Mean spherical approximation solvent models

Carnie and Chan and Blum and Henderson have calculated the capacitance for an idealized model of an electrified interface using the mean spherical approximation (MSA). The interface is considered to consist of a solution of charged hard spheres in a solvent of hard spheres with embedded point dipoles, while the electrode is considered to be a uniformly charged hard wall whose dielectric constant is equal to that of the electrolyte (so that image forces need not be considered). 

This was averaged over the total distribution of ionic and dipolar spheres in the solution phase. Parameters in the calculations were chosen to simulate the Hg/DMSO and Ga/DMSO interfaces, since the mean-spherical approximation, used for the charge and dipole distributions in the solution, is not suited to describe hydrogen-bonded solvents. Some parameters still had to be chosen arbitrarily. It was found that the calculated capacitance depended crucially on d, the metal-solution distance. However, the capacitance was always greater for Ga than for Hg, partly because of the different electron densities on the two metals and partly because d depends on the crystallographic radius. The importance of d is specific to these models, because the solution is supposed (perhaps incorrectly see above) to begin at some distance away from the jellium edge. 

Considerable progress has been made in going beyond the simple Debye continuum model. Non-Debye relaxation solvents have been considered. Solvents with nonuniform dielectric properties, and translational diffusion have been analyzed. This is discussed in Section II. Furthermore, models which mimic microscopic solute/solvent structure (such as the linearized mean spherical approximation), but still allow for analytical evaluation have been extensively explored [38, 41-43], Finally, detailed molecular dynamics calculations have been made on the solvation of water [57, 58, 71]. 

A number of theoretical models for solvation dynamics that go beyond the simple Debye Onsager model have recently been developed. The simplest is an extension of Onsager model to include solvents with a non-Debye like (dielectric continuum and the probe can be represented by a spherical cavity. Newer theories allow for nonspherical probes [46], a nonuniform dielectric medium [45], a structured solvent represented by the mean spherical approximation [38-43], and other approaches (see below). Some of these are discussed in this section. Attempts are made where possible to emphasize the comparison between theory and experiment. 

Models for solvation in water that allow for a structured solvent do indeed predict a multiexponential response. For instance, the dynamical mean spherical approximation (MSA) for water solvation predicts that solvation of an ion in water is well represented by two characteristic times [38]. Nonetheless, the specific relaxation times differ substantially from the observed behavior [33],... 

The latest models propose to represent electrolyte solutions as a collections of hard spheres of equal size, ions, immersed in a dielectric continuum, the solvent. For such a system, what is called the Mean Spherical Approximation, MSA, has been successful in estimating osmotic and mean activity coefficients for aqueous 1 1 electrolyte solutions, and has provided a reasonable fit to experimental data for dilute solutions of concentrations up to -0.3 mol dm". The advantage in this approach is that only one... 

The so-called mean spherical approximation (MSA) treatment of the solvation energy should also be mentioned. Within the frame work of that model the electrostatic energy of ions is given by a Born-like expression [25], where the effective radius of the ion is considered to be the sum of the ionic radius and a correction term which depends not only on the solvent molecule diameter but also on the dielectric permittivity. Thus, the effective radius is a function of the frequency of the electromagnetic field. 

Abstract Analytical solution of the associative mean spherical approximation (AMSA) and the modified version of the mean spherical approximation - the mass action law (MSA-MAL) approach for ion and ion-dipole models are used to revise the concept of ion association in the theory of electrolyte solutions. In the considered approach in contrast to the traditional one both free and associated ion electrostatic contributions are taken into account and therefore the revised version of ion association concept is correct for weak and strong regimes of ion association. It is shown that AMSA theory is more preferable for the description of thermodynamic properties while the modified version of the MSA-MAL theory is more useful for the description of electrical properties. The capabilities of the developed approaches are illustrated by the description of thermodynamic and transport properties of electrolyte solutions in weakly polar solvents. The proposed theory is applied to explain the anomalous properties of electrical double layer in a low temperature region and for the treatment of the effect of electrolyte on the rate of intramolecular electron transfer. The revised concept of ion association is also used to describe the concentration dependence of dielectric constant in electrolyte solutions. 

Continuum dielectric models of solvation can be generalized to include some aspects of the solvent molecularity. This has lead to the dynamic mean spherical approximation which improves the agreement between these kind of theories and experimental observations."... 

L. Blum, Solution of a model for the solvent-electrrrfyte interactions in the mean spherical approximation, /. Chem. Phys. 61,2129-2133 (1974). 

While the McMillan-Mayer theory (Section 4) prescribes the iiabir) as functionals of the Hamiltonian of a BO-level model, little has been learned from this sort of direct approach. The main contributions are an analytical study of charged hard spheres in an uncharged hard-sphere solvent by Stell, " Monte Carlo and molecular dynamics studies of somewhat more realistic models, " " and a study using the mean spherical approximation (Section 7.3). ... 

It is possible to take into account the short range ion-ion interaction effect on the volumetric properties of electrolytes by resorting to integral equation theories, as the mean spherical approximation (MSA). The MSA model renders an analytical solution (Blum, 1975) for the umestricted primitive model of electrolytes (ions of different sizes immersed in a continuous solvent). Thus, the excess volume can be described in terms of an electrostatic contribution given by the MSA expression (Corti, 1997) and a hard sphere contribution obtained form the excess pressure of a hard sphere mixture (Mansoori et al, 1971). The only parameters of the model are the ionic diameters and numerical densities. 

The DH model (eqs 8.45 and 8.46), can also be derived from statistical mechanics as the solution of the mean spherical approximation (MSA) for an electro-neutral mixture of point ions in a continuum solvent i.e. it is an example of a MM-level model). It represents a limiting behaviour of electrolyte solutions and breaks down quickly for concentrations higher than 0.01 mol dm the extended DH model is accurate to a 0.1 mol dm . For more concentrated solutions, it is natural to consider replacing the point ions with finite-size ions this leads to the consideration of so-called primitive models (PMs), which are MM models consisting of an electro-neutral mixture of charged hard spheres in a continuum solvent. The simplest PM is the restricted primitive model (RPM) consisting of an equimolar mixture of equal-diameter charged hard spheres in a dielectic continuum i.e. p+=p = pj2, <7+ = <7 = a). 

Percus-Yevick, and the mean spherical approximations. The last of these assumes that the solvent consists of hard spheres with a long-range attractive force. It is widely applied to the modeling of solvent effects. Generalizations to multi-component fluids are straightforward. ... 

The starting point for such analytical efforts is linear response theory. Different approaches include the dynamical mean spherical approximation (MSA), " generalized transport equations, and ad hoc models for the frequency and wavevector dependence of the dielectric response function e(k,w). These linear response theories are very valuable in providing fundamental understanding. However, they carmot explore the limits of validity of the imderlying hnear response models. Numerical simulations can probe nonlinear effects. They are very useful in the direct visualization and examination of the interplay between solvent and solute properties and the different relaxation times associated... 

This result was obtained by Chan et al. " and is similar to the second moment condition first derived by Stillinger and Lovett. However, it should be noted that Stillinger and Lovett consider only primitive model electrolytes, and the dielectric constant occurring in their formula is that of the pure solvent. For the mean spherical, LHNC, and QHNC approximations, all three formulas [(5.20b), (5.25), and (5.26)] must give consistently. 

In finite boundary conditions the solute molecule is surrounded by a finite layer of explicit solvent. The missing bulk solvent is modeled by some form of boundary potential at the vacuum/solvent interface. A host of such potentials have been proposed, from the simple spherical half-harmonic potential, which models a hydrophobic container [22], to stochastic boundary conditions [23], which surround the finite system with shells of particles obeying simplified dynamics, and finally to the Beglov and Roux spherical solvent boundary potential [24], which approximates the exact potential of mean force due to the bulk solvent by a superposition of physically motivated tenns. 

As with other three parameter systems, solvents are represented by points in a three dimensional model and polymer solubility by a volume. Solvents falling within this volume of solubility dissolve the polymer and those outside the volume do not. Hansen found that by doubling the scale of the d axis relative to the other axes, the volumes of solubility of most polymers were approximately spherical. This means that each polymer may be described in terms of the centre of this sphere having coordinates d o, po and and its radius (known as the radius of interaction), Rao- Values of the centre coordinates and radii of interaction for several polymers are given in Table 2.16. 

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