Percus-Yevick Approximation Research Articles

The reference system for the highly asymmetric electrolyte solutions: The analytical treatment

A multicomponent model of dimerizing hard spheres is proposed for the description of association and clusterization in highly asymmetric electrolyte solutions. The special type of association between small and large spheres permits the larger spheres to be bond with many smaller spheres. General expressions for the Wertheim-Baxter factor correlation functions for this model are obtained in the Wertheim's associative Percus-Yevick approximation. Some numerical calculations illustrate the influence of association on the structural properties of the model.

A small angle x-ray scattering study of the droplet–cylinder transition in oil-rich sodium bis(2-ethylhexyl) sulfosuccinate microemulsions

A method for nonlinear fitting of x-ray scattering data from polydisperse mixtures was developed. It was applied to the analysis of the structural changes in the droplet phase of oil-rich water-in-oil (w/o) sodium bis(2-ethylhexyl) sulfosuccinate (AOT) microemulsions with increasing temperature or upon addition of salt. Data were collected at different temperatures (15 to 60 °C) and salt concentrations (up to 0.6% NaCl) within the one-phase region of the L2 phase (w/o microemulsion) for different droplet sizes (water/AOT molar ratio wo=25 to 56) and concentrations (droplet weight fraction cw=2% to 20%). This allowed us to distinguish between contributions from individual scattering particles, e.g., droplets and cylinders to the total scattering intensity. The complete data set containing over 500 scattering curves could be interpreted by fitting the scattering of weighted sums of AOT covered water droplets, long cylinders, and inverse AOT micelles containing bound water only, to the experimental scattering curves. The polydispersity of the droplets and cylinders is described by Schulz distributions and the interactions between the droplets are calculated using a sticky hard-sphere potential in the Percus–Yevick approximation. The volume fractions of the components, their average sizes and polydispersity, and the stickiness of the water/AOT droplets are determined by a nonlinear fit to the experimental data.

An equation of state à la Carnahan–Starling for a five-dimensional fluid of hard hyperspheres

The equation of state for five-dimensional hard hyperspheres arising as a weighted average of the Percus–Yevick compressibility (35) and virial (25) equations of state is considered. This Carnahan–Starling-type equation turns out to be extremely accurate, despite the fact that both Percus–Yevick equations of state are rather poor.

Simulation studies of nearest-neighbor distribution functions and related structural properties for hard-sphere systems

Molecular dynamics simulations have been carried out to investigate nearest-neighbor distribution functions and closely related quantities for the system of hard-spheres. The nearest-neighbor distribution function and the exclusion probability function were computed to examine the density dependence on the structural ‘void’ and ‘particle’ properties. Simulation results were used to access the applicabilities of various theoretical predictions based on the scaled-particle theory, the Percus-Yevick equation, and the Carnahan-Starling approximation. For lower density systems the three different approximations give the nearest-neighbor distribution functions which are very close to one another and also to the resulting simulation data. Among those theoretical predictions, the Carnahan-Starling approximation gives remarkably good agreement with the simulation data even for higher density systems. Also calculated is the nth moment of the nearest-neighbor distribution functions, in which the corresponding length scale is directly related to the measurement of the characteristic pore-size distribution.

Direct correlation functions and bridge functions in additive hard-sphere mixtures

A method to obtain (approximate) analytical expressions for the radial distribution functions in a multicomponent mixture of additive hard spheres that was recently introduced is used to obtain the direct correlation functions and bridge functions in these systems. This method, which yields results practically equivalent to the generalized mean spherical approximation and includes thermodynamic consistency, is an alternative to the usual integral equation approaches and requires as input only the contact values of the radial distribution functions and the isothermal compressibility. Calculations of the bridge functions for a binary mixture using the Boublik-Mansoori-Carnahan-Starling-Leland equation of state are compared to parallel results obtained from the solution of the Percus-Yevick equation. We find that the conjecture recently proposed by Guzman and del Rio (1998, 98, Molec. Phys., 95, 645), stating that the zeros of the bridge functions occur approximately at the same value of the shifted distance for all pairs of interactions, is at odds with our results. Moreover, in the case of disparate sizes, even the Percus-Yevick bridge functions do not have this property. It is also found that the bridge functions are not necessarily non-positive.

Direct correlation functions and bridge functions in additive hard-sphere mixtures

A method to obtain (approximate) analytical expressions for the radial distribution functions in a multicomponent mixture of additive hard spheres that was recently introduced is used to obtain the direct correlation functions and bridge functions in these systems. This method, which yields results practically equivalent to the Generalized Mean Spherical Approximation and includes thermodynamic consistency, is an alternative to the usual integral equation approaches and requires as input only the contact values of the radial distribution functions and the isothermal compressibility. Calculations of the bridge functions for a binary mixture using the Boubl\'{\i}k-Mansoori-Carnahan-Starling-Leland equation of state are compared to parallel results obtained from the solution of the Percus-Yevick equation. We find that the conjecture recently proposed by Guzm\'{a}n and del R\'{\i}o (1998, {\em Molec. Phys.}, {\bf 95}, 645) stating that the zeros of the bridge functions occur approximately at the same value of the shifted distance for all pairs of interactions is at odds with our results. Moreover, in the case of disparate sizes, even the Percus-Yevick bridge functions do not have this property. It is also found that the bridge functions are not necessarily non-positive.

Theoretical Estimation of the Solubility of Oxygen in Silicon Melt

The solubility of oxygen in silicon melt has been calculated by applying the scaled particle theory of the Percus-Yevick equation for the liquid state. The calculated value is found to agree semi-quantitatively well with experimental data for oxygen solubility in silicon melt, when considering a crude but useful approximation in which an solute oxygen atom and the surrounding silicon atoms in the nearest neighbor shell are assigned as mixed hard sphere solvent atoms.

Local density enhancement in dilute supercritical solutions

We study the structure around a dilute Lennard-Jones (LJ) solute in a supercritical LJ fluid using integral equation theories and Monte Carlo simulation. We show that the Percus–Yevick approximation can be seriously in error in the near-critical region. The solute's coordination number as a function of the solvent density can be nearly linear, or quite non-linear, depending on the solute–solvent interaction. These results are rationalized as a crossover from a low-density energy-dominated regime to a high-density entropy-(packing-) dominated regime, and are discussed within the context of the local density enhancement concept.

Density functional for hard sphere crystals: A fundamental measure approach

A new free energy density functional for hard spheres is presented, along the lines of the fundamental measure theory, which reproduces the Percus-Yevick equation of state and direct correlation function for the fluid, with a tensor weighted density. The functional, based on the zero-dimension limit, is exact for any one-dimensional density distribution of the spheres. The application to the hard sphere crystals gives excellent results, solving all of the qualitative problems of previous density functional approximations, including the unit cell anisotropy in the fcc lattice and the description of the metastable bcc lattice.

Effects of polydispersity in quenched-annealed fluids: An integral equation approach

An extension of the replica Ornstein-Zernike (ROZ) equations for partly quenched polydisperse systems is presented. Explicit calculations have been performed for a monodisperse hard sphere fluid confined by a polydisperse hard sphere disordered matrix by using Percus-Yevick and hypernetted chain (HNC) approximations. The chemical potential of adsorbed fluid species has been evaluated. A numerical solution of the ROZ equations makes use of the orthonormal polynomials with the weight function corresponding to the distribution function of the diameters of matrix species. We have also compared the results of theoretical predictions with Monte Carlo simulation in a canonical ensemble. The result of this comparison suggests that the HNC approximation performs slightly better in predicting the structural properties of the system.

Phase behaviour and structure of model colloid-polymer mixtures

We study the phase behaviour and structure of model colloid-polymer mixtures. By integrating out the degrees of freedom of the non-adsorbing ideal polymer coils, we derive a formal expression for the effective one-component Hamiltonian of the colloids. Using the two-body (Asakura-Oosawa pair potential) approximation to this effective Hamiltonian in computer simulations, we determine the phase behaviour for size ratios q = p/c = 0.1, 0.4, 0.6, and 0.8, where c and p denote the diameters of the colloids and the polymer coils, respectively. For large q, we find both a fluid-solid and a stable fluid-fluid transition. However, the latter becomes metastable with respect to a broad fluid-solid transition for q0.4. For q = 0.1 there is a metastable isostructural solid-solid transition which is likely to become stable for smaller values of q. We compare the phase diagrams obtained from simulation with those of perturbation theory using the same effective one-component Hamiltonian and with the results of the free-volume approach. Although both theories capture the main features of the topologies of the phase diagrams, neither provides an accurate description of the simulation results. Using simulation and the Percus-Yevick approximation we determine the radial distribution function g(r) and the structure factor S(k) of the effective one-component system along the fluid-solid and fluid-fluid phase boundaries. At state-points on the fluid-solid boundary corresponding to high colloid packing fractions (packing fractions equal to or larger than that at the triple point), the value of S(k) at its first maximum is close to the value 2.85 given by the Hansen-Verlet freezing criterion. However, at lower colloid packing fractions freezing occurs when the maximum value is much lower than 2.85. Close to the critical point of the fluid-fluid transition we find Ornstein-Zernike behaviour and at very dilute colloid concentrations S(k) exhibits pronounced small-angle scattering which reflects the growth of clusters of the colloids. We compare the phase behaviour of this model with that found in studies of additive binary hard-sphere mixtures.

Fluids of hard ellipsoids: Phase diagram including a nematic instability from Percus-Yevick theory.

An important aspect of molecular fluids is the relation between orientation and translation parts of the two-particle correlations. Especially, a detailed knowledge of the influence of orientation correlations is needed to explain and calculate in detail the occurrence of a nematic phase. The simplest model system that shows both orientation and translation correlations is a system of hard ellipsoids. We investigate an isotropic fluid formed of hard ellipsoids with the Percus-Yevick theory. Solving the Percus-Yevick equations self-consistently and accurately in the high density regime gives, contrary to previous works, a clear criterion for a nematic instability. We calculate in detail the equilibrium phase diagram for a fluid of hard ellipsoids of revolution. Our results compare well with Monte Carlo simulations and density-functional theory.

A scaling approximation for structure factors in the integral equation theory of polydisperse nonionic colloidal fluids

The integral equation theory of pure liquids, combined with a new “scaling approximation” based on a corresponding states treatment of pair correlation functions, is used to evaluate approximate structure factors for colloidal fluids constituted of uncharged particles with polydispersity in size and energy parameters. Both hard sphere and Lennard-Jones interactions are considered. For polydisperse hard spheres, the scaling approximation is compared to theories utilized by small angle scattering experimentalists (decoupling approximation and local monodisperse approximation) and to the van der Waals one-fluid theory. The results are tested against predictions from analytical expressions, exact within the Percus–Yevick approximation. For polydisperse Lennard-Jones particles, the scaling approximation, combined with a “modified hypernetted chain” integral equation, is tested against molecular dynamics data generated for the present work. Despite its simplicity, the scaling approximation exhibits a satisfactory performance for both potentials, and represents a considerable improvement over the above mentioned theories. Shortcomings of the proposed theory, its applicability to the analysis of experimental scattering data, and its possible extensions to different potentials are finally discussed.

Integral equation theory for fluids ordered by an external field: separable interactions.

The structural and thermodynamical properties of classical fluids orientationally ordered by an external field are investigated by means of integral equation theories. A general theoretical framework for handling these theories is developed and detailed for the particular case of separable interactions between fluid particles. This approach is then illustrated for the case of two (off lattice) models: the ferromagnetic Heisenberg model and a simple liquid crystal model, for which the numerical solution of integral equations such as the Percus-Yevick, the hypernetted chain, and the reference hypernetted chain closure equations are presented and compared with Monte Carlo simulation results and the analytical solution of the mean spherical approximation. The zero-field case is also examined, and the spontaneous ordering is analyzed in detail, mainly in what concerns the appearance of infinite wavelength singularity in the Ornstein-Zernike equation and the relation with the one-body closure equations and the long range orientational ordering that occurs. In particular, it is shown that the Wertheim one-body closure equation appears as a sum rule compatible with the Ornstein-Zernike equation. The relation between the elastic constant and the long range tail of the pair correlation function is made explicit. In particular, the long range behavior of the various terms in the expansion of the pair correlation function is depicted. The numerical investigation of the two models shows that it is not possible to discriminate between the four integral equations, as to which one would be the most accurate in all cases. The general trends in the thermodynamical and structural properties seem to indicate that the Percus-Yevick approximation is generally better in the strong ordering case, whereas the reference hypernetted chain approximation might be better suited to the study of the isotropic phase and the low ordering regimes.

Hard-sphere colloids in chemically reacting solvent: Smith–Nezbeda model

The solvent mediated force between the hard sphere colloids in Smith–Nezbeda solvent is studied by means of the Percus–Yevick approximation. The oscillatory structure of the potential of mean force is found to be diminished with decreasing temperature. The agreement of the theory with simulation data is sufficiently well.

Pair and triplet correlation entropies of the hard sphere fluid

In this paper we have calculated the pair and triplet correlation entropies of the hard sphere fluid based on the Percus–Yevick approximation and the generalized mean spherical approximation. There is no difference between the two methods in the entropy calculations. Our results are in good agreement with Monte Carlo simulations over a range of thermodynamic states.

Connectedness-in-probability and continuum percolation of adhesive hard spheres: Integral equation theory

Integral equation theory was employed to study continuum percolation and clustering of adhesive hard spheres based on a “connectedness-in-probability” criterion. This differs from earlier studies in that an “all-or-nothing” direct connectivity criterion was used. The connectivity probability may be regarded as a “hopping probability” that describes excitation that passes from one particle to another in complex fluids and dispersions. The connectivity Ornstein–Zernike integral equation was solved for analytically in the Percus–Yevick approximation. Percolation transitions and mean size of particle clusters were obtained as a function of connectivity probability, stickiness parameter, and particle density. It was shown that the pair-connectedness function follows a delay-differential equation which yields analytical expressions in the Percus–Yevick theory.

Stability of Colloids in Chemically Reacting Network-Forming Solvent

We consider a model for a two-component solution which consists of a chemically reacting solvent and a chemically inert hard-sphere solute. A directional associative interaction between the solvent molecules promotes the formation of a network. A multidensity associative Ornstein–Zernike equation is solved analytically using the Percus–Yevick approximation. On the basis of the analysis of the osmotic compressibility, an instability curve for different parameters of the model dispersion is constructed. It is shown that concentration and size of colloids affect the formation of associative complexes in a solvent and directs in such a way the location and behavior of the spinodal curves. The effective interaction between colloids is calculated and discussed.

Effective interaction between hard sphere colloidal particles in a polymerizing Yukawa solvent

The effective interaction between colloidal hard sphere particles in a Yukawa solvent that can polymerize with the formation of chains and rings is studied and compared with the corresponding results for colloidal hard sphere particles in a solvent of polymerizing hard spheres. The attractive nature of the polymerizing Yukawa solvent particles induces significant changes in the effective interactions between the colloid particles as compared to a polymerizing solvent of hard spheres that was investigated in previous studies. The results for the colloid–solvent mixture are obtained using the associative Percus–Yevick approximation for Wertheim’s Ornstein–Zernike integral equation; the colloidal species are taken at a nonvanishing but very small concentration throughout this article. We present the effects of the size ratio of colloid spheres to solvent spheres, the degree of polymerization, and the solvent density on the effective interactions between colloid and solvent particles. The intercolloidal potential of mean force (PMF) is found to be highly dependent on these parameters for Wertheim’s model. It is found that the correlations between colloid particles obtained using the Yukawa solvent model are longer ranged and more attractive than those found using the hard sphere solvent model. A greater depletion of the solvent density around the colloidal particles is also observed for the Yukawa solvent model as compared to the hard sphere model; an increased polymer chain length also enhances the depletion of the solvent density. The PMF is found to be oscillatory in structure. The oscillatory structure also depends upon the average polymer chain length, specifically, the oscillatory structure in the PMF is strongly diminished as the average polymer chain length increases. Additionally, as the average polymer length increases, the attraction at the colloid–colloid contact distance decreases.

Equation of state of the hard-sphere solid: The modified weighted-density approximation with a static solid reference state

We have investigated the high-density hard-sphere solid through density-functional theory by applying our recent formulation of the modified weighted-density approximation (MWDA) with a static solid reference state. Our model utilizes both information about the fluid state, modeled by the Percus-Yevick approximation, as well as physical insight into the static solid. We find very good agreement with computer simulation data for the equation of state, as well as for the ideal and excess components of the pressure, and are able to improve the results from the original MWDA model over a wide range of packing fractions. The degree of particle localization and the value of the Lindemann parameter for the high-density hard-sphere solid are also improved over the original MWDA theory. Finally, we comment on the result that the excess pressure is negative for most packing fractions, suggesting possible physical means for this observation.