A full capacitive picture of the constant phase element: Estimations from impedance spectroscopy, distribution of relaxation times and fractional calculus

Impedance spectroscopy (IS) is classically used to study the dynamical properties of electrochemical systems but now is also often accompanied with chrono-based methods and the emerging analysis tool of fractional calculus to provide timescale information. From both measurement techniques, mathematical approaches have been derived for estimating the effective capacitance from the parameters of the ubiquitous constant-phase-element (CPE) in Cole impedance models. Nevertheless, there is a lack of global consensus on the ways in which capacitive properties of electrochemical materials should be assessed and reported. To address this issue, we here particularize the theoretical models developed in the most famous references of the CPE to provide numerical approximations on the total capacitance of this anomalous element through the non-parametric procedure of the distribution of relaxation times (DRT). We here draw parallels between the above-mentioned methods, indicating how one can estimate a real capacitance for the case of normal time-constant distributions. The current impulse-response function reveals that DRT and fractional calculus converge in capturing the adequate time scale to numerically estimate the total capacitive information about the electrochemical system dynamics, as well as the impedance does in the high-frequency region. Our work seeks to expand the concept of real capacitance and, in turn, provide an advanced strategy for connecting the current-controlled analytical techniques of IS, DRT and chronoamperometric experiments to clear the conscience of the electrochemistry's community in the use and interpretation of CPEs. • Transition from effective capacitance C eff to real capacitance C real of the CPE. • C real as a particular case of the famous formulas C eff = Q ω α − 1 or C eff = Q t 1 − α Γ α + 1 . • Complete capacitive landscape of the CPE in normal time-constant distributions. • A unified view of the capacitance of electrochemical capacitors via DRT analysis. • Comparison of the theoretical results by using numerical simulations.