Electric double layers with surface charge modulations: Novel exact Poisson-Boltzmann solutions

TL;DR

This paper derives new exact analytical solutions to the Poisson-Boltzmann equation for ions near macroions with modulated surface charges, revealing how charge patterns are screened in different geometries and salt conditions.

Contribution

It introduces novel exact solutions for modulated surface charge distributions in Poisson-Boltzmann theory, extending beyond homogeneous charge models.

Findings

Charge patterns decay exponentially for planar macroions without salt.

Charge patterns decay as an inverse power-law for cylindrical macroions without salt.

Salt presence causes exponential screening of charge patterns on planar macroions.

Abstract

Poisson-Boltzmann theory is the cornerstone for soft matter electrostatics. We provide novel exact analytical solutions to this non-linear mean-field approach, for the diffuse layer of ions in the vicinity of a planar or a cylindrical macroion. While previously known solution are for homogeneously charged objects, the cases worked out exhibit a modulated surface charge --or equivalently surface potential-- on the macroion (wall) surface. In addition to asymptotic features at large distances from the wall, attention is paid to the fate of the contact theorem, relating the contact density of ions to the local wall charge density. For salt-free systems (counterions only), we make use of results pertaining to the two-dimensional Liouville equation, supplemented by an inverse approach. When salt is present, we invoke the exact two-soliton solution to the 2D sinh-Gordon equation. This leads to inhomogeneous charge patterns, that are either localized or periodic in space. Without salt, the electrostatic signature of a charge pattern on the macroion fades exponentially with distance for a planar macroion, while it decays as an inverse power-law for a cylindrical macroion. With salt, our study is limited to the planar geometry, and reveals that pattern screening is exponential.