Electrostatic Force Between Two Colloidal Spheres

TL;DR

This paper derives an analytical solution for the electrostatic double-layer force between two charged colloidal spheres using the Linearized Poisson-Boltzmann Equation, validated through numerical simulations and approximations.

Contribution

It provides a new analytical multipole expansion solution for the electrostatic interaction between colloidal spheres with a developed Mathematica code for force calculation.

Findings

Analytical solution for electrostatic force derived from LPBE.

Numerical validation against LSA and PFA approximations.

Code enables force computation based on sphere separation.

Abstract

In this dissertation we analyzed the double-layer force interaction between two dielectric charged colloidal spheres of different radii immersed in a solution of ions in water and in thermal equilibrium. In the limit of low electrostatic potential the interaction is governed by the Linearized Poisson-Boltzmann Equation (LPBE). We obtained an analytical solution from the LPBE for the electrostatic interaction via multipole expansion, considering a uniform and fixed surface charge density on the spheres. A code in the Mathematica platform was developed, allowing us to calculate the force between the spheres as a function of their separation. In order to validate the code, we compared our numerical results with the analytical ones in the limits of Linear Superposition Approximation (LSA) (valid for sphere separations much greater than the Debye length) and Proximity Force Approximation (PFA) (valid for sphere separations and Debye length much smaller than the radii of the spheres).