Communications in Mathematical Sciences

Volume 15 (2017)

Number 4

Ion size and valence effects on ionic flows via Poisson–Nernst–Planck models

Pages: 881 – 901



Peter W. Bates (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)

Weishi Liu (Department of Mathematics, University of Kansas, Lawrence, Ks., U.S.A.)

Hong Lu (School of Mathematics and Statistics, Shandong University, Weihai, China)

Mingji Zhang (Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, N.M., U.S.A.)


We study boundary value problems of a quasi-one-dimensional steady-state Poisson–Nernst–Planck model with a local hard-sphere potential for ionic flows of two oppositely charged ion species through an ion channel, focusing on effects of ion sizes and ion valences. The flow properties of interest, individual fluxes and total flow rates of mixture, depend on multiple physical parameters such as boundary conditions (boundary concentrations and boundary potentials) and diffusion coefficients, in addition to ion sizes and ion valences. For the relatively simple setting and assumptions of the model in this paper, we are able to characterize, almost completely, the distinct effects of the nonlinear interplay between these physical parameters. The boundaries of different parameter regions are identified through a number of critical values that are explicitly expressed in terms of the physical parameters. We believe our results will provide useful insights for numerical and even experimental studies of ionic flows through membrane channels.


ionic flow, Poisson–Nernst–Planck models, ion size and valence effects

2010 Mathematics Subject Classification

34Axx, 34B16, 34D15, 37D10, 92C35

Published 16 May 2017