Revisiting the Poisson-Boltzmann theory: charge surfaces, multivalent ions and inter-plate forces

TL;DR

This paper revisits the Poisson-Boltzmann theory to analyze how boundary conditions and multivalent ions influence inter-plate electrostatic forces, revealing the conditions under which interactions switch from attractive to repulsive.

Contribution

It provides a detailed analysis of inter-plate forces under various boundary conditions and highlights the impact of multivalent counter-ions on the range of attractive interactions.

Findings

Range of attractive interactions increases with ion valency.

Boundary conditions significantly affect electrostatic force behavior.

Crossover from attraction to repulsion depends on ion valency and boundary conditions.

Abstract

The Poisson-Boltzmann (PB) theory is extensively used to gain insight on charged colloids and biological systems as well as to elucidate fundamental properties of intermolecular forces. Many works were devoted in the past to study PB related features and to confirm them experimentally. In this work we explore the properties of inter-plate forces in terms of different boundary conditions. We treat the cases of constant surface charge, constant surface potential and mixed boundaries. The interplay between electrostatic interactions, attractive counter-ions release, and repulsive van 't Hoff contribution are discussed separately for each case. Finally, we discuss how the crossover between attractive and repulsive interactions for constant surface charge case is influenced by the presence of multivalent counter-ions, where it is shown that the range of the attractive interaction grows with the valency.