Mean Spherical Model Research Articles

The free energy of spheres with dipoles: Monte Carlo with multistage sampling

This reports some Monte Carlo free energy results for a fluid of point dipoles embedded in hard spheres. The multistage sampling method is compared with thermodynamic integration techniques. The resulting free energies are also compared with recent theoretical results obtained from the mean spherical model approximation and from a perturbation theory approach.

Solution of the mean spherical model for a mixture exhibiting phase separation

We have solved analytically the Lebowitz-Percus Mean Spherical Model (MSM) approximation for a symmetric mixture of two species of particles, constrained to have an equal number of particles of species 1 and 2. In this mixture, like particles interact via a hard sphere potential for r < R, the hard core diameter, and a Yukawa potential −Ae−κ r/r for r > R; unlike particles interact via the same hard spheres potentials (same diameter) and an opposite Yukawa potential Ae −κr/r. We find that for A>0, corresponding to an attraction between particles of the same kind and a repulsion between particles of different kind, that there are no spatially homogeneous solutions of the MSM when θ>θc,θ≡βξε, where β is the reciprocal temperature, ξ the total reduced density of hard spheres and ε≡ Ae−κ R/R. We interpret this to mean the existence of phase separation when θ is above its critical value θc We are able to calculate θc analytically as a function of A, κ, and R. We also find that when A<0 there are always homogenous solutions of the MSM. Finally when κ→ 0 and A=−e2/D, we recover the Waisman-Lebowitz solution of the MSM for hard charged spheres of equal numbers of the +e and −e charged particles and dielectric constant D.

Exact solution of the mean spherical model for charged hard spheres in a uniform neutralizing background

The exact solution of the mean spherical model integral equation is found for a system of charged hard spheres in a uniform neutralizing background. This may be considered as a simple example of a fluid with nonadditive hard sphere diameters. Analytic expressions are given for the direct correlation function and for the Laplace transform of the radial distribution function. These, and the thermodynamic properties of the system, are compared with previous solutions of the mean spherical model.

Invariant expansion III: The general solution of the mean spherical model for neutral spheres with electostatic interactions

The partial solution for the mean spherical model of neutral spheres with electrostatic interactions obtained in a previous communication [L. Blum, J. Chem. Phys. 57, 1862 (1972).] is extended to the general case, in which the electrostatic interactions of odd parity, like the dipole-quadrupole interaction, are included. The solution is given in terms of a linear transform of the direct correlation function, which is shown to be a polynomial in r inside the hard core diameter. The coefficients of this polynomial are determined from a sufficient number of boundary conditions. The method of solution employs the so-called second Baxter form of the Ornstein-Zernike equation [Australian J. Phys. 21, 563 (1968).] which is derived for this particular case. Also, alternative forms of the Ornstein-Zernike equation in coordinate space are obtained. The case of molecules with linear dipole and quadrupole moments is summarily illustrated.

The radial distribution function for a fluid of hard spheres at high densities

A radial distribution function for hard spheres is obtained, which improves at high densities on the Percus-Yevick function. This is achieved by solving the mean spherical integral equation for a Yukawa form, adjusted to satisfy the Carnahan-Starling equation of state. The solution to the related problem of the mean spherical model approximation for a model fluid which interacts via the hard sphere potential plus the Yukawa potential is reported.

Dielectric constant of non-polar fluids

The theory of the static dielectric constant of non-polar fluids is re-examined using graph-theoretical techniques. A sample of arbitrary shape in an applied static electric field E 0 is considered. The non-local tensor which transforms from E 0 to the electrostatic polarization P is expressed as a sum of graphs. The relation appears simpler in terms of the inverse tensor, which is also obtained as a sum of graphs. Agreement with classical electrostatics is then manifest, and the shape-independence of the dielectric constant follows. An approximation suitable for numerical work is formulated, and a simplified version is solved analytically. The results are similar to those for fluids of rigid polar molecules in the Mean Spherical Model approximation. Reasons for both the similarities and the differences are given.

Mean Spherical Model for the 6:12 Potential

The mean spherical model (MSM) has been solved for the 6:12 potential. Comparison of the MSM results with the results of computer simulations and of the Percus-Yevick (PY) theory, shows the MSM thermodynamic properties to be inferior to those of the PY theory. On the other hand, the MSM radial distribution function (RDF) is somewhat better than that of the PY theory. In fact, for r>1.7σ, the MSM results are in excellent agreement with the simulation results.

The dielectric constant of water to high temperatures and pressures from the mean spherical model in the Wertheim solution

AbstractGeneral expressions for the static dielectric constant of a fluid of hard cores with embedded point dipoles have been presented by Wertheim [1], on the basis of a Mean Spherical Model (MSM) treatment. Numerical results are derived for water from these expressions, with the additional inclusion of polarizability effects, using only the dipole moment strength, polarizability coefficient and diameter of water‐molecules. These results are compared with new experimental data for water extending to 800 K and 5 kbar. The calculated values compare satisfactorily with measured data at temperatures between 500 and 800 K, for all densities up to 1 g/cm3. In the lower temperature region, at high densities, the predicted values are systematically too high.

Optimized Cluster Expansions for Classical Fluids. I. General Theory and Variational Formulation of the Mean Spherical Model and Hard Sphere Percus-Yevick Equations

Computationally convenient theoretical methods for calculating the thermodynamic properties and pair correlation functions of a classical multicomponent fluid are presented. The Mayer cluster series for the Helmholtz free energy and pair correlation function are transformed using topological reduction to more compact forms involving a renormalized potential. Then the convergence of the two series is improved by an optimal choice of the renormalized potential. The result is two rapidly convergent series which are useful both for ionic solutions and for simple liquids with short range intermolecular forces. When these series are truncated, very accurate and convenient approximations are obtained for both types of fluids. Another set of results is a variational formulation of the mean spherical model and hard sphere Percus-Yevick equations for the pair correlation functions of multicomponent fluids. The variational formulations greatly facilitate the process of solving the equations numerically. Each of these results can be extended to models for molecular fluids with only a moderate increase in computational effort.

Invariant Expansion. II. The Ornstein-Zernike Equation for Nonspherical Molecules and an Extended Solution to the Mean Spherical Model

The Ornstein-Zernike equation for fluids with nonspherical molecules obtained in a former publication [L. Blum and A. J. Torruella, J. Chem. Phys. 56, 303 (1972)] is written in coordinate space as a convolution matrix equation. A rather simple property of the angular coupling coefficients of our former work allows us to write the Ornstein-Zernike equation in irreducible form, as a set of uncoupled matrix equations, of rather small size. A generalization of Baxter's form of the Ornstein-Zernike equation to matrices allows us to write a formal solution to the mean spherical model of neutral hard spheres with almost arbitrary electrostatic multipoles. This is an extension of Wertheim's solution for dipoles [J. Chem. Phys. 55, 4291 (1971)]. The formal solution consists in showing that the direct correlation function inside the hard core is a polynomial in the interatomic distance r. The coefficients of the polynomials are obtained by solving a set of quadratic matrix equations. The class of potentials that admit this kind of a solution is larger than the electrostatic multipole interaction but does not include some cross interactions like the dipole-quadrupole interaction.

Nonexistence of Solidlike Solutions to the Mean-Spherical-Model Equations

We consider the mean-spherical-model (MSM) equation for systems composed of radially symmetric particles. It is shown that for all densities less than close packing no physically satisfactory solidlike (i.e., nonradial) solutions exist to the MSM equation. The Percus-Yevick equation for hard-core particles (i.e., hard disks in two dimensions and hard spheres in three dimensions) is a special case of the MSM equation.

Comparison of Two Theories for the Two Particle Distribution Function of Polar Fluids

Two recent theories for the two particle distribution function of polar fluids are compared. The formal theory of Nienhuis and Deutch [J. Chem. Phys. 55, 4213 (1971)] is valid for arbitrary sample geometry and arbitrary short-range forces accompanying the dipole-dipole interactions. The theory of M. S. Wertheim [J. Chem. Phys. 55, 4291 (1971)] presents an exact result for the mean spherical model (MSM) of a fluid of hard spheres with embedded dipole moment in an infinite system and in a finite spherical system. Perfect agreement is found when the approximations of the MSM are introduced into the Nienhuis-Deutch theory. This lends strong support to the conclusions that have been drawn about the long-range and shape dependent nature of the two particle distribution function for polar systems. Finally an expression is given for the excess free energy of a dipolar fluid in the MSM.

Mean Spherical Model for the Structure of Liquid Metals

Mean Spherical Model for the Structure of Liquid Metals

Mean Spherical Model Integral Equation for Charged Hard Spheres I. Method of Solution

The thermodynamic properties and radial distribution function of a ``primitive model electrolyte'' are calculated in the Mean Spherical Model (MSM) approximation. The system considered is formally described as a classical fluid of charged hard spheres (hard sheets in one dimension). The interaction potential between an ion of species i and an ion of species j, vij(r) is the sum of a Coulomb part and a hard core part qij(r)=∞ for r < Rij or 0 otherwise. The MSM approximation consists of supplementing the fact that the radial distribution function gij(r) is zero for r < Rij by equating the Ornstein—Zernike direct correlation function Cij(r) to -βvij(r) for r > Rij, β being the inverse temperature. In this paper, which is the first of two in this topic, we give the method of solution for this approximation and obtain Cij(r) for r < Rij as polynomials in r, which have a structure similar to that found in the PY approximation for a mixture of uncharged hard spheres. We give the solution up to a set of algebraic equations in the polynomial coefficients. In the second paper we discuss the explicit solution and derived thermodynamic quantities.

Mean Spherical Model Integral Equation for Charged Hard Spheres. II. Results

We continue our investigation of the solution of the mean spherical model integral equation for systems of charged hard spheres and charged hard sheets (in one dimension). The general method of solution was presented in Paper I of this series. This paper contains explicit expressions for the structure functions and thermodynamic properties of a variety of such systems in one and three dimensions. The results all have a very simple form and are in good agreement with various machine computations. When the charges on the particles vanish our results coincide with those obtained from the Percus—Yevick equation for hard spheres while in the limit of zero hard core diameters the results go over into those obtained from the linearized Debye—Hückel theory.

Calculations on the ``Restricted Primitive Model'' for 1–1 Electrolyte Solutions

This paper concerns the ``restricted primitive model'' of 1–1 aqueous electrolyte solutions at concentrations up to 2.0M. Monte Carlo data are compared with several recent theoretical treatments: those based on the hypernetted-chain (HNC) equation and the Percus—Yevick—Allnatt (PYA) equation, the mean spherical model, and the mode expansion and γ-ordering approaches. Some new theoretical calculations are reported. The hypernetted-chain treatment appears to be the most successful of the integral equation theories. Of particular interest is the observation of charge oscillation in the radial distribution functions in both the HNC and Monte Carlo calculations. It is also noted that the cube-root concentration dependence of the deviation from ideality, previously observed for real systems, occurs also in accurate treatments of the primitive model. When the diameter of the ions is sufficiently large, the thermodynamic properties of the restricted primitive model approach those of the corresponding uncharged system at very moderate concentrations.

Exact Solution of the Mean Spherical Model for Fluids of Hard Spheres with Permanent Electric Dipole Moments

The mean spherical model is solved in closed form for a fluid of hard spheres with permanent electric dipole moments. Both the pair distribution function g(12) and the direct correlation function c(12) consist of a spherically symmetric term and two other terms with different dependences on the orientations of the two dipole moments. The spherically symmetric part is the solution of the Percus-Yevick equation for hard spheres. The angle-dependent terms satisfy two coupled integral equations, which can be decoupled by appropriate changes of the dependent variables. The solutions are expressed in terms of the solution of the Percus-Yevick equation for hard spheres for both positive and negative densities. The effect on g(12) of the finite size of the sample is calculated for the case of a sphere. The correction term in g(12) is found to be of order 1/𝒱, where 𝒱 is the volume of the sample. It is a function not only of the relative distance vector of the two molecules, but also of their positions in the sample. The contribution to the polarization is, nevertheless, constant throughout the sample, in agreement with classical electrostatics. The dielectric constant ε, calculated by Kirkwood's formula, is obtained in closed form. It is a function of a single variable which does not contain the hard sphere diameter.

Some Critical Properties of Ornstein-Zernike Systems and Spherical Models

A recent work on the critical properties of Ornstein-Zernike (OZ) systems in which the direct correlation function $C(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ behaves as $\ensuremath{-}{(\mathrm{kT})}^{\ensuremath{-}1}w(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}g0$, where $w(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is the finite part of the pair potential between particles of a lattice gas with point hard cores, is extended to long-range interactions $w(r)\ensuremath{\propto}{r}^{\ensuremath{-}(d+\ensuremath{\sigma})}$, for $\ensuremath{\sigma}g0$, where $r=|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|$ and $d$ is the dimensionality. The relationship between OZ systems, the spherical model, and the mean spherical model is established, noting that the conditions defining an OZ system are precisely those of the mean spherical model. Explicit expressions for the asymptotic correlation functions in the critical region are obtained, and it is shown that, despite the long-range nature of the interaction, the relation between the exponent $\ensuremath{\eta}$ and the shape of the critical isotherm coincides with the relation predicted by the scaling theory, within a certain range of $\ensuremath{\sigma}$ for fixed $d$. The equation of state is considered both above and below the critical temperature ${T}_{c}$, and a breakdown of Widom's homogeneity argument is exhibited. The equation of state for $Tl{T}_{c}$ makes possible the determination of the coexistence curve for all values of $d$ and $\ensuremath{\sigma}$ for which the system has a critical point; and it is shown that, when applied to short-range interactions, the difficulties found previously by Stell no longer appear.

Remarks on some models for freezing

Some additional remarks are made about some interrelated models for freezing, and comparisons are made. Inconsistencies are pointed out in the treatment of the lattice gas model through the use of the approximated Percus-Yevick equation. The mean spherical model and the local mean spherical model resemble the former approximate model, but are exact; they are free from inconsistencies. These exact models, however, give a very unphysical picture of freezing.

Mean Spherical Model for Lattice Gases with Extended Hard Cores and Continuum Fluids

Mean Spherical Model for Lattice Gases with Extended Hard Cores and Continuum Fluids