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Understanding the system-level spectral immittance response of capacitiveenergy storage devices with analytically tractable physics-based models is notonly important for the progress of the technology, but also allows to developnew physical insights more easily. Here, we report a modifiedPoisson--Nernst--Planck (PNP) system describing charge concentration andelectric potential as a model of electro-kinetics for electrodes showing mixedresistive-capacitive behavior. This is done by (i) incorporating time shiftsbetween the current fluxes and both concentration gradients of charged speciesand the electric field, and (ii) introducing time fractional derivatives in thecontinuity equation. The aim is to characterize the deviation of immittancefrom that of ideal capacitors both at close-to-dc frequencies where theimpedance angle for example is larger than -90 deg., and also at mid-rangefrequencies where the system veers progressively toward resistive behavior.This latter tendency is important to model in order to identify the extend ofthe capacitive bandwidth of the device from the rest. Solution and simulationresults to the one-dimensional modified PNP system for symmetricelectrolyte/blocking electrode configuration are presented and discussed.

Modified Poisson-Nernst-Planck theory for low-to-mid frequency immittance of electric double-layer capacitors