Electrostatic Attraction between Overall Neutral Surfaces

Two overall neutral surfaces with positive and negative charged domains ("patches") have been shown in recent experiments to exhibit long-range attraction when immersed in an ionic solution. Motivated by the experiments, we calculate analytically the osmotic pressure between such surfaces within the Poisson-Boltzmann framework, using a variational principle for the surface-averaged free energy. The electrostatic potential, calculated beyond the linear Debye-H\"uckel theory, yields an {\it overall attraction} at large inter-surface separations, over a wide range of the system's controlled length scales. In particular, the attraction is stronger and occurs at smaller separations for surface patches of larger size and charge density. In this large patch limit, we find that the attraction-repulsion crossover separation is inversely proportional to the square of the patch charge-density and to the Debye screening length.