Porous electrodes are found in energy storage devices such as supercapacitors and pseudo-capacitors. 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) a region of large potential gradients at the mouth of the pore due to charge conservation. We develop a modified transmission circuit model that captures the effect of arbitrary pore sizes and demonstrate that a time-dependent interfacial capacitance needs to be included in the circuit. We also derive expressions for effective capacitance and charging timescale as a function of pore-size distribution, and show that the capacitance and charging timescale increase for narrower and less polydisperse distributions, resulting in a gain of energy density at a constant power density. Overall, our results advance the fundamental understanding of electrical-double-layer charging and will be useful for the electrode design of energy storage devices.
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