Near- and far-field expansions for stationary solutions of Poisson--Nernst--Planck equations

This work is concerned with the stationary Poisson--Nernst--Planck equation with a large parameter which describes a huge number of ions occupying an electrolytic region. Firstly, we focus on the model with a single specie of positive charges in one-dimensional bounded domains due to the assumption that these ions are transported in the same direction along a tubular-like mircodomain. We show that the solution asymptotically blows up in a thin region attached to the boundary, and establish the refined "near-field" and "far-field" expansions for the solutions with respect to the parameter. Moreover, we obtain the boundary concentration phenomenon of the net charge density, which mathematically confirms the physical description that the non-neutral phenomenon occurs near the charged surface. In addition, we revisit a nonlocal Poisson--Boltzmann model for monovalent binary ions and establish a novel comparison for these two models

On the asymptotic limit of steady state Poisson--Nernst--Planck equations with steric effects

When ions are crowded, the effect of steric repulsion between ions becomes significant and the conventional Poisson--Boltzmann (PB) equation (without steric effect) should be modified. For this purpose, we study the asymptotic limit of steady state Poisson--Nernst--Planck equations with steric effects (PNP-steric equations). By the assumptions of steric effects, we transform steady state PNP-steric equations into a PB equation with steric effects (PB-steric equation) which has a parameter $\Lambda$ and positive constants $\lambda_i$'s (depend on the radii of ions and solvent molecules). The nonlinear term of PB-steric equation is mainly determined by a Lambert type function which represents the concentration of solvent molecules. As $\Lambda=0$, the PB-steric equation becomes the conventional PB equation but as $\Lambda>0$, a large $\Lambda$ makes the steric repulsion (between ions and solvent molecules) stronger. This motivates us to find the asymptotic limit of PB-steric equation as $\Lambda$ goes to infinity. Under the Robin (or Neumann) boundary condition, we prove theoretically and numerically that the PB-steric equation has a unique solution $\phi_\Lambda$ which converges to solution $\phi^*$ of a modified PB (mPB) equation as $\Lambda$ tends to infinity. Our results show that the limiting equation of PB-steric equation (as $\Lambda$ goes to infinity) is a mPB equation which has the same form (up to scalar multiples) as those mPB equations in \cite{1942bikerman,1997borukhov,2007kilic,2009li,2009li2,2013li,2011lu}. Therefore, the PB-steric equation can be regarded as a generalized model of mPB equations.Comment: 21 pages, 4 figure