Measurement Models for Electrochemical Impedance...

J. Electrochem. Soc., Vol. 139, No. 7, July 1992 �9 The Electrochemical Society, Inc. 1917

samples, to Thomas J. Pisklak for cross-sectioning of the membranes, to J im Smith and co-workers for assistance in the analysis of water uptake of the membranes, to Larry Smith and Bob Aikman for supplying the membranes, and to Charlie Martin for helpful discussions.

Manuscript submit ted Dec. 23, 1991; revised manuscript received April 1, 1992.

Dow Chemical USA assisted in meeting the publication costs of this article.

REFERENCES 1. Z. Ogumi, T. Kuroe, and Z. Takehara, This Journal,

132, 2601 (1985). 2. M. R. Tant, K. P. Darst, K..D. Lee, and C. W. Martin, in

"Proceedings of the ACS Division of Polymcric Ma- terials: Science and Engineering," Vol. 60, Toronto, Ont., Canada, 1989 Spring Meeting.

3. M. C. Kimble, R. E. White, Y.M. Tsou, and R. N. Be- aver, This Journal, 137, 2510 (1990).

4. H. L. Yeager, B. O'Dell, and Z. Twardowski, ibid., 129, 85 (1982).

5. T.D. Gierke, G. E. Munn, and F. C. Wilson, in "Per- fluorinated Ionomer Membranes," A. Eisenberg and H. L. Yeager, Editors, ACS Symposium Series, Vol.

180, Chap. 10, Washington, DC (1982). 6. T. D. Gierke and W. Y. Hsu, ibid., Chap. 13. 7. T. Kim Gibbs and D. Pletcher. Electrochim. Acta., 25,

1105 (1980). 8. A. G. Loomis, in "International Critical Tables of Nu-

merical Data, Physics, Chemistry, and Technology," Vol. 3, E. W. Washburn, Editor, p. 255, McGraw Hill, Inc., New York (1938).

9. A.P. Colburn and R. L. Pigford, in "Chemical Engi- neering Handbook," 3rd ed., J .H. Perry, Editor, p. 540, McGraw Hill, Inc., New York (1950).

A. H. M. Cosjin, Electrochim. Acta., 3, 24 (1962). Y.-M. Tsou and F.C. Anson, This Journal, 131, 595

(1984). P. C. Lee and D. Meisel, J. Am. Chem. Soc., 102, 5477

(1980). T. Erdey-Gruz, "Transport Phenomena in Aqueous

Solutions," Adam Hilyer, London (1974). H. L. Yeager, Z. Twardowski, and L.M. Clarke, This

Journal, 129, 324 (1982). G. E. Boyd and D. P. Wilson, Macromolecules, 15, 78

(1982). M. Finauda and M. Milas, ibid., 6, 879 (1973). J. Hen and U.P. Strauss, J. Phys. Chem., 78, 1013

(1974). C. Tondre and R. Zana, ibid, 76, 3451 (1972).

10. 11.

12.

13.

14.

15.

16. 17.

18.

Measurement Models for Electrochemical Impedance Spectroscopy

I. Demonstration of Applicability

Pankaj Agarwal* and Mark E. Orazem** Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611

tuis H. Garcia-Rubio Department of Chemical Engineering, University of South Florida, Tampa, Fl~:,rida 33620

ABSTRACT

Use of measurement models is suggested as an intermediate step in the analysis of impedance data. In a manner analogous to the routine use of measurement models in the deconvolution of optical spectra, the measurement model could be used to guide development of physicoelectrochemical models by determining whether a data set is consistent with the Kramers-Kronig relations, by suggesting a form for the error structure of the data, and by providing an indication of the number and type of physical processes that can be resolved from the data. In this paper, a general measurement model is shown to apply for a wide variety of typical electrochemical impedance spectra. The application of the measurement models as a data filter will be addressed in subsequent papers.

Impedance spectroscopy has become a powerful tool for investigating the properties of electronic materials and electrochemical systems (1, 2). The response of a system to a sinusoidal perturbation can be used to calculate the impedance as a function of the frequency of the perturbation. Interpretation of electrochemical impedance spectra requires selection of an appropriate model which is regressed to the data. The most commonly used model is an electrical circuit analogue consisting of resistors, capacitors, inductors, and specialized distributed elements. De- velopment of detailed physicoelectrochemical models is becoming more common but is still less often pursued. Se- lection of an appropriate model can be time consuming, particularly if a detailed physicoelectrochemical model is desired. Some of the questions that must be addressed in developing an appropriate model are:

1. Was the experimental system stationary? One distinc- tion between electrochemical and electrical systems is that electrochemical processes are typically dynamic, i.e., the rate of electrochemical reactions can change in response to dissolution of the electrode, deposition of salt or oxide

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

films, and adsorption or absorption of reacting species. If the system changed significantly during the time in which data were collected, this nonstationary behavior should be incorporated into the model.

2. How many physical processes can be resolved from the data? Electrochemical systems rarely display well- resolved peaks in the impedance spectra. Since the ulti- mate goal of interpreting impedance spectra is to gain in- sight into the physical processes that control the system, the model developed must incorporate time constants for the relevant physical processes. Another way to phrase this question is to ask whether the. lack of complete agreement between the data and the model should be attributed to experimental error, to nonstationary processes, or to use of an inadequate or incomplete model.

Determination of stationary behavior.---A system can be determined to be stationary through carefhl experi- mentation, through successful regression of the data to electrical-circuit analogues, or through direct integration of the Kramers-Kronig relations. Stationary behavior can be identified experimentally by replication of the impedance spectrum. If the experimental frequency range is sufficiently broad the extrapolation of the impedance spec-

1918 J. Electrochem. Soc., Vol. 139, No. 7, July 1992 �9 The Electrochemical Society, Inc.

t rum to zero frequency can be compared to the corresponding values obtained from separate steady-state experiments. The Kramers-Kronig equations, e.g.

and

Zi(to) = - ( ~ ) f~; Zr(x) - Zr(t~ =w2 [1]

2 f ; xZi(x) - toZi(~o) Zr(to) = Zr( | + -~ x 2 w2 dx [2]

provide a relationship between the real and imaginary components of impedance [Zr(~) and Zi(to), respectively] that hold under the assumptions of stability, causality, and linearity (3-6). Lack of consistency with the Kramers-Kro- nig relations in electrochemical systems can often be at- tr ibuted to failure of the stability criterion. Since electrical-circuit models satisfy the Kramers-Kronig relations, a system can be judged to be stationary if a satisfactory fit to a circuit model is obtained. If a satisfactory fit is not obtained, the question remains as to whether the lack of fit is caused by use of an insufficient model or by nonstationary behavior. Direct application of the Kramers-Kronig relations eliminates the need to select a model. Use of the relations, however, requires integration from 0 to ~ in frequency, and most electrochemical impedance data do not adequately approximate the required frequency domain. Several approaches have been taken tbr extrapo- lating data into the unmeasured frequency domain (7-16). Additional problems are caused by imprecision in the integration associated with a poor signal to noise ratio at the extremes of the frequency domain. As a result of these is- sues, direct application of the Kramers-Kronig relations in experimental electrochemistry is not commonplace.

If a system is not stationary, the model for the impedance response could be developed to account for the time- dependent behavior. The current practice, however, is to revise the experimental conditions to reduce the time required for the experiment and thereby allow the system to be represented by a pseudo-steady state model.

Selection of an appropr ia te model . - -Lack of complete agreement between the data and a model could be attributed to the error structure (or noise) of the experimental data, to nonstat ionary processes, or to the use of an inadequate or incomplete model. Whether based on electrical- circuit analogues or on solution of conservation equations, models for impedance spectra are sufficiently constrained that they cannot provide a framework for determining the noise in the data. Several approaches have been taken to resolve this issue. Artificial-intelligence methods have been suggested to identify optimal electrical circuit analogues for regression to data (17-19). A model is built se- quentially until all features of the data are represented. In- terpretation of the resulting regressed circuit components in terms of physical processes may be difficult because several electrical-circuit models could be used to model a given impedance spectrum. On another extreme, exclu- sive use of a limited group of circuit models has been pro- posed for screening of "industrial" corrosion data (20). This approach resolves the issue by accepting the best of a limited number of regressions, and therefore information on relevant physical processes may be lost. Usually, trial and error selection of circuit models is made unti l the fit is deemed to be acceptable.

Measurement Models Models can be classified as being one of two types. Pro-

cess models are used to predict the response of a system from physical phenomena that are hypothesized to be im- portant. Regression of process models to data allows identification of physical parameters based upon the original hypothesis. In contrast, measurement models are built by sequential regression of line shapes to the data. The model can be used to identify characteristics of the data set that could facilitate selection of an appropriate process model.

Fig. 1. The Voight elechical circuit analogue corresponding to the measurement model used in this work (Eq. [3]).

The measurement model selected for this work was the Voight model, i.e.

~k Z(~) = Zo +~(1T + jxk~) [3]

An electrical circuit corresponding to the Voight model is presented in Fig. 1. Use of the Voight model was moti- vated by the routine use of measurement models in the deconvolution of optical spectra (21). To be useful, the measurement model must contain line shapes that correspond to physical processes. Since stationary optical (and impedance) spectra satisfy the Kramers-Kronig relations, the model must also be consistent with the Kramers-Kronig relations. The impedance response of a single Voight element is usually attributed to the electrical response of a linearized electrochemical reaction (1, 2). The term Zo corresponds to the ohmic resistance, the time constant ~k for element k is equivalent to RkCk in the Voight model, and hk is equivalent to Rk. Equation [3] is consistent with the Kra- mers-Kronig relations.

The tenets underlying the use of measurement models for impedance spectroscopy are:

1. By including a sufficient number of terms, a general measurement model based on Eq. [3] can fit impedance data for typical stationary electrochemical systems. The object of this paper is to demonstrate that this postulate holds for electrochemical systems that exhibit the influ- ence of mass transfer (through Warburg elements), pseudo-capacitive behavior, frequency dispersion (through constant-phase elements), and kinetic control.

2. Because the measurement model does fit stationary impedance data, an inability to predict the imaginary part of the impedance spectrum with parameters obtained by fitting the real part of the spectrum (or, conversely, to predict the real part of the impedance spectrum with parameters obtained by fitting the imaginary part of the spectrum) can be attributed to failure of the data to conform to the assumptions of the Kramers-Kronig relations rather than to failure of the model. Thus, as discussed in Ref. (22) and (23), the measurement model could be used to assess the consistency of impedance data with the Kramers-Kronig relations without integration.

3. Another result of the adequacy of the measurement model is that the model can be used to identify the frequency-dependent error structure of impedance spectra. The error structure can then be used to weight the data during regression and to provide a means of deciding whether a given regression provided a "good fit" (22, 23).

4. It may be possible to use an expanded form of the measurement model to provide an indication of the number and types of physical processes that can be resolved from the data. The measurement model could include mass-transfer effects by adding to Eq. [3] a line shape corresponding to the Warburg element. This approach is discussed in Ref. (24). The model is adjusted until a statistically good fit is obtained with the smallest number of elements.

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, therefore, be checked for consistency with the Kramers-Kronig relations without actually integrating the equations over frequency. The use of measurement models does not require extrapolation of the experimental data set; therefore, inaccuracies associated

J. Electrochem. Sot., Vol. 139, No. 7, July 1992 �9 The Electrochemical Society, Inc. 1919

Circuit h

Circuit 2:

Ci=cuit 3:

Circuit 4:

Circuit 5:

R= @ Ra

. . . . . . . . . . .

R~ Rs

R, I_ . [~} J

t . Cz

R=

Fig. 2. Electrical circuit analogues for: 1, hydrogen evolution on LaNis (16); 2, corrosion of carbon steel in 3 w/o NaCI solution near the corrosion potential (28); 3, corrosion of iron in 0.5M H2SO4 at the corrosion potential (29); 4, corrosion of a model pit electrode in 0.5M NaCI (30); and 5, corrosion of a painted metal (31).

w i t h a n i n c o m p l e t e f r e q u e n c y s p e c t r u m are resolved�9 Fo r t h e a p p l i c a t i o n to a p r e l i m i n a r y s c r e e n i n g of t h e data , t h e u se of m e a s u r e m e n t m o d e l s is s u p e r i o r to t h e use of m o r e specif ic e lectr ical-c~rcui t a n a l o g u e s b e c a u s e one can de ter - m i n e w h e t h e r t h e r e s i d u a l e r ro r s are d u e to a n i n a d e q u a t e mode l , to fa i lure of da t a to c o n f o r m to t he K r a m e r s - K r o n i g a s s u m p t i o n s , or to e x p e r i m e n t a l noise�9

Approach for Demonstration of Applicability T h e u s e f u l n e s s of t h e m e a s u r e m e n t m o d e l d e p e n d s o n

t h e e x t e n t to w h i c h t h e l ine s h a p e for e ach e l e m e n t corre- s p o n d s to t h a t g e n e r a t e d b y phys i ca l p h e n o m e n a � 9 Fo r example , t h e u t i l i ty of m e a s u r e m e n t m o d e l s in t h e field of op t ics is d u e to t he a c c u r a c y of t he Loren tz i an , Drude , a n d D e b y e m o d e l s of op t ica l e x c i t a t i o n (25-27). T he c la ims m a d e h e r e for t he m e a s u r e m e n t m o d e l s r e q u i r e t h a t t he l ine s h a p e u s e d b e a d e q u a t e to r e p r e s e n t typ ica l i m p e d - a n c e data . Whi le th i s a p p r o a c h h a s b e e n app l i ed d i rec t ly to e x p e r i m e n t a l da t a (23, 24), t he i n t e n t h e r e is to d e m o n - s t r a t e app l i cab i l i t y to p u b l i s h e d da t a for s o m e r a t h e r com- p l ex sys t ems . T h e a u t h o r s c i ted h a v e u sed e lect r ical - c i rcu i t a n a l o g u e s to m o d e l t h e i r resul t s , and, in th i s work , t h e m e a s u r e m e n t m o d e l s we re app l i ed to s y n t h e t i c da ta o b t a i n e d f r o m t h e a u t h o r s ' mode l s . T he m o d e l s we re se- l ec ted for th i s w o r k b e c a u s e t h e y e n c o m p a s s p h e n o m e n a typ ica l ly s e e n in e l e c t r o c h e m i c a l i m p e d a n c e spec t r a (i.e., di f fus ion , p s e u d o c a p a c i t i v e behav io r , a n d f r e q u e n c y dis- pers ion) . T h e c i rcu i t a n a l o g u e s are p r e s e n t e d in Fig. 2, a n d t h e a s soc i a t ed p a r a m e t e r v a l u e s are p r e s e n t e d in T a b l e I.

T h e " s y n t h e t i c d a t a " to w h i c h t h e m e a s u r e m e n t m o d e l was a p p l i e d w e r e ca l cu l a t ed in d o u b l e p r ec i s i on f r o m the m o d e l s p r e s e n t e d in Fig. 2 a n d Tab le I. T h e da ta were , the re fo re , i n h e r e n t l y c o n s i s t e n t w i t h t he K r a m e r s - K r o n i g re la t ions , a n d t he on ly " n o i s e " in t h e da ta was c a u s e d b y r o u n d - o f f e r ro r in t h e ca lcu la t ion �9 U s e of s y n t h e t i c da ta p r o v i d e s a good t e s t of t h e m e a s u r e m e n t m o d e l b e c a u s e t he e r ro r s a s s o c i a t e d w i t h e x p e r i m e n t a l no i se a n d non - s t a t i o n a r y b e h a v i o r a re avoided �9 A poo r fit o f t h e m e a s u r e -

Table I. Values for circuit parameters.

1 (a and b)

Circuit: (16) 2 (28) 3 (29) 4 (30) 5(31)

R~,[~ 23.22 - - 0.5 RI , ~ 418.3 150 10-O 15 C~, ~ F 18.82 1000 25 128 Rz, I~ 695.2 150 200 - - C2, ~F 92.58 - - 50 -- R 3 , ~ - - - - 100 - - L,H - - - - 100 - - Rw, ~ 848.4 500 - - - - �9 w, s 0.7055 5 - - - - Rp, Q, - - - - - - 1135 TCPE, S -- - - -- 10

- - - - - - 0.79

IOOO 2 x I0 s I x 10 -3 4 x lO s I x IO -z

m e n t m o d e l to s y n t h e t i c da ta can, the re fo re , be a t t r i b u t e d on ly to i n a d e q u a c i e s of t he model �9 Two n o n l i n e a r re- g r e s s ion p a c k a g e s [ L O M F P b y J. R. M a c d o n a l d (32) a n d a p r o p r i e t a r y p r o g r a m w r i t t e n by L. G a r c i a - R u b i o (33)] we re u s e d to fit t h e m e a s u r e m e n t m o d e l to t h e s y n t h e t i c data , a n d t h e fits o b t a i n e d we re ident ica l . T h e da ta we re no rma l - ized s u c h t h a t e a c h p o i n t h a d e q u a l w e i g h t in t h e re- g ress ion . Th i s n o r m a l i z a t i o n was u s e d to avo id exces s ive

2 0 0 0

, 5 0 ~

& ,,;. ~o0o I

500 / o~176176176 ~ ~ e~162

04 . . . . ~ I

0 2 0 0 0 4 0 0 0

~'r' Ohm=

40O0

2ooe' ~ _ _ _ .

0

- 2 0 0 0 _ ~ = a l _ _ ~ . , , , , , , I . ~ , , , , . , 1 ~ . . . , I . . . . . . . . I__, . . . . .

2 0 0 0 t . . . . . . . . I . . . . . . . . l - - ' . . . . . . . I = ~ C - - ~ = ~ l ~ r ~ . . . . . . ' t

1500 ~ j

C r 5 0 0 !

0

10 - 1 10 0 101 10 2 10 3 10 4 10 5

f r e q u e n c y , HZ

Fig. 3. Regression of o measurement model with one Voight element to synthetic data 1: )) obtained from Circuit 1 (hydrogen evolution on [aNis (16)]. Solid lines in all figures are the results of the model fit. The residual sum of squares for the fit was 37.15.


800

g

I 4OO

2 ~

~ . ~ . , i I _ _ 1 i + �9 1 . . . . 1 . . . .

0 500 1C00 1500 2000

Z r , Ohm=

g

5OO

0

- 5 0 0 . . . . u d ~ a ''''1 ~ . . . . . . d . . . . . . . + . . . . . . . . t . . ~

IOOO

6OO

2O0

0

10 - 1 IO 0 101 10 2 10 3 10 4 IC~ S

Frequer~cy, HZ

Fig. 4. Regression of a measurement model with two Voight elements to synthetic data (0) obtained from Circuit 1 [hydrogen evolution on LaNi~ (161]. The residual sum of squares for the fit was 0.5588. Solid lines in all figures are the results of the model fit. The lines presented in the lower two figures are the model fit and its deconvolution into the contribution of each Volght element.

weigh t ing on the l ow- f r equency data wh ich typica l ly have m u c h la rger va lues for t he impedance .

Results and Discussion The sequen t i a l m a n n e r in wh ich a m e a s u r e m e n t m o d e l

is cons t ruc t ed is d e m o n s t r a t e d in t he first subsect ion. The appl ica t ion of the m e a s u r e m e n t m o d e l is fur ther demon- s t ra ted for sys tems in f luenced by di f fus ional p rocesses and induc t ive behavior . The appl icabi l i ty of the measure - m e n t m o d e l is also d e m o n s t r a t e d for a sys tem that was m o d e l e d by a cons tan t -phase e l e m e n t and for a high- i m p e d a n c e sys tem. The in te rpre ta t ion of pa rame te r va lues is d i s cus sed in t he final subsect ion .

Sequential construction of a model: Circuit 1.--In accord- ance with Eq. [3], a m e a s u r e m e n t mode l is const ructed by sequen t i a l ly add ing k Voigh t e l e m e n t s w i th pa ramete r s hk and ~k unt i l the fit is no longer i m p r o v e d by the addi t ion of yet ano the r e lement �9 This p r o c e d u r e is i l lustrated for syn- the t ic data ob ta ined f rom the e lect r ica l -c i rcui t m o d e l pro- posed for e v o l u t i o n of h y d r o g e n on a LaNi~ ingot e lec t rode (16). A di f fus ional res i s tance was obse rved in this sys tem as a resul t o f t he incorpora t ion of h y d r o g e n into the metal . The e lec t r ica l -c i rcui t m o d e l p re sen ted as Circui t i in Fig. 2

600

6

1

2 0 0

0 a , =_,+ | i i i i I i l , , I , , , ,

0 5 0 0 t 0 0 0 1500 2 0 0 0

Zr 0hms

600

~. ,oo g

~ 2 0 0

0 ,

10 - 1 100 101 102 10 "3 tO 4 105

rreque~y. Xz

Fig. 5. Regression of a measurement model with three Voight elements to synthetic data (0) obtained from Circuit 1 [hydrogen evolution on LaNi5 (1611. The residual sum of squares for the fit was 2.11 2 x 10 4. Solid lines in all figures are the resuhs of the model fit. The lines presented in the lower two figures are the model fit and its deconvolufion into the contribution of each Voight element.

accoun ted for mass t ransfer of h y d r o g e n wi th a Warburg e lement , and th ree t i m e cons tan ts are ev iden t in the c i rcui t analogue. The c o m p l e x Warburg impedance , g iven by

t anh (X/jTw~) Zw = Rw ( ~ [4]

can be der ived f rom the solut ion of V2ci = 0 for a film of t h i ckness 5 (1, 2). The t ime cons tan t can be expres sed in t e rms of di f fus ivi ty and f i lm th ickness as ~w = 82/D~ - The Warburg e l e m e n t is s o m e t i m e s cal led a d i s t r ibu ted ele- m e n t because it can be m o d e l e d by a d is t r ibut ion of relax- a t ion t i m e (1).

T h e m a n n e r in w h i c h the m e a s u r e m e n t m o d e l behaves u p o n success ive incorpora t ion of l ine shapes is demon- s t ra ted in Fig. 3 t h rough 7. The d i f fe rence be tween the s imula ted data and a m e a s u r e m e n t mode l incorpora t ing one Voigh t e l e m e n t is p re sen ted in Fig. 3. T h e res idual s u m of squares ob ta ined by the fit was 37.15. The res idual s u m of squares was i m p r o v e d to 0.5588 by addi t ion of a second l ine shape as seen in Fig. 4. Inc lus ion of a th i rd l ine

8 ~ 800

500

~. 4 0 0

7

200

l I I 1 l ~ J J I I I I J i l l l ~ J 1

0 500 1000 1500 2000

Z t . O h m l

2 0 0 0 ~ , ,,,,,,i . . . . . . . '] . . . . . . . ~ . . . . . . . 'I . . . . . . . ' l . . . . . .

~

- ~ 0 0 1 . . . . , i . J . . . . . . . . I . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

I

800

600

400

200

. . . . . . . . I . . . . ~ ' " 1 " " ' ' ' " " 1 . . . . . . . . I ' ' ' ' " " 1 . . . . . . . .

0

- 2 0 0 . . . . . . . . ! . . . . . . . . f - - -> . . . . . . . I _. j _ u , , , , , t j _ x . L u - u ~ L . ~ _ , . . . . .

I0- I 10 0 101 10 2 10 3 10 4 10 ~

Frequency, HZ

Fig. 6. Regression of a measurement model with four Voight elements to synthetic data (()) obtained from Circuit 1 [hydrogen evolution on LaNis (16)]. The residual sum of squares for the fit was 1.772 • 10 -7. Solid lines in all figures are the results of the model fit. The lines presented in the lower two figures are the model fit and its deconvolution into the contribution of each Voight element.

shape, shown in Fig. 5, resu l ted in a three-order of magni- t u d e decrease in the res idual s u m of squares . The improve- m e n t of the fit ach ieved by add i t ion of a four th e l emen t (Fig. 6) is no t readi ly seen in a compa r i son of Fig. 5 and 6. The inc lus ion of a four th e l emen t was, however , con- s idered to be s tat is t ical ly s ignif icant because the res idual s u m of squares was i m p r o v e d by an addi t ional three-or- ders of magn i tude . A s ignif icant i m p r o v e m e n t in the fit was also obse rved for five Voigh t e l emen t s (Fig. 7), and the resul tan t res idual s u m of squares was 4.915 • 10 1..

1921

. . . . 1 ' ' " ' " 1 . . . . 1 . . . .

600

6 N-. 46)0 I

200

J. Electrochem. Soc., Vol . 139, No. 7, J u l y 1992 �9 The Electrochemical Society, Inc.

o 500 1ooo 1500 2000

Z r , O h m s

2 0 0 0 ~ - ~ ' " ' " 1 . . . . . . . . I . . . . . . . . I . . . . . . . . I . . . . . . . . I . . . . . . !

L\ 4

1 5 0 0 L ~

"

- 5 0 0 1 . . . . . . . . ! . . . . . . . J . . . . . . . . I . . . . . . . . I . . . . . . . . I j . . . . . .

800

500

400

l 200

. . . . . . . . I . . . . . . . q . . . . . . . . I . . . . . . . . i . . . . . . . . I . . . . . . . .

O'

- 2 0 0 . . . . . . . I . . . . . . . . I . . . . . . . . I ~ . ~ J ~ u . [ . . . . . . . . ! . . . . . . .

1 0 - 1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5

r r ~ r ~ e n ( : ) . . HZ

Fig. 7. Regression of a measurement model with five Voight elements to synthetic data (0) obtained from Circuit 1 [hydrogen evolution on LaNis (16)]. The residual sum of squares for the fit was 4.915 • 10 '1~ Solid lines in all figures are the results of the model fit. The lines presented in the lower two figures are the model fit and its deconvolufion into the contribution of each Voight element.

The res idual s u m of squares ob ta ined by fitting the m e a s u r e m e n t m o d e l to the synthe t ic data is p resen ted as a func t ion of the n u m b e r of Voight e l emen t s in Fig. 8. Fig- ure 8 can be used to show that a m a x i m u m of five Voight e l emen t s can be reso lved f rom the synthe t ic data because the s ix th Voigh t e l e m e n t did no t improve the qual i ty of the fit. F igu re 8 also shows that, whi le visual inspec t ion of Fig. 5 sugges ts tha t the fit ob ta ined wi th three e l emen t s was accep tab le (resul t ing in res idual errors less than 0.1%), addi t ion of the four th and fifth e le raent p rov ided signifi- cant i m p r o v e m e n t in the res idual error. The res idual error ob ta ined by regress ing the f ive- t ime-constant Voight

1922 J. Electrochem. Soc., Vol . 139, No. 7, Ju ly 1992 �9 The Electrochemical Society, Inc.

10;

10 G

10 4 !

1 0 - 6

i0 -8

10-10 . . . . . . . . I . , , , , ~ . . . . I . . . . . . . . . I . . . . . . . . . [ . . . . . . . . .

2 3 4 5

Number of Time Constonts

Fig. 8. Normalized sum of squares for the regression of a measurement model to synthetic data obtained from Circuit 1 [hydrogen evolution on LaNi5 (16)] as a function of the number of Voight time constants employed in the model: (O) regression to synthetic data; (A) regression to synthetic data with random noise added; and (dashed line) normalized noise level.

m o d e l to t he da ta is p r e s e n t e d as a func t ion of f r e q u e n c y in Fig. 9. The ave rage res idua l e r ror was less t h a n 2 x 10 -4 pe rcen t . Severa l po in t s m u s t be m a d e here . The measu re - m e n t m o d e l b a s e d on a s u m m a t i o n of Vo igh t e l e m e n t s p r o v i d e d an e x c e l l e n t fit, in sp i te of t h e fact t h a t t h e c i rcui t i n c l u d e d a Warburg e l e m e n t w h i c h p rov ides an imp ed - ance r e s p o n s e tha t d i f fers s igni f icant ly f rom tha t ob ta ined for a res i s to r a n d capac i to r in parallel . The m e a s u r e m e n t m o d e l fit t he syn the t i c da ta to w i th in s e v e n s ignif icant f igures; the re fo re , t he m e a s u r e m e n t m o d e l can be con- s i de r ed to be s ta t is t ica l ly a d e q u a t e to fit t he c o r r e s p o n d i n g e x p e r i m e n t a l i m p e d a n c e da ta for w h i c h only 3 or 4 signif- i can t f igures can be e x p e c t e d .

To exp lo r e t h e po ten t i a l in f luence of e x p e r i m e n t a l error, r a n d o m no i se w i t h a m a x i m u m a m p l i t u d e of 5% was a d d e d to t h e r e su l t s o f Circui t 1 (and r e p o r t e d as Circui t lb). The op t ima l r eg re s s ion of t h e m e a s u r e m e n t m o d e l y i e lded only t h r e e Vo igh t c i rcui t e l e m e n t s as s h o w n in Fig. 10. As s h o w n in Fig. 8, only t h r e e e l e m e n t s cou ld be r e so lved b e c a u s e the c o n t r i b u t i o n of t he fou r th c i rcui t ele- m e n t was ins ign i f ican t as c o m p a r e d to t he no ise of t he data. T h e n o r m a l i z e d s u m of squa re s ob ta ined by re- g r e s s ion of t h e m e a s u r e m e n t m o d e l was s l ight ly l ower t h a n t h e n o i s e level w h i c h is s h o w n as a d a s h e d l ine in Fig. 9. The res idua l e r rors are p r e s e n t e d as a f unc t i o n of f r e q u e n c y in Fig. 11 a n d are r a n d o m l y d i s t r i bu t ed wi th re-

l x 1 0 - 5

5x10 - 6

0

_5x10 - 6

' ' ' " ' " I ' ' i t ' " ' l ' ' ' " ' " I . . . . . iT

o

% 0o 2o\ ~ 1 7 6 _ nu~176176176 o

0 o o c, o o o o Q

oo

10 -1 10 0 101 10 2 10 3 10 4 10 5

Frequency, Hz

I ~ . . . . . . . . I . . . . . . . . I o . . . . . . . . I ' ' '~ '" ' i . . . . . . . . I 'o ' ' ' ~ 4xi0 6 0 0

2 x l 0 - 5 o o % o o _

oo o ~ oo o %00 o 0 o o -

. . . . noo o

f - ~

~._ 0 ~ o ~ o ~ . o ~ o o o o o

o o o

' I o o

2x10 - 6 o o o o o

o o

4x10 - 6

o

- 6 x 1 0 - 6 , , , , , , , ,( . . . . . . . . t , ,,,HU{ h . . . . . . . I . . . . . . . . I ~ Li,~

10 -1 10 0 101 10 2 10 5 10 4 10 5

Frequency, Hz

Fig. 9. Residual errors (O) as a function of frequency for the regression of a measurement model with five Voight elements to synthetic data obtained from Circuit 1 [hydrogen evolution on LaNis (16)].

spect to frequency. The parameter values obtained through regression of the measurement model to the LaNis system are presented in Table II. The parameter values obtained for the first three elements changed slightly (about 5%) to compensate for the noise and the lack of the fourth and fifth elements. As expected, only a rough equivalency is seen between the fit parameters and the parameters of the original circuit. For example, the largest time constant obtained in the fit (Table II) was 0.387 s; whereas, the largest time constant in the original circuit (Table I) had a value of 0.7055 s. Good agreement was seen between the ohmic resistance of the original circuit R~ and the high-frequency limit of the measurement model Z0.

Table II. The results of the regression of the Voight measurement model (Fig. 1) to the circuits shown in Fig. 2.

Circuit: la (Warburg) lb (Warburg) 2 (Warburg) 3 (Inductor) 4 (CPE) 5

Zo, 12 23.22 23.77 O 0 0.50036 1000 A1, 12 266.48 289.7 10.60 58,695 274.25 436,900 ~1, s 0.006404 0.006569 0.7954 0.002932 3.339 0.00442 As, 12 1,163 1,201 28.04 - 102.05 13.1 163,100 �9 2, s 0.388 0.3870 4.101 0.9975 0.001823 0.000181 h3, f~ 494.0 528.8 22.83 343.35 3.323 T3, s 0.0760 0.0731 0.1131 0.01709 0.01164 h4, 12 25.61 51.50 11.29 �9 4, s 9.02723 0.07162 0.09767 h~, ~ 9.126 8.004 49.91 %, s 0.00854 0.2877 0.6158 AS, ~ 0.8880 162.62 �9 6, s 0.03041 57.67 AT, f~ 0.01248 605.75 vT, s 0.906603 12.15 ha, ~ 0.00003352 26.01 Ts, s 0.0004966 492.9

1. Circuit 1: hydrogen evolution at a LaNis electrode in 30 weight percent (w/o) KOH solution at an applied potential of -1.1 V (Hg/HgO) (16). Circuit lb has 5% random noise added.

2. Circuit 2: corrosion of carbon steel in 3 w/o NaC1 solution near the corrosion potential (28). 3. Circuit 3: corrosion of iron in 0.5M H2SO4 at the corrosion potential (29). 4. Circuit 4: corrosion of a model pit electrode in 0.5M NaC1 (30). 5. Circuit 5: corrosion of a painted metal (31).

J. Electrochem. Soc., Vol. 139, No. 7, July 1992 �9 The Electrochemical Society, Inc.

800

600

:-_ 4 0 0 ?

200

. . . . I . . . . . I ' ' ' ' 1 ' ' ' '

I J i l l , i L ~ l l t t l [ , i J l

0 . 0 3 0

0.020

0 . 0 1 0

' ''''"'I ' '''""I

0

0 0 0

0 0 00000

0 0 0 ,0 0 0

0 500 1000 1500 2 0 0 0

Zr, Ohms

2 0 0 0 ~ , l ' ' " ' " t . . . . . . . . t . . . . . . . . t . . . . . . . . t

L \ 1 5 0 0 L ~

- 5 0 0 t . . . . . . . ~ . . . . . . . . t . . . . . . . . t ......

' . . . . . . . I ' ' ' . . . .

. f : o

I

800

600

400

200

' ' " " 1

/ 0

- 2 0 0 . . . . . . .

10 - 1 1 0 0

. . . . . . . . I ' ' ' " " ' ! ' ' . . . . . . I . . . . . . . . I . . . . . . .

, , . . . . . . 1 , , , , . , J . . . . . . . . I . . . . . . . . ! . . . . . . t

101 102 103 104 105

Frequency, HZ

Fig. 10. Regression of a measurement model with three Voight elements to synthetic data (O) obtained from Circuit lb with added random noise (5% maximum amplitude). The residual sum of squares for the fit was 2.424 x 10 -2. Solid lines in all figures are the results of the model fit. The lines presented in the lower two figures are the model fit and its deconvolution into the contribution of each Voight elements.

Application to diffusion systems: Circuit 2.--In general, the number of Voight elements needed to model a system which is affected by diffusion exceeds the number of t ime constants implicit in a model that incorporates a Warburg element. While the contribution of the additional Voight elements can, as shown above, be insignificant in compari- son to exper imental noise, the discrepancy remains because a single Voight e lement simply cannot account for the frequency dependence associated with mass transfer. If a measurement model is to be used to identify the correct number of t ime constants for a given system, additional model elements must be incorporated in much the

1923

r N ~ I' N L

0 . 0 0 0

- 0 . 0 1 0 ~

- 0 . 0 2 0

- 0 . 0 3 0

10 - 1

' ' ' ' " " 1

0 0

0 N

0 0

0 0

0 0

0

OI 1 J z t t ~ l ~ I i ~ J l l l i } , ,,,,,~

100 101 102

' ' ''""I

0

0

0 0

0

0

0

0 0

0 0

. i ,,.HI

lO 3

' ' '"'"I ' ' '""

0

0 0

0 0 0 0 %0

o o

0 0 0 0

0 0

, , , , , , , J , ,JJ,J

104 105

Frequency , Hz

0.030

0.020

0 . 0 1 0

N- 0 . 0 0 0

{( ~- -o.0~0

I

I :

N,_.- - -0 .020

- 0 . 0 3 0

- 0 . 0 4 0

10 - 1

I n I ~ n H V I I i I l l ~ l ~ I "= q I g ' ~ ' u I

O - 0 0

0 0 0 0

0 0 O 0

0 0 0 0

m ~ e

0 0 0 D

0 0 0

0 0

O

O

J l l l l l l i i zt,,,,l t , ,,,,n]

100 101 t 0 2

' ' ' " " 1 ' ' ' " ' " 1 ' ' ' " ' = .

0 O o

O O

O O o O O O

0 ~ o o

o o

o o O~ 0 o

o oo

p , I f H . ] , ~ J l * , , , I , I I

103 104 105

Frequency , Hz

Fig. 11. Residual errors as a function of frequency for the regression of a measurement model with three Voight elements to synthetic data 0 obtained from Circuit l b with added random noise (5% maximum amplitude).

same way as multiple models (Lorentzian, Drude, or Debye) can be used to develop measurement models for optical dispersion. The number of model elements used is, of course, irrelevant if the measurement model is to be used simply to check the consistency of data to the Kra- mers-Kronig assumptions.

The application of the Voight measurement model is presented in Fig. 12 for the corrosion of carbon steel in 3 weight percent (w/o) NaC1 solution near the corrosion potential (28). The corresponding electrical-circuit model (presented as Circuit 2 in Fig. 2) included a Warburg diffusion element to account for the mass-transfer resistance of the solution. Eight Voight elements were needed, even though only thre e t ime constants axe present in Circuit 2. The results of this regression do show that the measurement model can provide an adequate representation of experimental impedance data that can be modeled in terms of a Warburg impedance. Similar agreement was obtained for direct analysis of experimental data for systems influenced by mass transfer (23, 24).

Application to inductive systems: Circuit &--The appear- ance of a positive imaginary component of the impedance has been attributed at high frequencies to the stray capac- ity of the current-measuring resistor and at low frequencies to adsorption phenomena at the electrode surface (1). Transport-based models for this behavior are rare, and most electrical-circuit models account for this behavior by incorporating either an inductor or a capacitor with anega-


4 . 0

10

3O 8

~

I

2O

J e I

' + ' ' l ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' "

o 20 40 60 80 100 120 t40

Zr, Ohms

8

150

tOO=

50

-50

I i I

I J I , l , I ,t 50

40

3O r f~ - 20 !

10

0

-lC

10 -4

I I I I

, I ~ I , I , I m

10 -2 10 0 10 2 10 4 10 6

Frequency, HZ

Fig. 12. Regression of a measurement model with four Voight elements to synthetic data (O) obtained [Tom Circuit 2 [corrosion of carbon steel in 3 w / o NoCI solution near the corrosion potential (28)]. The residual sum of squares for the fit was 3.763 • 10 4. Solid lines in oil figures are the results of lhe model fit. The lines presented in the lower two figures are the model fit and its deconvolutlon into the contribution of each Voight element.

t i ve v a l u e for t h e capac i t ance . T h e m e a s u r e m e n t m o d e l was a p p l i e d to p s e u d o c a p a c i t i v e da t a r e p o r t e d b y L o r e n z a n d M a n s f e l d (29) for c o r r o s i o n of i r on in 0.5M H2SO+. T h e i r m o d e l , s h o w n as C i rcu i t 3 in Fig. 2, i n c o r p o r a t e d a n i n d u c t o r to fit t h e p s e u d o c a p a c i t i v e r e s p o n s e at low fre- q u e n c i e s . T h e V o i g h t m o d e l c a n a c c o u n t for i n d u c t i v e be- h a v i o r i f t h e m a g n i t u d e (Ak) is a l l ow ed to h a v e a n e g a t i v e s ign. T h e r e g r e s s i o n to C i rcu i t 3 is g i v e n in Fig. 13. T h e m e a s u r e m e n t m o d e l i n d i c a t e s t h a t t h r e e t i m e s c o n s t a n t s c a n b e r e s o l v e d f r o m t h e s y n t h e t i c da ta , a n d t h i s r e s u l t is fu l ly c o n s i s t e n t w i t h t h e c i rcui t . C o m p l e t e a g r e e m e n t was

400

2O0

-200

5 0

k._.,,

, I , l ,

0 200 400 600

400

30o

200

g ~00 ~

0

- t00

-200

200

150

IO0

g ~

o

-50

-100 i

10 -4

Zr~ Ohms

I

J + I ~ ,I , I

l I " '1 I "

J I , L , .I l _ _

10 - 2 10 0 10 2 10 4 10 6

t%eque~cy~ HZ

Fig. | 3. Regression of a measurement model with three Voight elements to synthetic data {O) obtained [Tom Circuit 3 [corrosion of iron in 0.SM H2SO4 at the corrosion potential (29)]. The residual sum of squares for the fit was 3.221 • 10 -14. Solid lines in all figures ore the results of the model fit. The lines presented in the lower two figures are the results of the model fit. The lines presented Jn Re lower two figures are the model fit and its deconvolutlon into lhe contribution of each Voight element.

o b t a i n e d b e c a u s e t h e c i r cu i t is c o m p o s e d of on ly pas s ive e l e m e n t s (one i n d u c t o r , t h r e e res i s tors , a n d two capaci - tors). T h e r e s i d u a l s u m of s q u a r e s for t h i s fit was f o u n d to b e 3.221 • 10 -14. T h e m e a s u r e m e n t m o d e l p r o v i d e s a good fit to da t a e x h i b i t i n g a n i n d u c t i v e r e sponse .

Application to constant phase elements: Circuit 4 . - -The b r o a d e n i n g or d e p r e s s i o n of a s emic i r c l e in t h e i m p e d a n c e p l a n e p lo t c a n b e a t t r i b u t e d to a va r i e t y of phys i ca l phe - n o m e n a . F o r e x a m p l e , m o d e l s h a v e b e e n d e v e l o p e d t h a t

1925

500

400

300

I

200

100

, , I , , �9 I , , , I , , , I , , , I J ,

' " " l " " " l " " " I " " " I ' " " I " ' "

d. Electrochem. Soc., Vol. 139, No. 7, July 1992 �9 The 51ectrochemical Society, Inc.

0 2 ~ 4 ~ 6 ~ B ~ 1 0 0 0 12~

1200

1000

BOO

600

~ 4 o o

o

- 200

41x~

300

[ 6

. 200 I

100

0

-100

10 -4

Z r , Ohms ' I " l ' I ' I

, I , ) , I , J ,

I ( ! I 1

10 -2 100 102 104 10 t5

F r e q u e n c y , H Z

Fig. 14. Regression of a measurement model with eight Voight elements to synthetic data (~)) obtained from Circuit 4 [corrosion of a model pit electrode in 0.5M NaCI (30)]. The residual sum of squares for the fit was 1.491 • 10 -2. Solid lines in all figures are the results of the model fit. The lines presented in the lower two figures are the results of the model fit. The lines presented in the lower two figures are the model fit and its deconvolution into the contribution of each Voight element.

associate depressed semicircles to multiple or coupled reaction sequences, to roughening of the electrode, and to f requency-dependent ohmic resistances caused by a nonuniform charging of the double layer (1). This type of impedance response cannot be simulated by simply incorporating a Warburg term in the model; therefore, generalized impedance elements termed "constant-phase elements (CPE)" are often incorporated into the model. The measurement model was applied to the results of a circuit pro- posed by Kendig et al. (30) for a model pit electrode. The

electrical circuit is presented as Circuit 4 in Fig. 2. The mathematical formulation used for the constant-phase element was

Rp Z c e E - [5]

(1 + (jVCPE~O)P)

where 13 has value between 0 and 1. As seen in Fig. 14, with addition of a sufficient number of elements, the Voight measurement model can provide a good representation of a constant-phase element. Eight Voight elements were used in the regression; whereas, only two t ime constants are evident in the circuit itself. As shown earlier, experimental noise will reduce the number of Voight elements that can be resolved.

Application to coated metals: Circuit &--The measurement model presented here is most effective in modeling systems that can be modeled otherwise with passive elements such as resistors, capacitors, and inductors. It is per- haps no surprise that, with only two elements, the measurement model can provide a good representation of a circuit developed for painted metals that does not include diffusional or constant-phase elements. The electrical circuit is presented as Circuit 5 in Fig. 2. The fit to synthetic data obtained from a model for a pmnted steel is presented in Fig. 15. The normalized sum of squares for this fit was found to be 2.055 • 10 -14.

Interpretation of parameter values.--It is not intended that use of measurement models replace the development of physicoelectrochemical models or circuit analogues specific to a given system. Instead, the use of a measurement model is intended to guide selection and development of more specific models. The only generalization that can be made for electrochemical systems is that the high- frequency asymptote Zo corresponds to the ohmic resistance Rs. It may be possible to derive a relationship between the parameters ~k and hk and physical parameters, but such a relationship must be developed for each system.

Conclusions A measurement model based on the Voight circuit was

shown in this work to provide a statistically adequate fit to models that represent typical impedance data. The measurement model can be used to check the consistency of electrochemical impedance data to the Kramers-Kronig relations because the model itself satisfies the Kramers- Kronig relations and because the model provides a statistically adequate fit to electrochemical impedance data. The approach taken would be to fit the raodel to either the real or the imaginary component of the spectrum and to predict the other component from the regressed model parameters. The applicability of the model established here allows use of the standard deviation of model parameters to establish a confidence window for the predicted component. Thus, a sound statistical basis can be established for rejection of data as being inconsistent with a stationary process. This approach will be presented in a separate paper (22).

The measurement model provided a one-to-one identification of the t ime constants for a class of processes that are described in terms of passive elements. Thus, the measurement model may allow correct identification of the number of physical processes that can be discerned from the impedance response of systems that are unaffect- ed by mass transfer or by the frequency dispersion associated with nonuniform current/potential distributions. Solid-state systems may, for example, be suited for this type of analysis. The measurement model presented here cannot, however, be generally used for identification of the largest number of t ime constants discernible in a given spectrum because several Voight elements were needed to model a distributed element. A fu]~ther extension of the measurement model approach is required. It may be possible, for example, to use a measurement model to identify the number and type of physical processes discernible in a given spectrum that is influenced by mass transfer, if

1926 J. Electrochem. Sot., Vol. 139, No. 7, July 1992 �9 The Electrochemical Society, Inc.

2 . 5 . 1 0 b

2 .0~10 5

1 ' 5N10 5

O

I l ' O x l O 5

5 . 0 x 1 0 4

- - " ' ' I ' ' 1 1

t i

2 x l O 5 4wlO S 6 x 1 0 5 Bx lO ~

& ,,j- i

[, &

N

8x10 S

6x10 S

4 x i O 5

2x10 5

~

2.5x105

2.0x105

1.5xI0 5

I00xi04

5.0wlO 4

-~.Ox100

i0 -I

' ' ' ' " " i

Z t , Ohm l

' ' "H ' l i ' ' . . . . . . I ' ' ' ' " " 1 ' ' ' " ' " l

. . . . . . . . I ' " ' '""1 . . . . . .

10 0 101 10 2 10 3 I0 # 10 5

Ff eqv l e r~cy . H i

Fig. 15. Regression of a measurement model with two Voight elements to synthetic data (~.)) obtained from Circuit 5 [corrosion of a painted metal (31)]. The residual sum of squares for the fit was 2.055 x 10 -14. Solid lines in all figures are the results of the model fit. The lines presented in the lower two figures are the model fit and its deconvolution into the contribution of each Voight element.

model e lements are included that account for diffusion phenomena. An e lement based, for example, on a Warburg (diffusional) impedance could represent diffusional processes with a single t ime constant. A measurement model would be built by summing elements represent ing differ- ent physical processes in a manner analogous to bui lding a measurement model for optical dispersion by summing Lorentzian, Drude, and Debye e lements (24).

Acknowledgments The assistance of J. Mat thew Esteban is greatly appreci-

ated. The port ion of this work performed at the Universi ty

of Flor ida (PAN and MEO) was conducted in suppor t of a series of projects suppor ted by the Office of Naval Re- search under Grant Number N00014-89-J-1619 (A. J. Sed- riks, Program Monitor), by Gates Energy Products, Gainesville, Florida, by the National Science Foundat ion under Grant No. EET-8617057, and by DARPA under the Optoelectronics program of the Flor ida Init iat ive in Ad- vanced Microelectronics and Materials. The work performed at the Universi ty of South Flor ida (L. H. G -R.) was suppor ted by the National Science Foundat ion under Grants No. RII-8507956 and INT-8602578.

Manuscr ipt submi t ted Ju ly 25, 1991; revised manuscr ip t received Jan. 10, 1992.

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ceschetti , in " Impedance Spect roscopy Emphasiz- ing Solid Materials and Analysis," J. R. Macdonald, Editor, Chap. 2, John Wiley & Sons, Inc., New York (1987).

2. C. Gabrielli , "Identif icat ion of Electrochemical Proc- esses by Frequency Response Analysis," Solartron Ins t rumenta t ion Group Monograph, The Solartron Electronic Group Limited, Farnborough, England (1980).

3. R. de L. Kronig, J. Opt. Soc. Am. Rev. Sci. Instrnm., 12, 547 (1926).

4. R. de L. Kronig, Physikal. Z., 30, 521 (1929). 5. H.A. Kramer, ibid., 522 (1929). 6. H. W. Bode, "Network Analysis and Feedback Ampli-

fier Design," D. Van Nostrand Company, Inc., New York (1945).

7. V. A. Tyagai and G. Ya. Kolbasov, Elektrokhimiya, 8, 59 (1972).

8. R. L. Van Meirhaeghe, E. C. Dutoit, F. Cardon, and W. P. Gomes, Electrochim. Acta, 20, 995 (1975).

9. D. D. Macdonald and M. Urquidi-Macdonald, This Journal, 132, 2316 (1985).

10. M. Urquidi-Macdonald, S. Real, and D. D. Macdonald, ibid., 133, 2018 (1986).

11. C. Haili, M. S. Thesis, Universi ty of California, Berke- ley (June 1987).

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

13. D. Townley, ibid., 137, 3305 (1990). 14. M. Urquidi-Macdonald and D. D. Macdonald, ibid.,

137, 3306 (1990). 15. J. M. Esteban and M. E. Orazem, ibid., 138, 67 (1991). 16. M. E. Orazem, J. M. Esteban, and O. C. Moghissi, Cor-

rosion, 47, 248 (1991). 17. B. A. van Hassel, I. C. Vinke, J. K. de Vries, B.A.

Boukamp, and A . J . Burggraaf, This Journal, 136, 420C (1989).

18. Z. B. Stoynov and B. S. Savova-Stoynov, J. Electro- anal. Chem., 170, 63 (1984).

19. Z. B. Stoynov and B. S. Savova-Stoynov, ibid., 209, 11 (1986).

20. D. C. Silverman, Corrosion, 46, 589 (1990). 21. "Deconvolution: with Applicat ions in Spectroscopy,"

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22. P. Agarwal, M. E. Orazem, and L. H. Garcia-Rubio, Paper presented at ASTM Internat ional Sympos ium on "Electrochemical Impedance: Analysis and Inter- pretat ion," San Diego, CA, Nov. 4-5, 1991.

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Time-Averaged Current Distribution for a Rotating-Disk Electrode under Periodic Current Reversal Conditions

K.-M. Jeong, C.-K. Lee,' and H.-J. Sohn* Depar tmen t of Mineral and Pe t ro leum Engineering, Seoul National University, Seoul, Korea 151-742

ABSTRACT

The time-averaged current distribution over the surface of a rotating-disk electrode is calculated under conditions of periodic current reversal by simultaneously solving the transient-convective-diffusion equation and Laplace's equation. The calculated results compare well with experiments performed using the copper/copper sulfate system. The grid search technique is used to determine the opt imum plating conditions in terms of uniform thickness of electrodeposits by vary- ing the duty factor and the ratio of anodic to cathodic current density.

Pulsed current (PC) technology has long been used in electroplating industries due to the improvements in the quality of electrodeposits, and its characteristics are well described in the monograph by Puippe and Leaman (1). But the current distribution over the electrode surface in pulse plating is generally less uniform than that in dc plating. Recently, Wan and Cheh (2) developed a mathematical model of the pulse plating of copper for the case below the limiting current density condition and calculated the ter- tiary current distr ibution on a rotating-disk electrode. Also Chin et al. (3), computed the secondary current distribution on a planar electrode neglecting the mass-transfer effect. They showed that the nonuniform ohmic potential drop in the solution outside the diffusion layer contributed to the nonuniform current distribution.

In order to minimize the nonuniformity of thickness distribution, organic additives as well as periodic current reversal (PCR) technique has been practiced in plating industries largely based on empiricism. The organic additives which act as levelers or brighteners generally tend to increase the polarization resistance, leading to an increase in throwing power (4). Popov et al. (5) showed that the macrothrowing power was considerably better with a PCR than with a PC. Also Granato and Sobral (6) found that dendritic growth of gold was not observed even at high current density in electrorefining with PCR technique.

Fedkiw and Brouns (7) developed a mathematical model of PCR plating on a planar electrode with an assumption of steady state. Recently, Pesco and Cheh (8, 9) presented a simulation of plated through-holes, and compared the current distribution between dc, PC, and PCR. They indi- cated that plating with PCR appeared to be a promising technique to obtain more uni ibrm metal-thickness distribution at low current densities. In addition, it was demon- strated that the PCR technique can decrease the energy consumption due to the low anodic overpotential at the lead electrode in the electrowinning of zinc in acidic media (11) and increase the current efficiency of zinc electrodep- osition in the presence of 5 mg/1 of lead (12). Mass6 and Piron (13) also showed that the beneficial effect of PCR in zinc electrowinning remained even in the presence of nickel.

In this study, a mathematical model to describe the current distribution on the rotating-disk electrode under periodic reversal conditions is presented with experimental confirmations followed by obtaining the opt imum plating condition in terms of process variables.

* Electrochemical Society Active Member. Present address: Korea Institute of Geology, Mining, and Mate-

rials, Daejeon, Korea.

Theoretical A mathematical model used in this study is similar to

that developed earlier by Wan and Cheh (2) except for the boundary condition for the anodic pulse current. Popov and Maksimovid (14) applied the same type of model for PCR plating without convective diflhsion. The transient material-balance equation together with Laplace's equation for the modeling of PCR plating will be described briefly.

Concentration profile in diffusion layer.--Electrodepo- sition with PCR technique on the rotating-disk electrode can be described in terms of the t ime-dependent material- balance equation, which can be expressed as

0C 0C ,]C 02C - - + v ~ - - + V z - - = D - - [1 ] Ot Or Oz oz 2

with the boundary conditions

C = C b O<-r, O<-z at t = O [2]

C = C b r 2 + z 2-+| at t > 0 [3]

aC i - D - z=O, f o r 0 ~ - r < r 0 at t > 0 [4]

Oz nF

aC - - = 0 r=O, for al lz at t > 0 [5] ~'Jr

where C is the concentration of species electrodeposited, Cb is the concentration of the same species in the bulk solution, D is the diffusivity, F is the Faraday constant, n is the number of electrons involved in the electrode reaction, t is time, vz and v~ are the fluid velocity in axial and radial directions, respectively, z is the distance from the disk in axial direction, and r is the radial direction coordinate.

The local current density, i in Eq. [4] is the periodic function of time as (9, 14)

ir for O<-t<t l , t 2< t< t3 , . . . i = - ia for t l - < t < t 2 , t3:-<t<t4,... [6]

where ic is the local cathodic currem~ density and i, is the local anodic current density, respectively. Figure 1 shows the form of periodic current reversal. The applied cathodic current density ipc and the anodic current density, ie, are related to the local current density

2 fr o iPc=r~oJo i c r d r for 0 - < t < t , , t 2 - - t< t~ . . . [7]

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