Beyond Poisson-Boltzmann: Fluctuations and Correlations

TL;DR

This paper develops a non-linear field theory for fluctuating counter-ion distributions, extending the Poisson-Boltzmann framework by including fluctuations and correlations through a loop expansion, and applies it to charged wall systems.

Contribution

It introduces a formalism that incorporates fluctuations and correlations beyond Poisson-Boltzmann theory using a loop expansion, providing explicit corrections and effective interactions.

Findings

Fluctuation corrections significantly affect electrostatic potential and ion density.

The importance of fluctuations depends on counter-ion valency and surface charge density.

Effective interactions between charged particles include correlation effects.

Abstract

We formulate the non-linear field theory for a fluctuating counter-ion distribution in the presence of a fixed, arbitrary charge distribution. The Poisson-Boltzmann equation is obtained as the saddle-point, and the effects of fluctuations and correlations are included by a loop-wise expansion around this saddle point. We show that the Poisson equation is obeyed at each order in the loop expansion and explicitly give the expansion of the Gibbs potential up to two loops. We then apply our formalism to the case of an impenetrable, charged wall, and obtain the fluctuation corrections to the electrostatic potential and counter-ion density to one-loop order without further approximations. The relative importance of fluctuation corrections is controlled by a single parameter, which is proportional to the cube of the counter-ion valency and to the surface charge density. We also calculate effective interactions between charged particles, which reflect counter-ion correlation effects.