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 κ a ≫ 1, where a is the radius of the colloid and κ 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 κ a ∼ 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.