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

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.

💡 Research Summary

The paper revisits the classic image‑charge technique for solving the Green’s function of the linearized Poisson‑Boltzmann (LPB) equation when a dielectric sphere is immersed in an ionic solution. By applying a finite Mellin transform to the LPB Green’s function, the authors convert the reaction potential generated by a point source inside the sphere into a one‑dimensional distribution of image charges. This distribution consists of two distinct components: a point image located at the Kelvin point (the inversion of the source with respect to the sphere) and a line image that extends from the Kelvin point to infinity. The line‑image charge density is not monotonic; it oscillates and decays as a function of distance, reflecting the Debye screening imposed by the electrolyte.

To make the line image usable in numerical simulations, the continuous charge density must be discretized. The authors achieve this by first approximating the complex exponential kernel of the line image with a Padé rational function. The Padé approximant captures both the oscillatory and decaying behavior with a modest order. Next, they apply a finite fraction expansion to the rational function, which decomposes it into a sum of simple fractions. Each fraction corresponds to a point charge; thus the continuous line is replaced by a small set of discrete image charges whose positions and magnitudes are analytically known. By adjusting the Padé order and the depth of the fraction expansion, the user can trade off between computational cost and desired accuracy. In practice, five to ten discrete charges are sufficient to reproduce the exact reaction potential with errors below 10⁻⁴.

The efficiency and accuracy of the method are demonstrated in a multiscale reaction‑field Monte Carlo (MRF‑MC) simulation of a 1:1 electrolyte surrounding a dielectric sphere. Compared with traditional image‑charge formulations that rely on infinite series or numerical quadrature, the Padé‑fraction discretization reduces the average CPU time by more than 30 % while maintaining sub‑millivolt potential accuracy. The approach remains robust across a wide range of Debye lengths and dielectric contrasts, confirming that the Mellin‑based representation naturally incorporates electrolyte screening.

Key contributions of the work are: (1) a rigorous derivation of the image‑charge representation for the LPB Green’s function in an electrolyte using finite Mellin transforms; (2) a practical discretization scheme that combines Padé approximation with finite fraction expansion, enabling fast and accurate evaluation of the line image; and (3) validation of the scheme within a realistic MRF‑MC framework, showing its suitability for large‑scale simulations of biomolecular or colloidal systems in ionic environments. The methodology opens the door to efficient electrostatic calculations in many contexts, including ion‑channel modeling, electrode–electrolyte interfaces, and nanoparticle‑electrolyte interactions, where both dielectric heterogeneity and ionic screening are essential.