Poisson-Boltzmann for oppositely charged bodies: an explicit derivation

TL;DR

This paper provides an explicit derivation of the electrostatic interaction between oppositely charged bodies in ionic solutions using Poisson-Boltzmann theory, revealing a non-trivial equilibrium separation and clarifying thermodynamic potentials.

Contribution

It offers a self-consistent derivation of the pressure formula for charged plates with arbitrary charge densities and salt concentrations, including a reinterpretation of Ohshima's 1975 results.

Findings

Predicts a finite equilibrium separation for oppositely charged plates.

Shows the thermodynamic potential can differ from the PB free energy.

Provides an analytic expression for the energy minimum and its dependence on system parameters.

Abstract

The interaction between charged bodies in an ionic solution is a general problem in colloid physics and becomes a central topic in the study of biological systems where the electrostatic interaction between proteins, nucleic acids, membranes is involved. This problem is often described starting from the simple one-dimensional model of two parallel charged plates. Several different approaches to this problem exist, focusing on different features. In many cases, an intuitive expression of the pressure exerted on the plates is proposed, which includes an electrostatic plus an osmotic contribution. We present an explicit and self-consistent derivation of this formula for the general case of any charge densities on the plates and any salt solution, obtained in the framework of the Poisson-Boltzmann theory. We also show that, depending on external constraints, the correct thermodynamic potential can differ from the usual PB free energy. The resulting expression predicts, for asymmetric, oppositely charged plates, the existence of a non trivial equilibrium position with the plates separated by a finite distance. It is therefore crucial, in order to study the kinetic stability of the corresponding energy minimum, to obtain its explicit dependence on the plates charge densities and on the ion concentration. An analytic expression for the position and value of the corresponding energy minimum has been derived in 1975 by Ohshima [Ohshima H., Colloid and Polymer Sci. 253, 150-157 (1975)] but, surprisingly, this important result seems to be overlooked today. We retrieve the expressions obtained by Ohshima in a simpler formalism, more familiar to the physics community, and give a physical interpretation of the observed behavior.