Modified Poisson--Nernst--Planck Model with Accurate Coulomb Correlation in Variable Media

Abstract

We derive a set of modified Poisson-Nernst-Planck (mPNP) equations for ion transport from the variation of the free energy functional which includes the many-body Coulomb correlation in media of variable dielectric coefficient. The correlation effects are considered through the Debye charging process in which the self energy of an ion is governed by the generalized Debye-Hückel equation. We develop the asymptotic expansions of the self energy taking the ion radius as the small parameter such that the multiscale model can be solved efficiently by numerical methods. It is shown that the variations of the energy functional give the self-energy-modified PNP equations which satisfy a proper weak formulation. We present numerical results from different asymptotic expansions with a semi-implicit conservative numerical method and investigate the effect of the Coulomb correlation.

Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains

“…Proof. Recall thatφ satisfies the Poisson's equation (27) for r ∈ (R 1 , R 2 ), t > 0, with Robin boundary condition (28) for t > 0. By integrating (27) in r over (…”

“…The last step is to estimate higher-order terms in (61). Recalling the Poisson's equation forφ, i.e., (27), we do integration by parts to get…”

“…There have been many results on wellposedness and long-time behavior of the solutions for the model [2,5,6,13,14,31]. Also, various versions of the PNP system are concerned for various applications [15,17,20,21,26,27,38,39]. In this paper, we consider the PNP system with variable dielectric coefficients, which takes the form…”

Optimal Rate Convergence Analysis of a Second Order Numerical Scheme for the Poisson-Nernst-Planck System

“…For simplicity, we denote by C n = λ D L D n , C p = λ D L D p , and κ = L 2 2λ 2 D . Recently, there has been growing interests in incorporating effects that are beyond the mean-field description to the PNP theory, such as the steric effect, ion-ion correlations, and inhomogeneous dielectric environment [12,13,30,19,22,23,16,27]. Various versions of modified PNP theory have been developed to account for such ignored effects within the framework of the PNP theory.…”

“…For instance, the steric effect of ions have been taken into account by including excess free energy of solvent entropy [13,14,22,32,15], hard-sphere interaction kernels [12,27], or the fundamental measure theory [23]. A modified PNP model has been proposed to consider Coulombic ion-ion correlations in inhomogeneous dielectric environment [19].…”