Communications in Mathematical Sciences

Volume 12 (2014)

Number 1

A new approach to the Lennard-Jones potential and a new model: PNP-steric equations

Pages: 149 – 173



Tai-Chia Lin (Department of Mathematics, National Taiwan University, Taida Institute for Mathematical Sciences (TIMS), Taipei, Taiwan)

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


A class of approximate Lennard-Jones (LJ) potentials with a small parameter is found whose Fourier transforms have a simple asymptotic behavior as the parameter goes to zero. When the LJ potential is replaced by the approximate LJ potential, the total energy functional becomes simple and exactly the same as replacing the LJ potential by a delta function. Such a simple energy functional can be used to derive the Poisson-Nernst-Planck equations with steric effects (PNP-steric equations), a new mathematical model for the LJ interaction in ionic solutions. Using formal asymptotic analysis, stability and instability conditions for the 1D PNP-steric equations with the Dirichlet boundary conditions for one anionic and cationic species are expressed by the valences, diffusion constants, ionic radii, and coupling constants. This is the first step to study the dynamics of solutions of the PNP-steric equations.


Lennard-Jones potentials, Poisson-Nernst-Planck equations, steric effects

2010 Mathematics Subject Classification

35B35, 35K57, 92C05

Published 20 August 2013