Measurement Models for Electrochemical Impedance Spectroscopy ...

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4152 J. Electrochem. Soc., Vol. 142, No. 12, December 1995 9 The Electrochemical Society, Inc.20 ' ' ' ' I ' ' ' ' I ' ' ' ' 1 ' ' '0.2 '"'"'~ '"'"'~ '"'"'q '"'"', '"'""1 '"""1157"-1o50 5 10 15 20Z~, kflFig. 3. Six successive impedance measurements for a copper diskrotating at 1000 rpm in an alkaline 1 M chloride solution. The datawere collected after 3 days of exposure.!oI. ot-0.2 , ...... 1 ...... .I ....... l ....... = i,..,J ...... J10 -1 10 0 101 10 2 10 3 10 4 10 5Frequency, Hzstrong function of frequency, and the real and imaginaryparts are indistinguishable at 320 K and almost indistinguishableat 400 K. The extent of the correlation of the realand imaginary standard deviations at 400 K is comparableto that reported by Zoltowski. 17 The correlation of real andimaginary noise and the observation of heteroskedasticitywith respect to frequency are important results, and theirsignificance is discussed in later sections.Nonstationary systems.--For most electrochemical systems,the nonstationary contribution to the bias error ens isnot equal to zero. Since the system evolves with time, enschanges from one experiment to another as a set of replicate(or consecutive) experiments are performed. Hence,Eq. 8 leads to inaccurate estimates for r because the contributionof end to the standard deviation cannot be ignored.One can identify three time scales over which an electrochemicalsystem can change during the course of consecutiveimpedance experiments in which data are collected atseveral frequencies for each scan. The smallest time scaleconsidered here is the time to collect a datum point at onefrequency. The time required to collect one complete impedancescan constitutes the next time scale, and the thirdtime scale is the total time elapsed from the start of the firstexperiment to the end of the last experiment.If the system is evolving rapidly, changes can occur duringthe time in which one datum point is collected.Impedance spectroscopy may not be a feasible experimentaltechnique for such systems. For systems showing aslower rate of change, the impedance at each frequencymay be measurable, but significant change can occur betweenthe start and end of a complete frequency scan.These types of nonstationarities result in the data beinginconsistent with the Kramers-Kronig relations. The issuesarising from these inconsistencies are discussed in the nextpaper in this series. 9The third time scale considered here appears when ssystem under investigation is evolving slowly. In this casethe change in the system during one complete scan is smalland can be ignored, but nonnegligible differences can beseen between successive spectra. Such pseudo-stationaryimpedance scans are typically observed for even the moststationary electrochemical systems. The need for a methodfor analysis of such pseudo-stationary systems is illustratedbelow.Six replicate (consecutive) impedance spectra obtainedfor a copper disk rotating at 1000 rpm in an alkaline (pH11.5) 1 M chloride aqueous solution are shown in Fig. 3.3The electrodes were held at the open-circuit condition. Theelectrical contact to the disk electrode (carbon brush andFig. 4. Real residual errors }or the regression o} a single-measurementmodel to the data shown in Fig. 3.stainless steel shaft) was cleaned with alcohol before andafter each impedance scan without interrupting the rotationof the disk. The long (1% closure error) autointegrationcycle of the frequency-response analyzer was used.Each spectrum shown in the figure was found by themethod presented in Ref. 9 to be consistent with theKramers-Kronig relations, but a careful analysis of thedata using the measurement-model approach showed thatthe data were not replicate, i.e., the system changed measurablyfrom one experiment to another2 To demonstratethat the impedance scans were not replicate, a measurementmodel with eight Voigt elements was regressed (usinga complex nonlinear least squares regression algorithm 7'8under modulus weighting) to the combined data set (i.e., asingle measurement model was regressed to all six data setssimultaneously). The normalized residual errors obtainedfrom the regression are shown in Fig. 4 and 5. The sinusoidalcharacter of the residual errors is caused by_ 0.0IL-0.2 510Frequency, HzFig. 5. Imaginary residual errors for the regression of a singlemeasurementmodel to the data shown in Fig. 3.
J. Electrochem. Soc., Vol. 142, No. 12, December 1995 9 The Electrochemical Society, Inc. 41531000I| , , , , .0.2 .... ''I ....... '1 " ...... I '"'"1 ' ....... I '"'""500r---500-1000: o o:. oo, , , I , , , ,~NLNIIL.N0 . 0 ~-1000 -500 0 500 1 )00%, QFig. 6. The imaginary departures from the mean residual errorere, i -- ere, for the regression shown in Fig. 4 and 5 as a function ofthe' corresponding real values.errors associated with the lack of fit of the model. The timevaryingcharacter of the measurements is not readily apparentin Fig. 3, but examination of the residual errorsshows that the system changed from one experiment to theother and that the residuals for the six experiments can bedistinguished from one another.The measurement-model analysis shown in Fig. 4 and 5shows that the standard deviation obtained using Eq. 8must contain, in addition to the desired stochastic contributionto the error structure, a significant contributionfrom a nonstationary bias component. The standard deviationsthat are corrupted by such bias errors show severaldistinctive features. The real and imaginary departuresfrom the mean residual error are correlated, as shown inFig. 6. The standard deviations calculated using Eq. 8 areshown in Fig. 7 as a function of frequency. Examination ofreal and imaginary values suggests that they are not equal.Both observations are counter to results obtained for sta-b104 ......... ' ........ ' ........ ' ........ ' ....... ' .10 2- -2--;'--I 0 - 4 _ 'I ....... I ........ j ........ i ........ i ......10 10 0 10 1 10 2 10 `5 10 4 10 5Frequency, HzFig. 7. Unfiltered standard deviation of the data shown in Fig. 3 asa function of frequency. Circles represent the real, and the trianglesrepresent the imaginary part of the standard deviation. The solid linerepresents the error structure for the n-GaAs sample held at 320 K(see Eq. 14). The dashed line represents the contribution that is proportionalto I Z.i I, the dashed-dot line represents the contribution thatis proportional to I Zrl, and2the dashed-dot-dot-dot line is the cantributionproportional to IZI /R,. The discontinuity apparent in thesolid line and the dash-dot-dot-dot line is caused by a change of thevalue of the current measuring resistor at 100 Hz.-0.2 ........ I .... ,,,I ....... J , ,,,,,,,I , ,,,,,.I ......10 -1 100 101 102 103 104 105Frequency,Fig. 8. Real residual errors for the separate regression of measurementmodels to the data shown in Fig. 3.tionary data, where the real part of the standard deviationwas observed to be equal to the imaginary part.For nonstationary systems, the stochastic component ofthe error cannot be calculated by taking the standard deviationof the residual errors directly. A procedure to filterout the nonstationary component of the error is outlined inthe next section. The significance of the lines in Fig. 7 isdiscussed in later sections.Identification of the noise modeL--For nonstationarydata, a measurement model is regressed to each data setusing the maximum number of parameters that can be resolvedfrom the data. For this work, a complex nonlinearleast squares program was used that was developed inhouse?'8 As a model for the stochastic contribution to theerror structure is not known a priori, modulus weighting istypically used for the regression. The regressed parametersfor the measurement model for each data set are slightlydifferent because the system changes from one experimentto the other. Hence, by regressing the measurement modelto individual data sets separately, the effects of the changeof the experimental conditions from one experiment to anotherare incorporated into the measurement modelparameters. The standard deviations of the real and imaginaryresidual errors therefore can be obtained as a functionof frequency and provide a good estimate for the standarddeviation of the stochastic noise in the measurement.The data set from Fig. 3 is used to illustrate the technique.The normalized real and imaginary residual errorsfor six regressions are shown in Fig. 8 and 9. The real andimaginary residual errors are randomly distributed aboutthe mean value at a given frequency. The plot of the imaginaryvs. real departures from the mean residual error,shown in Fig. 10, further suggests that the residual errors ata given frequency are not correlated. The cyclic behavior ofthe residual errors with frequency is caused by the lack offit of the model. The technique of estimating the standarddeviation of the stochastic contribution by calculating thestandard deviation of the residual errors is in effect a filterfor lack of fit. Using the measured error structure to weightthe regression of data that are free of bias errors, theresidual errors for the measurement model can be made tobe of the same order as the standard deviation of themeasurement. 9The real and imaginary standard deviations of the residualerrors are shown as a function of frequency in Fig. 11.Hz
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