Solution Of Mean Spherical Approximation Research Articles

Chemical potential of an ion in an asymmetric electrolyte within the mean spherical approximation (MSA)

We study the known solution of the mean spherical approximation (MSA) for the primitive model of an electrolyte consisting of charged hard spheres with unequal diameters. Especially, details of the non-trivial evaluation of chemical potentials are given. They can be obtained in various independent ways from the known MSA solution. The different ways of evaluation are utilised to show that consistent results are obtained. Details for other thermodynamic quantities are also presented. An additional purpose of this rederivation of results is to give a firmer basis and insight to obtain the correct and consistent thermodynamics for the more demanding and realistic situation with polar particles present in the MSA electrolyte.

A comprehensive comparison between thermodynamic perturbation theory and first-order mean spherical approximation: Based on discrete potentials with hard core

Using the TL (Tang and Lu, 1993) method, Ornstein-Zernike integral equation is solved perturbatively under the mean spherical approximation (MSA) for fluid with potential consisting of a hard sphere plus square-well plus square-shoulder (HS+SW+SS) to obtain first-order analytic expressions of radial distribution function (RDF), second-order direct correlation function, and semi-analytic expressions for common thermodynamic properties. A comprehensive comparison between the first-order MSA and high temperature series expansion (HTSE) to third-, fifth- and seventh-order is performed over a wide parameter range for both a HS+SW and the HS+SW+SS model fluids by using corresponding “exact” Monte Carlo results as a reference; although the HTSE is carried out up to seventh-order, and not to the first order as the first-order MSA the comparison is considered fair from a calculation complexity perspective. It is found that the performance of the first-order MSA is dramatically model-dependent: as target potentials go from the HS+SW to the HS+SW+SS, (i) there is a dramatic dropping of performance of the first-order MSA expressions in calculating the thermodynamic properties, especially both the excess internal energy and constant volume excess heat capacity of the HS+SW+SS model cannot be predicted even qualitatively correctly. (ii) One tendency is noticed that the first-order MSA gets more reliable with increasing temperatures in dealing with the pressure, excess Helmholtz free energy, excess enthalpy and excess chemical potential. (iii) Concerning the RDF, the first-order MSA is not as disappointing as it displays in the cases of thermodynamics. (iv) In the case of the HS+SW model, the first-order MSA solution is shown to be quantitatively correct in calculating the pressure and excess chemical potential even if the reduced temperatures are as low as 0.8. On the other hand, the seventh-order HTSE is less model-dependent; in most cases of the HS+SW and the HS+SW+SS models, the seventh-order HTSE improves the fifth- and third-order HTSE in both thermodynamic properties and RDF, and the improvements are very demonstrable in both the excess internal energy and constant volume excess heat capacity; for very limited cases, the seventh-order HTSE improves the fifth-order HTSE only within lower density domain and even shows a bit of inadaptation over higher density domain.

Liquid/vapor coexistence and surface tension of the Sutherland fluid with a variable range of interaction: Computer simulation and perturbation theory studies

The liquid-vapor phase diagram and surface tension for the Sutherland fluids with a variable range of interaction have been determined by canonical Monte Carlo simulations and compared with the augmented van der Waals analytic theory. The theory, based on a short-range Yukawa reference, performs well for medium-range models but its accuracy deteriorates with the shortening range due to deteriorating accuracy of the underlined analytic mean spherical approximation solution for the reference Yukawa potential. The simulation results are also analyzed from the point of the extended principle of corresponding states and it is shown that the surface tension and also, to a certain degree of accuracy, the vapor-liquid coexistence curve satisfy the principle.

Some Explicit Results for the Mean Spherical Approximation for Mixtures of Yukawa Fluids

The mean spherical approximation (MSA) is of interest because it produces an integral equation that yields useful analytical results for a number of fluids. One such case is the Yukawa fluid, which is a reasonable model for a simple fluid. The original MSA solution for this fluid, due to Waisman, is analytic but not explicit. Ginoza has simplified this solution. However, Ginoza's result is not quite explicit. Some years ago, Henderson, Blum, and Noworyta obtained explicit results for the thermodynamic functions of a single-component Yukawa fluid that have proven useful. They expanded Ginoza's result in an inverse-temperature expansion. Even when this expansion is truncated at fifth, or even lower, order, this expansion is nearly as accurate as the full solution and provides insight into the form of the higher-order coefficients in this expansion. In this paper Ginoza's implicit result for the case of a rather special mixture of Yukawa fluids is considered. Explicit results are obtained, again using an inverse-temperature expansion. Numerical results are given for the coefficients in this expansion. Some thoughts concerning the generalization of these results to a general mixture of Yukawa fluids are presented.

On the first-order mean spherical approximation

The general solution of the Ornstein–Zernike equation presented by Tang and Lu [J. Chem. Phys. 99, 9828 (1993)] is further discussed. By applying the Hilbert transform, the first-order factorization and direct correlation functions (DCF) are generally and analytically obtained, with emphasis on the mean spherical approximation (MSA) for Yukawa fluids. These analytical results are employed to produce a new DCF for hard spheres through integrating with the previous generalized mean spherical approximation [J. Chem. Phys. 103, 7463 (1995)]. The new DCF is of simple analytical form and remedies the deficiencies of its Percus–Yevick version at high densities. Comparisons between the first-order and full MSA solutions are also made. It is shown that the two solutions give very close results for thermodynamic properties in the phase stable region and phase coexistence curves away from the critical point. At unstable states, the first-order MSA looks more advantageous when applications go beyond homogeneous.

Low-Density Expansion of the Solution of Mean Spherical Approximation for Ion−Dipole Mixtures

A low-density expansion of the solution of the mean spherical approximation (MSA) for ion−dipole mixtures in the semi-restricted case (with equal size hard sphere cations and anions and different size hard sphere dipoles) is proposed. From the expansion, the explicit approximations of the three energy parameters (b0, b1, b2) and the thermodynamic properties are obtained. A reasonable explanation is offered as to why the dielectric constant of the vacuum cannot be used in the long-range term in the successful EOSs for real electrolytes solutions, although the dipolar solvent is treated as explicit molecule in the models. The energy of the solvation and the relationship between the primitive model (PM) and the nonprimitive model (NPM) are discussed in some limit cases. Comparisons of our approximation with the molecular simulation are also carried out. An EOS with rather simple expressions based on the expansion is applied to the calculations of activity coefficients, osmotic coefficients, and densities of the sodium chloride aqueous solutions.

Analytical solution of the Yukawa closure of the Ornstein-Zernik equation IV: the general 1-component case

In previous work we have studied the solution of the Ornstein-Zernike equation with a general multiyukawa closure. Here the direct correlation function is expressed by a rapidly converging sum of M (complex) exponentials. For a simple fluid the mathematical problem of solving the Ornstein-Zernike equation is equivalent to finding the solution of a linear algebraic equation of order M. The solution for the arbitrary case is given in terms of a scaling matrix Γ. For only one component this matrix is diagonal and the general solution using the properties of M-dimensional SOM Lie group is given. In the Mean Spherical Approximation (MSA) the excess entropy is obtained and expressed as a sum of 1-dimensional integrals of algebraic functions. We remark that the general solution of the M exponents-1 component case was found in our early work (Blum, L., and Hoye, J. S.,1978, J. stat. Phys., 19, 317) in implicit form. The present explicit solution agrees completely with the early one. Other thermodynamic properties such as the energy equation of state are also obtained, explicitly for 2 and 3 exponentials. The analytical solution of the effective MSA is also obtained from the simple variational form for the Helmholtz excess free energy δA (1) where (2) where both the excess energy ΔE(Γ) and the excess entropy ΔS(Γ) are functionals of Γ, which opens interesting possibilities that are discussed elsewhere. We remark that this is a non-trivial property, which is certainly true for the MSA (Chandler, D., and Andersen, H. C., 1972, J. chem. Phys., 57, 1930). It implies cross-derivative properties for the closure equations, which have been verified in all cases.

Thermodynamics and structural properties of the dipolar Yukawa fluid

We report computer simulations and a theoretical study of the thermodynamics and structure of a dipolar Yukawa system. A comparison between the analytical mean spherical approximation (MSA) solution, perturbation theory and Monte Carlo simulation data of pressure, internal energy and dielectric constant is given. In the perturbation theory, the MSA solution of hard core Yukawa fluid is used as a reference system. It was found that the MSA solution is reasonable only at lower dipole moments, while the perturbation theory gives good results at low and high values of dipole moment. Liquid–vapor coexistence data of dipolar Yukawa fluid are also obtained by Monte Carlo simulation and by both MSA and perturbation theory. It was found that at high dipole moments the liquid–vapor equilibrium disappears while chain-like structures appear in the low density fluid phase. The appearance of chain-like structures of dipolar Yukawa fluid is discussed in comparison with the Stockmayer fluid.

Static structure factor for a colloidal dispersion with size and “charge” polydispersities: Mean spherical approximation model in hard-sphere Yukawa fluids

An analytical model is presented of the static structure factor for a colloidal dispersion with size and ``charge'' polydispersities. The model is based on the mean spherical approximation solution of the Ornstein-Zernike equation in a hard-sphere Yukawa fluid mixture. With the use of the model, the size- and ``charge''-polydispersity effects are investigated on the structure, and characteristics of the effects are discussed.

Analytical model for thermodynamic properties of a colloidal dispersion with size and ‘charge’ polydispersities

An analytical model is presented for thermodynamic properties of a colloidal dispersion with size and ‘charge’ polydispersities. The model is based on the mean-spherical approximation solution of the Ornstein—Zernike equation in a multicomponent hard sphere Yukawa fluid, and is used to investigate the effects of polydispersity on the isotherm behaviour of the equation of state. The characteristics of the size and charge effects are discussed.

Analytical structure factors for colloidal fluids with size and interaction polydispersities

The hard sphere M Yukawa fluid is considered as a model for a colloidal fluid. On the basis of the mean spherical approximation solution of the Ornstein-Zernike equation, for the case of the closure relation consisting of the sum of the M Yukawa terms with the factorizable coefficients, compact and useful expressions of static structure factors are presented. The expressions are tractable for the hard sphere M Yukawa fluid with intrinsic size and interaction polydispersities as well as for the fluid with an arbitrary number of components.

Effective Interaction Potential between Colloidal Particles in a Solution

In a solution containing microions from an added electrolyte, the effective interaction potential between colloidal particles is studied on the basis of the primitive model. The approximation is employed that the correlations of solvent-particles with microions can be neglected, and the potential is calculated with the use of the mean spherical approximation solution of the Ornstein-Zernike equation with the Yukawa closure. In the case of the hydrophobic colloid, the theory is discussed in comparison with the DLVO theory and a role of the solvent-mediated potential is emphasized.

Scattering functions for multicomponent mixtures of charged hard spheres, including the polydisperse limit: Analytic expressions in the mean spherical approximation

We present a closed analytical formula for the scattering intensity from charged hard sphere fluids with any arbitrary number of components. Our result is an extension to ionic systems of Vrij’s analogous expression for uncharged hard sphere mixtures. Use is made of Baxter’s factor correlation functions within the mean spherical approximation (MSA). The polydisperse case of an infinite number of species with a continuous distribution of hard sphere diameters and charges is also considered. As an important by-product of our investigation, we present some properties of a particular kind of matrices (sum of the identity matrix with a dyadic matrix) appearing in the solution of the MSA integral equations for both uncharged and charged hard sphere mixtures. This analysis provides a general framework to deal with a wide class of MSA solutions having dyadic structure and allows an easy extension of our formula for the scattering intensity to different potential models. Finally, the relevance of our results for the interpretation of small angle neutron scattering experimental data is briefly discussed.

Solution of the polymer mean spherical approximation for the totally flexible sticky two-point electrolyte model

The general solution of the polymer MSA (mean spherical approximation) for an arbitrary mixture of charged hard spheres with multiple bonding obtained recently by Blum, Holovko and Protsykevytch [J. Stat. Phys,] is applied to the case of totally flexible linear polymer chains. The approximation is an extension of the associative MSA proposed by M.Holovko and Yu.Kalyuzhnyi, [Mol.Phys., 73, 1145(1991)] to study the effects of association in ionic systems. The model is defined by adding charges to the multicomponent version of the totally-flexible sticky two-point model of associating monomers introduced by Wertheim.

Model for Solvent-Mediated Force between Colloidal Particles in a Liquid: Application to Lyophobic Attraction

The solvent-mediated force between colloidal particles in a liquid is studied with the use of the model system consisting of the solvent spheres and the solute spheres (colloidal particles): spheres interact through Yukawa-type potentials with different damping factors. The study is based on the mean spherical approximation solution of the Ornstein-Zernike equation with the closure of three Yukawa terms. In the case of the lyophobic solute in the dilute limit, the dependence of the solvent-mediated force on the interaction between solvent spheres is investigated in relation to the exponential decay nature of the force measured in the range shorter than about 15 nm. The result is consistent with the experimental one.

Thermodynamic properties of an asymmetric fluid mixture with Yukawa interaction in the mean spherical approximation

The analytical solution of mean spherical approximation of the Ornstein–Zernike equation for a Yukawa fluid with factorizable coefficients is used to obtain a simple form of the equation of state for the mixture. These results are an extension of Ginoza’s work.

Mathematical comment on adhesive hard sphere models for fluids

A mathematical discussion is provided on the relationship between Baxter's adhesive hard sphere fluid and the hard sphere Yukawa fluid with sufficiently strong and short range attractive Yukawa tails. It is proved that the Percus-Yevick approximate solution of the Ornstein-Zernike equation for the former fluid can be included mathematically in the generalized mean-spherical approximation solution with the Yukawa closure, and the equivalence conditions of both solutions are presented.

Solution of the polymer MSA for the polymerizing primitive model of electrolytes

An analytical solution of the polymer MSA (mean spherical approximation) for the polymerizing primitive model of electrolytes is obtained. The approximation is an extension of the associative MSA proposed recently [M. Holovko and Yu. Kalyuzhnyi, Mol. Phys. 73 (1991) 1145] to study the effects of association in ionic systems. The model is defined by adding charges to the totally flexible sticky two-point model of associating monomers introduced by Wertheim. In the limiting case of an uncharged system, our solution reduces to the solution of the Wertheim polymer PY approximation solved recently [J. Chang and S.I. Sandler, J. Chem. Phys. 102 (1995) 437].

Phase equilibria and critical lines in model water/salt mixtures

The mean spherical approximation solution for the thermodynamics of a mixture of equal-sized dipolar hard spheres and charged hard spheres is used to calculate phase equilibria and critical lines. The binary system is characterized by the dimensionless ratio of the dipolar strength to the strength of the charges. At high values of this ratio, the critical curve is interrupted by a liquid–liquid equilibrium and the phase diagram (under the usual classification scheme) is type III. As the ratio is lowered, the critical curve becomes continuous; this is type I or II behavior. The continuous critical line is maintained until the critical temperature of the ionic component exceeds that of the dipolar component by a factor of approximately 3.56; such a lengthy unbroken critical line is unusual in real fluid systems but is observed in many mixtures of water with strong electrolytes. At sufficiently low values of the dimensionless ratio, the critical line is interrupted very near the critical point of the dipolar component; these systems exhibit type IV or V behavior. The simple model used here reproduces the qualitative features of the phase diagrams of real water/salt systems. Insights from the model suggest at least partial explanations for the unusually long unbroken critical lines observed in many of these systems.

Liquid theory for band structure in a liquid. III. The mean spherical approximation for p bands and the numerical solution of the mean spherical approximation for both s and p bands

Formally, the problem of calculating the single-electron energy levels for a liquid requires that one diagonalize a 1023×1023 matrix, but previous work has made it clear that precisely the same information is available from the solution of a simple classical liquid problem. We extend our previous applications of this idea in several ways here: (1) the mean spherical approximation (MSA) for liquids is used to provide an explicit route to the density of states for a band resulting from a basis of p orbitals, (2) the previous MSA solution for s bands and the new MSA solution for p bands are both generalized to allow for nonorthogonality in the basis, and (3) numerical procedures are described for solving the integral equations resulting from these MSA theories. These developments mean that it is now computationally feasible to find the band structure of almost any simple liquid within a tight-binding model. We illustrate this point by computing the s and p bands expected from a hard-sphere liquid with a minimal basis of hydrogenic orbitals on each atom.