Surface Charge--Potential Relation for Spherical Particles in Electrolyte Solutions
Abstract
Predicting the relationship between surface charge density and electrostatic potential for spherical particles remains a fundamental challenge in colloid science. Because the governing non-linear Poisson--Boltzmann equation lacks a general exact analytical solution, researchers typically rely on numerical calculations or various semi-empirical approximations. In this paper, we overcome these limitations by developing a dual-asymptotic framework that provides explicit, closed-form expressions for this surface charge--potential relationship across the entire spectrum of particle curvature. For weakly curved systems, where the Debye length λD is much smaller than the particle radius R (λD/R ≪ 1), a formal mathematical derivation rigorously establishes the Ohshima--Healy--White formula as an exact regular perturbation expansion. Conversely, for highly curved spheres (R/λD ≪ 1), we employ singular perturbation analysis using the scaled particle radius as the small parameter. This approach provides a first-principles mathematical justification for the spherical Debye--Hückel theory, proving that its leading-order expansion remains asymptotically exact within the full non-linear Poisson--Boltzmann framework due to the geometric deactivation of non-linearity. Crucially, we map the exact limits of this geometric regulation, demonstrating how non-linear screening re-emerges as the particle radius increases, with the breakdown threshold governed by the interplay between curvature and surface charge density.
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BibTeX
@misc{vinogradova2026surface,
title = {{Surface Charge--Potential Relation for Spherical Particles in Electrolyte Solutions}},
author = {Olga I. Vinogradova and Elena F. Silkina},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03613},
eprint = {2610.03613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03613},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Surface Charge--Potential Relation for Spherical Particles in Electrolyte Solutions
AU - Olga I. Vinogradova
AU - Elena F. Silkina
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03613
UR - https://arxiv.org/abs/2610.03613
AB - Predicting the relationship between surface charge density and electrostatic potential for spherical particles remains a fundamental challenge in colloid science. Because the governing non-linear Poisson--Boltzmann equation lacks a general exact analytical solution, researchers typically rely on numerical calculations or various semi-empirical approximations. In this paper, we overcome these limitations by developing a dual-asymptotic framework that provides explicit, closed-form expressions for this surface charge--potential relationship across the entire spectrum of particle curvature. For weakly curved systems, where the Debye length λ_D is much smaller than the particle radius R (λ_D/R ≪ 1), a formal mathematical derivation rigorously establishes the Ohshima--Healy--White formula as an exact regular perturbation expansion. Conversely, for highly curved spheres (R/λ_D ≪ 1), we employ singular perturbation analysis using the scaled particle radius as the small parameter. This approach provides a first-principles mathematical justification for the spherical Debye--Hückel theory, proving that its leading-order expansion remains asymptotically exact within the full non-linear Poisson--Boltzmann framework due to the geometric deactivation of non-linearity. Crucially, we map the exact limits of this geometric regulation, demonstrating how non-linear screening re-emerges as the particle radius increases, with the breakdown threshold governed by the interplay between curvature and surface charge density.
ER -