  Porous electrodes are found in energy storage devices such as supercapacitors
and pseudocapacitors. However, the effect of electrode-pore-size distribution
over their energy storage properties remains unclear. Here, we develop a model
for the charging of electrical double layers inside a cylindrical pore for
arbitrary pore size. We assume small applied potentials and perform a regular
perturbation analysis to predict the evolution of electrical potential and ion
concentrations in both the radial and axial directions. We validate our
perturbation model with direct numerical simulations of the
Poisson-Nernst-Planck equations, and obtain quantitative agreement between the
two approaches for small and moderate potentials. Our analysis yields two main
characteristic features of arbitrary pore size: i) a monotonic decrease of the
charging timescale with an increase in relative pore size (pore size relative
to Debye length); ii) large potential changes for overlapping double layers in
a thin transition region, which we approximate mathematically by a jump
discontinuity. We quantify the contributions of electromigration and charge
diffusion fluxes which provide mechanistic insights into the dependence of
charging timescale and capacitance on pore size. We develop a modified
transmission circuit model that captures the effect of arbitrary pore size and
demonstrate that a time-dependent transition-region resistor needs to be
included in the circuit. We also derive phenomenological expressions for
average effective capacitance and charging timescale as a function of pore-size
distribution. We show that the capacitance and charging timescale increase with
smaller average pore sizes and with smaller polydispersity, resulting in a gain
of energy density at a constant power density. Overall, our results advance the
mechanistic understanding of electrical-double-layer charging.
