The reduced electrophoretic mobility-reduced zeta potential relationship for a charged macroparticle is shown to be nonuniversal and to be highly nonlinear. In agreement with experimental results, a mobility reversal due to the macroion's charge inversion and a nonlinear dependence of the mobility on salt concentration is obtained.

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One of the purposes of this paper is to assess the degree of applicability of the nonlinear Poisson-Boltzmann equation. In order to do this we compare the thermodynamic properties calculated through this equation with Monte Carlo data on 1-1 and 2-2 electrolytes described by the restricted primitive model, in which the ions are modeled by hard spheres with a coulombic potential and the solvent is modeled as a continuum dielectric medium of uniform dielectric constant epsilon. We choose Monte Carlo data rather than real experimental data since all parameters are completely specified and there is no liberty for "adjustment." Thus this serves as a definitive test. In addition, we present a simple but numerically accurate alternative approximation scheme which is not only numerically superior to the Poisson-Boltzmann equation but avoids the necessity of solving a nonlinear partial differential equation which is approximate in the first place. The new approximation scheme that is presented here is suggested by recent developments in the statistical mechanical theories of ionic solutions which are reviewed in the Introduction. Although these theories themselves yield exceedingly good comparison with experimental (Monte Carlo) data, they involve fairly advanced theoretical and mathematical techniques and do not appear to be readily solvable for other than very simple geometries. The two approximations suggested here require only the solution of the linear Debye-Hückel equation, which has been solved for a variety of systems. These two approximations are simple to apply and yield good thermodynamic properties up to concentrations of 2 M for the restricted primitive model. In addition, they have a sound theoretical foundation and are offered as a substitute for the difficult-to-solve nonlinear Poisson-Boltzmann equation.

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In this paper we discuss a known variational principle for the Percus–Yevick integral equation and then generalize the procedure to obtain a variational principle for any integral equation associated with the Ornstein–Zernike equation. In particular we rederive a variational principle for the mean spherical model and then derive a variational principle for the hypernetted chain equation and compare these to the exact numerical results of Rasaiah and Friedman and to the Monte Carlo data of Rasaiah, Card, and Valleau for the restricted primitive model of electrolyte solutions.