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# Communications in Mathematical Sciences

## Volume 14 (2016)

### Number 1

### Mean-field theory and computation of electrostatics with ionic concentration dependent dielectrics

Pages: 249 – 271

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n1.a10

#### Authors

#### Abstract

We construct a mean-field variational model to study how the dependence of dielectric coefficient (i.e., relative permittivity) on local ionic concentrations affects the electrostatic interaction in an ionic solution near a charged surface. The electrostatic free-energy functional of ionic concentrations, which is the key object in our model, consists mainly of the electrostatic potential energy and the ionic ideal-gas entropy. The electrostatic potential is determined by Poisson’s equation in which the dielectric coefficient depends on the sum of concentrations of individual ionic species. This dependence is assumed to be qualitatively the same as that on the salt concentration for which experimental data are available and analytical forms can be obtained by the data fitting. We derive the first and second variations of the free-energy functional, obtain the generalized Boltzmann distributions, and show that the free-energy functional is in general non-convex. To validate our mathematical analysis, we numerically minimize our electrostatic free-energy functional for a radially symmetric charged system. Our extensive computations reveal several features that are significantly different from a system modeled with a dielectric coefficient independent of ionic concentration. These include the non-monotonicity of ionic concentrations, the ionic depletion near a charged surface that has been previously predicted by a one-dimensional model, and the enhancement of such depletion due to the increase of surface charges or bulk ionic concentrations.

#### Keywords

electrostatic interactions, concentration-dependent dielectrics, mean-field models, Poisson–Boltzmann theory, generalized Boltzmann distributions, non-convex free-energy functional, variational analysis, numerical computation

#### 2010 Mathematics Subject Classification

35J20, 35J25, 35Q92, 49S05, 92C05

Published 16 September 2015