Osmotic pressure between arbitrarily charged surfaces: a revisited approach

TL;DR

This paper introduces a new, efficient method to directly calculate osmotic pressure between arbitrarily charged surfaces in ionic solutions without solving for the electrostatic potential, applicable to generalized Poisson-Boltzmann theories.

Contribution

A novel analytical approach for osmotic pressure calculation that bypasses potential profile solutions, accommodating arbitrary boundary conditions and charge configurations.

Findings

Derived explicit formulas for osmotic pressure in terms of separation and salt concentration.

Demonstrated the method's applicability to sterically modified Poisson-Boltzmann theory.

Enhanced efficiency over traditional symmetry-based solutions.

Abstract

The properties of ionic solutions between charged surfaces are often studied within the Poisson-Boltzmann framework, by finding the electrostatic potential profile. For example, the osmotic pressure between two charged planar surfaces can be evaluated by solving coupled equations for the electrostatic potential and osmotic pressure. Such a solution relies on symmetry arguments and is restricted to either equally or oppositely charged surfaces. Here, we provide a different and more efficient scheme to derive the osmotic pressure straight-forwardly, without the need to find the electrostatic potential profile. We derive analytical expressions for the osmotic pressure in terms of the inter-surface separation, salt concentration, and arbitrary boundary conditions. Such results should be useful in force measurement setups, where the force is measured between two differently prepared surfaces, or between two surfaces held at a fixed potential difference. The proposed method can be systematically used for generalized Poisson-Boltzmann theories in planar geometries, as is demonstrated for the sterically modified Poisson-Boltzmann theory.