电解质溶液中球形颗粒的表面电荷-电势关系

中文摘要 AI 辅助

预测球形颗粒的表面电荷密度与静电势之间的关系，一直是胶体科学中的一个基本挑战。由于控制方程——非线性泊松-玻尔兹曼方程——缺乏通用的精确解析解，研究人员通常依赖数值计算或各种半经验近似。在本文中，我们通过发展一个双渐近框架来克服这些限制，该框架提供了在整个颗粒曲率范围内，表面电荷-电势关系的显式闭式表达式。对于弱弯曲系统，其中德拜长度 $\lambda_D$ 远小于颗粒半径 $R$（$\lambda_D/R \ll 1$），形式数学推导严格确立了Ohshima-Healy-White公式作为精确正则摄动展开。相反，对于高度弯曲的球体（$R/\lambda_D \ll 1$），我们使用以缩放颗粒半径为小参数的奇异摄动分析。这种方法为球形德拜-休克尔理论提供了第一性原理的数学证明，证明其领头阶展开在完整的非线性泊松-玻尔兹曼框架内保持渐近精确，这是由于非线性的几何失活。关键的是，我们描绘了这种几何调控的精确极限，展示了随着颗粒半径增加，非线性屏蔽如何重新出现，其失效阈值由曲率与表面电荷密度之间的相互作用所控制。

英文摘要

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 $R$ ($λ_D/R \ll 1$), a formal mathematical derivation rigorously establishes the Ohshima--Healy--White formula as an exact regular perturbation expansion. Conversely, for highly curved spheres ($R/λ_D \ll 1$), 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.

发表机构

• Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences（俄罗斯科学院弗鲁姆金物理化学与电化学研究所）

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