Elsevier

Solid State Ionics

Volume 288, May 2016, Pages 311-314
Solid State Ionics

Analysis of impedance spectroscopy of aqueous supercapacitors by evolutionary programming: Finding DFRT from complex capacitance

https://doi.org/10.1016/j.ssi.2015.11.008Get rights and content

Highlights

  • EIS data of aqueous-electrolyte supercapacitors were analyzed by a novel technique.

  • The analysis is done by transforming the data to capacitance representation.

  • A distribution function of relaxation times (DFRT) model is obtained.

  • ISGP is applied to obtain detailed quantitative analysis of SCs.

Abstract

Electrochemical impedance spectroscopy (EIS) of aqueous-electrolyte supercapacitors can shed light on the physical processes occurring within the cell, given a proper analysis technique. Impedance Spectroscopy Genetic Programming (ISGP) is a novel technique to analyze EIS data. ISGP finds a functional form of the distribution of relaxation times utilizing genetic algorithm and offers a detailed quantitative analysis.

In this work, EIS data of aqueous-electrolyte supercapacitors were analyzed by transforming the data to complex capacitance representation. A distribution function of relaxation times (DFRT) model was obtained, comprised of two peaks, each correlated with a different physical process (diffusion and EDLC formation). Two cell configurations were examined and the differences regarding capacitance distribution and diffusion rates are presented.

Introduction

Aqueous-electrolyte supercapacitors (SCs) exhibit high ionic conductivity, low viscosity and non-toxicity when compared to other electrolyte alternatives, e.g. organic solvent and ionic liquid SCs [1]. Although the electrochemical window of aqueous-electrolyte SCs is limited by the breakdown voltage of water, their low-cost and relatively high dielectric constant makes them suitable for high-power applications [2]. SCs are currently used in systems such as hybrid and electric vehicles, smart grids and backup power supplies [3]. Due to their growing use, there is a need for a comprehensive, efficient and easily performed diagnostic technique. Electrochemical impedance spectroscopy (EIS) fits this task adequately.

EIS is a widely used electrical measurement technique for electrochemical systems [4]. Performed on a wide range of frequencies, EIS has the ability to reveal physical processes that occur at different rates, e.g. mass diffusion, adsorption, chemical reaction, and interfacial phenomena. In the case of SCs, the most common model that depicts its EIS behavior is the one proposed by De Levie [5], which suggests the presence of ion diffusion and electrical double layer capacitance (EDLC) formation towards and within the porous electrode [6]. This diffusion bares a significant part of the total capacitance due to local polarization at intermediate frequencies. Ions migrate through oppositely-charged media and once the frequency is sufficiently low, the ions have enough time to penetrate deeper along pore length, and form the EDLC. Other proposed EIS models all share the idea of diffusion and EDLC formation, but may have a different element that depicts it, e.g. Randles Circuit [7].

One common method when analyzing EIS data of SCs is to transform the data to complex capacitance form, according to [8]:Cω=1iωZωwhere Z(ω) is the impedance, C(ω) is the complex capacitance and ω is the angular frequency. The complex capacitance can be defined as follows:C'ω=Z''ωωZω2;C''ω=Z'ωωZω2where Z′(ω) and Z″(ω) are the real and imaginary parts of the impedance, respectively; C′(ω) and C″(ω) are the real and imaginary parts of the complex capacitance, respectively. The real part of the complex capacitance corresponds to capacitance measured via DC (the lowest frequency point of C′(ω) is similar to the capacitance measured via constant-current discharge) and the imaginary part corresponds to energy losses via dissipation. This representation highlights the electrode–electrolyte interface and since the analysis is in terms of capacitance rather than resistance, it is found to be more valuable when dealing with SCs [8].

Information is usually extracted from the complex capacitance representation simply by observing the shape of the plotted graphs. Fig. 1 shows that C′(ω) reaches saturation at low frequencies, which signify the rate of the ions diffusion [9]. The frequency, f0, at which a peak in C″(ω) is observed is regarded as the point where the cell transforms from resistive behavior to capacitive one [10]. The reciprocal time of that frequency, τ0, is said to be a measure of the charge/discharge rate of the cell [11], and therefore is directly correlated to the cell's power. Although valuable information can be extracted using this approach, a quantitative evaluation that can shed light on the different physical processes within the SC can be beneficial.

Our analysis approach is also based on analyzing EIS of SCs in complex capacitance representation [12], [13], albeit providing a more detailed and quantitative result. The analysis is based on finding the distribution function of relaxation times (DFRT), according to:Cω=CeffΓlogτ1+iωτdlogτwhere Ceff is the total capacitance (serves as normalization factor), τ is the relaxation time and Γ is the DFRT. Eq. (3) depicts an ill-posed inverse problem; in order to the find the most suitable DFRT model we use Impedance Spectroscopy Genetic Programming (ISGP), a novel technique to find a functional form of the DFRT based on a genetic algorithm [14], [15], [16], [17]. ISGP analysis procedure starts with a ‘population’ of proposed DFRT models, each one comprised of one or more known mathematical peaks (e.g. Gaussian, Lorentzian). The next step involves grading each proposed model according to the discrepancy of the resulting capacitance from the measured data and its number of free parameters, among other aspects [16]. The highest graded models proceed to the next generation, where a new set of models are generated by ‘mutations’, i.e. replacement, removal or addition of a peak to an existing model. Then again, the best graded models are chosen to proceed to the next generation and the analysis moves forward. The outcome is an analytical function of the distribution of relaxation times, which fits best the measured data while ensuring the use of the least amount of free parameters. Each peak in the model is characterized by width, height and a central time constant. The area of each peak can be calculated analytically and, multiplied by the effective capacitance, serves as its contribution to the total capacitance of the cell.

We present here an analysis which follows the portrayed method on aqueous SCs with activated carbon (AC) electrodes. Two cases examined: Case I, where we investigated the influence of enhancing the solid content in the electrode on the expense of the electrolyte, i.e. increased amount of AC in the cell without changing the volume; and Case II where the volume of the electrode has been enlarged, i.e. increased amount of AC and electrolyte while maintaining a constant AC/electrolyte ratio. We correlated each peak in the obtained models with a different physical process (diffusion and EDLC formation) and were able to quantify each process' relative rate and contribution to the total capacitance of the cell.

Section snippets

Materials and methods

Single cell Elbit aqueous supercapacitors with thick electrodes (600–700 μm) of 6 cm2 area were assembled. The electrode material was prepared by mixing an activated carbon with KOH aqueous electrolyte. Polypropylene porous separator was placed between the electrodes. The cell was packed with a conductive polymer current collector. External copper current collectors were attached; the cell was pressed to a pressure of 10 kgf/cm2, tightened between isolated stainless steel plates and sealed with

Results and discussion

The obtained DFRT model for all examined cases is comprised of two peaks (Figs. 2c and 3c; the DFRTs are plotted as a function of frequency (f = 1/2πτ) for convenience); each peak can be associated with a physical process: (1) A wide and flat peak (peak 1 in the figures), taking place at frequencies higher than 0.5 Hz is correlated with the diffusion of electrolyte ions towards and within the porous electrode; and (2) a narrow peak (peak 2), occurring at lower frequencies, correlated with the

Conclusions

In this work we have shown detailed quantitative analysis of complex capacitance of SCs. By using our analysis approach, we are able to separate and identify the processes which take place within the cell, to calculate each process' contribution to the total capacitance and to determine their mean time constants or frequencies. Our findings correlate between structural changes within the SC and its electrical behavior, enabling rational design.

Acknowledgments

We thank TEPS consortium, the Nancy and Stephen Grand Technion Energy Program (GTEP) and the Israel Science Foundation grant No. 2797/11 for funding.

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