Local Hard-Sphere Poisson-Nernst-Planck Models for Ionic Channels with Permanent Charges

TL;DR

This paper develops a mathematical model using the Poisson-Nernst-Planck system with local hard-sphere potentials to analyze how ion sizes influence ionic flow through channels with permanent charges, providing detailed analytical results.

Contribution

It introduces a novel local hard-sphere potential model within the Poisson-Nernst-Planck framework and applies geometric singular perturbation theory to analyze ion size effects.

Findings

Derived nonlinear algebraic equations for ionic flow

Analyzed zero-current fluxes for small ion sizes

Examined ion size impact on flow rate at zero current

Abstract

The main goal of this work is to examine the qualitative effect of ion sizes via a steady-state boundary value problem. We study a one-dimensional version of a Poisson-Nernst-Planck system with a local hard-sphere potential model for ionic flow through a membrane channel with fixed boundary ion concentrations and electric potentials. A complete set of integrals for the inner system is illustrated that delivers information for boundary and internal layers. In addition, a group of simultaneous equations appears in the construction of singular orbits. The research aims to set up a simple formation defined by the profile of permanent charges with two mobile ion species, one positively charged, cation, and one negatively charged, anion. A local hard-sphere potential that depends pointwise on ion concentrations is included in the model to estimate ion-size impacts on the ionic flow. The analysis is built on the geometric singular perturbation theory, particularly on specific structures of this concrete model. For 1:1 ionic mixtures, we first conduct precise mathematical analysis and derive a matching system of nonlinear algebraic equations. We then extend the results by directing on a critical case where the current is zero. Treating the ion sizes as small parameters, we derive an approximation of zero-current fluxes for general values of permanent charge. We then focus on small values of permanent charge to obtain more concrete outputs. We will also examine ion size effects on the flow rate of matter for the zero-current case.