HUNGARIAN JOURNAL OF INDUSTRY AND CHEMISTRY Vol. 49(1) pp. 1–8 (2021) hjic.mk.uni-pannon.hu DOI: 10.33927/hjic-2021-01 THE EXPERIMENTAL VERIFICATION OF A GENERALIZED MODEL OF EQUIVALENT CIRCUITS ZOLTÁN LUKÁCS*1 , DÁVID BACCILIERI1 , AND TAMÁS KRISTÓF1 1Center for Natural Sciences, University of Pannonia, Egyetem u. 10, 8200 Veszprém, HUNGARY The determination of typical parameters of electrochemical systems, e.g. the polarization or charge transfer resistances, can be critical with regard to the application of Electrochemical Impedance Spectroscopy (EIS) if the lower frequency range is biased as a result of transport and/or adsorption/desorption processes. In such cases, the charge transfer resistance should be assessed from the higher frequency range which is typically inadequate in itself as an input for nonlinear parameter fitting. In earlier publications, an alternative mathematical treatment of both the Equivalent Circuit (EC) and of the parameter dispersion was provided using a generalized model of ECs and also a dispersion-invariant model of the electrochemical interface. In the present work, the previously presented experimental EIS results were crosschecked to verify the performance of the generalized model against a series of redox and corrosion systems. The results proved that the applied method is consistent and provides a fairly good correlation between the principal resistance data assessed by different methods. Keywords: Electrochemical Impedance Spectroscopy, parameter dispersion, linearized model, equivalent circuit, polarization resistance 1. Introduction Electrochemical Impedance Spectroscopy (EIS) is used to study systems in many electrochemical fields, e.g., electrode kinetics, the testing of bilayer systems, batter- ies, galvanic cells, corrosion, solid-state electrochemical processes, bioelectrochemistry, photovoltaic systems etc. In EIS, the studied system is perturbed from its equi- librium (or stationary) state by a small-amplitude sinu- soidal potential signal. For this AC potential signal, it is assumed that the transfer function of the electrochemical kinetic system can be represented by a so-called equiva- lent circuit (EC). The simplest circuit of this kind is the Voigt circuit, a parallel RC circuit, complete with a re- sistor representing the pure ohmic† solution resistance, which approximates the transfer function of the electro- chemical system well in many cases‡ (see Fig. 1A). The transfer function of Model A in Figure 1 is given by Z (ω) = RS + RCT 1 + iωτ , (1) where i denotes the imaginary unit, ω stands for the angu- lar frequency and τ = RCTCDL represents the time con- stant of the RC circuit. Eq. 1 describes an ideal theoretical *Correspondence: lukacs.zoltan@mk.uni-pannon.hu †Pure ohmic in the electrochemically relevant frequency range. ‡In this section, the effects of the transport and adsorp- tion/desorption steps are not considered. They are mentioned in suffi- cient detail in the section where the experimental results are discussed. model and the EIS curves plotted from it yield a perfect semicircle on the so-called Nyquist diagram. The value of the ohmic solution resistance (RS) is determined from the high-frequency data (∼ 10 kHz −1 kHz). If diffusion, adsorption or other processes do not influence the curve, then the sum of the solution and charge transfer resis- tances can be determined from the low-frequency data. In contrast with the theoretical expectations, in practice, the impedance spectroscopy diagrams measured have always been slightly flattened (‘depressed semicircles’). This de- pression can be modelled phenomenologically by using a power of less than 1 in the frequency-dependent term of Eq. 1: Z (ω) = RS + RCT 1 + (iωτ) 1−β , (2) where 0 < β � 1 denotes the constant phase coeffi- cient and the resulting element with the fractional power in Eq. 2 is referred to in the literature as a constant phase element. As the fractional power expression results in a constant phase impedance at high frequencies, this phe- nomenon is commonly referred to as the Constant Phase Element (CPE). Research into the CPE and related phenomena dates back many decades. Cole and Cole [1, 2] investigated the capacitance of solid and liquid dielectrics as well as inter- preted the appearance of the CPE as dependent on the fre- quency of the dielectric constant. The application of AC methods and EIS in particular to a wide range of electro- https://doi.org/10.33927/hjic-2021-01 mailto:lukacs.zoltan@mk.uni-pannon.hu 2 LUKÁCS, BACCILIERI, AND KRISTÓF Figure 1: Equivalent circuits corresponding to Models A- D and frequently applied in the relevant literature (see fur- ther comments in the text). chemical systems has become quite common in electro- chemical kinetics [3–8], electrode surface structure inves- tigations [9–14], corrosion studies [15–21], battery and fuel cell development [22–26], membrane studies [27,28] and many other fields in electrochemistry. The CPE appears in all the cited works and can be regarded with a high degree of certainty as an inherent characteristic of electrochemical systems. However, the origin and evaluation of the CPE has remained a contro- versial issue. Most works on the topics agree that the CPE can be interpreted as a consequence of the distribution of the time constant of RC circuits representing the char- acteristics with regard to the capacitance and conductiv- ity of the electrochemical interface. Cole [1] assumes a lognormal distribution of the relaxation time constants. A similar distribution function is proposed by Brug [3]. In subsequent works, the appearance of the CPE is fre- quently interpreted by means of equivalent circuits that include many RC elements in parallel or series. (Implic- itly, the distribution functions can also be recurred to such equivalent circuits.) The applicable equivalent circuits can be roughly divided into two categories. Some fall into the cate- gory of the so-called ‘3D’ (three-dimensional) dispersion models which correspond to the structures perpendicu- lar to the surface, contributing to the conventional equiv- alent circuit with a series of additional parallel RC cir- cuits [14, 15], as shown in Model B in Fig. 1. The 3D model (or its derivatives, see below) has been success- fully applied in the description of passive films and coat- ings [29–31]. The so-called 2D dispersion models [3, 14] are intended to describe the distribution of the EIS pa- rameters in two dimensions on the electrode surface. This dispersion can be attributed to the heterogeneity of the surface and the resulting deviations of the (intensive) ki- netic parameters, i.e., the time constants. Discussions concerning the 2D dispersion can lead, however, to an unexpected conclusion if this is conse- quently carried out. The 2D dispersion model, i.e., the dispersion of the time constants of the RC circuits, repre- senting the kinetic parameters of the individual active re- action sites connected in parallel to each other, as shown in Model C in Fig. 1, will degenerate to Model A. In or- der to avoid this, the effect of the solution resistance is taken into account to re-establish the ability of conven- tional EC patterns to interpret CPE behaviour (‘While an Ohmic resistance in physical systems cannot be avoided, the example illustrated in Fig. 1B illustrates the crucial role played by the Ohmic resistance in CPE behavior as- sociated with surface distributions.’ [14]). However, there are at least three very important reasons to have seri- ous reservations with regard to this concept. Firstly, Pa- jkossy has shown [10] that ‘capacitance dispersion due to irregular geometry appears at much higher frequencies than is usual in electrochemical methodologies’ which means that the irregular geometry, which acts on the EIS impedance function through the variation in the (local) solution resistance, cannot be the reason for the disper- sion in the generally applied and electrochemically rel- evant frequency range. Secondly, in highly conductive solutions and/or with high-resistance (i.e., slow) electro- chemical reactions, the contribution of the solution resis- tance to the overall impedance can be negligible com- pared to that of the charge transfer resistance, especially at lower frequencies. Thirdly, strictly speaking, the solu- tion resistance is not an inherent part of the impedance of the electrochemical impedance system. By moving the tip of the Luggin capillary closer or farther away, the value of the measured solution resistance can be varied signif- icantly, therefore, any calculations using it in the mod- elling or evaluation of the kinetic process are debatable. In conclusion, Models C and D in Fig. 1 are not ap- plicable in the interpretation of the CPE phenomenon. Consequently, by proceeding forwards on this path, it fol- lows that Model A in Fig. 1 is not and cannot be the ul- timate model (transfer function) of the electrochemical interface. The ultimate model is something more com- plex, which can, in certain cases, be simplified to Model A as far as the accuracy of measured data is concerned. In an earlier paper [32], this issue was already discussed in brief (see Fig. 3 and the relevant text in the cited paper). This contradiction was realized in the relevant lit- erature decades ago. In a noteworthy work, Agarwal et al. [33] approximated a number of ECs to Model B which obviously does not match the physical con- tent of the approximated ECs, however, the fitted curves match the experimental ones very well. These results are also confirmed from another point of view. It has been proven [34, 35] that some ECs, exhibiting quite different elements and connection patterns, have the same trans- fer function. These findings also show that the concept of the ‘equivalent circuit’ is by no means as solid nor unam- biguous as it would seem to be at first sight. After all, the question what is the minimum statement that can be both relevant and unambiguous concerning the EIS equivalent circuit and parameter determination in general is raised. A possible answer to this question was presented in two recent publications [36, 37]. It was assumed that Model A in Fig. 1 is approximately correct and devia- Hungarian Journal of Industry and Chemistry THE EXPERIMENTAL VERIFICATION OF A GENERALIZED MODEL OF EQUIVALENT CIRCUITS 3 Figure 2: Model E by applying the general scheme and Model F with the series of Maxwell circuits connected in parallel. tions from it are due to the parameter dispersions and/or mechanistic effects, moreover, all these effects can be fit- ted using a properly chosen series of EC elements. After due consideration, the model of a Voigt circuit (equivalent to Model A) and a series of Maxwell circuits connected in parallel was chosen (see Fig. 2, Model F). Model F has some important features which have practical advantages; the part of the EC corresponding to Model A, represent- ing the ‘ideal’ behaviour of the interface, is separated par- allel from the rest of the EC. This arrangement results in a linear separation in the compensated admittance: YC = YPR + iωCPR + K∑ k=1 iωCD,k 1 + iωτD,k (3) where YPR = 1/RPR and CPR denote the principal ad- mittance and capacitance, respectively, CD,k represents the capacitance and τD,k stands for the time constant of the kth Maxwell circuit connected in parallel. The phys- ical interpretation, performance and limitations of the principal parameters as well as the linearization in gen- eral are discussed in detail in previous papers [36, 37]. In Ref. [36], an equation to determine the principal admit- tance was published by our group (x = ω2): ∂ lnx ∂ lnY ′C = YPR ∂ lnx ∂Y ′C + 1 (4) The principal admittance which is, under certain condi- tions, equivalent to the reciprocal of the charge transfer or polarisation resistance, can be determined from a rel- atively narrow frequency range approximate to or higher than the critical frequency§. This is a serious advantage to any nonlinear model fitting which requires impedance data from a wider and, in particular, a lower frequency range (i.e., lower than the critical frequency) where the adsorption/desorption or transport (diffusion) processes may have a stronger impact on the measured impedance data. In a previous paper [36], Eq. 4 was tested in an §Critical frequency is understood as the frequency value where the imaginary part of the impedance has a (local) extremum at the ‘top’ of the ‘depressed semicircle.’ The accurate value of the critical frequency can be calculated via a quadratic fitting of the nearest points. In some EIS spectra, such an extremum cannot be established. experimental system (quinhydrone redox system in 10% HCl) and, in a more recent paper [37], other methods de- veloped to determine the principal admittance were tested in the quinhydrone as well as three other systems, namely Fe, Cu and COR (see their descriptions in the Experi- mental section below). In this paper, the testing of Eq. 4 against the latter three systems is presented and dis- cussed. 2. Experimental In order to gain a comprehensive overview concerning the performance of the proposed new parameter evaluation method based on Eq. 4, three test systems were created (the short names, used for identification in the paper, are in brackets): Fe3+/Fe2+ redox system (Fe): metallic iron was dis- solved in 10% m/v HCl at a concentration of 8 · 10−4 mol/dm3 and FeCl3 • 6H2O was added to set the same concentration of 8 × 10−4 mol/dm3 for FeCl3. Solutions were diluted using 10% HCl to the concentrations indi- cated in Table 1. The working, reference and counter electrodes were all composed of platinum for this system. The working electrode was a platinum plate with a surface area of 2 cm2, the reference electrode was a larger platinum sheet with a surface area of 5 cm2 and the counter electrode was a platinum net with an approximate surface area of 20 cm2. The potential of the platinum reference electrode was measured before and after the EIS measurements against a saturated Ag/AgCl electrode to check if the re- dox potential of the system was reasonably close to the expected, calculated values. Cu2+/Cu non-corrosive copper/copper ion system (Cu): the CuSO4 concentrations and the supporting elec- trolyte are included in Table 1. The working electrode was a cylindrical copper electrode 30 mm in length and with a diameter of 6 mm in a teflon holder, while the geometric surface area of the electrode was 5.94 cm2. The reference electrode was composed of copper with the same dimensions and the counter electrode was a copper sheet with a surface area of approximately 20 cm2. In the corrosive systems (COR), the working elec- trode was a DIN St 52-type cylindrical steel electrode 30 mm in length and with a diameter of 6 mm in a teflon holder, while the geometric surface area of the electrode was 5.94 cm2. The saturated Ag/AgCl electrode was used as the reference electrode and a platinum net with an ap- proximate surface area of 20 cm2 as a counter electrode. The electrolyte compositions are shown in Table 1. All the working electrodes were degreased in acetone. The platinum electrodes were kept in a 10% HCl solution for at least 3 hours before the experiment. The copper and steel electrodes were polished with #400, #600 and finally #1000 emery paper, while the copper electrodes were etched in a solution of 20% HNO3 for 5 minutes. The steel electrodes used for the corrosion experiments 49(1) pp. 1–8 (2021) 4 LUKÁCS, BACCILIERI, AND KRISTÓF Table 1: Summary of the parameters and identifiers of the systems and series. System Series Electrode Electrolyte Abbreviation Short description ID Definition Fe Fe(II)/Fe(III) redox 1E-4 Fe(II), Fe(III) concentration, M Platinum 10% HCl 2E-4 4E-4 8E-4 Cu Cu/Cu(II) reversible metal 1E-4 Cu(II) concentration, M Cu 1M H2SO4 2E-4 4E-4 8E-4 COR Corrosion system COR1 N/A St-52 steel 5% NaCl + 0.5% HAc* COR2 5% NaCl + 0.1% HCl COR3 1% HCl COR4 10% HCl *Acetic acid (system COR) were pre-treated in a 10% HCl solution for 5 minutes before being placed in the cell. All the experiments were carried out in a conventional three-electrode electrochemical cell with a volume of ap- proximately 700 cm3. The experiments were conducted at room temperature and the solutions were not deaer- ated. All EIS spectra were measured twice and both runs were evaluated. In all three systems (Fe, Cu and COR), the electrodes were prepared before the first run and only the electrolytes were replaced for the purpose of measur- ing the subsequent runs. The parameters and identifiers (used to denote the measurement systems and runs in the discussion of the results) are shown in Table 1. Consecu- tive runs using the same solution composition are denoted with a -1 or -2 suffix at the end of the IDs, e.g., 1E-4-1 and 1E-4-2 for the Fe3+ and Fe2+ concentrations in the Fe system, respectively, namely 10−4 mol/dm3 (M). The EIS spectra were measured with a Metrohm Au- tolab PGSTAT 302N-type potentiostat using NOVA 1.11 software. The amplitude of the applied potential signal was 10 mV (i.e., 20 mV p-p). All spectra were taken within the 10 kHz −100 mHz range and 20 frequency points were measured per decade over a logarithmically equidistant distribution. In all the experiments, two spec- tra were measured in order to assess the rate of impedance drifts over time. The linear fittings were generally very good which in- volved a high count of 9s in the correlation coefficient. In order to avoid the cumbersome counting of the digits, the Precision Factor PR was introduced according to PR = − lg ( 1 − R2 ) , (5) where R2 denotes the square of the correlation coefficient (e.g., if R2 = 0.999, then PR = 3). 3. Results and Discussion The Nyquist diagrams of the Fe, Cu and COR systems are shown in Figs. 3-5, respectively. The corresponding Bode plots are available in Ref. [37]. The three systems exhibit quite different characteristics with regard to how well they fit to the conventional EC parameters and non- linear least squares methods. The Fe system could be fit- ted with a Randles circuit including a CPE instead of a double layer capacitance if the diffusion-controlled low- frequency range was longer and consequently better sep- arated. This fitting procedure was tried but yielded uncertain results with high parameter errors and strong correlations in the Hessian matrix. In general, the analytic methods do not yield good results if the measurement data do not have ranges where they are mainly sensitive to one pa- rameter only. The measurements below 100 mHz last for quite a lengthy period of time and during this time range, the nonstationarity (‘time drift’) can produce a significant degree of bias with regard to the measurement data which should be avoided. The Cu system is a bit worse be- cause the low-frequency range data cannot be interpreted in terms of the conventional EC parameters at all. The COR system is, however, quite different since it can be fitted with a conventional CPE and charge trans- fer resistance. The common feature in all three systems is a relatively regular and similar arc in the frequency range higher than the critical frequency (i.e., the fre- quency where the imaginary part of the impedance has a maximum – this is somewhat unclear in the case of the Cu system but the regular behaviour exhibited in the high- frequency range is clearly visible). The similarities in and regularities of the higher-frequency range impedance data would suggest that only this range should be used to de- termine both the double layer capacitance, which is inher- Hungarian Journal of Industry and Chemistry THE EXPERIMENTAL VERIFICATION OF A GENERALIZED MODEL OF EQUIVALENT CIRCUITS 5 Figure 3: Nyquist plots of the Fe system. Figure 4: Nyquist plots of the Cu system. ently related to the high frequency range, and the charge transfer resistance, which is, in contrast, related to the lower frequency data range. This circumstance requires a more sophisticated approach, which has been outlined in Ref. [36] and is applied in this paper, for the three sys- tems discussed. Based upon the aforementioned considerations, the importance of developing methods that can determine the charge transfer resistance from the higher frequency data alone is indisputable. The testing of Eq. 4 with the dis- cussed experimental systems is shown in Figs. 6 to 8. The data points are obtained by calculating the gradient YPR according to Eq. 4 using the two adjacent points (Method (i)) and by fitting the gradients of the lines to 5 points of both greater and smaller frequencies at each frequency (Method (ii) in Ref. [36]). Therefore, the lines are formed Figure 5: Nyquist plots of the COR system. by effectively averaging out the points. The curves are strongly dependent on the frequency and, in this case, selecting the most appropriate value to characterize the charge transfer resistance of the system is always an is- sue. The optimal value of the principal admittance was selected on the basis of the linear-fitting correlation data, moreover, the data set providing the highest correlation coefficient was accepted as the ‘real’ value. The PR max- ima at very high frequencies were not taken into account (see Fig. 6). By comparing Figs. 3-5 with Figs. 6-8, it can be concluded that the more regular (depressed) semicircles are formed in the Nyquist diagram, the more established principal admittance data can be obtained from Eq. 4. However, if regular semicircles are formed by a sys- tem, then the practical usefulness of the proposed eval- uation is strongly limited because in such cases, con- ventional evaluation methods can generally be applied. This is the case with the COR system. The application of the method is necessitated in systems like that of Fe and Cu where conventional nonlinear fitting methods or any other simple methods (e.g. the graphical determination of RCT [36, 37]) cannot be used at all or with only a very limited degree of precision. In order to also test the versa- tility of the method in such systems, the principal admit- tance values determined via Method (ii) were compared to those calculated by Eq. 13 in Ref. [37]¶ for the three systems investigated in this paper and also for the quinhy- drone (QH) system published in the previous paper [36] on the subject, the results of which are presented in Fig. 9. According to the results, a fairly good correlation co- efficient is obtained which indicates that both methods, namely Method (ii) and Eq. 13, calculate the same phys- ¶The derivation leading to Eq. 13 (in Ref. [37]) is also based on Eq. 3 (in this paper) but includes specific mathematical transformations based on the elimination of the dispersion parameters from the respec- tive equations. 49(1) pp. 1–8 (2021) 6 LUKÁCS, BACCILIERI, AND KRISTÓF Figure 6: Principal admittance of the Fe system calculated by Eq. 4 using Method (i) (points) and Method (ii) (dotted lines). Solid lines indicate the PR values obtained using Method (ii). Figure 7: Principal admittance of the Cu system calculated by Eq. 4 using Method (i) (points) and Method (ii) (dotted lines). Solid lines indicate the PR values obtained using Method(ii). Figure 8: Principal admittance of the COR system calcu- lated by Eq. 4 using Method (i) (points) and Method (ii) (dotted lines). Solid lines indicate the PR values obtained using Method(ii). Figure 9: Correlation between the logarithms of the prin- cipal admittance obtained by Method(ii) using Eqs. 4 and 13 in Ref. [37]. The linear fit was calculated from the log- arithms of the principal admittance data. The QH (quinhy- drone) data were taken from Ref. [36]. ical quantity. The data in Fig. 9 prove that both methods are applica- ble for the determination of the principal admittance. The question that is typically raised in similar situations is which method is better. In our opinion, it is still too early to make such a judgement given that in Ref. [37] sev- eral additional equations were also published to calculate the principal admittance. Some of these equations differ in terms of their initial physical considerations (compare Eq. 4 in this paper and Eq. 13) in Ref. [37]], while others differ only as far as the weighting of data points is con- cerned (compare Eq. 8 with 8a or Eq. 13 with Eq. 14 in Ref. [37]). It is recommended that these methods should be ap- plied especially in those cases when the impedance data in the low-frequency range (typically below the critical frequency) are not interpretable or cannot be fitted for some reason (e.g., the applicable EC cannot be assessed unambiguously). In such cases, by applying two or more of the proposed equations, the uncertainty in this deter- mination can be decreased by only utilizing parameters, which are sufficiently close to one another. 4. Conclusions A recent method developed to determine the principal ad- mittance was tested on electrochemical redox and cor- rosion systems in order to check and compare the per- formances of the methods. The characteristics of the impedance with regard to such electrochemical systems exhibited features which often make conventional non- linear fitting methods unusable or at least inaccurate. The new method was applied to determine the principal ad- mittance from the impedance data obtained in the higher frequency range which exhibit a more regular shape but Hungarian Journal of Industry and Chemistry THE EXPERIMENTAL VERIFICATION OF A GENERALIZED MODEL OF EQUIVALENT CIRCUITS 7 are not applicable to conventional fitting methods alone. By comparing this new method with another also newly developed method (Eq. 13 in Ref. [37]), a strong corre- lation between the parameters measured using the two methods was observed, which proves that both of them are suitable for determining the principal admittance of systems which exhibit similarly regular behaviour within the higher frequency range. Acknowledgement Present article was published in the frame of the project GINOP-2.3.2-15-2016-0053 (‘Development of engine fuels with high hydrogen content in their molecular struc- tures (contribution to sustainable mobility)’). 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