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Nano-scale devices continue to challenge our theoretical understanding of microscopic systems. Of particular interest is the characterization of the interface electrochemistry of graphene-based sensors. Typically operated in a regime of high ion concentration and high surface charge density, dielectric saturation and ion crowding become non-negligible at the interface, complicating continuum treatments based upon the Poisson-Boltzmann equation. Using the Poisson-Boltzmann equation, modified with the Bikerman-Freise model to account for non-zero ion size and the Booth model to account for dielectric saturation at the interface, we characterize the diffuse layer capacitance of both metallic and graphene electrodes immersed in an aqueous electrolyte. We find that the diffuse layer capacitance exhibits two peaks when the surface charge density of the electrode is increased, in contrast with experimental results. We propose a self-consistent (and parameter-free) method to include the Stern layer which eliminates the spurious secondary peak in the capacitance and restores the correspondence of the model with experimental observations. This study sheds light on the interplay between the ion steric effects and the dielectric saturation in solvent, exposes the importance of quantum capacitance when graphene is used as an electrode, and demonstrates the importance of a self-consistent treatment of the Stern layer in continuum models of the electrode-electrolyte interface. Furthermore, the theoretical foundation provides a base upon which more detailed models of graphene-based sensors can be built.

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