Effective charge versus bare charge: an analytical estimate for colloids in the infinite dilution limit

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 kappaa >> 1, where a is the radius of the colloid and kappa is the inverse screening length. In practice, a careful comparison with effective charge parameters, obtained by numerically solving the full nonlinear PB theory, proves that our estimate is good down to kappaa similar to 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.