The Distribution of Capacitive Times for the Analysis of Electrochemical Impedance Spectroscopy Data from Supercapacitors
Authors: Py Baptiste , Ciucci Francesco 
Published: December 2024

Electrochemical impedance spectroscopy (EIS) is the experimental technique of choice for studying the properties of various electrochemical systems, including batteries, fuel cells, and supercapacitors [1]. In particular, EIS data collected from supercapacitors was successfully analyzed with the distribution of relaxation times (DRT) to elucidate charging and discharging mechanisms at the electrodes [2]. The DRT method is especially appealing as it represents a non-parametric alternative to the standard equivalent circuits and physical models [3].
The DRT method relies on the assumption that a blip of current linearly results in a voltage that decays with a characteristic timescale [4]. Deconvolving the DRT from measured impedances allows the identification of these characteristic timescales to characterize and optimize the performance of supercapacitors. Nevertheless, due to diffusional and mass-transport-related phenomena, the imaginary part of the impedance of supercapacitors generally diverges at low frequencies, which renders the DRT method less accurate [5]. In fact, the impedance modelled with the DRT tends to a finite real value at low frequencies. In other words, the DRT method is less suitable to identify timescales larger than a few seconds. To overcome this limitation, the distribution of capacitive times (DCT) was proposed by Brug et al. [6]. The DCT method models the admittance, i.e., the inverse of the impedance, in a similar fashion as the DRT describes the impedance. Recently, our group derived the theory underlying the DCT method to facilitate its application and enrich the framework of non-parametric distributions for EIS data analysis [7]. We also used artificial and real battery data to showcase the potential of the DCT method.
In this presentation, we will first summarize the main properties of the DCT, and in particular highlight the differences between the DCT and the DRT. Then, we will illustrate how to use the DCT to analyze real data from an electric double-layer supercapacitor with porous carbon electrodes. Moreover, we will explain how to recover the DCT from the experimental impedance, before showing how to compute physical parameters from the DCT spectrum. Overall, we envision that the DCT method will deepen the understanding of EIS data measured from supercapacitors.
References
[1] A. Lasia, Electrochemical impedance spectroscopy and its applications, Modern Aspects Electrochem. (2002) 143–248.
[2] W. Wu, Y. Zhuang, D. Yan, J. Huang, S. Peng, J. Wang, R. Zhuo, Z. Wu, P. Yan, G. Cao, Supercapacitive properties of MnO2 and underlying kinetics by distribution of relaxation time method, J. Power Sources. 474 (2020) 228667.
[3] F. Ciucci, Modeling electrochemical impedance spectroscopy, Curr. Opin. Electrochem. 13 (2019) 132–139.
[4] Y. Lu, C.-Z. Zhao, J.-Q. Huang, Q. Zhang, The timescale identification decoupling complicated kinetic processes in lithium batteries, Joule. 6 (2022) 1172–1198.
[5] E. Quattrocchi, T.H. Wan, A. Curcio, S. Pepe, M.B. Effat, F. Ciucci, A general model for the impedance of batteries and supercapacitors: The non-linear distribution of diffusion times, Electrochim. Acta. 324 (2019) 134853.
[6] G.J. Brug, A.L.G. Van Den Eeden, M. Sluyters-Rehbach, J.H. Sluyters, The analysis of electrode impedances complicated by the presence of a constant phase element, J. Electroanal. Chem. 176 (1984) 275–295.
[7] B. Py, A. Maradesa, F. Ciucci, From theory to practice: Unlocking the distribution of capacitive times in electrochemical impedance spectroscopy, Electrochim. Acta. (2023) 143741.