Gradient expansions to capture the nonlocal physics of the electrical double layer

A hallmark of highly charged surfaces or concentrated electrolytes is strong, nonlocal electrostatic and density correlations. In these regimes, the classical Poisson-Boltzmann (PB) mean-field theory for dilute electrolytes breaks down, and the predictions using PB can be qualitatively and quantitatively incorrect. For example, the influence of nonlocal electrostatic correlations lead to charge reversal and like-charge attraction, even though PB theory predicts an exclusively repulsive interaction between like-charged surfaces. Size correlations can lead to oscillatory density profiles that are not captured by PB theory. Here, we explore using a nonlocal Landau-Ginzburg-like free energy functional to describe the structure of an electrical double layer at a charged surface, as well as overlapping electrical double layers, at high charge density and concentration. We show that the correct boundary conditions of such a nonlocal model are given by a force balance at the interface, where the nonlocal effects must vanish. We apply the model to calculate surface forces, interactions between biological molecules, and electrokinetic flows.