Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries

The mathematical modelling and simulation of ion transport through biological and synthetic channels (nanopores) is a challenging problem, with direct application in biophysics, physiology and chemistry. At least two major effects have to be taken into account when creating such models: the electrostatic interaction of ions and the effects due to size exclusion in narrow regions. While mathematical models and methods for electrostatic interactions are well-developed and can be transferred from other flow problems with charged particles, e.g. semiconductor devices, less is known about the appropriate macroscopic modelling of size exclusion effects. Recently several papers proposed simple or sophisticated approaches for including size exclusion effects into entropies, in equilibrium as well as off equilibrium. The aim of this paper is to investigate a second potentially important modification due to size exclusion, which often seems to be ignored and is not implemented in currently used models, namely the modification of mobilities due to size exclusion effects. We discuss a simple model derived from a self-consisted random walk and investigate the stationary solutions as well as the computation of conductance. The need of incorporating nonlinear mobilities in high density situations is demonstrated in an investigation of conductance as a function of bath concentrations, which does not lead to obvious saturation effects in the case of linear mobility.