"Hidden" mechanisms for Gouy-Chapman layer and other critical features via Poisson-Boltzmann equations

TL;DR

This paper uses a dynamical system approach to analyze the Poisson-Boltzmann equation, revealing critical features like the Gouy-Chapman layer mechanism and phenomena such as charge saturation and force reversal, with implications for more complex models.

Contribution

It introduces a geometric phase space analysis of the PB equation, uncovering the mechanisms behind key electrostatic features and critical phenomena.

Findings

Revealed the equilibrium mechanism of the Gouy-Chapman layer.

Identified saturation of surface charge density and bounded electric pressure.

Discovered a critical length for electric force reversal.

Abstract

In this work a dynamical system approach is taken to systematically investigate the one-dimensional classical Poisson-Boltzmann (PB) equation with various boundary conditions. This framework, particularly, the phase space portrait, has a unique advantage of a geometric view of the dynamical systems, which allows one to reveal and examine critical features of the PB models. More specifically, we are able to reveal the mechanism of Gouy-Chapman layer: the presence of an {\em equilibrium} for the PB equation, including equilibrium-at-infinity for Gouy-Chapman's original setup as the limiting case. Several other critical, somehow counterintuitive, features revealed in this work are the saturation phenomenon of surface charge density, the uniform boundedness of electric pressure (given length) and of length (given electric pressure) in surface charge, and the critical length for a reversal of electric force direction. All have a common mechanism: the presence of an equilibrium for the PB equation. We believe that the critical features presented from the classical PB models persist for modified PB systems up to a certain degree. On the other hand, any qualitative change in these features as the sophistication of the model is increasing is an indication of new phenomena with new mechanisms.