete association limit. Explicit analytical expressions for the contact values of correlation functions are obtained. Furthermore, correlation functions beyond the hard-core region are calculated. The Helmholtz energy, Gibbs energy, and pressure of the heteronuclear dimer are obtained via the energy route. In addition, the critical temperature, critical density, and phase coexistence of the fluid are also computed.

Depletion interactions in suspensions of spheres and rod–polymers

Liquid-state integral equation methods are employed to study the thermodynamic and structural properties of ideal and repelling rigid rods mixed with hard spheres in the limits when one of the species is dilute. The role of rod aspect ratio and sphere/rod size asymmetry is explored over a wide range of system parameters encompassing the colloid, nanoparticle, and crossover regimes. Novel predictions are found for the polymer (sphere) mediated depletion potentials and second virial coefficients of particles (rods) in dense polymer (sphere) suspensions. The adequacy of the closure approximations employed is tested by comparison with available numerical calculations and more rigorous theories in special limits. The liquid-state theory appears to be accurate for all properties in the nanoparticle regime and for the insertion chemical potential of needles and spherocylinders. However, it significantly underestimates depletion attractions effects in the colloidal regime of short rods and large spheres due to nonlocal entropic repulsion effects between polymers and particles not captured by the classic Percus–Yevick approximation.

Modeling strategies of ultrasound backscattering by blood

Tissue characterization using ultrasound (US) scattering can allow the identification of relevant cellular biophysical information noninvasively. The characterization of the level of red blood cell (RBC) aggregation is one of the proposed applications. Different modeling strategies have been investigated by our group to better understand the mechanisms of US backscattering by blood, and to propose relevant measurable indices of aggregation. It could be hypothesized from these studies that the microstructure formed by RBC clusters is a main determinant of US backscattered power. The structure factor, which is related to the Fourier transform of the microscopic density function of RBCs, is described and used to explain the scattering behavior for different spatial arrangements of nonaggregated and aggregated RBCs. The microscopic density function was described by the Percus–Yevick approximation (nonaggregated RBCs), and for aggregated RBCs, by the Poisson distribution, the Neyman–Scott point process, and very recently by a flow-dependent rheological model. These statistical and microrheological models allowed the study of US backscattered power as a function of the hematocrit, scatterers’ size, insonification frequency, and level of RBC aggregation. Experimental results available from the literature were used to validate the different approaches. [Work supported by Canadian Institutes of Health Research (MOP-36467), HSFQ, FCAR, and FRSQ.]

The collective coordinates Jacobian

We develop an expansion for the Jacobian of the transformation from particle coordinates to collective coordinates. As a demonstration, we use the lowest order of the expansion in conjunction with a variational principle to obtain the Percus Yevick equation for a monodisperse hard sphere system and the Lebowitz equations for a polydisperse hard sphere system.