Reference Hypernetted Chain Equation Research Articles

Density-dependent interactions and thermodynamic consistency in integral equation theories

In this paper we present an alternative formulation of the well-known integral equation approximations designed to keep a consistent approach to the determination of thermodynamic properties in the case of density-dependent interactions. Obviously, residual inconsistencies inherent to the approximate character of the closure relations of the Ornstein–Zernike equation will not be corrected. In this connection, we will show how this approach is particularly successful when applied in conjunction with approximations in which the aforementioned inconsistencies are minimal, as is the case of the optimised Reference Hypernetted Chain equation. As a case study we will consider the Derjaguin–Landau–Verwey–Overbeek model of charged colloids which is one of the simplest realisations of density-dependent interactions.

Integral Equation Prediction of Reversible Coagulation in Charged Colloidal Suspensions

We exploit the accuracy of the reference hypernetted chain equation (RHNC) to study the stability of charged colloidal suspensions. To achieve this objective with confidence, we first check the performance of the RHNC integral equation by comparison with Gibbs ensemble Monte Carlo (GEMC) simulations for a simple DLVO potential. The results show that the theory describes accurately reversible coagulation into the secondary minimum for a typical intercolloidal potential. Once we have established this point, we make use of the theory to analyze the capability of the DLVO potential to describe appropriately certain experimental observations, in particular, the crossover from irreversible to reversible coagulation. In this regard we found that the inclusion of an empirical hydrophobic potential crucially affects the stability properties of the suspension and leads to an adequate prediction of the crossover diameter.

Integral equation study of liquid hydrogen fluoride

Liquid hydrogen fluoride is a well-known hydrogen bonded substance, in many aspects related to liquid water, and for which a wide variety of interaction models have recently been proposed. We have studied two of these models by means of a reference hypernetted chain equation in order to assess the ability of this latter approach to describe the properties of this highly associative system. Our calculations, when compared with molecular dynamic results, show that the integral equation reproduces quantitatively both the structure and the thermodynamics of liquid hydrogen fluoride over a wide range of thermodynamic states. However, the integral equation approach is apparently unable to produce estimates for the phase diagram since the low-density (gas phase) side of the binodal curve lies inside the nonsolution region of the equation. This failure can be understood as the result of the inability of standard integral equation theories to account for the behavior of low density strongly associative systems like highly charged electrolytes or, in this case, the gaseous phase of hydrogen fluoride.

The thermodynamics of molecules with discrete potentials

Fluids formed by molecules interacting with discrete potentials are examined in the context of perturbation theory and the reference hypernetted chain equation (RHNC) solution to the Ornstein—zernike equation. A perturbation theory for discrete-potential fluids (DPT) is presented, which only requires one to know the properties of a square-well fluid of variable range. Several potentials are studied: square-shoulder, a combination of a square-well and square-shoulder, and a discrete representation of a continuous potential model. We have found that the DPT approach reproduces the RHNC predictions in most of the cases.

Monte Carlo simulation and reference hypernetted chain equation results for structural, thermodynamic, and dielectric properties of polar heteronuclear diatomic fluids

We study fluids of heteronuclear two-center Lennard-Jones molecules with embedded point dipoles using both numerical simulation and integral equation theory. Extensive Monte Carlo simulations are performed for the structural, thermodynamic, and dielectric properties of two models of such fluids, with unusually long simulation runs to assure convergence of the dielectric constant. The results are used to test a generalization of a reference hypernetted chain approximation (RHNC-VM) used previously for nonpolar heteronuclear diatomics. Very good agreement is found between the two methods. We conclude that the RHNC-VM integral equation is a reliable method for studying both polar and nonpolar fluids of diatomic molecules.

Absence of criticality in the reference hypernetted chain equation for short ranged potentials

In this note the reference hypernetted chain equation is solved for a finite ranged potential. For the model considered no solution to the integral equation is found in a certain region of the phase diagram. To determine the presence of criticality in the no-solution line, the behaviour of the total correlation function h(r) at large distances is investigated. For finite ranged potentials, the decay of the function rh(r) at large distances can be either exponential (at low and intermediate densities) or exponentially damped oscillatory (at high densities). For the model considered, the no-solution line falls within the region where the function rh(r) shows exponential decay. The correlation length in the region where the decay of rh(r) is exponential is given by the inverse of the purely imaginary pole of the structure factor. The purely imaginary pole is analysed in the proximities of the no-solution line. It is shown that the correlation length remains finite when approaching the no-solution line. Therefore for the finite ranged potential considered in this work, the reference hypernetted chain equation does not exhibit criticality.

RESEARCH NOTE Absence of criticality in the reference hypernetted chain equation for short ranged potentials

RESEARCH NOTE Absence of criticality in the reference hypernetted chain equation for short ranged potentials

Structure of variable-width square-well fluids from the reference hypernetted chain equation

The problem of calculating the structure of square-well (SW) fluids in the context of non-conformal fluids is reviewed briefly. New and extensive results over a wide range of SW systems and thermodynamic states were obtained by means of the reference hypernetted chain (RHNC) integral equation. The results show that RHNC theory is in excellent agreement with the results of simulation, thus providing a reliable and fast procedure for obtaining the structure of fluids with discontinuous potentials. A few representative results are presented and compared with new Monte Carlo simulation data. The bridge functions obtained with the RHNC procedure are also compared with those calculated directly from the Monte Carlo radial distribution functions.

Structure and thermodynamics of heteronuclear two-centre Lennard-Jones fluids from Monte Carlo simulation and a reference hypernetted chain equation

We have performed extensive Monte Carlo (MC) simulations to evaluate structural and thermodynamic properties of heteronuclear two-centre Lennard-Jones fluids. Computations have been carried out for two uncharged molecular models which roughly describe liquid HCl and HBr. These simulations are used as test benchmarks for a reference hypernetted chain approximation (RHNC-VM), which is the natural extension to heteronuclear fluids of a previously developed approach fairly successful in the context of homonuclear diatomic fluids. The proposed theoretical approach leads to results in good agreement with MC simulation.

Integral equation and simulation studies of a realistic model for liquid hydrogen chloride

Liquid hydrogen chloride is modeled by a system of heteronuclear two-center Lennard-Jones particles with embedded point dipoles and quadrupoles. The effect of molecular polarizability is incorporated via an effective dipole approximation. The study is performed by Monte Carlo reaction field simulation and by hypernetted chain and reference hypernetted chain integral equations. Our simulation results yield dielectric properties in excellent agreement with experimental data for liquid HCl. As for the integral equation approach, we have experimented with an empirical choice of the reference system in the spirit of a recently proposed treatment which has proved extremely successful for pure and quadrupolar two-center Lennard-Jones fluids. The hypernetted chain equation performs slightly better when accounting for the multipolar contributions to the configurational energy, but as a whole the reference hypernetted chain equation, as introduced, here proves to be a more appropriate choice.

Hypernetted and reference hypernetted chain integral equations for Lennard-Jones diatomic fluids with quadrupolar interactions

The hypernetted chain and reference hypernetted chain integral equations are solved for quadrupolar two-center Lennard-Jones fluids and the computed fluid structure and thermodynamics are contrasted with computer simulation. A reference bridge function is determined through an empirical modification of Verlet’s approximation. The resulting reference hypernetted chain equation leads to good agreement with simulation data. On the contrary, the bare hypernetted chain approximation performs poorly, in particular as far as the equation of state is concerned, which is a well-known drawback of this closure when short ranged repulsive potentials come into play.

Bridge functions for models of liquid metals

The bridge function B(r) is extracted from pair-correlation data obtained by means of extensive molecular-dynamics simulations for two model potentials for liquid metals describing respectively aluminum and cesium. Insertion of the computed bridge functions in the reference hypernetted chain equation yields a pair-correlation function which exactly matches molecular-dynamics results. A comparison between these ‘‘experimental’’ bridge functions and the hard-sphere bridge function shows clear differences in the medium- and long-range behavior for the Al model potential and even more marked discrepancies for the softer Cs model potential.

Electronic spectra in polar fluids: Reference hypernetted chain theory

We treat the statically broadened absorption spectrum of a polar chromophore, whose dipole moment is assumed to change upon electronic excitation, at infinite dilution in a polar solvent. Calculations based on the reference hypernetted chain (RHNC) equation are compared to results from the mean spherical approximation (MSA) and to Monte Carlo simulations. The RHNC calculations are in quantitative agreement with simulation results for a strongly coupled solute–solvent system.

Improved integral equation for highly supercooled liquids: Numerical tests for soft-sphere fluids.

A new modified hypernetted-chain integral equation for highly supercooled soft-sphere fluids (MHNCS) is introduced. In the MHNCS approximation, the bridge function is approximated by an appropriate interpolation of the bridge function of the Percus-Yevick hard-sphere model and the leading term of the elementary diagrams, which was first successfully applied to classical one-component plasmas. We have obtained solutions of the MHNCS equation for highly supercooled fluids with sixth- and twelfth-inverse-power potentials. These results are compared with those of molecular-dynamics or Monte Carlo simulations and the solutions of two well-known thermodynamically consistent integral equations, i.e., the Rogers-Young (RY) equation and the reference hypernetted-chain (RHNC) equation. Below the freezing temperature, the solution of the MHNCS equation is found to show a splitting of the second peak of the pair distribution function (PDF) compatible with the computer simulations, while the RY or RHNC equation shows no such a splitting. Using the PDF thus obtained, thermodynamic and structural properties of the highly supercooled soft-sphere fluids are investigated.

A modified hypernetted-chain integral equation for the supercooled liquid of inverse power potentials

Using the integral equation of the theory of liquids, we discuss structural properties of highly supercooled soft-sphere liquids. Two different thermodynamic self-consistent (TC) integral equations, i.e., the Rogers-Young equation and the reference hypernetted-chain equation, have been tested for highly supercooled liquids. According to these results, we propose a new modified hypernetted-chain integral equation for highly supercooled liquids (MHNCS). The bridge function of the MHNCS equation is approximated by an appropriate interpolation of the bridge function of the Percus-Yevick hard-sphere model and that of the leading elementary diagram. We have obtained solutions of the MHNCS equation for sixth and twelveth inverse power potentials (soft-sphere model). Below the freezing point, the PDF calculated from the MHNCS approximation is found to show a splitting of the second peak, which is compatible with the results of computer simulations, while the TC equations show no such a splitting.

Monte Carlo simulation of an ion-dipole mixture

We demonstrate on a simple model of an ion-dipole mixture that the convergence of a Monte Carlo simulation at low ionic concentration may be quite slow. For a one molar concentration 105 trial moves per particle are needed for a 864 particle system to obtain a precise estimate of both the ion-dipole and ion-ion energies and the static pair correlation functions. Our results show that recently published Monte Carlo results (1989, Molec. Phys., 66, 299) on a similar system are far from being converged and question the conclusion drawn in this publication on the failure of the reference hypernetted chain equation.

An efficient procedure for solving the reference hypernetted chain equation (RHNC) for simple fluids

A rapidly convergent method for solving the Orstein-Zernike equation is used in conjunction with a recently proposed iteration strategy. This combination yields a very efficient and stable procedure, which saves a considerable amount of computer time when an integral has to be repeatedly solved for related systems. The procedure is applied to solving the optimized Reference Hypernetted Chain equation for a system of particles interacting through a pairwise Lennard-Jones 12-6 potential. The reference bridge function is calculated through an empirical formula. Several isotherms and isochores are calculated to map the region where no solutions of the integral equation are found. Computations of the chemical potential and pressure are performed for several densities along a number of isotherms. Applying the conditions of phase equilibrium, the phase diagram is obtained and is found to be in good agreement with recent simulation results.

The solution of the reference hypernetted chain equation for the dipolar hard diatomic fluid

The reference hypernetted chain equation (RHNC) for a fluid of dipolar hard diatomics was solved numerically. Three choices for the reference bridge function B0(12), were examined. The simplest, B0(12)=0 (HNC) and B0(12) corresponding to the uncharged homonuclear hard diatomic fluid, computed from simulation data and by the Percus–Yevick approximation. The computed fluid structure [i.e., the g(12) expansion coefficients] showed a remarkable quantitative agreement with the structure obtained from a mean reaction field Monte Carlo simulation. The same applied to the configurational energy. The values for the dielectric constant, however, furnished only qualitatively indications of the density dependence of this quantity. The probable origin of this discrepancy is analyzed.

A test particle approach to the zero separation theorems of molecular distribution functions

It is shown that the potential distribution of a strong test particle leads to the zero separation values of the cavity distribution functions yab(0) in a mixture. This relation furnishes a direct means of computing by Monte Carlo simulations the coincidence values of the cavity function ln yab(0) and the potential distribution 〈exp[−βΨa−βΨb]〉. Test particle simulations have been carried out for mixtures of Lennard-Jones molecules differing considerably in size [(σab/σbb)3 =0.25, 0.5, 0.75, 1.00, 1.25, 1.50, 1.75, and 2.00] and in strength of interaction (εab/εbb =0.5, 1.0, 1.5, and 2.0). Alternative Monte Carlo methods are employed to check the statistics. In order to predict the behavior of the potential distribution, a distribution function theory, the reference hypernetted chain (RHNC) equation, is solved based on the universality of the bridge functions. Hard sphere mixtures are taken as reference fluids. The criteria recently proposed by Rosenfeld and Blum are used to select the equivalent hard sphere diameters. Close agreement with MC results is attained for most of the states considered. This provides further evidence that the RHNC equation is a reliable theory for mixtures of simple fluids.

Reference-hypernetted chain equation with anisotropic bridge function for fluids of diatomic molecules

Modeling the bridge function in the reference-hypernetted chain equation with that of a hard-sphere fluid has proven to be highly successful for simple liquids, particularly when the reference-hard-sphere diameter is treated as an adjustable parameter. In this paper, we examine the generalization of this technique to liquids of diatomic molecules, with the reference bridge function taken from that of the corresponding hard-diatomic-molecule fluid. Specifically, the diatomic molecules of the system of interest interact through an atom–atom Lennard-Jones potential while the hard-diatomic-reference model is solved in Percus–Yevick approximation. We find that variation of the sphere diameter of the hard-diatomic molecule makes possible excellent agreement with simulation results for the two-center Lennard-Jones fluid, just as it does for simple fluids.