Non-linear screening of spherical and cylindrical colloids: the case of 1:2 and 2:1 electrolytes

TL;DR

This paper derives an analytical solution for the non-linear electrostatic screening of spherical and cylindrical colloids in 1:2 and 2:1 electrolytes using a multiple scale expansion, validated against numerical solutions.

Contribution

It introduces a new analytical approach based on a curvature expansion for solving Poisson-Boltzmann theory in colloids with asymmetric electrolytes.

Findings

Analytical solutions are accurate for $\, ext{κ}a > 1$.

The method shows good agreement with numerical results.

An overshooting effect of the effective charge is observed.

Abstract

From a multiple scale analysis, we find an analytic solution of spherical and cylindrical Poisson-Boltzmann theory for both a 1:2 (monovalent co-ions, divalent counter-ions) and a 2:1 (reversed situation) electrolyte. Our approach consists in an expansion in powers of rescaled curvature $1/(\kappa a)$, where $a$ is the colloidal radius and $1/\kappa$ the Debye length of the electrolytic solution. A systematic comparison with the full numerical solution of the problem shows that for cylinders and spheres, our results are accurate as soon as $\kappa a>1$. We also report an unusual overshooting effect where the colloidal effective charge is larger than the bare one.