Solvent mean spherical approximation

Secondly, the solvent has been introduced on the same footing as the ions. For low values of the charge the mean spherical approximation (MSA) has been extensively used, whereas some more complicated approximations are needed to describe the nonlinear behavior versus cr or 0. 

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. 

One should also mention the so-called mean spherical approximation (MSA) treatment of solvent reorganization [25]. McManis and Weaver [125] considered how the solvent radius and dielectric parameters affect the electron transfer within the frame of this theory. The frequency dependence of the effective radius should cause significant deviations from the Marcus expression for the activation energy of... 

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."... 

Ion-Solvent Interactions According to the Mean Spherical Approximation... 

A treatment for polar solvents on the basis of the mean spherical approximation was first given by Wertheim [24, 25]. The closure conditions are based simply on the dipole-dipole interaction energy between the polar molecules in the system. Neglecting molecular polarizability, these conditions are... 

Two points should be mentioned here. First, the effect of solutes on the solvent dielectric response can be important in solvents with nonlocal dielectric properties. In principle, this problem can be handled by measuring the spectrum of the whole system, the solvent plus the solutes. Theoretically, the spatial dependence of the dielectric response function, s(r, co), which includes the molecular nature of the solvent, is often treated by using the dynamical mean spherical approximation [28, 36a, 147a, 193-195]. A more advanced approach is based on a molecular hydrodynamic theory [104,191, 196, 197]. These theoretical developments have provided much physical insight into solvation dynamics. However, reasonable agreement between the experimentally measured Stokes shift and emission line shape can be... 

The mean-spherical approximation provides an adequate description of the PE star conformation under conditions of good or theta-solvent. However, in contrast to some early theoretical predictions, simulations give evidence that conformational transition related to the collapse of hydrophobic or thermosensitive PE stars is accompanied by the formation of various intramolecular structures of low symmetry (pearl necklaces, bundles). 

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). ... 

Fawcett WR, Tikanen AC (1996) Role of solvent permittivity in estimation of electrolyte activity coefficients on the basis of the mean spherical approximation. J Phys Chem 100 4251-4255... 

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. ... 

---