Spherical approximation

Equation (168) may be obtained from Eq. (156) or (160) as the asymptotic limit of Cab lor large r by simple substitution of the limiting form for gab, which is given by 

In conjunction with the Omstein-Zemike equation [Eq. (142)], the MSA defines an integral equation that has been solved exactly for a number of systems. For hard spheres, Eq. (168) is the same as the hard-sphere PY approximation, which has been solved by Thiele and Wertheim. For point charges, the MSA is equivalent to the DH approximation. Solutions have also been found for charged hard spheres of equal and disparate diameters, dipolar hard spheres, hard spheres with a Yukawa tail, charged hard spheres in a uniform neutralizing background,and hard nonspherical molecules with general electrostatic interactions.  

Application of these new developments has barely started. The analytical results for the MSA for various models might profitably be used as a reference system for perturbation-type calculations of the thermodynamic properties of ionic systems. A recent calculation perturbs the charged hard-sphere 

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This can be treated analytically within the mean spherical approximation for which... 

Another linearization of the HNC closure leads to the mean spherical approximation (MSA). For a fluid with a hard core, the MSA is... 

Once the degree of association is known, the structure of the bulk dimerizing fluid can be determined by implementation of the adequate closures, such as, for example, the extended mean spherical approximation (EMSA)... 

The EMSA requires the degree of dimerization A as an input parameter. This is quite disappointing. However, it ehminates the deficiency of the Percus-Yevick approximation, Eq. (38). The EMSA represents a simpHfied version, to obtain an analytic solution, of a more sophisticated site-site extended mean spherical approximation (SSEMSA) [67-69]. The results of the aforementioned closures can be used as an input for subsequent calculations of the structure of nonuniform associating fluids. 

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. 

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. 

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. 

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). 

Thus, we have derived the integral equation with respect to the function a. As in the case of Stokes problem it is possible to apply the spherical approximation, that is, the magnitude of the normal field at points of the surface S is... 

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. 

While Onsager s formula has been widely used, there have also been numerous efforts to improve and generalize it. An obvious matter for concern is the cavity. The results are very sensitive to its size, since Eqs. (33) and (35) contain the radius raised to the third power. Within the spherical approximation, the radius can be obtained from the molar volume, as determined by some empirical means, for example from the density, the molar refraction, polarizability, gas viscosity, etc.90 However the volumes obtained by such methods can differ considerably. The shape of the cavity is also an important issue. Ideally, it should be that of the molecule, and the latter should completely fill the cavity. Even if the second condition is not satisfied, as by a point dipole, at least the shape of the cavity should be more realistic most molecules are not well represented by spheres. There was accordingly, already some time ago, considerable interest in progressing to more suitable cavities, such as spheroids91 92 and ellipsoids,93 using appropriate coordinate systems. Such shapes... 

Figure 14.6 Incomplete filling of the h shell deduced from the spherical approximation and the resulting bond length alternation in neutral Qo-...

The long-range electrostatic term is expressed by mean spherical approximations which is a very promising method for describing the thermodynamic properties of electrolyte solutions [192,193] ... 

The Born equation, proposed in 1920, has been modified in various ways in order to get a single equation that can express the experimental ionic solvation energies. In recent years, the so-called mean spherical approximation (MSA) has often been used in treating ion solvation. In the MSA treatment, the Gibbs energy of ion solvation is expressed by... 

The concept of mean spherical approximation (MSA, 3) in Chapter 2) has also been used to reproduce the conductivity data of electrolytes of fairly high concentration [23]. The MSA method applies to both associated and non-associated electrolytes and can give the values of association constant, KA. Although not described here,... 

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. 

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. 

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],... 

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

From the various possible closures, the mean spherical approximation (MSA) [189] has found particularly wide attention in phase equilibrium calculations of ionic fluids. The Percus-Yevick (PY) closure is unsatisfactory for long-range potentials [173, 187, 190]. The hypemetted chain approximation (HNC), widely used in electrolyte thermodynamics [168, 173], leads to an increasing instability of the numerical algorithm as the phase boundary is approached [191]. There seems to be no decisive relation between the location of this numerical instability and phase transition lines [192-194]. Attempts were made to extrapolate phase transition lines from results far away, where the HNC is soluble [81, 194]. 

Here Veff (r, t) is the effective potential energy that is determined from mean spherical approximation and is written in terms of the two-particle direct correlation function c(r) ... 

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