Exact Screening-Ranged Expansions for Many-Body Electrostatics

TL;DR

This paper introduces an exact, convergent series framework for calculating electrostatic interactions among multiple charged spheres in electrolytes, extending classical models and enabling efficient, analytical computations of complex many-body effects.

Contribution

It develops a rigorous, spectral analysis-based series expansion for many-body electrostatics in electrolytes, unifying and extending existing approaches with analytical expressions.

Findings

Provides convergent screening-ranged series for potential, energy, and forces.

Enables analytical evaluation of many-body electrostatic interactions.

Captures complex effects like asymmetric screening and charge attraction.

Abstract

We present an exact many-body framework for electrostatic interactions among $N$ arbitrarily charged spheres in an electrolyte, modeled by the linearized Poisson--Boltzmann equation. Building on a spectral analysis of nonstandard Neumann--Poincar\'e-type operators introduced in a companion mathematical work arXiv:2512.08684, we construct convergent screening-ranged series for the potential, interaction energy, and forces, where each term is associated with a well-defined Debye--H\"uckel screening order and can be obtained evaluating an analytical expression rather than numerically solving an infinitely dimensional linear system. This formulation unifies and extends classical and recent approaches, providing a rigorous basis for electrostatic interactions among heterogeneously charged particles (including Janus colloids) and yielding many-body generalizations of analytical explicit-form results previously available only for two-body systems. The framework captures and clarifies complex effects such as asymmetric dielectric screening, opposite-charge repulsion, and like-charge attraction, which remain largely analytically elusive in existing treatments. Beyond its fundamental significance, the method leads to numerically efficient schemes, offering a versatile tool for modeling colloids and soft/biological matter in electrolytic solution.