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



Farjana Siddiqua (Department of Mathematics and Statistics, Florida International University, Miami, Fl., U.S.A.)

Zhongming Wang (Department of Mathematics and Statistics, Florida International University, Miami, Fl., U.S.A.)

Shenggao Zhou (Department of Mathematics and Mathematical Center for Interdiscipline Research, Soochow University, Suzhou, Jiangsu, China)


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.


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