Dynamics of Poisson-Boltzmann equations: “Hidden” mechanisms for Gouy-Chapman layer and beyond

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 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 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.By all means, these findings are by chance which is probably why the mechanism has been missing after the phenomenon of Gouy-Chapmann layer and alike had been discovered for very long. Only when viewed from the nonlinear dynamical systems theory, these results become rather apparent.