Ornstein-Zernike Integral Equation Research Articles

Calculation of the entropy for hard-sphere from integral equation method

The Ornstein-Zernike integral equation method has been employed for a single-component hard sphere fluid in terms of the Percus-Yevick (PY) and Martynov-Sarkisov (MS) approximations. The virial equation of state, excess chemical potential, and free energy have been computed in both approximations. The excess entropy has been obtained with both thermodynamic relation and free energy expression for reduced densities of 0.1 to 0.9. It has been shown that the entropy values from thermodynamic relation in the MS approximation are better than those from the PY approximation, especially for high densities, and presents a reasonable comparison with available data in the literature, while the values from free energy expression in both approximations are not only close to each other but also comparable to accurate ones.

Applicability of a central force water model to study adsorption in disordered hydrophobic matrices—replica Ornstein–Zernike theory

A simple central force water model to study adsorption in random Lennard-Jones-like matrices has been studied using the replica Ornstein–Zernike integral equation theory in hypernetted-chain (HNC) approximation and in HNC+bridge approximation. The structure of water in obstacle matrices of different sizes was studied, showing that the model appropriately accounts for the hydrophobic hydration. By calculating the chemical potential of water in the model adsorbent, we have constructed the adsorption isotherm. Except for the cases of highly dispersed matrices, water gets excluded from the crowded hydrophobic environment, as expected experimentally.

Analytical accurate expressions for radial distribution function and equation of state of hard sphere fluid based on the Percus-Yevick exact solution and a new unifying closure

ABSTRACT New accurate analytic expressions for the radial distribution function and equation of state of hard sphere fluid are obtained by use of its analytic solution from Percus-Yevick closure and by a new unifying closure of the Ornstein-Zernike integral equation. This will be possible by linearisation of the closure to get a set of one-parameter structural functions incorporating both Percus-Yevick and Hyper-Netted Chain closures. This will enable us to deliver very general expressions for the structural functions and equation of state for a potential and fluid. For the most important case of hard sphere fluid the method can be complemented with its known virial coefficients to get accurate analytical expressions for its radial distribution function and equation of state. The expressions are then successfully compared with those of computer simulations for hard sphere fluid at all densities with great accuracy.

A study of the structure and thermodynamics of non-ionic microemulsion droplets: Integral equation methods (IEs) and molecular dynamics simulation (MD)

This paper aims to explore the structural and thermodynamic properties of a bare neutral oil/water microemulsions (MEs), TX100 by using a combination of molecular dynamics simulations (MD) and Ornstein-Zernike integral equations (IEs) with the hypernetted chain closure relation (HNC), at different volume fractions ϕ (1.4%, 2.8%, 5%, 6.98%, 10%). The employed effective pair potential is a combination of a hard sphere, the van der Waals and a Yukawa type potential. Structural properties were studied by examining the radial distribution function (RDF), g(r), as a function of ϕ; increasing ϕ enhances order and correlation between droplets, manifesting as a narrow and pronounced correlation peak and a decrease in the average distance between MEs. Regarding the thermodynamic properties, as ϕ increase, the reduced pressure and internal energy increase exponentially.

Microscopic theory for the pair correlation function of liquidlike colloidal suspensions under shear flow.

We present a theoretical framework to investigate the microscopic structure of concentrated hard-sphere colloidal suspensions under strong shear flows by fully taking into account the boundary-layer structure of convective diffusion. We solve the pair Smoluchowski equationwith shear separately in the compressing and extensional sectors of the solid angle, by means of matched asymptotics. A proper, albeit approximate, treatment of the hydrodynamic interactions in the different sectors allows us to construct a potential of mean force containing the effect of the flow field on pair correlations. We insert the obtained pair potential in the Percus-Yevick relation and use the latter as a closure to solve the Ornstein-Zernike integral equation. For a wide range of either the packing fraction η and the Péclet (Pe) number, we compute the pair correlation function and extract scaling laws for its value at contact. For all the considered values of Pe, we observe a very good agreement between theoretical findings and numerical results from the literature, up to rather large values of η. The theory predicts a consistent enhancement of the structure factor S(k) at k→0, upon increasing the Pe number. We argue this behavior may signal the onset of a phase transition from the isotropic phase to a nonuniform one, induced by the external shear flow.

THERMODYNAMIC PROPERTIES OF DENSE HYDROGEN PLASMAS WITH PARTIALLY DEGENERATE SEMICLASSICAL IONS

In this article, we investigated the thermodynamic properties of a one-component dense non-ideal hydrogen plasma with semiclassical non-ideal ions against the background of degenerate quantum electrons. The effective potential of the ion-ion interaction that takes into account the quantum mechanical effects of diffraction was used as a model for the interaction. Ion-ion radial distribution functions were calculated based on the solution of the Ornstein-Zernike integral equation in the hypernetted-chain (HNC) approximation. In the calculations, the effective interaction potential was used as a micropotential since only the screening due to the uniform background (i.e., electrons) was taken into account, but not the screening due to the interaction between the ions themselves. In addition, the effective potential takes into account only the quantum effect of ion diffraction. Thermodynamic properties, such as the correlation energy and the non-ideal component of the equation of state, were calculated based on the obtained radial distribution functions and the interparticle interaction potential. Thermodynamic properties were presented as a percentage between the results based on the effective potential and the Yukawa potential depending on the density and coupling parameter values. The difference between the results is greater in denser plasma and at higher values of the coupling parameters.

Construction of a composite-sphere model for molecules of tetrahedral symmetry

We propose a ‘composite sphere’ model for molecules of tetrahedral symmetry. In particular, we choose as an example the hydrocarbon neopentane. We disassemble the molecule into five moieties: four methyl groups (species A) and one carbon atom (species B). By imparting pair potentials of spherical symmetry to the A–A (hard sphere type), B–B (hard sphere type), and A–B (square-well type) pairs, we are able to reconstitute the pentamer structure via Monte Carlo simulation. This is demonstrated for four density states: ρσ 3 = 0.03, 0.10, 0.15, and 0.20. We apply two criteria in assessing the per cent formation of pentamers: namely the proximity rule and the energy rule. The proximity rule gives 57% formation of pentamers, while the energy rule is able to give 99% pentamers. The structures are also calculated via the spherical Ornstein-Zernike integral equations with the Percus-Yevick (PY) closure. It is found that the PY closure is inadequate for predicting quantitatively the MC radial distribution functions. The root cause of the lack of agreement is attributed to the improper application of the isotropic version of the Ornstein-Zernike integral equations to a fluid that has become anisotropic.

Effect of hydrophobically modified PEO polymers (PEO-dodecyl) on oil/water microemulsion properties: in vitro and in silico investigations.

Microemulsions are excellent systems for transdermal delivery of multifunctional drugs because they have the potential to improve drug absorption/permeation and handling limitations. Biocompatible polymers are used as a coating of microemulsions to avoid the interactions that can occur between the microemulsions and the skin. Thus, they protect and lubricate these transporter nanovesicles. In this paper, we studied decane/water microemulsions covered with hydrophobically modified PEO polymer (PEO-m). To reveal the effect of hydrophobically modified PEO (PEO-m) polymer on the shape, the micro-arrangement and the dynamics of the microemulsions, we used an integrated strategy combining Molecular Dynamics simulation (MD), Small-Angle Neutron Scattering experiments (SANS), and the Ornstein–Zernike integral equations with the Hypernetted Chain (HNC) closure relation. We determined the microemulsion shape in vitro using the renormalized intensities spectra from SANS experiments. We discussed the micro arrangements of microemulsions, in vitro and in silico, employing the pair correlation function g(r) and the structure factor S(q), obtained from the three approaches with good agreement. Thus, we used the validated MD simulations to calculate the microemulsion's dynamics properties that we discussed using the mean-squared displacement (MSD) and the diffusion coefficients. We found that the presence of moderate quantities of PEO-m, from 4 to 12 PEO-m per microemulsion, does not influence the microemulsion shape, increases the stability of the microemulsion, and slightly decrease the dynamics. Our in vitro and in silico results suggest that polymer incorporation, which has interesting in vivo implications, has no disadvantageous effects on the microemulsion properties.

Integral equation theory of thermodynamics, pair structure, and growing static length scale in metastable hard sphere and Weeks-Chandler-Andersen fluids.

We employ the Ornstein-Zernike integral equation theory with the Percus-Yevick (PY) and modified-Verlet (MV) closures to study the equilibrium structural and thermodynamic properties of metastable monodisperse hard sphere and continuous repulsion Weeks-Chandler-Andersen (WCA) fluids under density and temperature conditions where the system is strongly overcompressed or supercooled, respectively. The theoretical results are compared to crystal-avoiding simulations of these dense monodisperse model one-component fluids. The equation of state (EOS) and dimensionless compressibility are computed using both the virial and compressibility routes. For hard spheres, the MV-based virial route EOS and dimensionless compressibility are in very good agreement with simulation for all packing fractions, much better than the PY analogs. The corresponding MV-based predictions for the static structure factor are also very good. The amplitude of density fluctuations on the local cage scale and in the long wavelength limit, and three technically different measures of the density correlation length, are studied with both closures. All five properties grow in a roughly exponential manner with density in the metastable regime up to packing fractions of 58% with no sign of saturation. The MV-based results are in good agreement with our crystal-avoiding simulations. Interestingly, the density dependences of long and short wavelength quantities are closely related. The MV-based theory is also quite accurate for the thermodynamics and structure of supercooled monodisperse WCA fluids. Overall our findings are also relevant as critical input to microscopic theories that relate the equilibrium pair correlation function or static structure factor to dynamical constraints, barriers, and activated relaxation in glass-forming liquids.

Investigation of Dusty Plasma Based on the Ornstein—Zernike Integral Equation for a Multicomponent Fluid

The electrostatic interaction between charged particles in a dusty plasma has been studied using the Ornstein—Zernike integral equation for a multicomponent plasma. The transition to the one-component approximation has been performed for the nonideal plasma subsystem. It has been shown that the interaction between charged plasma particles at the coupling parameter Γ of the dust subsystem less than unity is well described by the Debye potential with the complete screening constant. The static dielectric function in the region of low wavenumbers becomes negative at Γ > 1 and this region expands with increasing Γ. As a result, attraction between likely charged particles and repulsion between oppositely charged ones occur in a certain distance range. In this case, the total pressure, specific heat at constant volume, and isothermal compressibility of the dusty plasma remain positive in the entire studied range of the nonideality parameter Γ < 250, but the isothermal compressibility of only the dust nonideal subsystem becomes negative at Γ ≈ 2.

Bounded inverse power potentials: Isomorphism and isosbestic points.

The bounded inverse power (BIP) interaction pair potential, ϕ(r)=1/(aq+rq)n/q, where a and the exponent, n, are constants which control the interaction softness, q is a positive integer, and r is the pair separation, is shown to exhibit isomorphic scaling as does the well-known inverse power potential, i.e., where a = 0. If T is the temperature and ρ is the number density of particles, two state points are isomorphic if a reference state, ρ0, T0, a0 and another state, ρ, T, a are related through the relationships ρn/3/T=ρ0 n/3/T0 and a=a0ρ0/ρ1/3=a0T0/T1/n. The potential form is therefore density dependent along an isomorph. Molecular dynamics simulations and solutions of the Ornstein-Zernike integral equation for q = 2 demonstrate the existence of isosbestic points (IBPs) in the radial distribution function and structure factor for 6 ≤ n ≤ 18 and a wide range of a and ρ values. For the BIP potentials with not too small a values and over a wide density range, the IBP distance is insensitive to the number density and is equal to the distance, rT, defined through ϕ(rT) = T. For exponential potentials of the general form, ϕ(r) = C exp(-rm) with 1 ≤ m ≤ 3, there are also IBPs which are at r values that are typically ∼10-15% larger than predicted by the formula for rT.

Closure for the Ornstein-Zernike equation with pressure and free energy consistency.

The Ornstein-Zernike (OZ) integral equation theory is a powerful approach to simple liquids due to its low computational cost and the fact that, when combined with an appropriate closure equation, the theory is thermodynamically complete. However, approximate closures proposed to date exhibit pressure or free energy inconsistencies that produce inaccurate or ambiguous results, limiting the usefulness of the Ornstein-Zernike approach. To address this problem, we combine methods to enforce both pressure and free energy consistency to create a new closure approximation and test it for a single-component Lennard-Jones fluid. The closure is a simple power series in the direct and total correlation functions for which we have derived analytical formulas for the excess Helmholtz free energy and chemical potential. These expressions contain a partial molar volumelike term, similar to excess chemical potential correction terms recently developed. Using our bridge approximation, we have calculated the pressure, Helmholtz free energy, and chemical potential for the Lennard-Jones fluid using the Kirkwood charging, thermodynamic integration techniques, and analytic expressions. These results are compared with those from the hypernetted chain equation and the Verlet-modified closure against Monte Carlo and equations-of-state data for reduced densities of ρ^{*}<1 and temperatures of T^{*}=1.5, 2.74, and 5. Our closure shows consistency among all thermodynamic paths, except for one expression of the Gibbs-Duhem relation, whereas the hypernetted chain equation and the Verlet-modified closure exhibit consistency between only a few relations. Accuracy of the closure is comparable to the Verlet-modified closure and a significant improvement to results obtained from the hypernetted chain equation.

Electron transport in nonideal and degenerate plasmas

In this article, various methods of calculation for the Coulomb logarithm in the kinetic theory of electron transfer are considered. It is assumed that the interaction of electrons with ions is described by the Debye potential with a screening constant, which is determined either with or without the contribution of the ionic component. In order to take into account ion–ion correlations, the Ornstein–Zernike integral equation in the hypernetted chain approximation is solved numerically. The behavior of the collision cross section of electrons with ions has been studied in detail on the basis of the phase shift method.

Computation of static quantum triplet structure factors of liquid para-hydrogen.

The instantaneous and centroid triplet structure factors, , of liquid (one-center) para-hydrogen are computed on the crystallization line for temperatures T/K ≤ 33. The focus is on salient equilateral and isosceles features, and the methods utilized are path integral Monte Carlo (PIMC) simulations and Ornstein-Zernike (OZ) integral equations, which involve Jackson-Feenberg convolution (JF3) and other distinct closures. Long path integral simulation runs are carried out in the canonical ensemble, so as to obtain sufficiently accurate direct PI triplet results. Conclusions are drawn regarding general triplet structure features and the role and usefulness of the OZ closures. The equilateral features are studied in more detail, and one finds that (a) PIMC results point to the existence of regularity in the centroid main peak amplitudes; (b) some of the studied closures give qualitative descriptions for wave numbers below k ≈ 1 Å-1, but they all fail to describe the main peak amplitude regions (1.75 < k/Å-1 < 2.5); and (c) JF3 plays the role of a limit closure that is valid for increasing wave numbers (k ≥ 2.6 Å-1). In addition, representative isosceles PI features turn out to be reasonably bounded (within Δk = 0.1 Å-1) by those of some closures.

Integral equation theory based direct and accelerated systematic coarse-graining approaches.

Coarse-grained (CG) molecular dynamics (MD) simulations have become popular for investigating systems on multiple length and time scales ranging from atomistic to mesoscales. In CGMD, several atoms are mapped onto a single CG bead and the effective interactions between CG beads are determined. Iterative coarse-graining methods, such as iterative Boltzmann inversion (IBI), are computationally expensive and can have convergence issues. In this paper, we present a direct and computationally efficient theoretical procedure for coarse-graining based on the Ornstein-Zernike (OZ) and hypernetted chain (HNC) integral equation theory. We demonstrate the OZ-HNC-based CG method by coarse-graining a bulk water system, a water-methanol mixture system, and an electrolyte system. We show that the accuracy of the CG potentials obtained from the OZ-HNC-based coarse-graining is comparable to iterative systematic coarse-graining methods. Furthermore, we show that the CG potentials from OZ-HNC can be used to reduce the number of iterations and hence the computational cost of the iterative systematic coarse-graining approaches, like IBI and relative entropy minimization.

Calculating particle pair potentials from fluid‐state pair correlations: Iterative ornstein–zernike inversion

An iterative Monte Carlo inversion method for the calculation of particle pair potentials from given particle pair correlations is proposed in this article. The new method, which is best referred to as Iterative Ornstein-Zernike Inversion, represents a generalization and an improvement of the established Iterative Boltzmann Inversion technique (Reith, Pütz and Müller-Plathe, J. Comput. Chem. 2003, 24, 1624). Our modification of Iterative Boltzmann Inversion consists of replacing the potential of mean force as an approximant for the pair potential with another, generally more accurate approximant that is based on a trial bridge function in the Ornstein-Zernike integral equation formalism. As an input, the new method requires the particle pair correlations both in real space and in the Fourier conjugate wavenumber space. An accelerated iteration method is included in the discussion, by which the required number of iterations can be greatly reduced below that of the simple Picard iteration that underlies most common implementations of Iterative Boltzmann Inversion. Comprehensive tests with various pair potentials show that the new method generally surpasses the Iterative Boltzmann Inversion method in terms of reliability of the numerical solution for the particle pair potential. © 2018 Wiley Periodicals, Inc.

Screened Coulombic Orientational Correlations in Dilute Aqueous Electrolytes.

The ion-induced long-range orientational order between water molecules recently observed in second harmonic scattering experiments and illustrated with large scale molecular dynamics simulations is quantitatively explained using the Ornstein-Zernike integral equation approach of liquid physics. This general effect, not specific to hydrogen-bonding solvents, is controlled by electroneutrality conditions, dipolar interactions, and dielectric+ionic screening. As expected, all numerical theories recover the well-known analytical expressions established 40 years ago.

Coulomb Logarithm in Nonideal and Degenerate Plasmas

Various methods for determining the Coulomb logarithm in the kinetic theory of transport and various variants of the choice of the plasma screening constant, taking into account and disregarding the contribution of the ion component and the boundary value of the electron wavevector are considered. The correlation of ions is taken into account using the Ornstein–Zernike integral equation in the hypernetted-chain approximation. It is found that the effect of ion correlation in a nondegenerate plasma is weak, while in a degenerate plasma, this effect must be taken into account when screening is determined by the electron component alone. The calculated values of the electrical conductivity of a hydrogen plasma are compared with the values determined experimentally in the megabar pressure range. It is shown that the values of the Coulomb logarithm can indeed be smaller than unity. Special experiments are proposed for a more exact determination of the Coulomb logarithm in a magnetic field for extremely high pressures, for which electron scattering by ions prevails.

The influence of the poly(ethylene glycol) on the mean activity coefficients of NaCl aqueous solutions. The application of the MSA and HNC method

A primitive model electrolyte in mixture with uncharged hard spheres was used to model the concentration dependence of the mean activity coefficient of sodium chloride in aqueous solutions of poly(ethylene glycol) (PEG) of different degree of polymerization and concentration. Ornstein-Zernike integral equation was numerically solved using the hypernetted chain (HNC) closure and the results were compared with the results of an analytical solution valid within the mean spherical approximation (MSA) closure. Good agreement between the HNC and MSA activity coefficients was obtained. However, at higher NaCl concentrations the convergence of the numerical solution was not satisfied for large model PEG molecules. The presence of the neutral component in the solution increases the mean activity coefficients in the whole concentration range due to the confinement. For a given NaCl and PEG concentration, decreasing the dielectric constant of the solution diminishes the activity coefficient due to stringer interaction between the ions. Mean activity coefficients of NaCl in two aqueous solutions differing in the concentrations of PEG (molecular weight of 12,000gmol−1) were determined also experimentally and compared with MSA results. The agreement was semi-qualitative, mainly due to difficulties in estimating good model parameters for the PEG molecules and the potential decrease of the dielectric constant of the solvent.

A Parameterization of Empirical Sigma Enlarging Bridge Correction of Kovalenko-Hirata Closure in Ornstein-Zernike Theory for Lennard-Jones Fluids

Abstract We report the parameter values included in the sigma enlarging bridge (SEB) function for two-component Lennard-Jones fluids within the Ornstein-Zernike (OZ) integral equation framework, which was first proposed in our previous study [T. Miyata, Y. Ebato, J. Molec. Liquids, 217 (2016) 75] to improve the accuracy of the solvation free energy (SFE). In this article, we consider a wide range of thermodynamic states, with varying the solute size and the solute-solvent interaction strength. The SEB parameter was evaluated via the least square fitting of the first rising region of the radial distribution function obtained from OZ theory to that from molecular dynamics simulation. The SEB function was applied to both the hypernetted chain (HNC) and Kovalenko-Hirata (KH) closures. It is found that the SEB parameter increases monotonically with the solute size, whereas it hardly depends on the solute-solvent interaction strength. Also, the performance of bare HNC, bare KH, Percus-Yevick, and Verlet-modified closures are also examined, to report the relationship between the solute volume and the error of the SFE obtained from OZ theory. We found that the SFE errors under both HNC and KH closures are not necessarily proportional to the solute volume.