Poisson–Nernst–Planck equations with high-order steric effects

The Poisson–Nernst–Planck–Lennard-Jones (PNP–LJ) model is a mathematical model for ionic solutions with Lennard-Jones interactions between the ions. Due to the singular nature of the Lennard-Jones interaction kernel, however, the PNP–LJ model gives rise to an intractable analytic problem and a highly demanding computational problem. Previous works tackled this problem by replacing the LJ potential with a leading order approximation of the LJ potential giving rise to the steric PNP model. The steric PNP proved to be a successful model for the transport of ions in biological ionic channels. However, in other parameter regimes related to high concentrations, it is ill-posed. Presumably, since the steric PNP model is derived using only a leading-order approximation of the LJ potential. In this study, we go beyond the leading order approximation of the Lennard-Jones (LJ) interaction kernel and develop a new class of steric PNP equations that rely on a high-order approximation of the LJ potential. Surprisingly, we show that the introduction of high-order terms does not regularize the steric PNP model. Namely, at the limit of vanishing ionic sizes, high-order steric PNP equations, at all orders including the PNP–LJ model, are ill-defined at certain parameters regimes. Further analysis shows that this is an inherent property of PNP equations that account for steric effects and is related to pattern formation in these systems.