Effective charge versus bare charge for colloids in the infinite dilution limit

TL;DR

This paper introduces an analytical approximation for the effective charge of colloids based on Poisson-Boltzmann theory, valid across a range of colloid sizes and salt concentrations, especially relevant for colloidal suspensions and DNA modeling.

Contribution

It provides a new asymptotically exact analytical formula for effective charge that aligns well with numerical solutions, extending applicability to practical colloidal and biological systems.

Findings

Approximation is asymptotically exact for large $ppa a$

Good agreement with numerical PB solutions down to $ppa a \u2265 1$

Applicable to DNA molecules under physiological conditions

Abstract

We propose an analytical approximation for the dependence of the effective charge on the bare charge for spherical and cylindrical macro-ions as a function of the size of the colloid and salt content, for the situation of a unique colloid immersed in a sea of electrolyte (where the definition of an effective charge is non ambiguous). Our approach is based on the Poisson-Boltzmann (PB) mean-field theory. Mathematically speaking, our estimate is asymptotically exact in the limit $\kappa a\gg 1$, where $a$ is the radius of the colloid and $\kappa$ the inverse screening length. In practice, a careful comparison with effective charges parameters obtained by numerically solving the full non-linear PB theory proves that it is good down to $\kappa a\sim 1$. This is precisely the limit appropriate to treat colloidal suspensions. A particular emphasis is put on the range of parameters suitable to describe both single and double strand DNA molecules under physiological conditions.