Mellin Transform and Image Charge Method for Dielectric Sphere in an Electrolyte

TL;DR

This paper introduces a novel method combining Mellin transform and image charge techniques to efficiently solve the Poisson-Boltzmann equation for dielectric spheres in electrolytes, enabling accurate multiscale simulations.

Contribution

It develops a new analytical and numerical approach using Mellin transform and image charges, including a discretization algorithm, for better modeling of dielectric spheres in ionic solutions.

Findings

Efficient representation of reaction potential with image charges.

Accurate discretization of line images via Padé approximation.

Successful application in electrolyte Monte Carlo simulations.

Abstract

We revisit the image charge method for the Green's function problem of the Poisson-Boltzmann equation for a dielectric sphere immersed in ionic solutions. Using finite Mellin transformation, we represent the reaction potential due to a source charge inside the sphere in terms of one dimensional distribution of image charges. The image charges are generically composed of a point image at the Kelvin point and a line image extending from the Kelvin point to infinity with an oscillatory line charge strength. We further develop an efficient and accurate algorithm for discretization of the line image using Pad\'e approximation and finite fraction expansion. Finally we illustrate the power of our method by applying it in a multiscale reaction-field Monte Carlo simulation of monovalent electrolytes.