Quantitative Theory for Critical Conditions of Like-Charge Attraction Between Polarizable Spheres

TL;DR

This paper develops a new three-point image formula to accurately predict the critical conditions for like-charge attraction between polarizable spheres, validated by simulations and applicable to physical processes and force field calculations.

Contribution

It introduces a novel three-point image formula connecting Neumann's image principle with the incomplete beta function, providing precise critical conditions for LCA.

Findings

Critical conditions for LCA have less than 1% discrepancy with simulations.

The formula offers physical insights into self-assembly and phase separation.

Applicable to improve polarizable force field calculations.

Abstract

Despite extensive experimental and theoretical efforts, a concise quantitative theory to predict the occurrence of like-charge attraction (LCA) between polarizable spheres remains elusive. In this work, we first derive a novel three-point image formula, based on a key observation that connects the classical Neumann's image principle with the incomplete beta function. This approach naturally yields simple yet precise critical conditions for LCA, with a relative discrepancy of less than $1\%$ compared to numerical simulations, validated across diverse parameter settings. The obtained critical conditions may provide physical insights into various processes potentially involving LCA, such as self-assembly, crystallization, and phase separation across different length scales. Additionally, the new image formula is directly applicable to enhance the efficiency of polarizable force field calculations involving polarizable spheres.