A singular perturbation approach to the steady-state 1D Poisson-Nernst-Planck modeling

TL;DR

This paper develops an analytical singular perturbation approach to model steady-state ion transport in 1D Poisson-Nernst-Planck systems with small channel charge and length much greater than the Debye length, validated by numerical comparisons.

Contribution

It introduces a singular perturbation method for analyzing the steady-state PNP model in nanopores with small charge, providing analytical insights and boundary condition discussions.

Findings

Analytical solutions agree well with numerical results.

The model clarifies the influence of boundary conditions on ion transport.

The approach simplifies understanding of ion dynamics in nanopores.

Abstract

The reduced 1D Poisson-Nernst-Planck (PNP) model of artificial nanopores in the presence of a permanent charge on the channel wall is studied. More specifically, we consider the limit where the channel length exceed much the Debye screening length and channel's charge is sufficiently small. Ion transport is described by the nonequillibrium steady-state solution of the PNP system within a singular perturbation treatment. The quantities, 1/lambda -- the ratio of the Debye length to a characteristic length scale and epsilon -- the scaled intrinsic charge density, serve as the singular and the regular perturbation parameters, respectively. The role of the boundary conditions is discussed. A comparison between numerics and the analytical results of the singular perturbation theory is presented.