Analytical estimate of effective charges at saturation in Poisson-Boltzmann cell models

TL;DR

This paper introduces an analytical approximation for calculating the effective charge of highly charged colloids in salt solutions, extending existing models to finite concentrations and validating against numerical results.

Contribution

It presents a simple analytical scheme to estimate effective charges at saturation in Poisson-Boltzmann cell models, accounting for finite colloid concentrations.

Findings

Analytical expression for effective charge as a function of colloid volume fraction and salt concentration.

Good agreement with numerical effective charge calculations at saturation.

Extension of the infinite dilution limit to finite concentrations.

Abstract

We propose a simple approximation scheme to compute the effective charge of highly charged colloids (spherical or cylindrical with infinite length). Within non-linear Poisson-Boltzmann theory, we start from an expression of the effective charge in the infinite dilution limit which is asymptotically valid for large salt concentrations; this result is then extended to finite colloidal concentration, approximating the salt partitioning effect which relates the salt content in the suspension to that of a dializing reservoir. This leads to an analytical expression of the effective charge as a function of colloid volume fraction and salt concentration. These results compare favorably with the effective charges {\em at saturation} (i.e. in the limit of large bare charge) computed numerically following the standard prescription proposed by Alexander {\it et al.} within the cell model.