Contents Online
Communications in Mathematical Sciences
Volume 16 (2018)
Number 1
A modified Poisson–Nernst–Planck model with excluded volume effect: theory and numerical implementation
Pages: 251 – 271
DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n1.a12
Authors
Abstract
The Poisson–Nernst–Planck (PNP) equations have been widely applied to describe ionic transport in ion channels, nanofluidic devices, and many electrochemical systems. Despite their wide applications, the PNP equations fail in predicting dynamics and equilibrium states of ionic concentrations in confined environments, due to the ignorance of the excluded volume effect. In this work, a simple but effective modified PNP (MPNP) model with the excluded volume effect is derived, based on a modification of diffusion coefficients of ions. At the steady state, a modified Poisson–Boltzmann (MPB) equation is obtained with the help of the Lambert-$W$ special function. The existence and uniqueness of a weak solution to the MPB equation are established. Further analysis on the limit of weak and strong electrostatic potential leads to two modified Debye screening lengths, respectively. A numerical scheme that conserves total ionic concentration and satisfies energy dissipation is developed for the MPNP model. Numerical analysis is performed to prove that our scheme respects ionic mass conservation and satisfies a corresponding discrete free energy dissipation law. Positivity of numerical solutions is also discussed and numerically investigated. Numerical tests are conducted to demonstrate that the scheme is of second-order accurate in spatial discretization and has expected properties. Extensive numerical simulations reveal that the excluded volume effect has pronounced impacts on the dynamics of ionic concentration and flux. In addition, the effect of volume exclusion on the timescales of charge diffusion is systematically investigated by studying the evolution of free energies and diffuse charges.
Keywords
Poisson–Nernst–Planck equations, excluded volume effect, mass conservation, energy dissipation, diffusion timescale
2010 Mathematics Subject Classification
35Q92, 65M06, 92C05
Received 23 June 2017
Accepted 22 November 2017
Published 29 March 2018