Multicomponent fluids of hard hyperspheres in odd dimensions

Structural and thermodynamic properties of hard-sphere fluids

This Perspective article provides an overview of some of our analytical approaches to the computation of the structural and thermodynamic properties of single-component and multicomponent hard-sphere fluids. For the structural properties, they yield a thermodynamically consistent formulation, thus improving and extending the known analytical results of the Percus-Yevick theory. Approximate expressions linking the equation of state of the single-component fluid to the one of the multicomponent mixtures are also discussed.

Yuste S. Equation of State of Four- and Five-Dimensional Hard-Hypersphere Mixtures

New proposals for the equation of state of four- and five-dimensional hard-hypersphere mixtures in terms of the equation of state of the corresponding monocomponent hard-hypersphere fluid are introduced. Such proposals (which are constructed in such a way so as to yield the exact third virial coefficient) extend, on the one hand, recent similar formulations for hard-disk and (three-dimensional) hard-sphere mixtures and, on the other hand, two of our previous proposals also linking the mixture equation of state and the one of the monocomponent fluid but unable to reproduce the exact third virial coefficient. The old and new proposals are tested by comparison with published molecular dynamics and Monte Carlo simulation results and their relative merit is evaluated.

Five dimensional binary hard hypersphere mixtures: A Monte Carlo study

Additive binary mixtures of five dimensional hyperspheres were investigated by Monte Carlo simulations. Both equal packing fraction and equal mole fraction systems with diameter ratios of 0.4 and 0.5 were examined. A range of total densities were studied, spanning low to moderate density fluids. The pair correlation functions and the equations of state were determined and compared with molecular dynamics data and a variety of theoretical predictions. A significant result of the equal packing fraction simulations was the discovery of how quickly the larger hyperspheres reorganized into a dense fluid after a random initial placement. In the equal mole fraction case, the pair correlation functions for the larger hypersphere agree with the pair correlation function of a pure fluid at an appropriately scaled density. The theoretical results for the equation of state compare well to the Monte Carlo calculations for all but the highest densities studied.

Equation of state for five-dimensional hyperspheres from the chemical-potential route

We use the Percus-Yevick approach in the chemical-potential route to evaluate the equation of state of hard hyperspheres in five dimensions. The evaluation requires the derivation of an analytical expression for the contact value of the pair distribution function between particles of the bulk fluid and a solute particle with arbitrary size. The equation of state is compared with those obtained from the conventional virial and compressibility thermodynamic routes and the associated virial coefficients are computed. The pressure calculated from all routes is exact up to third density order, but it deviates with respect to simulation data as density increases, the compressibility and the chemical-potential routes exhibiting smaller deviations than the virial route. Accurate linear interpolations between the compressibility route and either the chemical-potential route or the virial one are constructed.

Nonionic amphiphile nanoarchitectonics: self-assembly into micelles and lyotropic liquid crystals

Amphiphiles, molecules that possess both hydrophilic and hydrophobic moieties, are architecturally simple molecules that can spontaneously self-assemble into complex hierarchical structures from lower to higher dimensions either in the bulk phase or at an interface. Recent developments in multifunctional nanostructure design using the advanced concept of nanoarchitectonics utilize this simple process of assembly. Amphiphilic self-assemblies involving lipids or proteins mimic the structure of biological systems, thus highlighting the necessity of a fundamental physical understanding of amphiphilic self-assembly towards a realization of the complex mechanisms operating in nature. Herein, we describe self-assembled microstructures of biocompatible and biodegradable tetraglycerol lauryl ether (C12G4) nonionic surfactant in an aqueous solvent system. Temperature-composition analyses of equilibrium phases identified by using small-angle x-ray scattering (SAXS) provide strong evidence of various spontaneously self-assembled mesostructures, such as normal micelles (Wm), hexagonal liquid crystal (H1), and reverse micelles (Om). In contrast to conventional poly(oxyethylene) nonionic surfactants, C12G4 did not exhibit the clouding phenomenon at higher temperatures (phase separation was not observed up to 100 °C), demonstrating the greater thermal stability of the self-assembled mesophases. Generalized indirect Fourier transformation (GIFT) evaluation of the SAXS data confirmed the formation of core-shell-type spherical micelles with a maximum dimension ca. 8.7 nm. The shape and size of the C12G4 micelles remained apparently unchanged over a wide range of concentrations (up to 20%), but intermicellar interactions increased and could be described by the Percus-Yevick (PY) theory (after Carnahan and Starling), which provides a very accurate analytical expression for the osmotic pressure of a monodisperse hard sphere.

Multicomponent fluid of nonadditive hard spheres near a wall

A recently proposed rational-function approximation [Phys. Rev. E 84, 041201 (2011)] for the structural properties of nonadditive hard spheres is applied to evaluate analytically (in Laplace space) the local density profiles of multicomponent nonadditive hard-sphere mixtures near a planar nonadditive hard wall. The theory is assessed by comparison with NVT Monte Carlo simulations of binary mixtures with a size ratio 1:3 in three possible scenarios: a mixture with either positive or negative nonadditivity near an additive wall, an additive mixture with a nonadditive wall, and a nonadditive mixture with a nonadditive wall. It is observed that, while the theory tends to underestimate the local densities at contact (especially in the case of the big spheres) it captures very well the initial decay of the densities with increasing separation from the wall and the subsequent oscillations.

Chemical-potential route: a hidden Percus-Yevick equation of state for hard spheres

The chemical potential of a hard-sphere fluid can be expressed in terms of the contact value of the radial distribution function of a solute particle with a diameter varying from zero to that of the solvent particles. Exploiting the explicit knowledge of such a contact value within the Percus-Yevick theory, and using standard thermodynamic relations, a hitherto unknown Percus-Yevick equation of state, p/ρk(B)T = -(9/η) ln(1-η)-(16-31η)/2(1-η)(2), is unveiled. This equation of state turns out to be better than the one obtained from the conventional virial route. Interpolations between the chemical-potential and compressibility routes are shown to be more accurate than the widely used Carnahan-Starling equation of state. The extension to polydisperse hard-sphere systems is also presented.

Conditional pair distributions in many-body systems: exact results for Poisson ensembles

We introduce a conditional pair distribution function (CPDF) which characterizes the probability density of finding an object (e.g., a particle in a fluid) to within a certain distance of each other, with each of these two having a nearest neighbor to a fixed but otherwise arbitrary distance. This function describes special four-body configurations, but also contains contributions due to the so-called mutual nearest neighbor (two-body) and shared neighbor (three-body) configurations. The CPDF is introduced to improve a Helmholtz free energy method based on space partitions. We derive exact expressions of the CPDF and various associated quantities for randomly distributed, noninteracting points at Euclidean spaces of one, two, and three dimensions. Results may be of interest in many diverse scientific fields, from fluid physics to social and biological sciences.

Monte Carlo study of four dimensional binary hard hypersphere mixtures

A multithreaded Monte Carlo code was used to study the properties of binary mixtures of hard hyperspheres in four dimensions. The ratios of the diameters of the hyperspheres examined were 0.4, 0.5, 0.6, and 0.8. Many total densities of the binary mixtures were investigated. The pair correlation functions and the equations of state were determined and compared with other simulation results and theoretical predictions. At lower diameter ratios the pair correlation functions of the mixture agree with the pair correlation function of a one component fluid at an appropriately scaled density. The theoretical results for the equation of state compare well to the Monte Carlo calculations for all but the highest densities studied.

Exact solution of the Percus-Yevick integral equation for fluid mixtures of hard hyperspheres

Structural and thermodynamic properties of multicomponent hard-sphere fluids at odd dimensions have recently been derived in the framework of the rational function approximation (RFA) [Rohrmann and Santos, Phys. Rev. E 83, 011201 (2011)]. It is demonstrated here that the RFA technique yields the exact solution of the Percus-Yevick (PY) closure to the Ornstein-Zernike (OZ) equation for binary mixtures at arbitrary odd dimensions. The proof relies mainly on the Fourier transforms c(ij)(k) of the direct correlation functions defined by the OZ relation. From the analysis of the poles of c(ij)(k) we show that the direct correlation functions evaluated by the RFA method vanish outside the hard core, as required by the PY theory.

Nonadditive hard-sphere fluid mixtures: a simple analytical theory

We construct a nonperturbative fully analytical approximation for the thermodynamics and the structure of nonadditive hard-sphere fluid mixtures. The method essentially lies in a heuristic extension of the Percus-Yevick solution for additive hard spheres. Extensive comparison with Monte Carlo simulation data shows a generally good agreement, especially in the case of like-like radial distribution functions.