Solutions of Poisson-Nernst Planck Equations with Ion Interaction

Abstract

This paper develops a mixed Finite Element Method (mFEM) based on both classical rectangular elements (with equal nodal points for all degrees of freedom) and Taylor-hood elements to solve Poisson-Nernst-Planck (PNP) equations with steric effects. The Nernst-Planck (NP) equation for ion fluxes is modified to incorporate finite-size effects in terms of hard-sphere repulsion. The resultant modified NP and Poisson equation representing electrostatic potential is then coupled to form a system of the equation that describes a realistic transport phenomenon in an ion channel. Consequently, we apply mFEM based on both Taylor-hood and classical rectangular elements to discretize the 2D steady system of equations with iterative linearization for the nonlinear components. The numerical scheme is first validated using a 2D analytical solution for PNP equations, the steric components varied and the effects on concentration analyzed and compared against PNP and modified PNP for two monovalent ion species. Classical rectangular elements presented a better comparable approximate solution than Taylor-hood.