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Power-law charge relaxation of inhomogeneous porous capacitive electrodes

Porous electrodes{made of hierarchically nanostructured materials{are omnipresent in various electrochemical energy technologies from batteries and supercapacitors to sensors and electrocatalysis. Modeling the system-level macroscopic transport and relaxation in such electrodes given their complex microscopic geometric structure is important to better understand the performance of the devices in which they are used. The discharge response of capacitive porous electrodes in particular do not necessarily follow the traditional exponential decay observed with at electrodes, which is good enough for describing the general dynamics of processes in which the rate of a dynamic quantity (such as charge) is proportional to the quantity itself. Electric double-layer capacitors (EDLCs) and other similar systems exhibit instead power law-like discharge profiles that are best described with differential equations involving non-integer derivatives. Using the fractional-order integral in the Riemann-Liouville sense and superstatistics we present a treatment of the macroscopic response of such type of electrode systems starting from the mesoscopic behavior of sub-parts of it. The solutions can be in terms of the Mittag-Leffer (ML) function or a power law-like function depending on the underlying assumptions made on the physical parameters of initial charge and characteristic time response. The generalized three-parameter ML function is found to be the best suited to describe experimental results of a commercial EDLC at different time scales of discharge.