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Effective charge versus bare charge 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 $κa\gg 1$, where $a$ is the radius of the colloid and $κ$ 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 $κa\sim 1$. 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.