Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for HardSphere Potential: I–V Relations and Critical Potentials. Part II: Numerics
 Weishi Liu,
 Xuemin Tu,
 Mingji Zhang
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We consider a onedimensional steadystate 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 integrodifferential 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 hardsphere 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.
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 Title
 Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for HardSphere Potential: I–V Relations and Critical Potentials. Part II: Numerics
 Journal

Journal of Dynamics and Differential Equations
Volume 24, Issue 4 , pp 9851004
 Cover Date
 20121201
 DOI
 10.1007/s108840129278x
 Print ISSN
 10407294
 Online ISSN
 15729222
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Ion flow
 PNP–DFT
 Hardsphere
 I–V relation
 Critical potentials
 34D15
 45J05
 65L10
 78A35
 92C35
 Authors

 Weishi Liu ^{(1)}
 Xuemin Tu ^{(1)}
 Mingji Zhang ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd., Room 405, Lawrence, KS, 66045, USA