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Application of Measurement Models to Impedance Spectroscopy

III. Evaluation of Consistency with the Kramers-Kronig Relations

Pankaj Agarwal *'a and Mark E. Orazem** D e p a r t m e n t o f Chemica l .Engineering, Univers i ty o f Florida, Gainesvi l le , Flor ida 32611, U S A

Luis H. Garcia-Rubio D e p a r t m e n t o f Chemica l Engineer ing , Univers i t y o f S o u t h Florida, Tampa, Flor ida 33620, U S A

ABSTRACT

The Kramers-Kronig equations and the current methods used to apply them to electrochemical impedance spectra are reviewed. Measurement models are introduced as 'a tool for identification of the frequency-dependent error structure of impedance data and for evaluating the consistency of the data with the Kramers-Kronig relations. Through the use of a measurement model, experimental data can be checked for consistency with the Kramers-Kronig relations without explicit integration of the Kramers-Kronig relations; therefore, inaccuracies associated with extrapolation of an incomplete frequency spectrum are resolved. The measurement model can be used to determine whether the residual errors in the regression are due to an inadequate model, to failure of data to conform to the Kramers-Kronig assumptions, or to noise.

This paper is part of a series intended to present the foun- dation for the application of measurement models to impedance spectroscopy. The basic premise behind this work is that determination of measurement characteristics is an essential aspect of the interpretation of impedance spectra in terms of physical parameters. The importance of the error structure identification for interpretation of impedance measurements has been recognized for some time (see, e.g., Ref. 1-11), but experimental assessment of the error structure was complicated by the difficulty of obtaining truly replicate impedance measurements. Re- cently, measurement models have been demonstrated to be useful tools for identification of both stochastic and bias contributions to the error structure of impedance spectra, 11-18 and other groups have begun employing the concept for assessing consistency with the Kramers-Kronig relations. 19,2~

In the first paper of this series, 13 it was shown that a measurement model based on Voigt circuit elements can provide a statistically significant fit to typical electrochemical impedance spectra. In the second paper, 21 a method was demonstrated in which the measurement model is used to identify the stochastic component of the frequency-dependent error structure of impedance data, and a preliminary model for the stochastic component of the error was proposed. In this paper of the series we address the use of the measurement model for identification of the bias component of the error structure. This method is placed in context of the current methods used to assess the consistency of impedance data with the Kramers-Kronig relations.

Background In principle, the Kramers-Kronig relations can be used to

determine whether the impedance spectrum of a given system has been influenced by bias caused, for example, by instrumental artifacts or t ime-dependent phenomena, b Al- though this information is critical to the analysis of impedance data, the Kramers-Kronig relations have not found widespread use in the analysis and interpretation of

* Electrochemical Society Student Member. ** Electrochemical Society Active Member.

Present address: Department of Materials, Swiss Federal In- stitue of Technology (Lausanne), Lausanne, Switzerland.

b A distinction is drawn in this work, as in Ref. 21, between errors caused by a lack of fit of a model and experimental errors that are propagated through the model. The bias errors, as referred to here, may be caused by a changing base line or by instrumental artifacts, but do not include errors associated with model inadequacies.

J. Electrochem. Soc., Vol. 142, No. 12, December 1995 �9

electrochemical impedance spectroscopy data due to dif- ficulties with their application. The integral relations require data for frequencies ranging from zero to infinity, but the experimental frequency range is necessarily constrained by instrumental limitations or by noise at tr ibut- able to the instabili ty of the electrode.

The Kramers-Kronig relations have been applied to electrochemical systems by direct integration of the equations, by experimental observation of stability and ]inearity, and by regression of electrical circuit models to the data. Each of these approaches has its merits and its disadvantages. The Kramers-Kronig equations and the methods used to apply them to electrochemical impedance spectra are reviewed here. The disadvantages associated with current methods used to check experimental data for consistency with the Kramers-Kronig relations can be circumvented by application of measurement models to impedance spectra.

The Kramers-Kronig Relations The Kramers-Kronig relations, developed for the field of

optics, are integral equations which constrain the real and imaginary components of complex quantities for systems that satisfy conditions of causality, linearity, and stability. ~2-25 Bode 2~ extended the concept to electrical impedance and tabulated various forms of the Kramers-Kronig relations. Several transformations used in electrochemical literature are given below (see, e.g., Ref. 27). The imaginary part of the impedance can be obtained from the real part of the impedance spectrum through

i

z~(~o) = - Jo x 2 ~~ 2 dx [1]

where Z~(r and Zj(r are the real and imaginary components of the impedance as functions of frequency ~. The real part of the impedance spectrum can be obtained from the imaginary part through

( 2 ) fo x ~ ' - ~Z~(~' Zr(o~) = Zr(~) + ' x 2 _ r176 2 ' d x [2]

if the high-frequency asymptote for the real part of the impedance is known, and through

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4160 J. Electrochem. Soc., Vol. 142, No. 12, December 1995 �9 The Electrochemical Society, Inc.

if the zero frequency asymptote for the real part of the impedance is known.

In response to the integration limits, a fourth constraint, that the impedance approach finite values at frequency limits of zero and infinity, is commonly added. This constraint, sometimes claimed to prevent application of the Kramers-Kronig relations to capacitive systems, is not needed because a simple variable substitution 2~ can be used if the imaginary part of the impedance tends to infinity according to 1/to as co --> 0. 28

Review of Methods for Determining Consistency The usual approach in interpreting impedance spectra is

to regress a model to the data. The models employed are based on the use of a perturbing signal that has an amplitude sufficiently small that the process can be linearized about a dc polarization point. It is important, therefore, that the impedance response be characteristic of a system that is causal, linear, and stable. The condition of linearity can be achieved by using sufficiently small amplitude per- turbations. The condition of stability requires that the system return to its original condition when the perturbing signal is terminated. An additional implied constraint of stationary behavior may be difficult to achieve in electrochemical systems such as corrosion where the electrode may change significantly during the time required to col- lect impedance data. It is, therefore, of practical importance to the experimentalist to know whether the data satisfy the Kramers-Kronig relations. The approaches for ascertaining the degree of consistency include direct integration of the Kramers-Kronig relations, experimental replication of data, and regression of electrical circuit analogues to the data.

Direct integration of the Kramers-Kronig relations.- The Kramers-Kronig relations provide a transformation that can be used to predict one component ot the impedance if the other is known over the frequency limits of zero to infinity. The usual way of using the Kramers- Kronig equations, therefore, is to calculate the imaginary component of impedance from the measured real component using, for example, Eq. 1, and to compare the values obtained to the experimental imaginary component. Alter- natively, the real component of impedance can be calculated from the measured imaginary values using Eq. 2 or 3. The major difficulty in applying this approach is that the measured frequency range is typically not sufficient to allow integration over the frequency limits of zero to infinity. Therefore, discrepancies between experimental data and the impedance component predicted through application of the Kramers-Kronig relations could be attributed to the use of a frequency domain that is too narrow, as well as to the failure to satisfy the constraints of the Kramers-Kronig equations. The Kramers-Kronig relations, in principle, can be applied with a suitable extrapolation of the data into the unmeasured frequency domain. Several methods for extrapolation have appeared in electrochemical literature.

Kendig and Mansfeld 29 proposed extrapolating an impedance spectrum into the low frequency domain under the assumption that the imaginary impedance is symmetric, thus, the polarization resistance Rp otherwise obtained from

Rp = Zr(OC) __ Z r ( 0 ) = (2) fo Zi(x)'x dx [4]

is obtained from

Rp = Z~(o~) - Z~.(O) = (4) f:,~ Z~(X)x dx [5]

where ton~ is the frequency at which the maximum in the imaginary impedance is observed. This approach is limited to systems which can be modeled by a single relaxation time constant. 3~ The limitation to a single time constant is severe because multiple elementary processes with different characteristic time constants are usually observed in electrochemical impedance spectra.

Macdonald and Urquidi-Macdonald 3~ have presented an approach based on extrapolating polynomials fit to impedance data. The experimental frequency domain was divided into several segments, and the individual impedance components Zr(r and Zj(to) were fit to a polynomial expression given by~

Z~ = ~ a~to ~ [6] k=O

and

Z~ = ~ bkto k [7] k=O

which was extrapolated into the unmeasured frequency domain. The Kramers-Kronig equation (e.g., Eq. 1, 2, or 3) was integrated numerically using the extrapolated piecewise polynomial fit for either the real or the imaginary component of the impedance, respectively. 3~ The extrapolation algorithm was applied to various of systems (including synthetic impedance data derived from equivalent electrical circuits and experimental systems such as TiO2-coated carbon steel in aqueous HC1/KC1 solutions). While piecewise polynomials are excellent for smoothing, the best example being splines, they are not reliable for extrapolation and result in relatively many parameters.

Haili 34 provided an alternative approach based on the expected asymptotic behavior of a typical electrochemical system. For extrapolation to to = 0 the imaginary component Zj(to) was assumed to be proportional to to as to ---> 0, consistent with the behavior of a Randles-type equivalent circuit. This approach would apply as well to a finite War- burg impedance, which has an imaginary component that is also proportional to to as to ~ 0. The real impedance approaches a constant limit which is the sum of the ohmic solution resistance and the polarization resistance. The extrapolation in this region involves only one adjustable parameter whose value approaches Zr(tom~) if tomin is sufficiently small. At high frequencies, the imaginary component was assumed to be inversely proportional to frequency as to -~ ~, and the real component was assumed to approach a constant equivalent to the ohmic solution resistance R~. The method of Haili guarantees well-behaved extrapolation of the impedance spectrum at upper and lower frequency limits with only five adjustable parameters. Haili's work confirmed the importance of extrapolating impedance data to both zero and infinite frequency when evaluating the Kramers-Kronig relations.

Esteban and Orazem 35'~6 presented an approach which circumvented the problems associated with extrapolations of polynomials and yet avoided making a priori assumption of a model for asymptotic behavior. Esteban and Orazem suggested that, instead of predicting the imaginary impedance from the measured real impedance using Eq. 1 or, al- ternatively, predicting the real impedance from the measured imaginary values using Eq. 2 or 3, both equations could be used simultaneously to calculate the impedance below the lowest measured frequency tomin. A low-frequency limit too was chosen for integration of the Kramers- Kronig relations that was typically three or four orders of magnitude smaller than tom,n- The calculated impedance, in the domain too -< to < tom~, forced the experimental data set to satisfy the Kramers-Kronig relations in the frequency domain r ~ ~o < tom~. The parameter too was chosen to satisfy the requirements that the real component of the impedance spectrum attains an asymptotic value and that the imaginary component approaches zero as co --~ too- Inter- nal consistency between the impedance components requires that the calculated functions be continuous with the experimental data at to~n- These requirements cannot be satisfied simultaneously by data from systems that do not satisfy the constraints of the Kramers-Kronig relations; therefore, discontinuities between experimental and extrapolated values were attr ibuted to inconsistency with the Kramers-Kronig relations. The approach described by Es- teban and Orazem 35'36 is different from other algorithms presented above because the Kramers-Kronig relations

J. Electrochem. Soc., Vol. 142, No. 12, December 1995 �9 The Electrochemical Society, Inc. 4161

themselves were used to extrapolate data to frequencies below the lowest measured frequency. Extrapolation of polynomials or a pr ior i assumption of a model was thereby avoided.

While each of the algorithms described here have been applied to some experimental data with success, any approach toward extrapolation can be applied only over a small frequency range and cannot be applied at all if the experimental frequency range is so small that the data do not show a maximum in the imaginary impedance. The extrapolation approach for evaluating consistency with the Kramers-Kronig relations cannot be applied, therefore, to a broad class of experimental systems for which the unmeasured portion of the impedance spectrum at low frequencies is not merely part of a tail but instead represents a significant portion of the impedance spectrum.

E x p e r i m e n t a l checks for c o n s i s t e n c y . - - E x p e r i m e n t a l methods can be applied to check whether impedance data conform to the Kramers-Kronig assumptions. A check for linear response can be made by observing whether spectra obtained with different magnitudes of the forcing function are replicate or by measuring higher order harmonics of the impedance response. Stationary behavior can be identified experimentally by replication of the impedance spectrum. Spectra are replicate if the spectra agree within the expected frequency-dependent measurement error. If the experimental frequency range is sufficient, the extrapolation of the impedance spectrum to zero frequency can be compared to the corresponding values obtained from separate steady-state experiments. The experimental approach to evaluating consistency with the Kramers-Kronig relations shares constraints with direct integration of the Kramers- Kronig equations. Because extrapolation is required, the comparison of the dc limit of impedance spectra to steady- state measurement is possible only for systems for which a reasonably complete spectrum can be obtained. Experi- mental approaches for verifying consistency with the Kramers-Kronig relations by replication are limited fur- ther in that, without an a priori estimate for the confidence limits of the experimental data, the comparison is more qualitative than quantitative. Therefore, a method is needed for evaluating the error structure, or frequency-dependent confidence interval, for the data that obtained in the absence of nonstationary behavior.

Regress ion o f c ircui t a n a l o g u e s . - - E l e c t r i c a l circuits consisting of passive and distributed elements satisfy the Kramers-Kronig relations (see, for example, the discussion in Ref. 37-39). Therefore, successful regression of an electrical circuit analogue to experimental data implies thai the data satisfy the Kramers-Kronig relations. 5'4~ This approach has the advantage that integration over an infinite frequency domain is not required, therefore a portion of an incomplete spectrum can be identified as being consistent without use of extrapolation algorithms.

Perhaps the major problem with the use of electrical circuit models to determine consistency is that interpretation of a "poor fit" is ambiguous. A poor fit is not necessarily the result of an inconsistency of the data with the Kramers- Kronig relations. A poor fit also may be attributed to use of an inadequate model or to regression to a local rather than global minimum (caused perhaps by a poor initial guess). A second unresolved issue deals with the regression itself, i.e., selection of the weighting to be used for the regression, and identification of a criterion for a good fit. A good fit could be defined by residual errors that are of the same size of the noise in the measurement , but, in the absence of a means of determining the error structure of the measurement, such a criterion is speculative at best.

Use of the Measurement Model for Evaluation of Consistency with the Kramers-Kronig Relations

Some progress has been made over the past five years in the development of measurement models as tools for assessing the consistency of impedance data with the Kramers-Kronig relations. The use of measurement models

to check for the consistency of the experimental data with the Kramers-Kronig relations was proposed in 1992 by Agarwal et al. is In 1991 12 and in 1993, 14 Agarwal et al. demonstrated the use of a measurement model based on a series of Voigt elements to assess consistency of experimental data with the Kramers-Kronig relations. Modulus weighting was used to regress the measurement model to the data, and a Monte Carlo calculation was used to provide a quantitative basis on which to reject inconsistent data. In ] 993, Agarwal et al. 15 used the measurement model to assess the extent of replicacy of repeated measurements and to check data for consistency with the Kramers-Kronig relations. The weighting was based on a preliminary model for the error structure of the impedance measurements which represented a refinement to the use of modulus weighting.

In 1993, Orazem et al. n demonstrated that weighting by the stochastic contribution to the error structure greatly enhanced the quality and quantity of information obtainable from impedance measurements. Spectra, obtained at temperatures between 320 and 420 K for an n-type GaAs/Ti Schottky diode, were interpreted through a Maxwell circuit (with components related to physical parameters), which was treated as a measurement model. Modulus-, proportional-, and no-weighting options yielded only one time constant for these data; whereas weighting by the measured error structure yielded four time constants. The number of electron transitions and their activation ener- gies were verified by independent deep-level transient spectroscopic (DLTS) measurements.

In 1994, Boukamp and Macdonald I9 described the use of distributed relaxation time models (DRT) as measurement models for assessing consistency of data with the Kramers- Kronig relations. The approach taken to assess consistency was fundamentally the same as that used in the preliminary work of Agarwat et al. 12,1~ with exceptions that proportional weighting rather than modulus weighting was used in their regressions and that a quantitative criterion for rejection of data was not provided. Their observation that the measurement mode] provided good but somewhat ap- proximate fits to continuous distributions can be attributed to the use of proportional weighting rather than weighting by the variance of the measurement. As was shown in the preliminary work of Agarwal et aI., 1~.~4 a rough assessment of consistency with the Kramers-Kronig relations can be obtained using a suboptimal weighting strategy, but in the absence of independent assessment of the stochastic contribution to the error structure, one cannot know whether the residual errors for the regression fall within the noise level of the data or whether a degree of inconsistency found by regressing the measurement model is statistically significant. More recently, a ]inearized application of measurement models has been suggested by Boukamp 2~ which eliminates the need for sequential in- crease in the number of line shape parameters by using one line shape for every frequency measured. The application of such a ]inearized model is constrained by the need for an independent assessment of the level of noise in the measurement.

A quantitative assessment of the consistency of experimental data with the Kramers-Kronig relations must be made in the context of the overall error structure of the measurement. The stochastic contribution to the error structure of the data set chosen to illustrate the approach was identified in Ref. 21, and the bias contribution is identified here.

From the perspective of the approach proposed here, the use of measurement models to identify consistency with the Kramers-Kronig relations is equivalent to the use of Kramers-Kronig transformable circuit analogues, discussed in the previous section. An important advantage of the measurement-model approach is that it identifies a small set of model structures which are capable of repre- senting a large variety of observed behaviors or re- sponses, n-~ The problem of model discrimination therefore is reduced significantly. The inability to fit an impedance


spectrum by a measurement-model can be attributed to the failure of the data to conform to the assumptions of the Kramers-Kronig relations rather than the failure of the model. The measurement-model approach, however, does not eliminate the problem of model multiplicity or model equivalence over a given frequency range. The reduced set of model structures identified for the measurement model makes it feasible to conduct studies aimed at identification of the error structure, the propagation of error through the model and through the Kramers-Kronig transformation, and issues concerning parameter sensitivity and correlation.

Kramers-Kronig relations as a statistical observer.--A significant advantage of the measurement model approach is that the resulting models can be transformed analytically (in the Kramers-Kronig sense). This means that, in contrast to the other approaches for evaluation of consistency (e.g., fitting to polynomials), the real and imaginary parts of the impedance are related through a finite set of common parameters. The measurement models therefore can be used as statistical observers, 41 that is, adequate identification and estimation of the model parameters over a given experimental region, e.g., a range of frequencies in the imaginary domain, allow the description (or observation) of the behavior of the system over another region, the real domain. The selection of the experimental region used for this evaluation takes advantage of the relative parameter sensitivity i t / the real and imaginary domains (as discussed in the Algorithm section, below).

The method proposed here for assessing the consistency of experimental data with Kramers-Kronig relations is developed in conjunction with an overall assessment of the error structure for impedance measurements. The error structure can be expressed as

Z -- Z • Ei, esidua I : E]o f + Ebias + Estochaslic [8 ]

where the caret signifies the model value for the complex impedance Z, and the residual error contains contributions from the lack of fit of the model, a systematic bias, and stochastic noise. The bias error may include contributions from nonstationary behavior (%~) and instrumental artifacts (ein~), i.e.

Eblas --= Ens + Ein s [9]

but does not include errors associated with a lack of fit. The nonstationary contribution to the bias usually is observed most easily at low frequencies. Instrumental artifacts may be seen at high frequency resulting from equipment limitations.

Discussion of bias errors must be made in the context of the overall error structure of the measurement. Recently, Macdonald has described the manner in which the integral Kramers-Kronig relations transform errors, and has suggested a frequency ---> time -+ frequency domain transformation that acts as a filter for both bias and stochastic errors) ~ If regression techniques are used, the experimentally determined level of stochastic noise can be used both for weighting and as a criterion for assessing the quality of the fit. ~-18'21 In the second paper of this series, 21 a procedure was presented for assessing the standard deviation of the stochastic contribution to the error structure. By using the measurement model as a filter for the lack of replicacy of sequential impedance measurements, an accurate estimate can be obtained for the level of noise in the measurement. In the proposed algorithm, the properties of the measurement model are used to assess the bias contribution to the error structure.

A l g o r i t h m . - - W h i l e in principle a complex fit of the measurement model may 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 constrained by the observation that the standard deviation of the noise in the real and imaginary part of the impedance is the same; 11'~8 there-

fore, little information is contained in the asymptotic limits of the imaginary impedance where the imaginary impedance tends toward zero. As a result, selection of the preferred approach should be guided by use of the real part of the measurement for assessing consistency in the frequency limit (high or 10w) where the asymptotic behavior of the imaginary impedance is seen. Conversely, the imaginary part of the measurement should be used for assessing consistency in the frequency limit where the asymptotic behavior of the imaginary impedance is not seen.

The influence of noise on the relative information content of the real and imaginary components of impedance data is illustrated by the following example. Experimental data obtained for corrosion of copper in 0.5 M C1- solution with pH of 11.5 42 are shown in Fig. i. The regression of a measurement model to the data was weighted by the variance of the stochastic contribution to the error structure as determined in Ref. 2 i. A Voigt circuit was used for a measurement model (see, e.g., Fig. i of Ref. 13). Issues associated with the quality of this fit also are discussed later. The influence of noise on the relative information content of the real and imaginary components of impedance data can be seen by examination of the frequency dependence of the line shapes which make up the measurement model, given in Fig. 2. The solid lines represent the deconvolution of the measurement model into its six component line shapes. The model value, obtained by summation of the line shapes, passes through the data given by open circles.

Since the standard deviations of the stochastic error are the same for the real and the imaginary parts of the impedance, the noise in the imaginary impedance as a per- centage of signal tends toward infinity at frequencies that are high or low. For this example, both real and imaginary components of the impedance appear to approach finite asymptotic values at high frequency. The approach of the imaginary impedance toward a finite value at high frequency could be caused by a high-frequency process, suggested by the rightmost line shapes in Fig. 2, or by instrumental artifacts. Generally, for cases where the real part of the impedance approaches a constant value at high frequency, the imaginary part of the impedance contains less information as compared to the real part of the impedance. The data collected do not show asymptotic behavior at low frequency. At the low frequency end, both the real and imaginary data may be considered to be roughly equally reliable with respect to the signal-to-noise ratio. Examina- tion of Fig. 1 shows that at low frequency the real part of

15000

10000 O9 E 0

5000

0 i t i

5 0 0 0 1 0 0 0 0 1 5 0 0 0

Zr, Ohms

Fig. 1. Results of regression of a measurement model to impedance data obtained for a copper disk electrode in 0.5 M CI- solution of pH | 1.5. The circles represent the experimental date. The middle line represents the complex fit to the data. The upper and lower lines represent the 95.4% confidence interval for the prediction.


10 s . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . . . 10 5 . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . .

10 4

! 10 3

r

~ 1 0 2

10 ~

10 o I(10

-1 10 1 1010-1 10 o 101 102 103 104 105 10 -1

Frequency, Hz

10 4

10 2

10 ~

1(10 101 102 1(13 10'* 10 s Frequency, Hz

Fig. 2. Deconvolution of a measurement model into its components. The circles represent the experimental data. The solid lines represent the measurement model and its components. (a, left) Real part of the impedance; (b, right) imaginary part of the impedance.

the deconvolution of the measurement model does not have significant frequency dependence. However, a significant change is observed in the imaginary part of the deconvolution. Hence, the imaginary part of an impedance spectrum has higher information content at the lower frequencies presented in Fig. 2.

The algorithm presented in Table I is proposed to check for the consistency of the impedance data of-Fig, i. The application of the algorithm to experimental data is described in subsequent sections.

Monte-Carlo simulation for confidence intervaL--Mac- d o n a l d ha s de sc r ibed the use of M o n t e Car lo s i m u l a t i ons to assess the in f luence of a n a s s u m e d e r ro r s t r u c t u r e on mode l i d e n t i f i c a t i o n 8 a n d on the conf idence in te rva l s for re - gressed pa ramete r s . /~ In con t ras t , M o n t e Car lo s i m u l a t i ons are used he re to exp lo re the m a n n e r in w h i c h t he uncer - t a i n t y in p a r a m e t e r e s t ima te s is p r o p a g a t e d t h r o u g h the model . Ca l cu l a t i on of the 95.4% conf idence i n t e rva l for a mode l p r e d i c t i o n was used to p r ov i de a q u a n t i t a t i v e c r i te - r i on for r e j ec t ion of e x p e r i m e n t a l da t a c o r r u p t e d by b ias errors . This c a l c u l a t i o n was done t h r o u g h the use of M o n t e - Car lo s i m u l a t i o n as o u t l i n e d b e l o w Y

The p re sence of r a n d o m or s tochas t i c e r rors gives r ise to a n u n c e r t a i n t y in the p r e d i c t i o n of p a r a m e t e r s in a regression. Regress ion of a mode l to e x p e r i m e n t a l d a t a Z resu l t s in t he p a r a m e t e r e s t ima te ~ w i t h the s t a n d a r d d e v i a t i o n vec to r a(a) a n d a c o r r e s p o n d i n g mode l v a l u e Z. U n d e r t he a s s u m p t i o n t h a t s tochas t i c e r rors are n o r m a l l y d i s t r i bu t ed , a ser ies of p a r a m e t e r s ap are g e n e r a t e d such t h a t the p r o b a - b i l i ty distribution for ~ - ap follows a normal distribution with a standard deviation of ~(a). For each ap there is a simulated Zp. A plethora of these simulations is performed, and the standard deviation a(Z) of these simulations is

Table I. The proposed algorithm for using measurement models to check data presented in Fig. 1 for consistency

with the Kramers-Kronig Relations.

Step 1: Check for consistency at the high-frequency end. (a) Regress the measurement model to the real part of the data. (b) Predict the imaginary part and the 95.4% confidence interval

through Monte-Carlo simulation. (e) High frequency data that lie outside the 95.4% confidence

intmwal are deemed inconsistent�9 Delete the inconsistent data points.

Step 2: Check for consistency at the low-frequency end. (a) Regress the measurement model to the imaginary part of the

truncated data set. (b) Predict the real part and the 95,4% confidence interval

through Monte-Carlo simulation, (c) Low-frequency data that lie outside the 95.4% confidence

interval are deemed inconsistent. Delete the inconsistent data points.

calculated. Ztrue lies with 95.4% confidence in the range 2 _+ 2~(Z).

The regress ion of a m e a s u r e m e n t mode] to e x p e r i m e n t a l d a t a yields t he m e a n a n d the c o r r e s p o n d i n g s t a n d a r d devi - a t i on of t he p a r a m e t e r s . A va lue for t he impedance , a t one f requency, and, for one M o n t e - C a r l o s imula t ion , was ca lcu- l a t ed by

N~ Ak, p Zp((D) = Rsol, p -F ~ 1 + jTk,pO) [1O]

k = l

where Nolo is the n u m b e r of l ine shapes in the m e a s u r e - m e n t model . The p a r a m e t e r s R~oLp , hk,p, a n d Tk.p were ob- t a i n e d f rom

R~o], p = Rso 1 + Ran<p(0, 1)~[RsoJ

hk.p = ~k + Rank+l.p(0,l)O-[Ak] [] l]

%,p = % + Rank.2,p(0, 1)~[%]

w h e r e 2No,~ + 1 u n i q u e G a u s s i a n r a n d o m dev ia tes Ranj.p(0,1) w i t h zero m e a n a n d a s t a n d a r d d e v i a t i o n of one were ca l cu l a t ed for each e v a l u a t i o n of Zp(~O). The ca l cu l a - t i on a t e ach f r equency was r e p e a t e d Nr~d t imes. A la rge Nr~d is de s i r ab l e for a n accu ra t e s imula t ion . The m e a n a n d s t a n d a r d dev i a t i on of the i m p e d a n c e were c a l c u l a t e d by

Nrand E z~(~)

Z(~o) = p-1 [12] Nrand

a n d

Nrand ~E ( z ~ ( ~ o ) - ~(o,)) ~

(~(Zp(e))) = [13] Nr~nd - 1

respectively. The 95.4% conf idence l imi t was t hus ca lcu- l a t ed by

Z~p(~) = z(~) + 2~[zp(~o)] [14]

Zio(~o) = Z(~o) - 2~[Zp(r

whe re Zup(O~) a n d Zlo(r are the u p p e r a n d lower conf idence in te rva l s , respectively.

Application to experimental data.--The above a l g o r i t h m was app l i ed to e l ec t rochemica l i m p e d a n c e d a t a o b t a i n e d for cor ros ion of a r o t a t i n g c o p p e r d isk e lec t rode in a 0.5 M C1- aqueous so lu t ion w i t h pH a d j u s t e d to 11.5 by a d d i t i o n of NaOH. The da t a were co l lec ted u s ing a S o l a r t r o n 1286 p o t e n t i o s t a t a n d a S o l a r t r o n 1250 f r e q u e n c y - r e s p o n s e a n a - lyzer (FRA). D a t a were co l lec ted f r e q u e n c y - b y - f r e q u e n c y using the long-channel integration feature of the FRA,


0 1 . . . . . . . . , . . . . . . . . , . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . . .

rb 0.05

U.I

~ 0

~ - 0 . 0 5 u._

o

o o

o o

0

o o o

o o

.o

% o

o

o

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o o

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1 1 10 rJ 101 102 103 104 0 Frequency, Hz

10 s

0.1

005

W

r i -

g -005

0 0

O o ( 3 o

o o

o o

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D 0 000 0 0 0 0

0 0 0

o ~ 0 ~ 0

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o

-0 10 1 100 1 01 102 103 104 105

Frequency, Hz

Fig. 3. Residual errors for the regression presented in Fig. 2. The solid lines at values of roughly _+0.005 correspond to the 2~ values for the measured stochastic component of the impedance spectra. The jog (barely visible) in these lines at 100 Hz is caused by a change in the value of the current-measuring resistor. The jog is seen more clearly in Fig. 9. (a, left) Real part of the impedance; (b, right) imaginary part of the impedance.

which completed a measurement at each frequency on reaching a 1% closure error. The current measuring resistor was changed from 1,000 to 10,000 t~ at a frequency of I00 Hz.

Examination of the data given in Fig. I and 2 shows that, as is frequently seen for measurements on slowly corroding electrodes, the imaginary part of the impedance does not show a maximum in the measured frequency range. There- fore, as discussed in the previous sections, methods based on direct integration of the Kramers-Kronig relations cannot be applied to check for consistency. The algorithm de~ veloped by Esteban et al 35,35 was applied to the data set, and it indicated that most of the data set is inconsistent. This is an erroneous conclusion as is shown later.

Results of a complex regression of a measurement model to the experimental data are shown in Fig. 1 to 3. The fit, shown here, was obtained by weighting the data with the model for the error structure proposed in Ref. 21 and by using six Voigt circuit elements in series with a solution resistance. This was the best fit obtainable for the data. Use of seven Voigt elements gave a confidence interval for the solution resistance that included zero. The circles in Fig. 1 and 2 represent the experimentM data points, and the solid line represents the regression of the measurement model to the experimental data. In Fig. 3, the circles represent the relative residual errors obtained from the regression. Qual- itative examination of the fit suggests that the regression may be acceptable. The average absolute magnitude of the residual errors is only 3.4%, and the largest errors are about 8 and 10% for the real and imaginary parts, respectively. The average 95.4% confidence intervals for the regressed parameters was about 15 % with a maximum value of 37% and a minimum value of 6.5%. The residual sum of squares normalized by the sum of squares of the impedance was equal to 0.27%. Therefore, based on the traditional approach of regression of circuit analogues to impedance data, one may conclude that the fit to the data is good and, therefore, the data are consistent.

The quality of the fit, however, must be assessed by comparison to the measured error structure for the impedance data. For example, the residual errors shown in Fig. 3 are, considerably larger than the stochastic noise of the measurement, obtained by the techniques presented in Ref. 21. Another indication that the fit can be improved is provided by observation that the residual sum of squares divided by the sum of squares of the standard deviation of the measurement was equal to 594. The residual errors therefore were significantly larger than the noise level of the experi- ment. One possible explanation for the deficiencies of this fit is that the measurement model cannot provide an adequate fit to data that are clearly influenced by mass-trans-

fer processes. However, the measurement model has been shown to provide statistically adequate fits to synthetic data obtained from circuits including Warburg elements. 13 The more likely explanation is that the data are not consistent with the Kramers-Kronig relations.

To check for consiw of the data, the measurement model was regressed first to the real part of the data. The best fit was obtained by using eight Voigt circuits, two more than could be obtained with the fit described above to both real and imaginary components of the data. The results of this regression to the real part of the data are shown in Fig. 4 and 5. The middle line in Fig. 4a represents the regression of the measurement model to the real part of the data, and the middle line in Fig. 4b represents the model predictions using parameters obtained from the regression. The two outer lines in Fig. 4a and b represent the 95.4% confidence interval lines calculated by Monte-Carlo simulation. The residual errors are shown in Fig. 5. The circles in Fig. 5 represent the normalized residual errors. Examina- tion of the high frequency end of the imaginary residuals (Fig. 5b) shows that the data points with frequencies greater than 15,000 Hz lie outside the 95.4% confidence interval. Hence these points were deemed to be inconsistent and were deleted.

The measurement model was then regressed to the imaginary part of the truncated data set, and the real part of the impedance and the 95.4% confidence interval were calculated. The best fit was obtained with eight Voigt circuits. The results of the regression are shown in Fig. 6 and 7. Examination of the residual errors and the 95.4% confidence interval for the real part in Fig. 7a shows that two low frequency data points lie outside the confidence interval, and, hence, these points were deemed to be inconsistent and were deleted.

Results for complex regression of a measurement model to the truncated data set are shown in Fig. 8 and 9. The best fit for this regression was obtained by using ten Voigt circuits. Use of II Voigt elements gave confidence intervals for some parameter estimates that included zero. Examina- tion of the residual errors shows that the average residual error has been reduced to about 0.4% for both real and imaginary parts of the impedance and that the residual errors are comparable to the 2~r level for the noise of the measurement, given by solid lines in Fig. 9. Figures 8 and 9 for regression of the truncated data set should be compared to Fig. 1 and 3 where regression was made for the complete data set. The statistical measures of the fit are also improved with respect to the fit obtained for Fig. 1 and 3. The residual sum of squares normalized by the sum of squares of the impedance was equal to 0.001% for the truncated data set as compared to 0.27 %. The residual sum of squares

J. Electrochem. Soc., Vol. 142, No. 12, D e c e m b e r 1995 �9 The Electrochemical Society, Inc. 4165

divided by the sum of squares of the standard deviation of the measurement was equal to 2.5 for the truncated data set as compared to 594. The mean value for the magnitude of the residual errors for the regression was 0.20 and 0.23% for the imaginary and real parts, respectively.

Only six Voigt circuits and a solution resistance (13 parameters) could be regressed to the complete data set; but, by deleting the inconsistent data points, ten Voigt circuits and a solution resistance (21 parameters) could be resolved from the data. Deletion of data that were strongly influenced by bias errors increased the amount of information that could be extracted from the data. The bias in the complete data set induced correlation in the model parameters which reduced the number of parameters which could be identified. Removal of the biased data resulted in a better conditioned data set that enabled reliable identification of a larger set of parameters. The procedure presented here cannot be used to guarantee that the remaining data are perfectly free of bias errors. The procedure is used to identify data that fall outside the 95.4% confidence interval of the model and can be rejected as being inconsistent with the Kramers-Kronig relations.

Comments on the regression procedure.- -When the measurement model consisting of seven Voigt elements and a solution resistance was regressed to the data shown in Fig. 1, the confidence interval for the solution resistance

2 . 0 x l O 4 .. . . . . ~ . . . . . ~ . . . . . . q . . . . . . q . . . . . m . . . . . .

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(Reo~ Fit)

Fig. 4. Results of regression of a measurement model to impedance data obtained for Cu in 0.5 M CI- solution of pH 11.5. The circles represent the experimental data. The middle line represents the real fit to the data. The outer lines represent the 95.4% confidence interval for the prediction. (a, tap} Real part of the impedance; (b, bottom) imaginary part of the impedance.

| N

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Fig. 5. Residual errors for regression presented in Fig. 4. The solid lines represent the 95.4% confidence interval. (a, tap) Real part of the impedance; (b, bottom) imaginary part of the impedance.

included zero. A model consisting of six Voigt elements in series with a solution resistance, therefore, was treated as being the best model for fitting of the complete data set. This result is somewhat unusual as compared to the results obtained for other data sets. When the number of parameters exceeds the maximum number obtainable within a given confidence interval, the solution resistance is usually well determined, and one or more of the Voigt parameters has a confidence interval that includes zero. In the usual case, there is no ambiguity as to whether a model with many parameters should be pursued. In contrast, for the data shown in Fig. 1, many parameters and better regression statistics could be obtained by setting the solution resistance to zero and increasing the number of Voigt elements in the model. The model containing a solution resistance was chosen for the regression procedure presented here partly on physical grounds, as a solution resistance on the order of 10 ~ was expected for the disk electrode, and partly because it proved to be more sensitive than the alternative in detecting bias errors.

Regression of a model consisting of ten Voigt elements to the entire data set yielded a residual sum of squares divided by the sum of squares of the standard deviation of the measurement of 3.4, larger than the best value obtained by fitting a measurement model with ten Voigt elements and a solution resistance to the truncated data set, but a substan-

4166 J. Electrochem. Soc., Vol. 142, No. 12, D e c e m b e r 1995 �9 The Electrochemical Society, Inc.

tial improvement over the best value that could be obtained by fitting a measurement model using a nonzero value for the solution resistance to the entire data set. The mean value for the magnitude of the residual errors for the regression was 0.40 and 0.47% for the imaginary and real parts, respectively. The principal suggestion that portions of the data were inconsistent with the Kramers-Kronig relations was that the residual errors, while small, were not distributed about zero. The mean value for the residual errors was -0.25 and 0.18% for the real and imaginary parts, respectively. The corresponding mean values for the fit shown in Fig. 8 and 9 were -0.11 and 0.03% for the real and imaginary parts, respectively.

The fitting procedure described here was repeated with a measurement model without a solution resistance. Regres- sion to the real part of the impedance revealed that 5 data points at the high-frequency end could be rejected within the 95.4% confidence interval for the model. The confidence interval for the fit to the imaginary part of the impedance was so broad that no data could be rejected at the low-frequency end. The use of a measurement model with a nonzero value for the solution resistance was more sensitive to bias errors than the use of a model with more parameters but without a solution resistance.

For the calculation of the 95.4% confidence interval, 5000 Monte-Carlo simulations were performed at each fre-

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Fig. 6. Results of regression of a measurement model to impedance data obtained for a copper disk electrode in 0.5 M CI- solution of pH 11.5. The circles represent the experimental data. The middle line represents the imaginary fit to the data. The outer lines represent the 95.4% confidence interval for the prediction. (a, top) Real part of the impedance; (b, bottom) imaginary part of the impedance.

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quency to ensure that impedance values follow a Gaussian distribution. Results of simulations at a frequency of 1,632.7 Hz are shown in Fig. 10 to illustrate that the calculated impedance has a Gaussian distribution. Comparison of the two curves suggests that 5000 simulations were sufficient to give a good approximation to a theoretical Gaus- sian distribution.

Discussion and Conclusion As used here, the measurement model provides much

more than a preliminary analysis of impedance data in terms of the number of resolvable time constants and asymptotic values as suggested, for example, by Zol- towski. 44"4~ As shown in the previous paper, Z1 the measurement model can be used as a filter for lack of replicacy that allows accurate assessment of the standard deviation of impedance measurements. This information is critical for selection of weighting strategies for regression, provides a quanti tat ive basis for assessment of the quality of fits, and can guide experimental design. The measurement model is used here to assess the bias component of the error structure. This work is part of an overall assessment of measurement errors. The next step in the interpretation of these data is the development of deterministic models which can account for the physical phenomena associated with this system. The analysis presented here can be used to

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Fig. 8. Results of regression of a measurement model to truncated impedance data. The circles represent the experimental data. The middle line represents the complex fit to the data. The upper and lower lines represent the 95.4% confidence interval for the prediction. This figure can be compared to Fig. 1, obtained for regression of the measurement model to the entire data set.

ensure that the data used for comparison to the model are not corrupted by bias errors, thus facilitating interpretation in terms of physical parameters.

The use of measurement models is superior to the use of polynomial fitting because fewer parameters are needed to model complex behavior and because the measurement model satisfies the Kramers-Kronig relations implicitly. Experimental data can be checked for consistency with the Kramers-Kronig relations without actually integrating the equations over frequency, avoiding the concomitant quadrature errors. Claims that the use of measurement models eliminates the need for extrapolation of the Kramers-Kronig relations, 14'19 are not accurate. The use of measurement models requires an implicit extrapolation of the experimental data set, but the implications of the extrapolation procedure are different from extrapolations re- ported in the literature. The extrapolations done with measurement models are based on a common set of parameters for the real and imaginary parts and on a model structure that represents the observations adequately. Con-

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Fig. 10. The stepped line represents the histogram of the 5000 Monte Carlo simulations performed at 1632.7 Hz, and the continuous solid line represents the theoretical normal distribution. The ab- scissa is given in units of the standard deviation of the model value.

fidence in extrapolation using measurement models is, therefore, higher. For the application to a preliminary screening of the data, the use of measurement models is superior to the use of more specific electrical circuit analogues because one can determine whether the residual

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Frequency, Hz

Fig. 9. Residual errors for regression presented in Fig. 8. The solid lines at values of roughly +0.005 correspond to the 2(~ values for the measured stochastic component of the impedance spectra. The jog seen in these lines at 100 Hz is caused by a change in the value of the current measuring resistor. (a, left) Real part of the impedance; (b, right) imaginary part of the impedance. This figure can becompared to Fig. 3, obtained for regression of the measurement model to the entire data set.


errors are due to an inadequate model, to failure of data to conform to the Kramers-Kronig assumptions, or to experimental noise. The algorithm proposed here, in conjunction with error structure weighting, provides a robust way for checking for consistency of impedance data.

A quantitative criterion was established by which data that are influenced by the bias error could be rejected. This approach is appropriate if, as is usually the case, the models used to interpret the data are stationary. An alternative to rejection of data is to incorporate the bias error into the weighting rather than weighting the regression by the square of the bias-free value for the stochastic noise. The frequency and/or time dependence of the bias error may provide useful insight into the experimental system. As- sessment of experimental or instrumental artifacts may guide changes in the experimental system. The nonstationary contribution to the bias error may be interpreted in terms of physical processes if the time-varying character of the system is sufficiently slow that reliable impedance measurements may be made at each frequency (see, for example, Ref. 46). Under such conditions, it may be possible to develop a process model that accounts explicitly for the time-varying character of the system. Another direction for development of regression models for impedance spectroscopy is to adapt the error-in-variables models, ~7 which are particularly effective when the variance of the independent parameter (in this case, frequency) is not insignificant with respect to the variance of the measured impedance components.

Finally, by weighting the regression by the measured error structure and by eliminating data that are corrupted by bias errors, the measurement model can provide a fit to experimental data that yields residual errors that are of the same size as the noise in the measurement. This work can be used to establish a standard for the quality of fit that can be achieved by a process model which describes the phys- ics of the system. In future work we will show that, for some experimental systems, process models can be found that can fit impedance spectra to within the noise of the measurement.

Acknowledgment The work performed at the University of Flor ida (P.A.

and M.E.O.) was supported by the Office of Naval Research and by Gates Energy Products. The work performed at the University of South Flor ida (L.H.G.R.) was supported by the National Science Foundat ion under Grants No. RII- 8507956 and INT-8602578. The assistance of Professor Os- car Crisalle is also gratefully acknowledged. This paper was wri t ten while one author (M.E.O.) was on sabbat ical leave at the UPR 15 du CNRS "Physique des Liquides et Electrochimie," in Paris, France. The use of their facilities to prepare portions of this work and the helpful suggestions of Dr. Claude Deslouis and Dr. Bernard Tribollet are greatly appreciated.

Manuscript submit ted Apri l 10, 1995; revised manuscript received Aug. 11, 1995.

University of Florida assisted in meeting the publication costs of this article.

REFERENCES 1. C. Gabrielli , E Huet, M. Keddam, and J. E Lizee, J.

Electroanal. Chem., 138, 201 (1982). 2. R Zoltowski, J. Electroanal. Chem., 178, 11 (1984). 3. C. Gabrielli , M. Keddam, and J. E Lizee, ibid., 163,419

(1984). 4. B. A. Boukamp, Solid State Ionics, 20, 31 (1986). 5. J. R. Macdonald and L. D. Potter, Jr., ibid., 23, 61

(1987). 6. B. Robertson, B. Tribollet, and C. Deslouis, This Jour-

nal, 135, 2279 (1988). 7. R Zoltowski, J. Electroanal. Chem., 260, 269, 287

(1989). 8. J. R. Macdonald, Electrochim. Acta, 35, 1483 (1990). 9. J. R. Macdonald and W. J. Thompson, Commun. Statist.

Simula., 20, 843 (1991).

10. J. R. Macdonald, Electrochim. Acta, 38, 1883 (1993). 11. M. E.Orazem, P. Agarwal, A. N. Jansen, R T. Wojcik,

and L. H. Garcia-Rubio, ibid., 38, 1903 (1993). 12. R Agarwal, M. E. Orazem, and A. Hiser, in Hydrogen

Storage Materials, Batteries, and Chemistry, D. A. Corrigan and S. Srtnivasan, Editors, PV 92-5, p. 120, The Electrochemical Society Proceedings Series, Pennington, NJ (1992).

13. R Agarwal, M. E. Orazem, and L. H. Garcia-Rubio, This Journal, 139, 1917 (1992).

14. R Agarwal, M. E. Orazem, and L. H. Garcia-Rubio, in Electrochemical Impedance: Analysis and Interpre- tation, ASTM STP 1188, p. 115, J. Scully, D. Silver- man, M. Kendig, Editors, American Society for Test- ing and Materials, Phi ladelphia (1993).

15. P. Agarwal, O. C. Moghissi, M. E. Orazem, and L. H. Garcia-Rubio, Corrosion, 49, 278 (1993).

16. M.E. Orazem, P. Agarwal, L. H. Garcia-Rubio, J. Elec- troanal. Chem. Interfac. Electrochem., 378, 51 (1994).

17. P. Agarwal, Ph.D. Thesis, University of Florida, Gainesville, FL (1994).

18. M.E. Orazem, R Agarwal, C. Deslouis, and B. Tribollet, Submitted.

19. B.A. Boukamp and 5. R. Macdonald, Solid State Ionics, 74, 85 (1994).

20. B. A. Boukamp, This Journal, 142, 1885 (1995). 21. R Agarwal, M. E. Orazem, O. Crisalle, and L. H. Gar-

cia-Rubio, ibid., 142, 4149 (1995). 22. R. de L. Kronig, J. Opt. Soc. Am. Rev. Sci. Instrum., 12,

547 (1926). 23. R. de L. Kronig, Physikal. Z., 39, 521 (1929). 24. H. A. Kramers, ibid., 39, 522 (1929). 25. B. Davies, in Applied Mathematical Series, 2nd ed.,

Vol. 25, Springer-Verlag, New York (1985). 26. H. W. Bode, Network Analysis and Feedback Amplifier

Design, D. Van Nostrand Co., Inc., New York (1945). 27. M. Urquidi-Macdonald, S. Real, and D. D. Macdonald,

This Journal, 133, 2018 (1986). 28. C. Gabrielli , M. Keddam, and H. Takenouti, in Electro-

chemical Impedance: Analysis and Interpretation, ASTM STP 1188, p. 140, J. Scully, D. Silverman, and M. Kendig, Editors, American Society for Testing and Materials, Phi ladelphia (1993).

29. M. Kendig and E Mansfeld, Corrosion, 39, 466 (1983). 30. D. D. Macdonald and M. Urquidi-Macdonald, This

Journal, 132, 2316 (1985). 31. F. Mansfeld, M. W. Kendig, and W. J. Lorenz, ibid., 132,

290 (1985). 32. M. Urquidi-Macdonald, S. Real, and D. D. Macdonald,

Electrochim. Acta, 35, 1559 (1990). 33. B. J. Dougherty and S. Smedley, Electrochemical Im-

pedance: Analysis and Interpretation, ASTM STP 1188, p. 154, J. Scully, D. Silverman, and M. Kendig, Editors, American Society for Testing and Materials, Phi ladelphia (1993).

34. C. Haili, M.S. Thesis, University of California, Berke- ley, CA (1987).

35. J. M. Esteban and M. E. Orazem, This Journal, 138, 67 (1991).

36. M.E. Orazem, J. M.Esteban, and O. C. Moghissi, Corro- sion, 47, 248 (1991).

37. D. D. Macdonald and M. Urquidi-Macdonald, This Journal, 137, 515 (1990).

38. D. Townley, ibid., 137, 3305 (1990). 39. M. Urquidi -Macdonald and D. D. Macdonald, ibid.,

137, 3306 (1990). 40. M. K. Brachman and J. R. Macdonald, Physica, 20,

1266 (1956). 41. D.G. Luenberger, IEEE Trans. Automatic Control, AC-

16, 596 (1971). 42. O. C. Moghissi, Ph.D. Thesis, University of Florida,

Gainesville, FL (1993). 43. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T.

Vetterling, Numerical Recipes, The Art of Scientific Computing, Cambridge University Press, Cambridge (1989).

44. P. Zoltowski, Pol. J. Chem., 68, 1171 (1994). 45. P. Zoltowski, J. Electroanal. Chem., 375, 45 (1994). 46. B. Stoynov and B. S. Savova-Stoynov, ibid.,183, 133

(1985). 47. R M. Reilly and H. Pat ino-Leal , Technometrics, 23, 221

(1981).

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