Mean spherical approximation “Pairing

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. 

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). 

We start with a model of polar molecules in which the effects of polarizability are neglected. More precisely, we assume that in the absence of external fields, the potential energy associated with N particles is a sum of pair potentials < >( /), each of which depends on the positions r, and tj and orientations S2, and itj of particles / and j. Thus the particles are regarded as rigid, with no internal coordinates, and we assume for simplicity that they are all identical. Extensions of the results of Section II to mixtures are for the most part straightforward, as discussed by Hoye and StelP and in references they cite. Pertinent references to the mean spherical approximation generalized to mixtures are also given at an appropriate point in this chapter. 

The mean spherical approximation (MSA) is defined by three equations relating the pair correlation function h 2) and the direct correlation function c(12). These are ... 

In the last two decades, new extended laws have been obtained for the concentration dependence of transport properties. It was possible [ 15,16] to use the Fuoss-Onsager theory together with new, more accurate equilibrium pair distribution functions as obtained with the help of the hypemetted chain (HNC) or mean spherical approximation (MSA). 

Bernard O, Blum L (1996) Binding mean spherical approximation for pairing ions an exponential approximation and thermodynamics. J Chem Phys 104 4746 754... 

MSA is the extension of Debye-Hiickel theory to high electrolyte concentrations using the same continuum model [397, 398). Combined with the law of mass action for ion- pair formation, MSA-MAL (mass action law approach), and finally as associative mean spherical approximation (AMSA) [399-401], it permits us to take into account any ion-complex formation. Equations for electrolyte conductivity are given by Blum et al. [402, 403]. However, the complexity of battery electrolytes hinders the application of a high concentration continuum approach. 

Figure 7.5 exhibits structure factors calculated within the mean spherical approximation for states marked by diamonds in Figure 7.4 (left panel). The S q) exhibits a principal refraction peak as known from other simple liquids. It is caused by the hard-core-driven excluded volume phenomenon. The high-temperature curves 1 and 2 exhibit peaks, which are only slightly smaller and somewhat broader than the peaks of an HS system at the same densities. The attraction modifies the pair correlations and thus the excluded volume effects, as can be inferred by comparing the curves 1 and 3. Lowering the temperature, the short-ranged attraction causes the particles to move closer ... 

Tikanen AC, Fawcett WR. (1996) The role of solvent permittivity in estimation of electrolyte activity coefficients for systems with ion pairing on the basis of the mean spherical approximation. Ber Buns Phys Chem 100 634—640. 

Earlier [9] we simplified the expressions for as given by the first-order mean spherical approximation by Tang et al. [20, 21], which is based upon pair-wise additivity of the interaction. The result is ... 

An exact statistical theory of smectics based on the pair potential, Eq. [12], is extremely difficult to accomplish. Therefore we derive a mean-field approximation to the theory. For this purpose we require the mean-field version of the single molecule potential function. In a previous chapter this problem was examined for the case of the nematic phase. A perfectly general form oiV 2 was assumed and expanded in a series of spherical harmonics. A new coordinate system was then chosen such that the polar axes coincided with the director. The single molecule potential was then obtained by averaging V 2 over all possible positions and orientations of molecule 2 consistent with the structure of the nematic phase. The resulting single molecule potential had the form... 

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