Influence of Relative Debye Length on Electric-Double-Layer Charging Inside a Nanopore

The operating principle of energy storage devices, such as supercapacitors and hybrid capacitors, is the accumulation of charge onto double layers inside their porous electrodes. Typically, the relative Debye length, i.e., the ratio of Debye length to pore diameter, varies by orders of magnitude inside a porous electrode. However, theoretical models in the literature assume either thin or overlapping double layers. Here, we develop a perturbation expansion theory for arbitrary relative Debye lengths to solve the Poisson-Nernst-Planck (PNP) equations in the limit of small potentials. We derive analytical expressions for the charge distribution, potential profile and total current, and show that our analytical results compare favorably with direct numerical simulations of the PNP equations. We show that an arbitrary relative Debye length results in a jump discontinuity in the electric potential at the pore mouth, and also modifies the nanopore charging timescale. Finally, we derive an expression for effective capacitance that is able to qualitatively explain the pore-size dependence of nanopore capacitance results reported in the literature.