Non-linear screening of spherical and cylindrical colloids: the case of 1:2 and 2:1 electrolytes

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From a multiple scale analysis, we find an analytic solution of spherical and cylindrical Poisson-Boltzmann theory for both a 1:2 (monovalent co-ions, divalent counter-ions) and a 2:1 (reversed situation) electrolyte. Our approach consists in an expansion in powers of rescaled curvature 1/(κa), where a is the colloidal radius and 1/κ the Debye length of the electrolytic solution. A systematic comparison with the full numerical solution of the problem shows that for cylinders and spheres, our results are accurate as soon as κa>1. We also report an unusual overshooting effect where the colloidal effective charge is larger than the bare one.