Communications in Mathematical Sciences

Volume 21 (2023)

Number 2

On the equilibrium of the Poisson–Nernst–Planck–Bikermann model equipping with the steric and correlation effects

Pages: 485 – 515



Jian-Guo Liu (Departments of Mathematics and Physics, Duke University, Durham, North Carolina, U.S.A.)

Yijia Tang (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China)

Yu Zhao (Department of Mathematics, Faculty of Science, Beijing University of Technology, Beijing, China)


The Poisson–Nernst–Planck–Bikermann (PNPB) model, in which the ions and water molecules are treated as different species with non-uniform sizes and valences with interstitial voids, can describe the steric and correlation effects in ionic solution neglected by the Poisson–Nernst–Planck and Poisson–Boltzmann theories with point charge assumption. In the PNPB model, the electric potential is governed by the fourth-order Poisson–Bikermann (4PBik) equation instead of the Poisson equation so that it can describe the correlation effect. Moreover, the steric potential is included in the ionic and water fluxes as well as the equilibrium Fermi-like distributions which characterizes the steric effect quantitatively.

In this work, we analyze the self-adjointness and the kernel of the fourth-order operator of the 4PBik equation. Also, we show the positivity of the void volume function and the convexity of the free energy. Following these properties, the well-posedness of the PNPB model in equilibrium is given. Furthermore, because the PNPB model has an energy dissipated structure, we adopt a finite volume scheme which preserves the energy dissipated property at the semi-discrete level. Various numerical investigations are given to show the parameter dependence of the steric effect to the steady state.


Poisson–Nernst–Planck–Bikermann fourth-order Poisson–Bikermann equation, free energy, steady state, steric and correlation effects

2010 Mathematics Subject Classification

35Q92, 65M08, 65M12, 92E20

Received 4 January 2022

Accepted 7 June 2022

Published 1 February 2023