The electrical impedance of hierarchical carbon electrodes

Many electrodes used in energy storage devices are produced by grinding a micro- and meso- or macro-porous material into a powder and depositing it on a flat substrate, which leads to a hierarchical structure. At the largest scale the structure is that of a flat layer; the intermediate structure is that of porous grains; the smallest scale is that of the microporous skeleton that makes up the grains.

Here, we use electrical impedance spectroscopy to investigate both experimentally and through mathematical modelling how electric charges are stored and transported in such structure. We focus on electrodes produced with a variety of carbon xerogels soaked with KCl solutions. Nonane adsorption is used during the electrode preparation to selectively block the micropores, and discriminate their contribution from that of larger pores. The impedance data are analysed with a mathematical model built on the Poisson–Boltzmann equation for charge storage and on the Nernst–Planck equation for ion transport, which accounts for the hierarchical structure of the electrodes.

The equilibrium charge storage in large pores is found to be quantitatively described by the classical electric double layer. In micropores, however, non-ideality of the electrolyte has to be invoked to account for the measured capacitance, hinting at ion desolvation. As for the charge transport, each structural level is found to have a characteristic frequency, above which the storage of electric charges is transport-limited at the considered scale. The hierarchical structure of the electrode is responsible for salient characteristics of
the impedance. In particular the Warburg regime (with impedance scaling with the square root of frequency) is observed when transport is rate-limiting at one scale only. In hierarchical electrodes, an additional regime is observed with impedance scaling with the frequency to the power 1/4, when transport limitations are present at two scales simultaneously.