Exact asymptotic expansions for the cylindrical Poisson-Boltzmann equation

TL;DR

This paper derives exact asymptotic expansions for the Poisson-Boltzmann equation around long rods, providing improved analytical tools for understanding electrostatic potentials in polyelectrolytes, especially at low salinity.

Contribution

It introduces new asymptotic expansions and practical expressions for the Poisson-Boltzmann equation solutions, enhancing analysis of electrostatics in polyelectrolytes.

Findings

Derived analytical expansions for the Manning radius.

Provided accurate expressions for electrostatic potential across all distances.

Analyzed behavior in low salinity regimes.

Abstract

The mathematical theory of integrable Painleve/Toda type systems sheds new light on the behavior of solutions to the Poisson-Boltzmann equation for the potential due to a long rod-like macroion. We investigate here the case of symmetric electrolytes together with that of 1:2 and 2:1 salts. Short and large scale features are analyzed, with a particular emphasis on the low salinity regime. Analytical expansions are derived for several quantities relevant for polyelectrolytes theory, such as the Manning radius. In addition, accurate and practical expressions are worked out for the electrostatic potential, which improve upon previous work and cover the full range of radial distances.