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Abstract
Determination of the distribution function of relaxation times (DFRT) is an approach that gives us more detailed insight into system processes, which are not observable by simple electrochemical impedance spectroscopy (EIS) measurements. DFRT maps EIS data into a function containing the timescale characteristics of the system under consideration. The extraction of such characteristics from noisy EIS measurements can be described by Fredholm integral equation of the first kind that is known to be ill-posed and can be treated only with regularization techniques. Moreover, since only a finite number of EIS data may actually be obtained, the above-mentioned equation appears as after application of a collocation method that needs to be combined with the regularization. In the present study, we discuss how a regularized collocation of DFRT problem can be implemented such that all appearing quantities allow symbolic computations as sums of table integrals. The proposed implementation of the regularized collocation is treated as a multi-parameter regularization. Another contribution of the present work is the adjustment of the previously proposed multiple parameter choice strategy to the context of DFRT problem. The resulting strategy is based on the aggregation of all computed regularized approximants, and can be in principle used in synergy with other methods for solving DFRT problem. We also report the results from the experiments that apply the synthetic data showing that the proposed technique successfully reproduced known exact DFRT. The data obtained by our techniques is also compared to data obtained by well-known DFRT software (DRTtools).
Highlights
In electrochemical impedance spectroscopy (EIS), the experiments are usually interpreted by fitting complex-valued impedance measurements Z j = Z j + iZ j, j = 0, 1, ... , N − 1, against chosen equivalent electrical circuit (EEC) models
The first software to extract distribution function of relaxation times (DFRT) from EIS data is based on Fourier transform technique (Kobayashi and Suzuki 2018)
In this work we are focused on the regularization techniques that are embedded in FTIKREG, DRTtools and DFRT-Py software (Table 2); and there are several facts that should be discussed
Summary
In the first approach, which has been studied in Dion and Lasia (1999), Gavrilyuk et al (2017) and Renaut et al (2013), the integral operators A1, A2 in (4) are discretized by means of quadrature formulas This approach can subsume the methods (Boukamp 2015; Boukamp and Rolle 2017; Schichlein et al 2002) in which the Eq (2) is reduced to a deconvolution problem by a suitable change of variables, after which a numerical Fourier transform is employed. In both previously mentioned approaches the level of additional discretization, governed by the number of knots of a quadrature formula or by the number of basis functions, should be properly tuned Such tuning is especially crucial in the case of noisy impedance measurements when the application of regularization techniques avoids numerical instabilities in solving (4). This discretization issue does not even appear in (4) as no additional discretization of
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