Mean-spherical approximation integral equations

Lupkowski and Monson have also discussed the connection between the approximations described above and the integral equation approximations based upon the CSL equation described in Section III.B. They show that the ISF-ORPA is the same as a reference mean spherical approximation (RMSA) to the CSL integral equation, following the work of Madden " in connection with the corresponding approximations for simple fluids. This relationship provides the basis of the computational methods that have been used so far. The approach has been applied to the calculation of the influence... 

During the last 20 years, a quantitative description of conductance and self-diffusion up to 1-M solutions has been achieved by the use of modern gij functions coming from integral equation techniques such as the hypernetted chain (HNC) equation or mean spherical approximation... 

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. 

Theory nowadays overcomes the limitation of concentration range by integral equation and simulation methods. Mean spherical approximation (MSA) and hypernetted chain approximation (HNC) are the most important features yielding modern analytical transport equations over extended concentration ranges. Nevertheless, the IcCM expressions maintain their importance. They are reliable expressions for the determination of limiting values of the transport properties at infinite dilution of the electrolyte as a convenient basis for the provision of... 

Von Solms, N., Chiew, Y.C. Analytical integral equation theory for a restricted primitive model of polyelectrolytes and counterions within the mean spherical approximation. II. Radial distribution functions. J. Chem. Phys. 118, 4321 (2003). doi 10.1063/l.1539842... 

Integral Equation and Eield-Theoretic Approaches In addition to theories based on the direct analytical extension of the PB or DH equation, PB results are often compared with statistical-mechanical approaches based on integral equation or density functional methods. We mention only a few of the most recent theoretical developments. Among the more popular are the mean spherical approximation (MSA) and the hyper-netted chain (HNC) equation. Kjellander and Marcelja have developed an anisotropic HNC approximation that treats the double layer near a flat charged surface as a series of discrete layers.Attard, Mitchell and Ninham have used a Debye-Hiickel closure for the direct correlation function to obtain an analytical extension (in terms of elliptic integrals) to the PB equation for the planar double layer. Both of these approaches, which do not include finite volume corrections, treat the fluctuation potential in a manner similar to the MPB theory of Outhwaite. 

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

In this chapter we have seen how all angular dependencies can be integrated out analytically because of the spherical symmetry of the central field potential of an atom. We are now left with the task to determine the yet unknown radial functions, for which we derived the self-consistent field equations based upon the variational minimax principle. We now address the numerical solution of these equations. The mean-field potential in the set of coupled SCF equations is the reason why we cannot solve them analytically. Hence, the radial functions need to be approximated in some way in... 

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