Poisson-Nernst-Planck systems for ion flow with density functional theory for hard-sphere potential: I-V relations and critical potentials. II: Numerics
In this paper, the one dimensional steady-state Poisson-Nernst-Planck (PNP) type model for ionic flow through membrane channels is considered. Improving the classical PNP 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 (DFT). The resulting problem is a singularly perturbed boundary value problem of an integro-differential system. It is investigated 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 considered. The first one is a numerical approach of solving the boundary value problem and obtaining I-V curves. The second task is to numerically detect two critical potential values \(V_c\) and \(V^c\) which existence was theoretically proved early. It is proposed an algorithm of 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. 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, the proposed algorithm can still be applied to find \(V_c\) and \(V^c\) numerically. Part I, see J. Dyn. Differ. Equations 24, No. 4, 955--983 (2012; Zbl 1270.93011).
scientific article; zbMATH DE number 6132840
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