Poisson-Nernst-Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I-V Relations and Critical Potentials. Part II: Numerics

We consider a one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flow through membrane channels. Improving the classical Poisson-Nernst-Planck models where ion species are treated as point charges, this model includes ionic interaction due to finite sizes of ion species modeled by hard sphere potential from the Density Functional Theory. The resulting problem is a singularly perturbed boundary value problem of an integro-differential system. We examine the problem and investigate the ion size effect on the current-voltage (I-V) relations numerically, focusing on the case where two oppositely charged ion species are involved and only the hard sphere components of the excess chemical potentials are included. Two numerical tasks are conducted. The first one is a numerical approach of solving the boundary value problem and obtaining I-V curves. This is accomplished through a numerical implementation of the analytical strategy introduced by Ji and Liu in [Poisson-Nernst-Planck systems for ion flow with density functional theory for hard-sphere potential: I-V relations and critical potentials. Part I: Analysis, J. Dyn. Differ. Equ. (to appear)]. The second task is to numerically detect two critical potential values V (c) and V (c) .The existence of these two critical values is first realized for a relatively simple setting and analytical approximations of V (c) and V (c) are obtained in the above mentioned reference. We propose an algorithm for numerical detection of V (c) and V (c) without using any analytical formulas but based on the defining properties and numerical I-V curves directly. For the setting in the above mentioned reference, our numerical values for V (c) and V (c) agree well with the analytical predictions. For a setting including a nonzero permanent charge in which case no analytic formula for the I-V relation is available now, our algorithms can still be applied to find V (c) and V (c) numerically.