A systematic approach toward error structure identification for impedance spectroscopy

Transcription

1 Journal of Electroanalytical Chemistry Journal of Electroanalytical Chemistry 572 (2004) A systematic approach toward error structure identification for impedance spectroscopy Mark E. Orazem * Department of Chemical Engineering, University of Florida, P.O. Box , Gainesville, FL , USA Received 24 April 2003; received in revised form 15 September 2003; accepted 20 November 2003 Available online 26 February 2004 Abstract The state-of-the-art is reviewed for the use of measurement models for assessing the stochastic and bias error structure of impedance measurements. The methods are illustrated for published impedance data that contain both capacitive and inductive components. This systematic error analysis demonstrates that, in spite of differences between sequential impedance scans and the appearance of inductive and incomplete capacitive loops, the individual data sets represented a pseudo-stationary system and could be interpreted in terms of a stationary model. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Measurement model; Regression; Frequency-response analysis; Impedance; Admittance; Immittance 1. Introduction Much of our collaboration with the CNRS group headed by Michel Keddam was dedicated toward understanding the application of measurement models for assessing the error structure of impedance measurements. In collaboration with members of the CNRS group, these concepts were refined and applied to a variety of transfer-function measurements. The measurement model was shown to be directly applicable to electrohydrodynamic impedance spectroscopy (EHD), in which the speed of rotation is modulated [1]. In particular, it was shown that the use of modulus weighting previously employed for such measurements was inappropriate, because, to a first approximation, the standard deviation of the spectra was roughly independent of frequency. The use of an experimentally determined error structure to weight the regression was shown to increase the amount and quality of information that could be extracted from impedance spectra. * Tel.: ; fax: address: meo@che.ufl.edu (M.E. Orazem). The statistical result that the variances of real and imaginary parts of the impedance were equal was investigated and verified for systems containing a large electrolyte resistance [2]. The error analysis approach was used to facilitate development of interpretation models for PANI free-standing polyaniline membranes [3] and to establish a model for interpreting impedance data for corrosion of cast iron in drinking water [4]. The error analysis was combined with development of mathematical models [5] to demonstrate the influence of current distribution and poisoning on the impedance response associated with the reduction of ferricyanide on a platinum rotating disk electrode [6]. The error analysis facilitated discrimination among mathematical models for convective diffusion to a disk electrode under a submerged impinging jet for corrosion of steel in brines containing CO 2 [7]. The objective of this paper is to provide a cohesive summary of the progress made in the application of measurement models for assessing errors in impedance measurements. The approach is illustrated by application to published data provided by de Melo (see [8]) which reveals a significant inductive component. The following section provides a review of the measurement model approach /$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi: /j.jelechem

2 318 M.E. Orazem / Journal of Electroanalytical Chemistry 572 (2004) The measurement model concept The measurement model was introduced as a means to resolve the following recurring issues in regression of impedance data [9 11]: 1. Identification of the most appropriate weighting strategy for regression. 2. Assessment of the noise level in the measurement. 3. Identification of the frequency range that was unaffected by instrumental artifacts of non-stationary behavior. The relevance of the measurement model to these issues is presented in this section Definition of errors The errors in an impedance measurement can be expressed in terms of the difference between the observed value ZðxÞ and a model value bzðxþ as e res ðxþ ¼ZðxÞ bzðxþ ¼ e fit ðxþþe bias ðxþþe stoch ðxþ; ð1þ where e res represents the residual error, e fit ðxþ is the systematic error that can be attributed to inadequacies of the model, e bias ðxþ represents the systematic experimental bias error that cannot be attributed to model inadequacies, and e stoch ðxþ is the stochastic error with expectation Efe stoch ðxþg ¼ 0. A distinction is drawn, following Agarwal et al. [9 11], between stochastic errors that are randomly distributed about a mean value of zero, errors caused by the lack of fit of a model, and experimental bias errors that are propagated through the model. The experimental bias errors, assumed to be those that cause lack of consistency with the Kramers Kronig relations [12 14], may be caused by non-stationarity or by instrumental artifacts. The problem of interpretation of impedance data is therefore defined as consisting of two parts: one of identification of experimental errors, which includes assessment of consistency with the Kramers Kronig relations, and one of fitting, which entails model identification, selection of weighting strategies, and examination of residual errors. The error analysis provides information that can be incorporated into regression of process models Complex nonlinear regression Macdonald [15] provides a perspective on historical trends in the application of regression techniques to impedance spectroscopy. The regression of models to impedance data generally employs a complex nonlinear application of the method of least squares [16 18]. Complex nonlinear least-squares (CNLS) regression techniques were developed in the late 1960s as an extension of nonlinear least-squares (NLS) regression techniques [19,20]. The use of CNLS is consistent with the expectation that real and imaginary components of impedance data satisfy the constraints of the Kramers Kronig relations [12 14]. The CNLS approach provides an improvement over NLS techniques because a combined model parameter set is estimated by simultaneous regression of the model to both real and imaginary parts of the measured spectrum. Weighted CNLS was first applied to impedance data by Macdonald et al. [21,22]. The concept of weighting is critical for impedance spectroscopy because the impedance is a strong function of frequency [23,24]. The regression of a complex function bz to complex data Z can be expressed in a least-squares sense as the minimization of TV S ¼ Z bzðxjpþ 1 Z bzðxjpþ ; ð2þ where V is a symmetric positive-definite variancecovariance matrix of the experimental stochastic errors, Z represents the complex impedance data measured at frequencies x, and bz ðxjp Þ represents the complex model calculated for frequency x as a function of a parameter vector P [25,26]. Under the assumption that the covariance terms in V can be neglected, i.e., that V is a diagonal matrix, and under the additional assumption that residual errors are uncorrelated, Eq. (2) can be replaced by S ¼ XN Z r ðx k Þ bz r ðx k jpþ Z 6 j ðx k Þ bz j ðx k jpþ 7 4 þ 5; k¼1 V r;k V j;k ð3þ where V r;k and V j;k represent the real and imaginary components, respectively, of the variance of the stochastic errors, Z r ðx k Þ and Z j ðx k Þ represent the real and imaginary parts of the impedance data measured at frequency x k,andbz r ðx k jpþ and bz j ðx k jpþ represent the real and imaginary parts of the model value calculated for frequency x k as a function of a parameter vector P. The statement that the covariance terms in V can be neglected implies both that stochastic errors at one frequency are uncorrelated with errors at another frequency and that errors in real and imaginary parts of the impedance at a given frequency are not correlated. Use of Eq. (3) is therefore predicated on the assumption that errors in real and imaginary parts of the impedance are not correlated. If Eq. (3) is used under conditions where the error covariance terms cannot be neglected, the incorrect error structure will be reflected in the parameter estimates. Carson et al. [27] used numerical simulations to show that, when a frequency-response analyzer algorithm is used to obtain the impedance response from time-domain signals containing normally distributed

3 M.E. Orazem / Journal of Electroanalytical Chemistry 572 (2004) errors, the real and imaginary parts of the impedance were uncorrelated. In contrast, use of a phase-sensitivedetection algorithm yielded correlation between real and imaginary components of the impedance [28]. Assessment of the error structure, therefore, plays an integral role in parameter estimation for impedance models The search for appropriate weighting for regression Three approaches have been documented in the literature for incorporating the error structure of impedance data into interpretation strategies. One approach has been to assume a standard form for the stochastic errors. Two models are commonly used. Zoltowski [29] and Boukamp [30,31] advocated use of modulus weighting under the assumption that the standard deviation is proportional to the frequency-dependent modulus jzðxþj of the impedance. Macdonald et al. [23,32,33] advocated use of a modified proportional weighting strategy. The rationale behind the use of a modified proportional weighting strategy is summarized succinctly by Macdonald [15]. The drawback in using an assumed error structure is that it may not provide the correct weighting matrix, and the use of an incorrect error structure will influence the parameter estimates. A clear example of the influence weighting strategies has on the parameter estimation is provided by Orazem et al. [24] who show that the number of relaxation processes that can be identified is smaller when an assumed error structure is employed as opposed to one that is experimentally measured. The parameter estimates found using the experimentally determined error structure were consistent with independent measurements [34]. A second approach has been to use the method of maximum likelihood [35], in which the regression procedure is used to obtain a joint estimate for the parameter vector P and the error structure of the data [23,33]. The maximum likelihood method is recommended under conditions where the error structure is unknown [35], but the error structure obtained by simultaneous regression is severely constrained by the assumed form of the error-variance model. In addition, the assumption that the error variance model can be obtained by minimizing the objective function ignores the differences among the contributions to the residual errors shown in Eq. (1). The third approach is to use experimental methods to assess the error structure. Independent identification of error structure is the preferred approach, but even minor non-stationarity between repeated measurements introduces a significant bias error in the estimation of the stochastic variance. Dygas and Breiter [36] report on the use of intermediate results from a frequency-response analyzer to estimate the variance of real and imaginary components of the impedance. Their approach allows assessment of the variance of the stochastic component without the need for replicate experiments. The drawback is that their approach cannot be used to assess bias errors and is specific to a particular commercial impedance instrumentation. Measurement models, developed for impedance spectroscopy by Agarwal et al. [9 11], are generally applicable and can be used to estimate both stochastic and bias errors of a measurement from imperfectly replicated impedance measurements. The drawback is that replicated measurements are required. The objective of this work is to report the progress made in the understanding of measurement models and their implementation Measurement models The measurement model method for distinguishing between bias and stochastic errors is based on using a generalized model as a filter for non-replicacy of impedance data. The measurement model is composed of a superposition of line-shapes which can be chosen arbitrarily, subject to the constraint that the model satisfies the Kramers Kronig relations. The model composed of Voigt elements in series with a solution resistance shown in Fig. 1, i.e., Z ¼ R 0 þ XK R k ð4þ 1 þ jxs k¼1 k has been shown to be a useful measurement model. With a sufficient number of parameters, the Voigt model was able to provide a statistically significant fit to a broad variety of impedance spectra [9]. Other transfer-function models can be considered. For example, Eq. (4) can be generalized to allow constant-phase-element behavior, i.e., Z ¼ R 0 þ XK R k k¼1 1 þðjxþ 1 a k : ð5þ s k Correlation among parameters may create difficulties when using the CPE-modified model. Pauwels et al. [37] have proposed that a transferfunction formulation P K k¼0 Z ¼ b kððjxþ n Þ k P M m¼0 b mððjxþ n Þ m ; ð6þ Fig. 1. A schematic representation of a Voigt circuit used by Agarwal et al. [9 11] as a measurement model.

4 320 M.E. Orazem / Journal of Electroanalytical Chemistry 572 (2004) may be a more parsimonious model for certain classes of impedance measurements. The concepts outlined below apply for any measurement model which satisfies the Kramers Kronig relations. Pauwels et al. [37] noted that, if n ¼ 1, Eq. (6) shows a frequency behavior similar to that of a Voigt model (Eq. (4)). The special feature of the transfer-function formulation was that, for n ¼ 1=2, Eq. (6) yields a frequency-dependence consistent with diffusion impedance. A recent comparison between the Voigt and transfer-function formulations revealed that the error structure obtained using the two models were identical and that, while the transfer-function formulation required fewer parameters to obtain a good fit to the data, the confidence intervals for the parameter estimates were larger [38]. While the line-shape parameters may not be unequivocally associated with a set of deterministic or theoretical parameters for a given system, the measurement model approach has been shown to represent adequately the impedance spectra obtained for a large variety of electrochemical systems [9]. The line-shape models represent the low-frequency stationary components of the impedance spectra (in a Fourier sense). Regardless of their interpretation, the measurement model representation can be used to filter and thus identify the non-stationary (drift) and high frequency (noise) components contained in an impedance spectrum. 3. Method for implementation The use of measurement models for analysis of errors requires replicate measurement of the impedance response using the same measurement frequencies for each replicate. The discussion here is centered on a series of impedance measurements reported by Aoki et al. [8] for a pure aluminum disk rotating at 120 rpm in a 0.75% citric acid solution. Three sequential impedance spectra obtained at the open-circuit potential are presented in Fig. 2. The impedance measurements were carried out under potentiostatic modulation with a 1255 Solartron frequency-response analyzer connected to a 1287 Solartron electrochemical interface. The frequency range for the measurement was 10 khz to 0.1 mhz with five frequency points per decade. Owing to the very low frequency limit, each scan took roughly 12 h to complete, and the interval between scans was 24 h. The sequence of measurements indicates that there was a substantial change from one measurement to another. This lack of reproducibility, in itself, raises the question of whether each individual measurement was corrupted by non-stationary phenomena. In addition, the combination of inductive and capacitive loops could be associated with failure to be consistent with the -Z j / kω Hz Data Set #1 Data Set #2 Data Set # mhz Z r / kω Kramers Kronig relations. Separate experiments were used to demonstrate that the impedance measurements were taken after the system had reached a steady state [8]. The objective of this work is to illustrate the manner in which a measurement-model-based analysis of errors could be used to assess whether the measurements were in fact stationary Time scales for impedance measurement 0.01 Hz Fig. 2. Sequential impedance diagrams obtained by Aoki et al. [8] for a pure aluminum disk rotating at 120 rpm in a 0.75% citric acid solution at the open-circuit potential. As described by Agarwal et al. [10], there exist three time scales over which an electrochemical system can change during the course of consecutive impedance experiments. The smallest time scale is the time to collect a data point at one frequency. The time required to collect one complete impedance scan constitutes the next time scale, and the third time scale is the total time elapsed from the start of the first experiment to the end of the last experiment. It is evident that the system described by Fig. 2 was changing on the time scale of a complete set of replicated scans. Indeed, the lack of replication from one scan to another complicates the direct assessment of the stochastic errors. The standard deviations of real and imaginary parts of the impedance spectra presented in Fig. 2 obtained by direct calculation are presented in Fig. 3. The direct estimation of standard deviation includes contributions from the non-stationary behavior between scans as well as contributions from stochastic errors. The additional question, addressed in a subsequent section, is whether, in spite of the observation that non-negligible differences are seen between successive spectra, the system was evolving sufficiently slowly that the change during one complete scan was insignificant. This answer must be obtained in the context of the assessment of stochastic errors in the measurement.

5 M.E. Orazem / Journal of Electroanalytical Chemistry 572 (2004) σ / Ω real part imaginary part Fig. 3. Standard deviations for real and imaginary parts of the impedance obtained by direct calculation from the sequential impedance diagrams presented in Fig Evaluation of stochastic errors To eliminate the contribution of the drift from scan to scan, a measurement model was regressed to each scan using the maximum number of parameters that could be resolved from the data. Modulus weighting was employed. As shown in Table 1, seven Voigt elements could be identified. The fit of the model to the data is presented as dashed lines in Fig. 2. The inductive loop evident in Fig. 2 was modeled by element #6, which has a negative resistance value and a positive time constant. The parameter values listed in Table 1 were sorted in increasing values of time constant s k to facilitate comparison of system parameters across the rows. The parameters for the measurement model for each data set differ slightly because the system changes from one experiment to the other. Hence, by regressing the measurement model to individual data sets separately, the effects of the change of the experimental conditions from one experiment to another were incorporated into the measurement model parameters. Significant changes were seen in parameter values for element #7, which accounts for the unfinished capacitive loop at the lowest frequencies measured. The distribution of time constants and corresponding resistance values is presented in Fig. 4. The error bars correspond to 1r, calculated using a linear approximation. The error bars for the time constants s k are drawn, but were too small to be seen on the scale of Fig. 4. A discussion of the Voigtelement distribution approach to visualizing impedance data is presented by Orazem et al. [39]. The change from scan to scan is reflected in minor shifts in characteristic time constant and resistance for each Voigt element. The relative residual errors for the three data sets are presented in Fig. 5. Lines were included to facilitate identification of the trend in the residual errors. Note that the appearance of very large relative residual errors in Fig. 5(b) for the imaginary part of the impedance is an artifact of the scaling used to obtain a relative residual error. The relative residual error is equal to infinity when the imaginary impedance is equal to zero. The relative residual error is very large in the frequency range ( mhz) where the imaginary part of the impedance crosses the abscissa twice. The scale of the ordinate in Fig. 5(b) was therefore chosen to emphasize the residual errors in the higher frequency region. As indicated by Eq. (1), the residual errors include contributions from lack of fit, bias errors associated with instrument artifacts or non-stationary behavior, and stochastic events. Examination of Fig. 5 confirms that the residual errors themselves cannot be interpreted to be an estimate for the stochastic error because, as indicated by the trend in the residual errors, the error associated with lack of fit is significant. Table 1 Model parameters for the fit of a Voigt measurement model to the impedance data presented in Fig. 2 Variable Data set #1 Data set #2 Data set #3 Value r Value r Value r R s R s R s R s R s R s R 6 ) ) ) s R

6 322 M.E. Orazem / Journal of Electroanalytical Chemistry 572 (2004) R k / Ω Data Set #1 Data Set #2 Data Set #3 negative value for R k 10-5 τ k / s Fig. 4. Distribution of Debye (or Voigt) relaxation time constants for the regression summarized in Table 1. The error bars correspond to 1r, calculated using a linear approximation. Fig. 5. Residual errors for the fit of a Voigt measurement model to the impedance data presented in Fig. 2. The model parameters for the three spectra are given in Table 1: (a) real part; (b) imaginary part. Following Agarwal et al. [10], the variance of real and imaginary residual errors can be obtained as a function of frequency, i.e., r 2 Z r ðxþ ¼ 1 X N ðe res;zr;kðxþ e res;zr ðxþþ 2 ; ð7þ N 1 k¼1 where N is the number of replicates, e res;zr;k represents the residual error at frequency x for scan k, obtained from its unique model, and e res;zr represents the mean value for the residual errors at frequency x. A similar expression is used for the imaginary part of the impedance. Eq. (7) can provide a good estimate for the variance of stochastic errors under the following set of assumptions: 1. The model parameters account for the drift from one scan to another. 2. The frequency-dependent systematic errors associated with the lack of fit are unchanged from one scan to another. This assumption is justified only if the same number of statistically significant parameters are used for each scan and if the same features are evident in the successive impedance scans. 3. The systematic errors associated with instrument artifacts are unchanged from one scan to another. This assumption is justified under the conditions that Assumption (2) is justified. 4. The systematic error associated with non-stationary behavior is unchanged from one scan to another. Assumption (4) represents the most serious restriction to the measurement model approach for estimation of error structure. It can be anticipated that the influence of non-stationarity may be largest for the first of a sequence of impedance measurements. Under conditions that the data sets do not satisfy the Kramers Kronig relations, the measurement model approach may overestimate the standard deviation of stochastic errors. The incorrect error structure may bias the regression used to check consistency with the Kramers Kronig relations. This difficulty can be addressed in an iterative approach in which the data identified as being inconsistent with the Kramers Kronig relations are removed from the data set used to obtain the stochastic error structure estimate. In any case, the estimate obtained using the measurement model approach will be more accurate than would be obtained by direct calculation of the standard deviation. Thus, the variance of the real and imaginary residual errors provides a good estimate for the frequency-dependent variance of the stochastic noise in the measurement. Under the assumptions detailed above, the standard deviation of the stochastic errors can be identified from the standard deviation of the residual errors. The resulting standard deviations for the real and imaginary parts of the impedance are presented in Fig. 6(a). The standard deviation is a function of frequency and, in keeping with previous results, statistical inference tests indicated that the standard deviation for real and imaginary parts of the impedance were indistinguishable. The model for the standard deviation followed the form proposed by Orazem et al. [2]

7 M.E. Orazem / Journal of Electroanalytical Chemistry 572 (2004) σ / Ω (a) 100 σ / Z (b) Fig. 6. Standard deviations for the data presented in Fig. 2, obtained from the residual errors presented in Fig. 5. (a) Standard deviation for real and imaginary M parts of the impedance. The line represents the model given as Eq. (8). (b) Standard deviations normalized to the magnitude of the impedance. Shaded symbols indicate the standard deviations calculated directly from the impedance data shown in Fig. 2. r Zr ¼ r Zj ¼ az j þ b j Zr jþ c jzj2 þ d; ð8þ R m where R m is the current measuring resistor used for the experiment and a, b, c, and d are constants to be determined. For the experiments presented here, R m ¼ 10 4 X, a ¼ 0:00458, b ¼ 0, c ¼ 5: , and d ¼ 0. The standard deviations were normalized to the magnitude of the impedance as shown in Fig. 6(b). Shaded symbols indicate the values for standard deviations calculated directly from the impedance data shown in Fig. 2. The standard deviation of the stochastic part of the measurement range from 2% of the modulus at low frequency to 0.1% of the modulus at higher frequencies. The standard deviation is not proportional to the modulus of the impedance, but, for this case, modulus weighting provides a first-order estimate for the error structure. The standard deviations calculated directly from the impedance data are much larger at low and high frequencies, indicating that the non-stationary character of the measurements are caused by processes having large and small time constants, respectively. There are several cautions for the use of measurement models to filter impedance data for determination of the stochastic error structure: 1. The CNLS regression is sensitive to the initial parameter values. It is best to ensure that the initial value for the time constant of a Voigt element corresponds to the frequency of a feature to be modeled. This is especially important when inductive loops are to be modeled. 2. Eq. (7) requires that the replicated measurements be made at the same frequencies. 3. While it is not necessary that the spectra be identical, the replicated measurements should show roughly the same features. 4. Error analysis methods using measurement models are sensitive to data outliers. Often, outliers can be attributed to external influences. Data collected within 5 Hz of the line frequency and its first harmonic (e.g.,50 and 100 Hz in Europe or 60 and 120 Hz in the United States) should be deleted. Start-up transients cause some systems to exhibit a detectable artifact at the first frequency measured. This point, too, should be deleted. Dygas and Breiter [36] described methods that could be used to identify outliers at arbitrary frequencies but, in our work, it has been sufficient to avoid or delete the first frequency measured and measurements at the line frequencies Evaluation of bias errors Since the Voigt measurement model is itself consistent with the Kramers Kronig relations, the ability to fit this model to within the noise level of the measurement should indicate that the data are consistent. A refined approach was developed to resolve the ambiguity that exists when the model does not provide a good fit to the data. The lack of fit of the model could be due to causes other than inconsistency with the Kramers Kronig relations. For example, the number of frequencies measured might be insufficient to allow regression with a sufficient number of Voigt parameters, the noise level of the measurement might be too large to allow regression with a sufficient number of Voigt parameters, or the initial guesses for the nonlinear regression could be poorly chosen. Thus, while in principle a complex fit of the measurement model could be used to assess the consistency of impedance data, sequential regression to either the real or the imaginary provides greater sensitivity to lack of consistency. The optimal approach is to fit the model to the component that contains the greatest amount of information. Our work suggests that the imaginary part of the impedance is much more sensitive to contributions of minor line-shapes than is the real part of the

8 324 M.E. Orazem / Journal of Electroanalytical Chemistry 572 (2004) impedance. Typically, more Voigt line-shapes can be resolved when fitting to the imaginary part of the impedance than can be resolved when fitting to the real part. The solution resistance cannot be obtained by fitting the measurement model to the imaginary part of the impedance. The solution resistance is treated as an arbitrarily adjustable parameter when fitting to the imaginary part of the impedance. The application of measurement models to assess consistency with the Kramers Kronig relations is demonstrated for the first scan shown in Fig. 2. A measurement model was fit to the imaginary part of the spectrum using the experimentally determined error structure to weight the regression. The number of Voigt elements was increased until the maximum number of statistically significant parameters was obtained. The result is presented in Fig. 7(a). The parameter set obtained by the regression to the imaginary part of the impedance was used to predict the real part of the impedance, as shown in Fig. 7(b). The prediction tracks the Z j / kω data very well except at the lowest frequencies measured. To determine whether the discrepancy seen at the lowest frequencies is significant, a Monte Carlo simulation was used to identify the frequency-dependent confidence interval for the model prediction. The results are presented as dashed lines in Figs. 7(a) and (b). The data fall within the confidence interval for the model. The broad model confidence interval evident at low frequency reflects the uncertainty in the parameter estimate for the incomplete capacitive loop. Plots of relative residual errors provide a much more sensitive perspective on the Kramers Kronig consistency check. The residual errors for the fit shown in Fig. 7 are presented in Fig. 8(a) for the imaginary part of the impedance. The dotted lines in Fig. 8(a) provide the (Z j,dat -Z j,mod ) / Z j,mod (a) (a) Z r / kω (b) Fig. 7. Results for the fit of a Voigt measurement model to the imaginary part of the first impedance spectrum presented in Fig. 2. Dashed lines represent the 95:4% confidence interval for the model obtained by Monte Carlo simulation using the calculated confidence intervals for the estimated parameters. (a) fit to the imaginary part; (b) prediction of the real part. (Z r,dat -Z r,mod ) / Z r,mod (b) Fig. 8. Relative residual errors for the fit of a Voigt measurement model to the imaginary part of the first impedance spectrum presented in Fig. 2. Dashed lines represent the 95:4% confidence interval for the model obtained by Monte Carlo simulation using the calculated confidence intervals for the estimated parameters. Dotted lines represent the 2r bound for the stochastic error structure determined in the previous section. (a) fit to the imaginary part: represents the relative residual error for the regression. (b) prediction of the real part: represents the relative error for the prediction of the real part of the impedance for an assumed solution resistance of 74:3 X. represents the relative error for the prediction of the real part of the impedance for an assumed solution resistance of 78:3 X.

9 M.E. Orazem / Journal of Electroanalytical Chemistry 572 (2004) % bound for the stochastic errors. Most of the residual errors fall within the 2r noise level of the measurement. The weighted v 2 =m statistic for the regression, ideally equal to unity, had a value of 3.5. The value of the weighted v 2 =m statistic was larger than usually obtained because the lowest-frequency loop was so far from being completed. This larger value of the v 2 =m statistic cannot be attributed to lack of consistency between real and imaginary parts of the measurement, caused, for example, by non-stationary behavior, because the fit was made to only the imaginary component. The influence of incomplete spectra on regression statistics has been explored by Orazem et al. [40]. Examination of the residual errors for the real part of the impedance in Fig. 8(b) confirms the assessment that the discrepancy between data and model prediction at low frequencies was insignificant when compared to the confidence interval for the prediction. The dashed lines in Fig. 8(b) provide the 95.4% bound obtained by Monte Carlo simulation for the model prediction. The dotted lines in Fig. 8(b) provide the 95.4% bound for the stochastic errors. A discrepancy seen at high frequencies can be corrected by adjusting the assumed solution resistance. The circles indicate the residual errors for the prediction when the solution resistance was assumed to be equal to 74:3 X. An adjustment by 4 X, indicated by, was sufficient to place the high-frequency residual errors within the confidence interval without adversely affecting the agreement at intermediate and lower frequencies. The resulting solution resistance was in good agreement with the value of 78:5 2:9 X reported in Table 1 for the initial regression results. Thus, within the confidence of the measurement model technique, the data presented in Fig. 2 cannot be determined to be inconsistent with the Kramers Kronig relations. If some of the real impedance values at low frequencies were to have fallen outside the predicted confidence interval, such data would be removed from the spectrum used for subsequent regression studies using stationary models. There are several cautions for the use of measurement models to asses the consistency of impedance data with the Kramers Kronig relations: 1. The number of line-shapes that can be determined in a complex fit should increase when Kramers Kroniginconsistent data are removed. Deletion of data that are strongly influenced by bias errors increases the amount of information that can be extracted from the data. In other words, the bias in the complete data set induces correlation in the model parameters which reduces the number of parameters that can be identified. Removal of the biased data results in a better conditioned data set that enables reliable identification of a larger set of parameters. 2. In general, inconsistencies at low frequencies may be attributed to non-stationary behavior, and inconsistencies at high frequencies may be attributed to instrument artifacts. It should be noted that some instrument artifacts may be evident over a broad frequency range and that some artifacts may result in impedance features that are consistent with the Kramers Kronig relations. Thus, the inability to identify inconsistent data does not prove that the measured spectrum reflects only the properties of the electrochemical system under study. 3. The measurement model procedure described here provides a statistical basis by which one can decide whether data satisfy the Kramers Kronig relations. The method may not be sufficiently sensitive to identify inconsistencies. The sensitivity of the method can be improved by decreasing the noise level of the measurement and by increasing the number and range of frequencies sampled. 4. In our experience, the procedure described in the present work is not effective for removing data in the middle of a spectrum. Inconsistencies in the middle of a frequency range can arise due to switching of current measurement circuitry and/or switching from one impedance measurement technique (i.e., multisine Fourier transform to phase sensitive detection). As discussed in the previous section, regression of measurement models is sensitive to such discontinuities. Dygas and Breiter [36,41] have described methods that account for discontinuities caused by switching of current measurement circuitry. 5. The measurement model method for assessing the consistency of impedance data with the Kramers Kronig relations cannot be applied to impedance data collected by Fourier analysis of multi-sine signals because the spectra so obtained inherently satisfy the Kramers Kronig relations [42]. A correlation coefficient must be used to determine whether an impedance spectrum collected using multi-sine modulation was influenced by instrumental or non-stationary artifacts [15]. 4. Conclusions The analysis of errors presented here confirms that the impedance data of Aoki et al. [8] satisfied the Kramers Kronig relations. This result supports their assertion that, in spite of the differences between sequential impedance scans and the appearance of inductive and incomplete capacitive loops, the individual data sets represented a pseudo-stationary system and could be interpreted in terms of a stationary model. The measurement model approach for evaluating consistency with the Kramers Kronig relations is superior to direct integration, as described, for example, by Macdonald and co-workers [43 46], because it provides a rational approach to extrapolation of the data set into the unmeasured frequency regimes.

10 326 M.E. Orazem / Journal of Electroanalytical Chemistry 572 (2004) Experimental data can, therefore, be checked for consistency with the Kramers Kronig relations without actually integrating the equations over frequency, avoiding the concomitant quadrature errors. The measurement model approach is superior to direct use of Voigt models to assess consistency, as suggested, for example, by Boukamp and Macdonald [47] and Boukamp [48], because the weighting used is based on an experimentally measured rather than an assumed error structure. While the Kramers Kronig relations are useful for validation of impedance data, it should be noted that these relations provide a necessary-only, not a necessary-and-sufficient, condition. Failure to satisfy the Kramers Kronig relations provides unequivocal evidence of non-causal, nonlinear, or non-stationary influences, yet satisfaction of the Kramers Kronig relations does not prove that the data are representative of a specific electrochemical cell. For example, some instrument artifacts may appear as Kramers Kronigconsistent features in impedance spectra. The concept that underlies the measurement model approach for error analysis is not limited to the Voigt model presented here. Any generalized model that satisfies the Kramers Kronig relations can be used. The choice between models is driven by convenience, parsimony, and flexibility. Acknowledgements The experimental data were provided by H.G. de Melo. This work benefited from collaboration with many members of the Laboratory Physique des Liquides et Electrochimie, UPR 15 du CNRS, Universite Pierre et Marie Curie, including C. Deslouis, T. El. Moustafid, I. Frateur, H. Takenouti, and B. Tribollet. 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