Exact Screening-Ranged Expansions for Many-Body Electrostatics

TL;DR

The paper addresses exact many-body electrostatics for $N$ arbitrarily charged dielectric spheres in an electrolyte within the linearized Poisson–Boltzmann framework ($\text{LPBE}$ / $\text{DH}$). It develops a convergent screening-ranged Neumann-type expansion based on the spectral properties of nonstandard Neumann–Poincaré-type operators, yielding a Neumann-series solution $\vec{\tilde{\mathbb G}} = \sum_{\ell=0}^{+\infty} (-1)^{\ell} \mathbb{K}^{\ell} \vec{\tilde{\mathbb S}}$, where each $\ell$-th term encodes Debye screening factors $\frac{e^{-\kappa R_{ij}}}{R_{ij}}$ along interaction paths. The method provides explicit expressions for the potentials, interaction energy, and forces, applicable to general fixed-charge distributions including heterogeneously charged and Janus particles, with the total energy and force decomposed as $\mathcal{E}=\sum_{\ell\ge0} \mathcal{E}^{(\ell)}$ and $\mathbf{F}_i=\sum_{\ell\ge1} \mathbf{F}_i^{(\ell)}$, reproducing DLVO at leading order and revealing higher-order many-body and multipolar corrections. It yields analytical conditions for phenomena such as opposite-charge repulsion (OCR), like-charge attraction (LCA), and asymmetric dielectric screening (ADS), connects to image-charge and effective-dipole results, and provides a computationally efficient, convergent framework suitable for coarse-grained modeling of colloids and soft/biological matter in electrolytic solutions.

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--Poincaré-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--Hückel 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.

Figures (1)

• Figure 1: a) Electrostatic potential and iso-potential contour lines obtained for a three-sphere system. $\varepsilon_{1}=2$, $\varepsilon_{2, 3}=3$, $a_1+a_{2,3}=35$ Å, $a_1=10 a_{2,3}$, $\varepsilon_\text{sol}=80$, $q_1=3e$, $q_{2,3} = -2e$, $\kappa^{-1}=8.07$ Å. The arrows represent the forces acting on the spheres. Spheres $2$ and $3$ are identical, but located at a different distance from sphere 1. The most significant correction introduced by the inclusion of higher order screened terms consists in an excess negative potential located inside sphere 3. This potential is more pronounced in the region facing sphere 1. This induces a difference in the particle polarization, reflected by the difference in surface charge distribution shown in panel b). c) Two spheres interaction energy w.r.t. $R=R_{1 2}$, $R\ge36$ Å; $a_1+a_2=35$ Å, $a_1=10 a_2$, $\varepsilon_1=2$, $\varepsilon_2=3$, $q_1=3 e$, $q_2 = -2 e$, $\varepsilon_\text{sol}=80$, $\kappa^{-1}=40$ Å. Lines F, 1 and 2 depict the full $\mathcal{E}^\text{Int}$, $\mathcal{E}^{(1)}$ (i.e. DLVO) and $\mathcal{E}^{(1)}+\mathcal{E}^{(2)}$, respectively. Lines 2.0, 2.1 and 2.1' correspond to one-term (i.e. MZ), two-term and simplified two-term (i.e. FLL) approximations of $\mathcal{E}^{(2)}$, respectively. (Embedded inset shows a close-up view). d) Example of $\sigma_i^\text{f}$ for a generalized Janus particle: free charges $q_{i,1}$ and $q_{i,2}$ are uniformly distributed over spherical caps of polar angles $[0,\, \theta_{i,1}]$ and $[\pi-\theta_{i,2},\, \pi]$, respectively, while the intermediate surface (i.e. $(\theta_{i,1},\, \pi-\theta_{i,2})$) has no fixed charge.