Spread-charge model for electric double layers

Electric double layers in concentrated electrolytes exhibit a range of complex phenomena, including steric crowding, overscreening, and voltage-dependent capacitance. Mean-field theories offer a mathematically and conceptually simple framework for describing these phenomena and relating them to experimentally relevant observables. Here, we develop a spread-charge theory for the double layer in which ions carry finite-width charge distributions. In this formulation, nonlocal electrostatic response arises from the spatial extent of ionic charges, without altering the underlying Coulomb operator. The resulting model reproduces the key features of concentrated electrolyte double layers without any phenomenological electrostatic terms or ad hoc boundary conditions. Charge spreading produces a Kirkwood crossover from monotonic to damped oscillatory decay in bulk solutions, which in turn governs the long-range decay of the electrostatic potential, differential capacitance, and disjoining pressure in confined geometries. We show that in systems with no external charge, the Bazant-Storey-Kornyshev (BSK) theory emerges as the long-wavelength limit of the spread-charge model. This correspondence gives the phenomenological nonlocal electrostatic correction in BSK a microscopic interpretation in terms of the finite spatial extent of ionic charge. The spread-charge framework provides a physically interpretable mean-field route for connecting bulk screening, interfacial structure, and measurable double-layer response in concentrated electrolytes and ionic liquids.