Analysis of the Charging Current in Cyclic Voltammetry and Supercapacitor's Galvanostatic Charging Profile Based on a Constant-Phase Element.

We investigated the charging current in cyclic voltammetry and the galvanostatic charging/discharging behavior of a controversial constant-phase element (CPE) to describe an electrical double layer used only in electrochemical impedance spectroscopy. The linear potential sweep in the time domain was transformed into the frequency domain using a Fourier transform. The current phasor was estimated by Ohm's law with the voltage phasor and a frequency-dependent CPE, followed by an inverse Fourier transform to determine the current in the time domain. For galvanostatic charging/discharging, the same procedure, apart from swapping the voltage signal with the current signal, was applied. The obtained cyclic voltammetry (CV) shows (1) a gradual increase in the charging current, (2) a higher charging current at a low scan rate, and (3) a deviation from the linear relationship between the charging current and the scan rate. For galvanostatic charging/discharging, the results demonstrate (1) curved charging/discharging behavior, (2) a higher voltage in the early stage, and (3) a lower voltage during longer charging periods. In contrast to a previous approach based on solving a differential equation with a simple RC circuit, our Fourier transform-based approach enables an analysis of electrochemical data with an arbitrary and complex circuit model such as a Randles equivalent circuit. The CPE model is more consistent with previous experimental results than a simple ideal capacitor, indicating a ubiquitous CPE in electrochemistry and a fair figure of merit for supercapacitors.

Introduction

Cyclic voltammetry (CV), a powerful technique used to investigate electrochemical processes, has been attracting much attention in relation to renewable energy.[1,2] CV studies provide information about (1) the formal potential and its close link to the standard reduction potential of redox couples, (2) the charge-transfer kinetics of electrodics, (3) the concentration of redox couples, (4) chemical reaction mechanisms, and (5) the diffusion coefficient of redox couples, among others.[1] In spite of the popularity of CV, there are several difficulties with this process, such as a voltage drop by uncompensated solution resistance (iR drop) and background signals. At a high electrolyte concentration and in conventional three-electrode systems, the iR drop is negligible in most experiments. The latter case, however, has been a practical issue in interpretations of CVs because the precise estimation of the faradic current is essential either for the determination of concentrations in analytical chemistry or for the figure of merit of electrocatalysts or supercapacitors. The dominant portion of the background current in CV originates from the electrical double layer (EDL), which is modeled as a Gouy–Chapman–Stern (GCS) double layer consisting of a serial connection between two capacitances corresponding to the outer Helmholtz layer (OHP) and the capacitance, which originates from the diffuse layer.[3] At a high electrolyte concentration, the capacitance of the diffuse layer increases and the total capacitance therefore converges to the capacitance of the OHP. Assuming that the capacitance is independent of the applied potential, the background current in CV for the applied potential (Figure A) is explained by assessing the constant charging current with the transient current at the vertex point of the potential sweep, as shown in Figure C. In reality, this behavior of ideal polarized electrodes is rare, meaning that this system has been verified based on the simple capacitance model. Thus, CV in the absence of a redox couple is often used as a background, which is not suitable for quantitative analysis and not practical for several processes in electrochemistry such as a hydrogen evolution reaction (HER).

Figure 1

(A) Time versus the applied potential of CV with scan rate = 100 mV/s and (B) corresponding power spectrum of the potential in the frequency domain. CVs obtained by an inverse Fourier transform at different scan rates with CPE of (C) n = 1.0 (pure capacitor), (D) n = 0.9, (E) n = 0.8, and (F) n = 0.7. (G) Dependence of the charging current upon n of CPE. Equivalent circuits are composed of 100 Ω as a solution resistance and Q0 = 10–3 F/s as a CPE.

Electrochemical impedance spectroscopy (EIS) is another electrochemical technique concomitant with CV.[4] EIS is based on a small-amplitude AC signal at a constant DC bias. EIS data have been analyzed based on equivalent circuits composed of passive elements such as resistance, capacitance, and Warburg impedance. A physical meaning is assigned to each element, including the charge-transfer resistance, double-layer capacitance, and diffusion impedance. Experimental data have shown a systematic deviation from a simple model, especially for double-layer capacitance as a nonfaradic process. While the GCS model predicts the response of an ideal capacitor for a small AC signal whose phase shift is π/2, the experimental current compared to the applied AC voltage is shifted by n × π/2, where n is a value between 0 and 1. This new passive element, called a constant-phase element (CPE) in EIS, has been widely used in equivalent circuits for EIS. The CPE can be written aswhere i is an imaginary unit (i2 = −1), ω is an angular frequency, and Q0 and n (0–1) are the characteristic values corresponding to the double layer.[5,6] The Q0 parameter represents ideal capacitor behavior when n = 1 or intermediate characteristics between a capacitor and a resistor when n < 1.[7] Although the origin of the CPE is still not clear,[8−12] it remains a viable empirical element for use in EIS techniques.

Galvanostatic charging/discharging has also been used to characterize supercapacitors and secondary batteries due to its relatively direct physical relationship to the capacitive charge. For an ideal capacitor, the triangular response upon a constant current with a small deviation at the vertex caused by the solution resistance can be predicted. However, the empirical results deviate severely from the theoretical model, thus raising issues with regard to the validity of the simple capacitor model for interpretation of results.

Questions remain on how the charging current will flow in CV and galvanostatic charge–discharge for the CPE instead of the simple capacitance. On the other hand, the ideal capacitor explains the charging current, which is inconsistent with the empirical charging current in both CV and galvanostatic charging/discharging. On the other hand, the CPE instead of a capacitor has been successfully adopted to interpret EIS data for small perturbations. This raises the question: why do we not apply the CPE model to other electrochemical techniques to predict the charging current for larger perturbations? The bottleneck is that the CPE is described by a function of a complex variable depicted in a phasor diagram due to the AC characteristic, while CV is expressed as a function of a real variable. Specifically, the charging current of CPE shown in eq upon linear sweep voltage or galvanostatic charging/discharging is not straightforward. Q0 of the CPE is in units of F/s instead of F from the real capacitance. Sadkowski published the first effort to address this question based on a Laplace transform with a simple CPE.[13−15] A similar approach was also applied to estimations of galvanostatic charge–discharge.[16] Previous reports investigated the electrochemical response of a simple equivalent circuit consisting of a serial connection of the resistance and the CPE because solving a general equation for a more general model, such as the Randles equivalent circuit, is difficult. To circumvent the mathematics involved in solving a differential equation for a complex model, our group recently investigated the relationships between voltammetry, chronoamperometry, and EIS using the Fourier transform (FT), in which the frequency domain was used to predict the electrochemical response in the time domain.[17] In the present paper, we applied the Fourier transform to determine a bundle of harmonic waves for CV and the corresponding current signal to observe the charging current of the CPE in CV. A similar process was also applied to galvanostatic charging/discharging, where our model returns the voltage signal corresponding to the harmonic current signal at each frequency with the effective capacitance to estimate the ability of a capacitor. The results show that the charging behavior is more consistent with the experiments and with previous reports on the CPE, demonstrating the validity of the CPE model for use in various electrochemical techniques based on our FT-based approach.

Theory and Experiments

All results were obtained using a house-built MATLAB code to perform the discrete Fourier transform of triangular potential sweep and the corresponding current signal. A simple equivalent circuit for an electrochemical cell consisted of a serial connection of solution resistance (Rs) and CPE. The current signal at the frequency domain was calculated based on simple Ohm’s lawwhere V and I are complex numbers at a specific frequency, Z is the total impedance, and other parameters are the same as in eq . For all calculations, solution resistance and Q0 were 100 Ω and 10–3 F/s(1–, respectively. Then, the data array of I (complex number of current) at each frequency was transformed into the time domain by the inverse Fourier transform to obtain the current signal at the time domain.

Cyclic voltammetry was measured in a three-electrode system cell using a CHI 900B potentiostat (scan rate = 50 mV/s). A Au disk electrode (BASi, 1.6 mm diameter), a mercury–mercurous sulfate electrode (MSE, sat. K2SO4), and a Pt wire served as the working electrode, reference electrode, and counter electrode, respectively. A 0.105 M KClO4 electrolyte was purged with argon gas for 15 min before the experiment. Both the voltage signal and the current signal were transformed into the frequency domain based on a previously reported method,[17] followed by impedance data. It should be mentioned that this impedance from large perturbations is not the same as the conventional electrochemical impedance from small perturbations. Then, the low-frequency region of impedance was fitted with Randles equivalent circuits shown in Figure A whose impedance is written as followswhere Z is the total impedance, Rs is the solution resistance, Rct is the charge-transfer resistance, Q1 and n1 (0–1) are the characteristic values corresponding to CPE, Q2 and n2 (near 0.5) are the characteristic values corresponding to Warburg impedance, and other parameters are the same as in eq . The house-built MATLAB code was used to find optimized parameters for eq based on nonlinear curve-fitting in least-squares sense from split real parts and imaginary parts from experimental CV.

Figure 4

(A) Schematic model of the Randles equivalent circuit. (B) Nyquist plots of impedance data from cyclic voltammetry (black dot) and from fitting with the Randles circuit (red dot). (C) Cyclic voltammogram of the bare Au disk electrode in 0.105 M KClO4 with a scan rate of 50 mV/s (black solid line) and the fitted voltammogram from the equivalent circuit after the inverse Fourier transform to the time domain (red solid line).

Results and Discussion

Scheme illustrates the method used to estimate the charging current in a cyclic voltammogram (CV) with two different equivalent circuits. The first circuit, shown in Scheme A, was composed of the solution resistance and the conventional double-layer capacitance (Cdl). Instead of Cdl, the empirical CPE was adopted. It is expressed here as eq (Scheme B). The total impedance in both cases was the vector sum of Rs and the capacitive element depicted in the corresponding phasor plot. Note that the phase of the CPE differs from that of Cdl. Owing to the nature of the capacitive element being dependent on the frequency, expressed here as eq , the prediction of the current in the time domain is not straightforward. To circumvent this problem, the frequency domain was adopted, as illustrated in Scheme C. First, the triangular waveform of the CV was transformed into the frequency domain, resulting in complex numbers of the voltage phasor for each frequency. The complex numbers for the current phasor were then calculated simply with Ohm’s law, i.e., by dividing the complex number of the total impedance into the complex number of the voltage, as described in eq . Then, the inverse Fourier transform was applied to determine the current in the time domain. Finally, the CV was plotted as the voltage versus the current in the time domain. To assess the galvanostatic charge–discharge, voltage as a stimulus for an electrochemical cell was swapped with the current as a response, while other procedures were identical to those used with the CV.

Scheme 1

Illustration of Estimation of the Charging Current for Cyclic Voltammetry

Equivalent circuits composed of solution resistance and either (A) conventional double-layer capacitance or (B) constant-phase element (CPE) with corresponding phasor diagrams. (C) Procedures for predicting cyclic voltammogram by the Fourier transform. Details are described in the main article.

CVs were studied using the equivalent circuits shown in either Scheme A or Scheme B with Rs = 100 Ω and Q0 = 10–3 F/s(1– and with various n values. Figure A,B shows the representative linear potential program at 100 mV/s in the time domain and the corresponding power spectrum in the frequency domain after the Fourier transform operation, respectively, demonstrating the contribution of the relatively low-frequency harmonic wave during the conventional electrochemical measurement. For an ideal capacitor with n = 1.0, the CV shown in Figure C is in good agreement with the previous theoretical result. Specifically, transient currents at both vertices dependent on the RC time with the steady-state charging current are clearly observed, indicating the validity of our procedure based on the Fourier transform. Figure D–F shows the CVs of CPE with n = 0.9, 0.8, and 0.7, respectively. Compared to Figure C, showing the ideal capacitor, first, the CVs of the CPE demonstrate a gradual increase in the current in the charging region, whereas transient behavior is still shown at both vertices. The smaller n is, the more CV tilts apparently. Empirically, the charging current in the majority of electrochemical cells follows the CVs of the CPE, as shown in Figure D,E as compared to an ideal CV for a capacitor. Therefore, the CPE is a better model than a conventional capacitor to explain the electrical double-layer in CVs. Second, the charging currents for the CPE are much larger than that for a conventional capacitor at a slow scan rate, while the charging currents of both systems converge to similar values at a high scan rate. To clarify this point, the current at 0 V was used as an estimator of the charging current and was plotted depending on the scan rates, as shown in Figure G. At a slow scan rate of 10 mV/s, a charging current of 43 μA for n = 0.7 is double that of 20 μA for n = 1.0. In contrast, a charging current of 206.6 μA for n = 0.7 is nearly identical to that of 200.2 μA for n = 1.0 at a higher scan rate of 100 mV/s. The smaller the n is, the larger the charging current is when observed at a slow scan rate. Third, the linear relationship between the charging current and the scan rate is no longer valid for the CPE. The charging current of the ideal capacitor (black dashed line in Figure G) increases linearly depending on the scan rate. For the CPE, however, there is a deviation from the linear line indicated by the green dashed line for n = 0.7. This tendency has been widely observed in electrochemical experiments and in previous reports,[13,15,16] indicating the validity of our FT-based approach for CV with the CPE model. In summary, the CPE model strongly affects the charging behavior in CV compared with an ideal simple capacitor model, coinciding with empirical experiments for CVs well.

The CPE model is superior to the conventional capacitor model used in the CV; hence, we assume that this model will also be useful in characterization of supercapacitors. To investigate the effect of the CPE model on a supercapacitor, the galvanostatic charge–discharge was applied. Charging is the core operation in a supercapacitor. Figure A shows a time versus current curve applied to an electrochemical cell as a means of excitation. Here, constant currents of 10 μA were applied to the equivalent circuit shown in Figure as the charging current and discharging current. This current in the time (i) domain was transformed into the frequency domain (I), similar to that in Scheme C, after which the voltage phasor (V) was calculated by the multiplication of a complex number to the current (I) and the total impedance (Z) at each frequency. The voltage in the time domain was then generated by the inverse Fourier transform. Figure B shows the galvanostatic charge–discharge curves for both the ideal capacitor and the CPE. The triangular response with the small iR drop was in good agreement with the predicted behavior of the ideal capacitor with a constant current, demonstrating the validity of our method in the galvanostatic mode. For the CPE, the response is not linear but instead appears to be curved. As distortion in CV depends on the n value of the CPE, the smaller the n is, the larger the curvature becomes. The initial region of CPEs shows a higher voltage than the ideal capacitor, demonstrating that the apparent capacitance of the CPE in this region is smaller than in the ideal case. In the extended charging region, however, the magnitude of apparent capacitance is reversed from the low values at the early stage, indicating lower efficiency as an energy storage device. These results are in good agreement with a previous report on the CPE based on the Laplace transform.[16] Interestingly, the majority of papers on supercapacitors demonstrate CPE-like behavior in experiments, but the characterization of supercapacitors has been conducted based on an ideal capacitor model. Thus, both Q0 and n are more reliable indicators than the traditional capacitance value when characterizing a supercapacitor. These two values do not represent the capacitance, but one can estimate the apparent capacitance, also referred to as the effective capacitance. Several models have been used to estimate the effective capacitance from the CPE in EIS corresponding to small perturbations.[7,18] For large perturbations such as galvanostatic charging/discharging and cyclic voltammetry, the electrochemical response is much more complex. The following equation describes the effective capacitance (Ceff) for a simple equivalent circuit consisted of the solution resistance and the CPE[16]where Γ is the γ function and t is the time. One can estimate this time-dependent Ceff after the estimation of Q0 and n. For instance, Ceff is 2.23 × 10–3 F in Figure B with Q0 = 1.0 × 10–3 F/s–0.3, n = 0.7, and t = 20 s. The effective capacitance of the CPE at a short time is lower than that of an ideal capacitor (Supporting Information) such that the larger potential in the initial region of the CPE in Figure B is reasonable. It is also important to note that similar behaviors for the charging–discharging of hybrid supercapacitors have been reported.[19] Unfortunately, we do not have any ideal method to identify the two different processes of either a hybrid supercapacitor or the CPE. Nonetheless, the suggested CPE model with our FT-based approach is a very promising model applicable to estimations and for explaining the behaviors of supercapacitors.

Figure 2

(A) Time versus the applied current in galvanostatic charging–discharging. (B) Galvanostatic charge and discharge profiles of the equivalent circuit consisted of CPE with various n values.

Both CV and galvanostatic charging/discharging show the unique features from the CPE model compared to the behaviors of a conventional capacitor. Although the origin of the CPE has not yet been clearly revealed, the CPE model represents the empirical experimental results from electrochemistry well. To understand the physical meaning of the CPE in an electrochemical system, Bode plots of equivalent circuits were devised, as shown in Figure . Respectively, Figure A,B shows the amplitude and phase of equivalent circuits depending on the frequency, respectively. At a high frequency, the amplitude and phase of the total impedance approach the solution resistance, as expected. The phase, however, still shows a difference between the simple capacitor and the CPE. The phase of the simple capacitor passed the inflection point approaching 0 of the resistor, while that of the CPE with n = 0.7 does not reach the inflection point. This implies that the CPE with a smaller n possesses capacitive characteristics even at a high frequency. The effect of the CPE is more dramatic in the low-frequency region. Both the amplitude and phase of the total impedance are smaller than those of the simple capacitor. For the CV, larger charging current levels were observed, as shown in Figure , an outcome consistent with the smaller amplitude of the total impedance because the lower frequency contributes more to the slower potential scan. In addition, the remaining resistive characteristics of the CPE at a low frequency from the phase cause a gradual increase in the charging current. For galvanostatic charging/discharging, the phase of the total impedance from the CPE shows more capacitive behavior at a high frequency and more resistive behavior at a low frequency. Therefore, the initial response of the galvanostatic measurement depending on the high frequency tends to increase the potential, while the longer charging causes the dissipation of the energy due to the nature of the resistor.

Figure 3

Bode plot of equivalent circuits in Scheme B for the amplitude (A) and the phase (B).

Our FT-based approach can be applied not only to a simple serial RC circuit model but also to a general equivalent circuit model, which cannot easily be solved by the Laplace transform. The CV was obtained from a Au working electrode, as indicated by the black solid line in Figure C, to get the practical charging current. Its impedance was analyzed by our FT-based approach, as shown in Scheme C. The black dots in Figure B represent the impedance data for this large-amplitude perturbation, which is symmetric most likely due to the symmetry of the current in the CV. Half of the experimental data for the low-frequency region appear to consist of a hemisphere and a linear line with ca. 45°, similar to the Randles equivalent circuit shown in Figure A in spite of the considerable difference in the amplitude of the perturbation. It should be noted that both charge-transfer resistance (Rct) and the Warburg impedance are not expected for an ideal polarized electrode in an electrolyte without redox species, whereas they were observed experimentally.[20] The simple RC circuit models previously reported do not represent the practical CV; thus, we assumed a Randles equivalent circuit based on the trend shown in Figure B. Although the physical meaning and the origin of Rct and the Warburg impedance are not obvious, we speculate that the heterogeneity of the electrode/electrolyte interface,[7] liquid bulk phenomena,[20] or the variance in the capacitance depending on the electrode potential[21] may evoke these components. In contrast to solving a complex differential equation, we applied the lsqcurvefit function for nonlinear fitting in MATLAB based on the Randles equivalent circuit. The red dots in Figure B show the fitting result from the FT-based approach. Even this equivalent circuit shows some deviation from the experimental data in both the impedance and CV because the Au/electrolyte interface is not a perfectly ideal polarized electrode. Although the charging current for the electric double layer is dominant, experimental background current can come from various sources, though this is out of the scope of this manuscript. Note that the experimental impedance can be fitted with any equivalent circuit by our method. Then, the inverse Fourier transform results in the CV are expressed by the red line in Figure C. These results demonstrate that the experimental electrochemical data in the time domain, such as the CV and galvanostatic charging/discharging data, can be modeled with an arbitrary circuit model to find the physical phenomena.

Conclusions

We investigated the charging current in the CV and galvanostatic charging/discharging behavior of the CPE, which has been used in only electrochemical impedance spectroscopy. To circumvent the frequency-dependent reactance of the CPE, the response for specific excitation is analyzed based on the harmonic wave at each frequency using the Fourier transform combined with Ohm’s law. The response in the time domain is acquired by the inverse Fourier transform. Compared with CV based on a conventional simple capacitor model, the equivalent circuit with CPE shows (1) a gradual increase in the charging current, (2) a higher charging current at a low scan rate, and (3) a deviation from the linear relationship between the charging current and the scan rate. For galvanostatic charging/discharging, the results demonstrate (1) the curved charging/discharging behavior, (2) the larger voltage in the early stage, and (3) the lower voltage in longer charging. From empirical experimental results, CPE seems to model the real system better than the simple capacitor, indicating the ubiquitous CPE in electrochemistry. The CPE model may also apply to the characterization of various electrochemical systems with an arbitrary circuit model, especially the effective capacitance of supercapacitors with Q0 and n.