Inhomogeneous 2D Lennard-Jones Fluid: Theory and Computer Simulation

In this work, thermodynamical properties of a two-dimensional (2D) Lennard-Jones (LJ) fluid are studied. Here, to increase the accuracy of our theoretical calculations, the correlation functions in three-particle level (triplet) are applied. To obtain the triplet correlation functions, the Attard's source particle method is extended to 2D systems. In the Attard's procedure, the inhomogeneous Ornstein-Zernike (OZ) equation is solved using the Treizenberg-Zwanzwig (TZ) expression and a closure relation like the hypernetted-chain (HNC) approximation. In the present work, we also have performed the Monte Carlo (MC) simulation. The theoretical results are in fairly agreement with the MC simulation. Also, our results show that the approach proposed here is suitable to study the 2D LJ fluid.     

Inhomogeneous 2D Lennard-Jones Fluid: Theory and Computer Simulation

In this work, thermodynamical properties of a two-dimensional (2D) Lennard-Jones (LJ) fluid are studied. Here, to increase the accuracy of our theoretical calculations, the correlation functions in three-particle level (triplet) are applied. To obtain the triplet correlation functions, the Attard's source particle method is extended to 2D systems. In the Attard's procedure, the inhomogeneous Ornstein-Zernike (OZ) equation is solved using the Treizenberg-Zwanzwig (TZ) expression and a closure relation like the hypernetted-chain (HNC) approximation. In the present work, we also have performed the Monte Carlo (MC) simulation. The theoretical results are in fairly agreement with the MC simulation. Also, our results show that the approach proposed here is suitable to study the 2D LJ fluid.     

二元高斯核模型的相不稳定性研究

从考虑巨势Ω在平衡点附近随密度的微小涨落δρ_α(r)的变化出发，由系统的关联函数可以研究描述高分子链二元系统的相不稳定性.在超网链(HNC)近似下研究了二元高斯核模型的相不稳定性，得到了它的相不稳定线，同时确定此相变是由浓度涨落为主的相不稳定性引起的退混合相变.将HNC近似下的结果与随机相近似下的结果进行了比较. 关键词： 高斯核模型 关联函数 相不稳定性 高分子链     

An Analytical Calculation of the Equation of State and the Critical Point in a Dense Classical Fluid of Charged Hard Spheres

A theoretical expression for the equation of state is developed which is based on the mass action law, the Carnahan-Starling expression for the hard-sphere contributions and the mean-spherical approximation for the Coulombic interactions. Numerical calculations of the pressure and the critical data show good agreement with other methods (HNC, MC etc.).     

Properties of the functional formalism and their application to the theory of classical fluids

A general formalism is derived relating any generating functional of a hierarchy of functions to some other functionals yieldingUrsell, Husimi, and similar expansions of the original hierarchy and vice versa. There are two expansions starting with an equation of the O.-Z. type. This formalism is applied to the grand partition function with an external potential which is a generating functional for the molecular distribution functions. When the external potential is induced by adding particles to the system we obtain several hierarchies of integral equations related to each other in a simple fashion. As the Kirkwood-Salsburg, Mayer-Montroll, Green equations, the P. Y., HNC and a HNC similar approximation with their extensions are special cases of these hierarchies the relations between them become transparent. At the same time the heuristic feature in the choice of functionals and independent functions in earlier derivations of some of these equations is removed.     

Chemical potential of electrolytes adsorbed in porous media with charged obstacles: application of the continuum replica methodology

A theoretical study is reported of a quenched-annealed system where both components were modelled as size symmetric +1: ?1 primitive model electrolytes. The partly quenched system was studied by using the replica Ornstein-Zernike (ROZ) integral equation theory in the hypernetted chain (HNC) approximation and grand canonical Monte Carlo (GCMC) simulations. The primary interest was the excess Gibbs free energy (logarithm of the mean activity coefficient) of the adsorbed electrolyte and an expression for this quantity, valid within the ROZ/HNC formalism, was derived. The effects of the concentration of matrix ions, pre-quenching conditions, and the electrolyte and solvent conditions (concentration, temperature, dielectric constant) on the structure and thermodynamics of the adsorbed electrolyte were examined. The numerical results indicated that the mean activity of the adsorbed electrolyte differs substantially from the corresponding quantity for the bulk electrolyte. The excess chemical potential depends strongly on the concentration of charged obstacles and matrix preparation, and also on the temperature and dielectric constant of the annealed electrolyte solution. Newly generated computer simulation results for the structural and thermodynamic parameters, obtained by the grand canonical Monte Carlo method, were used to assess the validity of the ROZ/HNC approximation. It was shown that the ROZ/HNC theory yields good agreement with the computer simulations.     

Structural and thermodynamic properties of the restricted primitive model electrolyte in a mixture with uncharged hard spheres: a grand canonical Monte Carlo simulation and integral equation study

A restricted primitive model electrolyte in a mixture with uncharged hard spheres was studied at room temperature using grand canonical Monte Carlo computer simulation and Ornstein–Zernike integral equation theory in the hypernetted chain approximation (HNC). The mean spherical approximation results are also presented for a few cases. We obtained the pair distribution functions of species of the system, the dependencies of the total fluid density and the ionic fraction on the chemical potentials, the excess internal energy and the heat capacity at constant volume for a wide range of chemical potentials of the species from the simulations and HNC theory. In the majority of cases, good agreement between the theoretical predictions and simulation data is obtained. The composition of the mixture is determined by the chemical potentials of both species. The pair distribution functions have a Debye-like shape at low densities for various values of the ion fraction. By increasing the chemical potential of the uncharged component, weak trends for structuring of the solution are observed with the formation of ion-hard sphere-ion complexes. At high densities, a tendency for in-phase oscillations of ion–ion functions is observed similar to the pure electrolyte in the restricted primitive model. We analysed the chemical potential–density and the chemical potential–ion fraction projections of the equation of state in detail. Also, the heat capacity at constant volume has been calculated for the first time. The model and the results are useful for the development of the theory of inhomogeneous fluid mixtures.     

Phase Separation of Penetrable Core Mixtures

A two-component system of penetrable particles interacting via a gaussian core potential is considered, which may serve as a crude model for binary polymer solutions. The pair structure and thermodynamic properties are calculated within the random phase approximation (RPA) and the hypernetted chain (HNC) integral equation. The analytical RPA predictions are in semi-quantitative agreement with the numerical solutions of the HNC approximation, which itself is very accurate for gaussian core systems. A fluid-fluid phase separation is predicted to occur for a broad range of potential parameters. The pair structure exhibits a nontrivial clustering behaviour of the minority component. Similiar conclusions hold for the related model of parabolic core mixtures, which is frequently used in dissipative particle dynamics (DPD) simulations.     

Saturation and condensate fraction reduction of cold alpha matter

The ground state energy of ideal α  -matter at T=0$T=0$ is analyzed within the framework of variational theory of Bose quantum liquids. Calculations are done for three local α–α potentials with positive volume integrals and two-body correlation functions obtained from the Pandharipande–Bethe equation. The energy per particle of α matter is evaluated in the cluster expansion formalism up to four-body diagrams, and using the HNC/0 and HNC/4 approximation for a Bose liquid. At low densities the two methods predict similar EOS whereas at higher densities they are sensitively different, the HNC approximation providing saturation at lower density, bellow the saturation value of nuclear matter. Inclusion of higher-order terms in the cluster expansion of the condensate fraction is leading to a stronger depletion of the alpha condensate with the density compared to the two-body approximation prediction.     

Symmetry properties of integral equations in the theory of classical fluids

The functional formalism described in a preceding paper, leads to a great number of integral equations for molecular distribution functions. Some of them have a certain symmetry property which, in the case of the pair distribution function includes symmetry with respect to the permutation of particles. This symmetry condition is necessary for any self consistent approximation. Among all integral equations known to date only the PY and the HNC equations satisfy this condition. In this paper we derive some new symmetric equations. Integral equations which are obtained with the help of the Percus method, involvingn-particle distribution functions (n > 2), cannot be symmetric with respect to the interchange of particles.     

A variational theory of fermion matter with spin dependent correlations

A variational wave function is constructed for a system of fermions interacting with a spin dependent potential. The correlations due to the repulsive core of the potential are described by a spin independent Jastrow product ansatz and the correlations due to the longer range part of the potential, which are assumed to be spin dependent, are described by an independent pair ansatz. A cluster expansion is derived for the variational energy and a set of hypernetted chain (HNC) equations obtained to sum the cluster series in terms of the elementary diagrams. Neglecting the elementary diagrams, the HNC equations are solved numerically for the spin dependent potential, V3, in neutron matter. The introduction of spin dependent correlations is found to give a small lowering of the variational energy in the HNC approximation. The results are very sensitive to an accurate treatment of the many-body terms within the HNC approximation, however, and it is shown that additional approximations can easily lead to an exaggeration of the effect of the spin dependent correlations.     

Euler-Lagrange equations and hypernetted chain calculations of boson matter

The use of two-body distribution functions which minimize the HNC approximation to the Jackson-Feenberg energy of boson matter is shown to lead to convenient expressions for the calculation of $\text{?g}\text{?ρ}$, pressure and compressibility. Study of the Pandharipande-Bethe energy of boson matter does not reveal a minimum and raises questions regarding the utility of this energy functional.     

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.     

Helmholtz free energy of electron screened plasmas in the hypernetted-chain (HNC) approximation

A new formula for the Helmholtz free energy for plasma of ions with electron screening described by linear response theory is obtained in the hypernetted-chain (HNC) approximation. HNC results for the free energy are compared with equivalent Monte Carlo simulations with good agreement.     

Clustering of continuous distributions of charge

An investigation is carried out of the association and clustering of equimolar mixtures of oppositely charged Gaussian charge distributions (CDs) of the form ～ exp ( ? r²?/2α²), where r is the separation between the centres of charge and α governs the extent of charge spreading (α→0 is the point charge limit). The results of molecular dynamics (MD) and Ornstein–Zernike integral equation with the mean spherical approximation (MSA) and hypernetted-chain (HNC) closures are compared for these systems. The MD and HNC radial distribution functions, g(r), agree very well for not too small α. The MD and MSA, g(r), also agree well for α ≈ 1 and greater. The potential energy per particle for the three methods also agrees well over a wider range of α values, better than might be expected from inspection of the radial distribution functions, because the dominant contributions to U come predominantly from intermediate and long distance ranges where there is good agreement between the g(r) from the MSD and HNC closures. The nature of the association and clustering of the charges as a function of α is explored through the mean nearest neighbour distance for unlike and like species and the mean and root-mean-square force. The velocity and force autocorrelation functions are also calculated; they show increasingly oscillatory behaviour in the small α limit, originating in vibrations of a pair of CDs of opposite sign.     

Configurational entropy of adsorbed rigid rods: Theory and Monte Carlo simulations

The configurational entropy of straight rigid rods of length k (k-mers) adsorbed on square, honeycomb, and triangular lattices is studied by combining theory and Monte Carlo (MC) simulations in grand canonical and canonical ensembles. Three theoretical models to treat k-mer adsorption on two-dimensional lattices have been discussed: (i) the Flory-Huggins approximation and its modification to address linear adsorbates; (ii) the well-known Guggenheim-DiMarzio approximation; and (iii) a simple semi-empirical model obtained by combining exact one-dimensional calculations, its extension to higher dimensions and Guggenheim-DiMarzio approach. On the other hand, grand canonical and canonical MC calculations of the configurational entropy were obtained by using a thermodynamic integration technique. In the second case, the method relies upon the definition of an artificial Hamiltonian associated with the system of interest for which the entropy of a reference state can be exactly known. Thermodynamic integration is then applied to calculate the entropy in a given state of the system of interest. Comparisons between MC simulations and theoretical results were used to test the accuracy and reliability of the models studied.     

Integral equation theory of a one dimensional hard-ellipse fluid between hard walls

Density functional theory is used to study the structure of a one dimensional fluid model of hard-ellipse molecules with their axes freely rotating in a plane, confined between hard walls. A simple Hypernetted chain (HNC) approximation is used for the density functional of the fluid and the integral equation for the density is obtained from the grand potential. The only required input is the direct correlation function of the one dimensional hard-ellipse fluid. For this model, the pressure, sum rule and the density at the walls are obtained. The Percus Yevick (PY), for lower density, and HNC, for higher density, integral equations are also solved to obtain the direct correlation function of hard-ellipse model introduced here. We obtain the average density at the wall as well as the radial density profile. We compare these with Monte Carlo simulations of the same model and find reasonable agreement.     

Extended mean spherical approximation for electrolyte solutions

The range of applicability of the mean spherical approximation to a primitive-model electrolyte is extended to allow for stronger Coulomb coupling. The new terms in the approximation are extracted from the HNC formalism, which is adequate for strongly coupled Coulomb systems. The new approximation satisfies the Stillinger-Lovett conditions. The new integral equation thus obtained is presented.     

Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions

The phase function for solar light scattering by large particles such as cloud droplets is strongly anisotropic due to very strong peaking in the forward direction. This creates numerical difficulties when attempting to calculate accurate reflected and transmitted radiances, which are important for remote sensing of atmospheric and surface properties. A popular approach uses the delta function to approximate the forward-scattering peak in a fraction of energy and a limited number of polynomial terms or a geometrically truncated function for the remaining fraction (so-called truncation approximations). This article compares and discusses several methods for fast and accurate calculations using truncation approximations. When using a single truncation approximation for all scattering orders, large biases appear in directions near the solar and anti-solar points. As shown here, high accuracy can be obtained using different truncation approximations depending on the order of scattering. Of particular importance is the use of phase functions close to the exact phase functions for the first few orders of scattering. Applying the method in combination with the Monte Carlo (MC) method, in which the truncation fraction for a scattering order depends on the scattering angle at the previous scattering event, obtains accurate radiance calculations under almost all geometrical and optical conditions, including in directions near the solar point. Because the method also reduces computational noise due to the MC sampling of radiance, it is useful for fast and accurate radiance calculations for cloudy atmospheres.