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

Understanding the system-level spectral immittance response of capacitive energy storage devices with analytically tractable physics-based models is not only important for the progress of the technology, but also allows to develop new physical insights more easily. Here, we report a modified Poisson--Nernst--Planck (PNP) system describing charge concentration and electric potential as a model of electro-kinetics for electrodes showing mixed resistive-capacitive behavior. This is done by (i) incorporating time shifts between the current fluxes and both concentration gradients of charged species and the electric field, and (ii) introducing time fractional derivatives in the continuity equation. The aim is to characterize the deviation of immittance from that of ideal capacitors both at close-to-dc frequencies where the impedance angle for example is larger than -90 deg., and also at mid-range frequencies where the system veers progressively toward resistive behavior. This latter tendency is important to model in order to identify the extend of the capacitive bandwidth of the device from the rest. Solution and simulation results to the one-dimensional modified PNP system for symmetric electrolyte/blocking electrode configuration are presented and discussed.