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Poisson-Boltzmann theory is the cornerstone for soft matter electrostatics.We provide novel exact analytical solutions to this non-linear mean-fieldapproach, for the diffuse layer of ions in the vicinity of a planar or acylindrical macroion. While previously known solution are for homogeneouslycharged objects, the cases worked out exhibit a modulated surface charge --orequivalently surface potential-- on the macroion (wall) surface. In addition toasymptotic features at large distances from the wall, attention is paid to thefate of the contact theorem, relating the contact density of ions to the localwall charge density. For salt-free systems (counterions only), we make use ofresults pertaining to the two-dimensional Liouville equation, supplemented byan inverse approach. When salt is present, we invoke the exact two-solitonsolution to the 2D sinh-Gordon equation. This leads to inhomogeneous chargepatterns, that are either localized or periodic in space. Without salt, theelectrostatic signature of a charge pattern on the macroion fades exponentiallywith distance for a planar macroion, while it decays as an inverse power-lawfor a cylindrical macroion. With salt, our study is limited to the planargeometry, and reveals that pattern screening is exponential.

Electric double layers with surface charge modulations: Novel exact Poisson-Boltzmann solutions