Communications in Mathematical Sciences

Volume 9 (2011)

Number 2

A mathematical model for the hard sphere repulsion in ionic solutions

Pages: 459 – 475



Bob Eisenberg (Department of Molecular Biophysics & Physiology, Rush Medical Center, Chicago, Illinois)

Yunkyong Hyon (Institute for Mathematics and Its Applications, University of Minnesota)

Chun Liu (Department of Mathematics, Pennsylvania State University, University Park)


We introduce a mathematical model for the finite size (repulsive) effects in ionic solutions. We first introduce an appropriate energy term into the total energy that represents the hard sphere repulsion of ions. The total energy then consists of the entropic energy, electrostatic potential energy, and the repulsive potential energy. The energetic variational approach derives a boundary value problem that includes contributions from the repulsive term with a no flux boundary condition for charge density which is a consequence of the variational approach, and physically implies charge conservation. The resulting system of partial differential equations is a modification of the Poisson-Nernst-Planck (PNP) equations widely if not universally used to describe the drift-diffusion of electrons and holes in semiconductors, and the movement of ions in solutions and protein channels. The modified PNP equations include the effects of the finite size of ions that are so important in the concentrated solutions near electrodes, active sites of enzymes, and selectivity filters of proteins. Finally, we do some numerical experiments using finite element methods, and present their results as a verification of the utility of the modified system.


finite size effects, energetic variational approach, Poisson-Nernst-Planck equations, hard sphere, Lennard-Jones repulsive potential, hard sphere potential in density functional theory

2010 Mathematics Subject Classification

65C30, 76A05, 76Mxx

Full Text (PDF format)