Mean-spherical approximation theory

Ruas, A. Bernard, O. Caniffi, B. Simonin, P.J. Turq, P. L. Moisy, P. (2006) Uranyl(VI) Nitrate Salts Modeling Thermodynamic Properties Using the Binding Mean Spherical Approximation Theory and Determination of "Fictive" Binary Data, IPhys.Chem. B., 110 3435 - 3443. 

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

Recent developments of the chemical model of electrolyte solutions permit the extension of the validity range of transport equations up to high concentrations (c 1 mol L"1) and permit the representation of the conductivity maximum Knm in the framework of the mean spherical approximation (MSA) theory with the help of association constant KA and ionic distance parameter a, see Ref. [87] and the literature quoted there in. 

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. 

An important advance in the understanding of microscopic solvation and Onsager s snowball picture has recently been made through the introduction of the linearized mean spherical approximation (MSA) model for the solvation dynamics around ionic and dipolar solutes. The first model of this type was introduced by Wolynes who extended the equilibrium linearized microscopic theory of solvation to handle dynamic solvation [38]. Wolynes further demonstrated that approximate solutions to the new dynamic MSA model were in accord with Onsager s predictions. Subsequently, Rips, Klafter, and Jortner published an exact solution for the solvation dynamics within the framework of the MSA [43], For an ionic solute, the exact results from these author s calculations are in agreement with Onsager s inverted snowball model and the previous numerical calculations of Calef and Wolynes [37]. Recently, the MSA model has been extended by Nichols and Calef and Rips et al. [39-43] to solvation of a dipolar solute. 

Onsager inverted snowball theory, 34 linearized mean spherical approximation in, 35... 

In a parallel series of developments, starting with the Mayer theory and continuing with the so-called mean spherical approximation, the effects of hydration and ion association were arbitrarily removed from consideration, in spite of their undeniable presence in nature. 

Another approach to the conductance of electrolytes, which is less complex than that of Lee and Wheaton, is due to Blum and his co-workers. This theory goes back to the original Debye-Hiickel-Onsager concepts, for it does not embrace the ideas of Lee and Wheaton about the detailed structure around the ion. Instead, it uses the concept of mean spherical approximation of statistical mechanics. This is the rather portentous phrase used for a simple idea, which was fully described in Section 3.12. It is easy to see that this is an approximation because in reality an ionic collision with another ion will be softer than the brick-wall sort of idea used in an MSA approach. However, using MSA, the resulting mathematical treatment turns out to be relatively simple. The principal equation from the theory of Blumet al. is correspondingly simple and can be quoted. It runs... 

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... 

In addition to the short-range interactions between species in all solutions, long-range electrostatic interactions are found in electrolyte solutions. The deviation from ideal solution behavior caused by these electrostatic forces is usually calculated by some variation of the Debye-Huckel theory or the mean spherical approximation (MSA). These theories do not include terms for the short-range attractive and repulsive forces in the mixtures and are therefore usually combined with activity coefficient models or equations of state in order to describe the properties of electrolyte solutions. 

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. 

In the physical picture ion-pairs are just consequences of large values of the Mayer /-functions that describe the ion distribution [22], The technical consequence, however, is a major complication of the theory the high-temperature approximations of the /-functions applied, e.g. in the mean spherical approximation (MSA) or the Percus-Yevick approximation (PY) [25], suffice in simple fluids but not in ionic systems. 

In addition to the repulsive part of the potential given by Eq. (4), a short-range attraction between the macroions may also be present. This attraction is due to the van der Waals forces [17,18], and can be modelled in different ways. The OCF model can be solved for the macroion-macroion pair-distribution function and thermodynamic properties using various statistical-mechanical theories. One of the most popular is the mean spherical approximation (MSA) [40], The OCF model can be applied to the analysis of small-angle scattering data, where the results are obtained in terms of the macroion-macroion structure factor [35], The same approach can also be applied to thermodynamic properties Kalyuzhnyi and coworkers [41] analyzed Donnan pressure measurements for various globular proteins using a modification of this model which permits the protein molecules to form dimers (see Sec. 7). 

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."... 

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... 

Marcus theory 37 Maxwell construction 46, 50, 51 mean spherical approximation (MSA) 8,49-51,171, 172,178-180 -closure 9,49,50,171,180 mechanical instability 51 melittin 122... 

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). ... 

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