J. Electrochem. Soc., Vol. 138, No. 1, January 1991 �9 The Electrochemical Society, Inc. 67

27. P. Furchhammer and H. J. Engell, Werkst. Korros., 20, 1 (1969).

28. Y. M. Kolotyrkin, This Journal, 108, 209 (1961). 29. H. P. Leckie and H. H. Uhlig, ibid., 113, 1262 (1966);

116, 906 (1969). 30. J. O'M. Bockris, D. Dra~id, and A. Despid, Electrochim.

Acta, 4, 325 (1961). 31. S. Asakura and K. Nobe, This Journal, 118, 13, 19

(1971). 32. C. L. McBee and J. Kruger, in "Localized Corrosion,"

R.W. Staehle, B. F. Brown, and J. Kruger, Editors, p. 252, NACE, Houston, TX (1974).

33. T. P. Hoar and W. R. Jacob, Nature, 216, 1299 (1967). 34. T. P. Hoar, Corros. Sci., 7, 355 (1967). 35. N. Sato, Electrochim. Acta, 16, 1683 (1971). 36. N. Sato, This Journal, 129, 255 (1982). 37. G. Okamoto and T. Shibata, Corros. Sci., 10, 371 (1970). 38. W. E. O'Grady and J. O'M. Bockris, Surf. Sci., 10, 249

(1973). 39. G. Okamoto, Corros. Sci., 13, 471 (1973).

40. W. E. O'Grady, This Journal, 127, 555 (1980). 41. J. Kruger, in "Passivity and Its Breakdown on Iron

and Iron-Based Alloys," R. W. Staehle and H. Okada, Editors, p. 91, NACE, Houston, TX (1976).

42. M. A. Heine, D. S. Keir, and M. S. Pryor, This Journal, 112, 29 (1965).

43. M. J. Pryor, in "Localized Corrosion," R.W. Staehle, B.F. Brown, and J. Kruger, Editors, p. 2, NACE, Houston, TX (1974).

44. C. Y. Chao, L. F. Lin, and D. D. Macdonald, This Jour- nal, 128, 1187, 1194 (1981).

45. J. A. Richardson and G. C. Wood, Corros. Sci., 10, 313 (1970); This Journal, 120, 193 (1973).

46. K. Hashimoto and K. Asami, Corros. Sci., 19, 251 (1979). 47. R. Nishimura, M. Araki, and K. Kudo, Corrosion, 40,

465 (1984). 48. B. MacDougall, D. F. Mitchell, G. I. Sproule, and M. J.

Graham, This Journal, 130, 543 (1983). 49. M. Urquidi-Macdonald and D. D. Macdonald, ibid.,

134, 41 (1987).

On the Application of the Kramers-Kronig Relations to Evaluate the Consistency of Electrochemical Impedance Data

J. Matthew Esteban* and Mark E. Orazem*

Department of Chemical Engineering, University of Florida, Gainesville, Florida 326I t

ABSTRACT

The use of the Kramers-Kronig (KK) relations to evaluate the consistency of impedance data has been limited by the fact that the experimental frequency domain is necessarily finite. Current algorithms do not distinguish between the re- sidual errors caused by a frequency domain that is too narrow and discrepancies caused by a system which does not sat- isfy the constraints of the KK equations. A new technique is presented which circumvents the limitation of applying the KK relations to impedance data which truncate in the capacitive region. The proposed algorithm calculates impedance values below the lowest experimental frequency which "force" the data set to satisfy the KK equations. Internally consist- ent data sets yield low-frequency impedance values which are continuous at the lowest measured experimental fre- quency. A discontinuity between the calculated low-frequency values and the experimental data indicates inconsistency which cannot be attributed to the finite experimental frequency domain.

To facilitate the interpretation of impedance meas- urements, an investigator should know whether the exper- imental data is characteristic of a linear and stable system. It has been suggested that the Kramers-Kronig (KK) rela- tions can be employed to evaluate and analyze complex impedance data of electrochemical systems (1-3). These equations, developed for the field of optics, constrain the real and imaginary components of complex physical quan- tities for systems that satisfy the conditions of causality, linearity, stability, and finite impedance values at the fre- quency limits of zero and infinity (4-6). Bode (7) extended the concept to electrical impedance, and has tabulated var- ious forms of these equations. Macdonald and Urquidi- Macdonald (8) have demonstrated analytically that equiva- lent electrical circuits involving passive elements (R, C, L), the Warburg impedance, the pore diffusion model, R--C transmission lines, R--L transmission lines, and R--C--L transmission lines obey the KK relations. Consequently, experimental data that can be fitted to the analytical re- sponse of an equivalent circuit described above through nonlinear regression analysis satisfy the conditions of the KK relations.

There is, however, controversy over the extent to which the KK relations can be used to validate electrochemical impedance data. The relationships are held to be valid, but Mansfeld and Shih (9-11) have argued that they are useless for data that do not include all the time constants for the system. Since the KK relations involved integrals over fre- quencies ranging from zero to infinity, valid data can ap- pear to be invalid if the measured frequency range is insuf- ficient. Macdonald et al. (2, 3, 12) have repeatedly emphasized the importance of collecting experimental

* Electrochemical Society Active Member.

data over a sufficiently wide frequency range in order to evaluate the KK relations with satisfactory accuracy, but this is not always possible. While the upper limit of mod- ern frequency analyzers (65 kHz to 1 MHz) is sufficient for most electrochemical systems, the lower measureable fre- quency limit for systems exhibiting large time constants is often governed by noise. It is this value that currently re- stricts the utility of the KK transforms.

There is, therefore, a need to address the problem of ap- plying the KK relations to data sets with finite frequency domains which do not extend to a sufficiently low value. Once this problem is properly resolved then the sensitivity of the KK relations to evaluate experimental impedance data for violations of the causality, linearity, and stability conditions can be individually investigate d . Previous au- thors have approached the problem differently:

Mansfeld and Shih (9-11) stated that the KK transforms yield valid results only when the impedance data have reached a dc limit within the experimental frequency do- main. Application of a KK algorithm to valid data sets which truncate in the capacitive region result in discrepan- cies that may erroneously lead to the conclusion that the data sets are invalid. These discrepancies, however, are due to the neglected contributions to the integrals associ- ated with the inaccessible frequency domain (12). In one specific case, they were able to validate the experimental data using a KK algorithm only after having extrapolated the data below the lowest measured frequency using the fitting parameters of an equivalent circuit (10). This shows the importance of performing the integration over the widest frequency domain possible. It should be pointed out that, since the impedance response of electrical cir- cuits satisfy the KK relations, a "good fit" between the ex- perimental data and the analytic response of an equivalent

68 J. Electrochem. Soc., Vol . 138, No. 1, J a n u a r y 1991 �9 The Electrochemical Society, Inc.

circuit (i.e., which exhibit randomly distributed residuals) indicates that the data set satisfies the constraints associ- ated with the KK relations (8, 13).

Macdonald et al. (3) suggested extrapolating the experi- mental data beyond the frequency extremes using polyno- mial expressions obtained from a regression analysis of the data set. Although this procedure minimizes the prob- lem of the "the inaccessible 'tails' of the integrals" (12), ex- trapolation of higher order polynomials in this manner is dangerous and may be justifiable only within a narrow fre- quency domain.

It is evident from the discussions of Shih and Mansfeld and Macdonald and Urquidi-Macdonald that the applica- tion of the KK relations to a broad range of experimental impedance data requires a method to eliminate errors as- sociated with the unmeasured frequency domain. The ob- ject of this work was to develop such an algorithm that can distinguish between errors due to a finite frequency do- main that is too narrow and discrepancies due to data that are truly inconsistent with the KK relations. This al- gorithm is intended to reduce the possibility for false re- jection of a valid data set caused by an insufficient experi- mental frequency domain. The utility of the proposed technique is illustrated using synthetic data derived from equivalent circuits. Since the conditions of the KK rela- tions are necessarily satisfied, the sensitivity of the tech- nique for checking the violation of the stability criterion was investigated. The authors are currently investigating the application of this procedure to t ime-dependent pro- cesses such as the corrosion of copper, a luminum alloys, and hydrogenation of metal hydrides, and the extension of the method for application to purely capacitive systems. These will be addressed in subsequent papers.

Zr(W)

Concept Through the KK relations, the value of one impedance

component (real or imaginary) can be calculated at any given frequency if the other component is known over the entire frequency range. However, experimental difficulties constrain the acquisition of data within a finite frequency domain ~ i , -< to < to . . . . The apparent lack of agreement between the experimental data and the corresponding KK transformations can be attributed to two factors: (i) the problem of residuals due to the unmeasured impedance values in the frequency ranges o~ < ~,.m and ~o > t o~ , and, (ii) to processes associated with a nonstationary, non- linear, and/or noncausat system.

The contributions to the integrals of the KK relations predominantly arise in the region near the frequency being evaluated (1). This means that if an experimental data Set truncates in the capacitive region the problem of residuals described by factor (i) will be much more signifi- cant at the low-frequency range than at the high-frequency range. This is evident from the examples given in Ref. (9-11) where the greatest discrepancies between the experimental data and the calculated impedance values occurred at low frequencies.

The concept behind this work is that functions Z~(to) and Z~(~o) can be found for the low-frequency domain coo <- to < to~, which, when appended to the experimental data set, force the data to satisfy the KK equations over the frequency range o~0 to ~O~,x. Residual errors at the high-fre- quency extreme do not influence this calculation since the major contribution to the integrals of the KK relations is from the lower frequency domain. The consequence of this assumption will be discussed in a later section. The frequency range to0-(Omax is chosen to satisfy the require- ment that the impedance spectrum does not terminate in the capacitive region; i.e., the real component attains an asymptotic value and the imaginary component ap- proaches zero as r --~ too. Internal consistency between the impedance components also requires that the calculated functions be continuous with the experimental data at torero. These requirements cannot simultaneously be satis- fied for data from systems that do not satisfy the con- straints of the KK relations. Discontinuities at tom~, can therefore be attributed to properties unrelated to the use of a finite frequency range in the collection of data.

] [ T I 92 t Ne o o o o o o o O O O o t ~g~.t s~ ~ S~gm~t

: ~ I ol m Experimental

7 ~ .~. z, k ,l~ Frequency k-o . I ~ in terva l

\ Low F requency Interval

S~;ment Segment I ~ 8 ~ e ~ 4~ 1 I ' NC o o o o e

~t (Drnin (Drnax

I K t I NC 2 N e

Low Frequenc Experimental Interval a I ~, Frequency

I ~ Interval M a I z.= o 0ooo,r J \

oa i o a 1

s I Q)~ q GJmi n COma x

log ca

Fig. 1. A representation of the algorithm to calculate the low-fre- quency impedance values for an experimental data set that truncates in the capacitive region. The mathematical development of the algorithm is given in Appendix A.

A graphic representation of the algorithm is given in Fig. 1, and the mathematical development is presented in Appendix A.

Results and Discussion To test the approach, the algorithm was applied to sev-

eral synthetic data sets. The objectives were: 1. To show that consistent data truncated in the capaci-

tive region will be deemed to be consistent. Internally con- sistent data were generated within finite frequency ranges for the various equivalent circuits illustrated in Fig. 2. The impedance spectra of these equivalent circuits necessarily satisfy the KK relations; therefore, the ability of the al- gorithm to calculate the low-frequency impedance values which are continuous with the experimental data at tomm is demonstrated. The analytic impedance equations for the electrical circuits are listed in Appendix B.

2. To show that the method can discriminate between errors due to a finite frequency range that is too narrow and discrepancies due to data that are truly inconsistent with the KK relations. A nonstationary system was simu- lated by changing the value of a circuit element through the course of the "experiment ." The appearance of a dis- continuity at ~Omin is shown to be characteristic of inconsis- tent impedance data.

In ternal ly consistent da ta se t s . - -Syn the t i c data are gen- erated for the equivalent circuits shown in Fig. 2. Ten "ex- perimental" data points were obtained per logarithmic decade of frequency. The truncation frequencies were chosen such that the data sets did not exhibit a dc limit in the low-frequency range. The impedance values below tomm were calculated using the algorithm presented in this paper, and the results are shown in Fig. 3 to 7; the analytic impedance of the electrical circuits are shown for compari- son. These figures show that the calculated low-frequency impedance functions, Zr(r and Zi(~ol), are continuous with the experimental data at tomi, and closely approximate the theoretical values. It can be shown that the "complete" im- pedance data sets (i.e., the calculated low frequency im- pedance values appended to the experimental data set) satisfy the KK relations using the algorithms suggested by Macdonald et at. (3).

The utility of the approach described here is that an equivalent circuit fit is not required to determine the con- sistency of the impedance data. Moreover, for data sets

J. Electrochem. Sac., Vol . 138, No. 1, Janua ry 1991 �9 The Electrochemical Society, Inc. 69

C 1 = 100#F 12.o

R s = I0 I'~ lO.O

V V V ~r 6.0

R~ = 105s

3 4.0 C~ = IO/~F L

b) O v v v JR~^~OOn \A/J O 00

- - V V V ~ v v ~ -~ ,0

% = 2 0 0 0

Tw= 1.5sec

Im,I~ Jm,4 I.,I~ l i.J,,, I IHHI, I *.,,~ I.,,4 JH.,~ IH,, a

~ A . : Th . . . . t t c . ) Imped . . . . b ~ f t b: Calculated Impedanc.__

2 -

0.0 Umln

sIIIIl~ lltH+m~ IIIHI~ r+i1slJ~ iilS,~ rillllll SHH~ ImlIl,i lUl l.o 10 -s 10 -s t0 -4 10 -3 10 -a 10 -t t00 10 t 102 103

FPequency Hz

-6 .0

- 5 .0

-4 .0

- 3 .0

-2 .0 ~

N - t . 0

c) o

Yo = 1100#F CPE

ItS] n =O.71 Rs = 1.87fl r - - - - l ~ - - I

i IDOl i

R 1 = 5.80[1

C~ = 19,uF

R,: - L-A/VV- R2= 6951"]

Fig. 2. The electrical circuits employed to generate synthetic imped- ance data. The analytic expressions for the impedance components are presented in Appendix B.

which do not exhibit a dc limit, employment of the KK re- lations no longer requires extrapolation below the lowest measure frequency using the fitting parameters of an equivalent circuit (11) or through polynomial expressions obtained within the experimental frequency domain (3). The proposed KK algorithm will calculate accurately the lower-frequency impedance (i.e., within 5% of the theoreti- cal values) if the experimental data provides enough infor- mation about the dominant t ime constant in the lower-fre- quency domain. It should be pointed out that the data set obtained from the electrical circuit of Fig. 2a was previ- ously employed (11) to illustrate the shortcomings of the KK algorithm suggested by Macdonald et al. (3). This same data set is deemed to be consistent using the KK algorithm presented in this paper, as shown in Fig. 3.

The truncation frequencies of the data sets in Fig. 3 and 4 were sufficiently low to provide enough information on all the t ime constants of the equivalent circuits. The effect of increasing r is illustrated in Fig. 5 to 7. The low fre- quency impedance values could not be accurately calcu- lated as the experimental data moved further into the ca- pacitive region. This reflects the lack of information about the impedance behavior of the system, particularly of the dominant t ime constant within the low-frequency region. It has been observed for consistent data sets that the conti- nuity of the calculated real impedance at carom is an indica- tion of consistency, while the appearance of a disconti- nuity for the calculated imaginary impedance reflects the missing t ime constant(s) in the lower frequency domain (see, for example, curve c of Fig. 6 and 7).

Cons is ten t d a t a sets w i t h s c a t t e r . - - T h e applicability of the algorithm to data sets with scatter is presented in this

Fig. 3. Calculated low-frequency impedance values for the simple Randles circuit of Fig. 2a. The truncation frequency co=i. is 10 -z Hz. a, Theoretical impedance response; b, calculated impedance values in the frequency range 10 -6 to 10 -z Hz.

section. Two "noisy" data sets (hereafter referred to as "good-noise" and "bad-noise") were simulated using the analytic impedance response of the equivalent circuit of Fig. 2d and a random number generator. Random "noise" was added or subtracted up to 5% of the impedance value at a given frequency. These data sets were fitted to the equivalent circuit using LOMFP (14), a complex nonlinear least squares immittance fitting program. The goodness of fit between the experimental data and the analytic re- sponse can be visually ascertained from the relative resid- ual plots shown in Fig. 8 and 9, for good-noise and bad- noise, respectively. A "good" fit will exhibit a random dis- tribution for the residuals about the analytic impedance values, such as that shown in Fig. 8, and verifies the consis- tency of the data set called good-noise. The residual plots for bad-noise displays a nonrandom distribution for the real impedance, and provides a measure of the ill-fit of the data set to the chosen equivalent circuit.

The low-frequency impedance values below r were calculated using the KK algorithm, and the results are pre- sented in Fig. 10 and 11 for good-noise and bad-noise, re- spectively. The calculated lower frequency impedance values for good-noise are shown to be continuous with the experimental data at the truncation frequency era, in, and is

350

30O

250

200

0 t50

100

50

0

5 0

I I L

b

IIII1~ I IIllrl~ I lll i i~ I Il~l~ I IIIIIl~ I lil l l~ i llliill t lJlllJ~ I fllllt~ I IHI& a

C 0: Theoret~ce) Impedance

b: u O - iO -5 HZ

C: W e - t0 -3 HZ

t00

6O

40

N I

0

t Umtn

-IIIIIIII I I I I I I l l IIII111~ I I I I I I I I I I I I I I l l IIIIIIIII IIItt1~ III111111 Jllllll~ I I I I -20

jO -5 40 -4 10 -3 10 -2 10 -t t00 t0 t dO 2 103 104 t05 Frequency. Hz

Fig. 4. Calculated low-frequency impedance values for the Randles circuit with Warburg impedance shown in Fig. 2b. The truncation fre- quency is 0.1 Hz. The accuracy of the algorithm was improved for curve b because the problem of residuals was minimized by using a lower value for COo. a, Theoretical impedance response; b, calculated imped- ance values in the frequency range 10 -s to 0.1 Hz; c, calculated im- pedance values in the frequency range 10 -3 to 0.1 Hz.

20

70 J. Electrochem. Soc., Vol . 138, No. 1, J a n u a r y 1991 �9 The Electrochemical Society, Inc.

~.0 l,liilll I I'lNlli~ ,liillit-illlill I ,illilll I ,,lilii I lilili, I ililiil I lllilill-Hl'll ~.o

6,0 - - b a

- C - ~ I: Theor'etlcal -- l.S

7.0

_ b: I t i l i n - l HI

6.0 - - C: ~.1, " i0 Ha - - t . 2

5.0 0 0.8

4.0 L

3,0 0.4

t , 0

o.o u,r I =Hind ~H~lal IHI,~ ~u,~ ~IHI~ Im~ ~und ~mld ~ml -~.4 i0 -~ ~0 -~0 -~ t0 ~ ~0 ~ ~0 ~ ~0 ~ ~104 t0 ~ tO s t07

Frequency. Hz

Fig. 5. Calculated low-frequency impedance values for the Rondles/ CPE electrical circuit of Fig. 2c. This figure illustrates the effect of in- creasing truncation frequency, while maintaining the value of r o at 10 -z Hz. a, Theoretical impedance response; b, calculated impedance values in the frequency range 10 -~ to 1 Hz; c, calculated impedance values in the frequency range 10 -3 to 10 Hz; d, calculated impedance values in the frequency range 10 -~ to 50 Hz.

SO0

500 II1\1 ~ O . , n - O , . ,

i i i t ~ ~ 1 7 6 400 / J / ~ d : Umj n - 5 .0 H i

8

i ' ~ 200

o

7 -zoo ~n~ ~md ,m~,~ ~n~ ~H~ ~lm~ ~n~ ~HH~I ~H~

I0 -4 ~0 -s I0 -~ I0 -i I0 ~ lO ~ I0 z ~0 ~ jO 4 IO s

Frequency, Hz

Fig. 7. Calculated low-frequency imaginary impedance values for the electrical circuit of Fig. 2d. This figure illustrates the effect of increas- ing truncation frequency, while maintaining the value of <Oo at 10 -4 Hz. a, Theoretical impedance response; b, calculated imaginary impedance values in the frequency range 10 -4 to 0.1 Hz; c, calculated imaginary impedance values in the frequency range 10 -4 to 1 Hz; d, calculated imaginary impedance values in the frequency range 10 -4 to 5 Hz.

an indication of the internal consistency of the data set. The polynomial expressions defined in Eq. [A-3] and [A-4], Appendix A, effectively smoothed the scattered imped- ance data contained in good-noise. The results for bad- noise show the calculated impedance below the truncation frequency to be discontinuous with the experimental data at r and suggest that bad-noise does not satisfy the KK relations. The conclusions about the validity of these noisy data sets obtained from the KK algorithm are consistent with the residual-plot analyses derived from LOMFP.

Invalid data sets.---The examples presented in this sec- tion are for nonstat ionary systems. Synthetic data were generated for the equivalent circuit shown in Fig. 2d by making the value of the resistor R, or the capacitor C, vary with t ime (see the section on simulation of inconsistent in- dependence data in Appendix B). The "experimental" fre- quency range was 0.1 Hz-65 kHz. Ten data points were col- lected per logarithmic decade of frequency and the number of delay cycles at each frequency N was six. In-

2 . 5 _Ira,, I J lHIII~ I H m , I I BHHI~ J l l t l l~ J lHI]~ I IHHI~

;>.O El, b 8 : Theore t i ca l ~mpedanc~_

C " ~ b: vei n - 0.5 HZ -

,.~ d _ \ o: =.,o . , . o . , - %

"~ ~ r v.,. SOHz ~K O3 E c" l.O 0

E o.e p.,i

o.o _ --- i

-o.5 - I IIl,~ J lHII~ I lllll "l I IIlil~ I lilll ''l I IIIlll~ I nilll~ i ltl,~ I I I ]0 -4 ]0 -~ ]0 -a ]0 -~ ~0 ~ ]0 J ]0 a ]0 ~ 10 ~ t0 s

Frequency . Hz

Fig. 6. Calculated low-frequency real impedance values for the elec- trical circuit of Fig. 2d. This figure illustrates the effect of increasing truncation frequency, while mointoinlng the value of o~o at 10 -4 Hz. a, Theoretical impedance response; b, calculated real impedance values in the frequency range ~0 -~ to O.i Hz; C, coiculated real impedance values in the frequency range 10 -4 to 1 Hz; d, calculated real imped- ance volues in the frequency range 10 -4 to 5 Hz.

creasing degrees of inconsistency were obtained by chang- ing the value of the "inconsistency factor" f. For a variable RI, the values of f equal to -+0.I, -+0.175, +_0.25, and -+0.5 correspond to a percent difference of -+7.0%, -+12.2%, + 17.4%, and -+35.0%, respectively, between the initial and final values of RI (see Eq. [B-9] Appendix B). For a variable CI, the values of fequal to + i0 -8, -+ 5 • 10 -8, and _+ 10 -~ cor- respond to a percent difference of -+ 15.5%, -+77.5%, and -+ 155%, respectively, between the initial and final values of C~.

Observation of a discontinuity at ~mi~ for inconsistent data.--The calculated low frequency impedance values for a system with a variable R, are presented in Fig. 12 and 13, while those for a variable C~ are shown in Fig. 14 and 15. The algorithm has "forced" consistency of the data sets with the KK equations at the expense of continuity at tornin between the calculated values and the experimental data. The degree of inconsistency is evident from the extent of discontinuity at the truncation frequency. It was observed that the calculations were very sensitive to the value of the lowest frequency (o 0. For a given invalid data set, the dis- continuity at COm~ became more pronounced as (% was de-

O.iO ~111HI I I J I I I I I I I I I I IIIIIIIJ I II111111 I I I I I I I I I I I I I I o,10

~ - ee e �9 I ~ 0.05 '-~ ro 0.05 �9 �9

c_ r U a a o a �9 ~5 * o ~ �9 o" f -

�9 ~ . ' . ~ " . ~ . . ~ . .

'-' ~ o �9 ~ a "%0 �9 [] o .

[] (3 ~ [] 0 -0.05

-o~o I I l l l l l l l I l l l l l f f l I I IHIIII I I l l l l l d I IIIlff lJ f l I I IH~ ~ t o

iO -~ I0 ~ ~0 i I0 2 ~0 3 10 4 i0 ~

Frequency. Hz

Fig. 8. Relative residual plots between the data set good-noise and the equivalent circuit fit of Fig. 2d using LOMFP (14). The random dis- tribution of the residuals indicates a good equivalent circuit fit. ( 0 ) Relative residuals for the real impedance; ([]) relative residuals for the imaginary impedance.

J. Electrochem. Soc., V o l . 1 3 8 , N o . 1, J a n u a r y 1991 �9 The E lec t rochemica l Society, Inc. 7 1

0 . i 5 0.15 L I t l t i l i i I I I IIIIII t i I I I I i l l 1 I I l i l iJJ t I i i l i lJ i I I i l l t l i

d �9 �9 �9

r0 0.10 - - �9 O.iO r _ �9 � 9

- - - o % �9 oeO �9

>m o.o5 ~ c o o o ~ a a a - n 0.o5 c~ [3 0 >

.~ _ % �9 o ~ �9 o . . ~ o o

o �9 cI �9 �9 "~

C.. -- ~ �9 II~ 0 O n o.oo - ; - - - �9 . . . . E-~- . . . . . . . [] . . . . . . ~ . . . . . . ~ - ~ 0"00 3

, g o o o . o o o o o - N~ : ~ 1 7 6 oo ~ ~

-0.05 Eli li#~l i 1Hrml r riH~ril~ l t iH~l i IHl -0.05 10 -I 10 0 10 ~ 10 2 10 3 10 4 10 5

Frequency, Hz

Fig. 9. Relative residual plots between the data set bad-noise and the equivalent circuit fit of Fig. 2d using LOMFP (14). The nonrandom dis- tribution of the real impedance residuals about the analytic impedance values suggests a poor equivalent circuit fit. ( 0 ) Relative residuals for the real impedance; ([-I) relative residuals for the imaginary im- pedance.

creased. This is in contrast to the results obtained for con- sistent data; i.e., the calculated impedance at the low-frequency range approached the theoretical imped- ance as the value of ~o0 was lowered.

By employing equivalent electrical circuits to simulate nonstationary systems, the sensitivity of the algorithm to test for the violation of the stability criterion is illustrated in Fig. 12 to 15. Even at the lowest degree of inconsistency, the data sets were deemed to be invalid by the appearance of a distinct discontinuity at ~Om~. The sensitivity of the proposed KK algorithm to test for violations of the cau- sality and linearity conditions were not investigated in this paper because the impedance data of the electrical circuits listed in Fig. 2 are inherently linear and causal. Macdonald and Urquidi-Macdonald (12) have stated that their KK al- gorithm was insensitive to the violation of the linearity condition for the specific case of iron in sulfuric acid. Ap- plication of the proposed KK algorithm to real experimen- tal data with large ampli tude voltage perturbations is an obvious extension of the present paper in order to investi- gate its sensitivity to the linearity constraint.

Determination of acceptable degree of discontinuity.--It is expected that most experimental systems will show some

2soo j ililt] I I lllill I I III!111 I illllll"lil]llll IIIIIII t l llllll~ i llilii I t Ililtl t I It1111 -soo I 1:3

~000 �9 -GO0

O O I I q l I I O O � 9 j i - p i

1500 - - i

~ 6 0 lO00

U -200 i

500 I I I

Ill I Ilmlllliill ~ 0 0 - -

�9 I i

-500 lllllll llllllIIl IIIIIII~ III~ llllllll IIIIIIUl llllllll ! lllllIl~ llllllll IIII 200

t 0 -5 10 -4 10 . 3 1 0 -2 t 0 - t 10 ~ 10 t 10 2 l O s 10 4 1 0 5

F r e q u e n c y , Hz

Fig. 11. Calculated low-frequency values for bad-noise: a data set which exhibited nonrandomly distributed relative residuals for the equivalent circuit fit. The data set is deemed to be inconsistent with the KK relations. (--) Theoretical impedance response; (�9 noisy real im- pedance data; ([]2) noisy imaginary impedance data; ( 0 ) calculated real impedance values in the frequency range 10 -s to 0.1 Hz; (I ) calcu- lated imaginary impedance values in the frequency range 10 -s to 0.1 Hz.

degree of discontinuity since the algorithm is very sensi- tive to even slight deviations from stationary behavior. To determine bounds for acceptable amounts of disconti- nuity, the inconsistent data sets of Fig. 12 were fitted to the equivalent circuit of Fig. 2d using LOMFP (14). The fitted values for the circuit elements of Fig. 2d are given in Table I. The extent to which these parameters are affected depends on the magnitude of the time constants associ- ated with the circuit elements.

The parameters Rw and Tw changed more as compared to the other circuit elements because the dominant t ime con- stant in the 10w-frequency region is associated with the Warburg impedance. This is the same frequency region where the value of the resistor R~ changed the most; how- ever, the t ime constant R~C~ was significantly less than Tw at any given frequency. Similar results were obtained for other inconsistent data sets (where either C~ or Rw were al- lowed to be functions of time) or when the relative magni- tude of the t ime constants was altered. These results were more evident when the dominant time constant was about

2500 -800 I I1ff1~ I IIli11~ I IHIII~ I IIIIli~ I IHII~ I llllll~ IHIlil~ IHIllq I IIIlfl~ I IHnl

- i

2000 L ~ i l � 9 1 4 9 1 4 9 1 4 9 1 4 9 e 0 [ -600

-" I - i

J500 --

6 ,ooo - , ~ 6

j ~ -2oo N "

.o= . . \., ~ / i n i l m l n u i i l i i l i i i, : .............. 0

0 - - ~ 1

-soo IIIIIllt IIIIIIli IIIIIIll IIIIIIlt IIIIII1~ l i l l l l l l IIIIII1~ ii i i i i id i i i i i i l l I I I l l l~ 200

10 -5 1 0 -4 10 -~ t 0 - 2 1 0 - I 1 0 0 1 0 t 1 0 2 1 0 3 1 0 4 1 0 5

Frequency, HZ

Fig. 10. Calculated low-frequency impedance values for good noise a data set which exhibited randomly distributed relative residuals for the equivalent circuit fit. This data set is shown to satisfy the KK relations. ( - - ) Theoretical impedance response; ( 0 ) noisy real impedance data; (I-I) noisy imaginary impedance data; ( 0 ) calculated real impedance values in the frequency range 10 -s to 0.1 Hz; ( I ) calculated imaginary impedance values in the frequency range 10 -s to 0.1 Hz.

~5oo ~ ,,,,,,,~ , , , ,q , , , , , , ,~ ,,,,~,,I ,,,,,,l~ , , , , , I ,,',1,'I ,,,I,, Boo

/J . , o ,

ooo_ J / \ - , ooE- / / \ \ ~ -,oo

o = W~ --,o0 -50o IlJillJ~ IIIII1~ IIIIIIId IIIII1~ IIIIIII1~ IIIIIII1~ IIIt11~ IIIIII1~ IIIIIII -0oo

] 0 - 4 ~ 0 - 3 t 0 - ~ j 0 - t j 0 0 1 0 t 1 0 2 J 0 s 1 0 4 10 5

Frequency, Hz

Fig. 12. Calculated low-frequency impedance values for a simulated nonstationary system in the frequency range 10 -4 to 0.1 Hz. The elec- trical circuit of Fig. 2d was used with a variable resistor R1 and positive values of "inconsistency factor" f. The value of the resistor increased during the course of the "experiment." Curve a: f = 0.1 (ARI = 7 .0%) , b: f = 0 .175 (AR1 = 12.2%), c: f = 0.25 (AR1 = 17.4%), d: f = 0.5 (AR1 = 35 .0%) .

72 J. Electrochem. Soc., Vol . 138, No. 1, J a n u a r y

Table I, Fitted parameters for the equivalent circuit of Fig, 2d with a variable resistor R~ using f = 0.0. The nonstotionary systems with various degrees of inconsistency ore given

1991 �9 The Electrochemical Society, Inc.

FP (10). The consistent data set is given for by f = 0.1, 0.175, 0.25, and 0.5

f R~, I~ R~, ~ C1, F R~, 12 C~, F Rw, 12 Tw,~

0.0 23.22 418.3 1.88E-5 695.2 9.26E-5 846.4 0.7055 0.100 23.22 419.1 1.87E-5 699.0 9.24E-5 857.3 0.7141 0.175 23.41 420.2 1.87E-5 701.7 9.22E-5 865.8 0.7204 0.250 23.48 421.0 1.86E-5 704.3 9.20E-5 874.3 0.7265 0.500 23.74 423.3 1.84E-5 712.3 9.15E-5 903.73 0.7459

two orders of magni tude greater than the other time con- stants.

Acceptable bounds for levels of discontinuity at the truncation frequency were obtained by comparison to changes in the calculated values of the circuit parameters. The results are presented in Fig. 16, which illustrates the percent difference between the values of the circuit ele- ments for the stationary and nonstationary systems as a function of the percent difference between the calculated and the experimental values for the real impedance at o~, . For these nonstationary systems, a discontinuity of less than 15% represents a deviation of less than -+5% for the Circuit elements not associated with the dominant t ime constant. In other words, regression of data showing a 15% discontinuity can be expected to yield circuit parameters associated with electrochemical reaction rate constants within 5% of the values that would apply at the beginning of the experiment. The Warburg component would show a large error. This establishes an error bounds for the circuit parameters of impedance data sets deemed to be incon- sistent.

The effect of truncating high-frequency impedance da ta . - The algorithm incorporated an assumption that the high- frequency impedance data have negligible contributions to the integral when the KK equations are evaluated at low frequencies. There is a limit to the truncation of high-fre- quency data, however, for this assumption to be valid.

Low-frequency impedance values were calculated for the equivalent circuit shown in Fig. 2d with decreasing (0max. The results are presented inF ig . 17. The Bode plots for ~om~ = 10 kHz show that the calculated low-frequency impedance is continuous with the available data at 0.1 Hz. However, when O~max = 1 kHz, there is a distinct disconti- nuity at r due to the residual errors at the high-fre- quency extreme.

The max imum frequency r must be sufficiently large that the impedance data at the high-frequency range do not truncate in the capacitive region. This requirement can be relaxed by assuming polynomial expressions for the

real and imaginary impedance in the range to > coax. Addi- tional summation terms and polynomial coefficients would be introduced into Eq. [12] and [13] to account for the new intervals at the high-frequency extreme.

Numerical constraints.--The accuracy of the calcula- tions is influenced by the choice of the adjustable parame- ters of the algorithm (see Appendix A). The polynomial or- ders K and M must be sufficient to establish the behavior of the impedance components within the low-frequency range; values for K and M between 6 and 10 were deemed to be adequate. As illustrated in Fig. 4, the problem of re- siduals was minimized by setting coo to be about three or four orders of magnitude less than COmic. The parameters N~ and N~ were specified by dividing the entire frequency range into logarithmic decades, and assigning 300 to 1000 ~i divisions per decade. There was minimal difference be- tween the results obtained when ~i : 1000 and when ~ = 300 (note that the optimal value for 8~ can be deter- mined if an adaptive quadrature method were employed). A noticeable difference was observed when double preci- sion (8-byte floating-point number) was employed as com- pared to extended precision (10-byte floating-point num- ber); the discrepancy is due to round-off errors when the summations are performed.

When Nc = 4, N~ = 6, ~i = 1000, and K = M = 9, there are 10,000 evenly spaced frequency divisions, and the sum- mation routine is performed at 20 frequencies within ~o0 --- to < O~m~ .. The average computational t ime required for implement ing the algorithm under these conditions was approximately 5 min on a 16 MHz 80386 personal com- puter with a math co-processor.

Conclusion An approach has been presented which successfully re-

solves the problem of applying the KK relations to imped- ance data which do not exhibit a dc limit. This method dif- fers from current KK algorithms such that it does not require extrapolation of the experimental data beyond the measurable frequency domain in order to evaluate the in-

3000

2500

2000

t500

~ t000

500

0

-500

-lO00

- t500

: - ~ ~ll.,~ 1.1,~ I J,,,~ i~.l,q I IH,, I ~,,,~ IH,,~ IH = - _--- ~ a: f - -0. I

~,- a " ~ b: f " -0 .175

d: f - -0.5

_ z o , . ,

~ . -zl i,.,) o

:l lI l l l~ [Itllill~ IIIIII,] llilfilll flfllll~ IIIIII~ lllfll~ lIIll(d l l l f -2oo

10 -4 1.0 -s :10 -2 10 - t iO 0 ~0 t :10 a 10 3 10 4 :10 5

Frequency. Hz

1600 3000

1400 2500

t200 2000

1000 t500

.00 g g ,ooo

oo0

400 0

200 -500

- t000

-1500

Fig. 13. Calculated low-frequency impedance values for a simulated nonstationary system in the frequency range 10 -4 to 0.1 Hz. The elec- trical circuit of Fig. 2d was used with a variable resistor R~ and negative values of "inconsistency factor" f. The value of the resistor decreased during the course of the "experiment." Curve a: f = - 0 . 1 (AR1 = - - 7 . 0 % ) , b: f = - 0 . 1 7 5 (AR~ = - 1 2 . 2 % ) , c: f = - 0 . 2 5 (ARI = - 17.4%), d: f = - 0 . 5 (~R1 = -35.0%).

~ ~ ~ ~ F T ~ ,s00

- --- 14oo f - to-S

b: f - 5 x i 0 - s ..-~_- ,200

~ ~ 6OO .

400

200

o

:lL.,~ i i.,d , . . l l i.,Id ~Hml i. , id J.l,d I . ,~ i . t~ -~oo I 0 -4 I 0 -3 i 0 .2 I 0 - I I 0 ~ IO ~ :10 2 10 3 I 0 4 IO S

Frequency, Hz

Fig. 14. Calculated low-frequency impedance values for a simulated nonstationary system in the frequency range 10 -4 to 0, | Hz. The elec- trical circuit of Fig. 2d was used with a variable capacitor CI and posi- tive values of "inconsistency factor" f. The value of the capacitor in- creased during the course of the "experiment." Curve a: f = 10 -s ( A C I = 15.5%), b: f = 5 • 10 -s ( A C 1 = 7 7 . 5 % ) , c: f = 10 -7 (ACI = 155%).

J. Electrochem. Soc., Vol. 138, No. 1, January 1991 �9 The Electrochemical Society, Inc. 7 3

,ooo \ , ,._o,o,

~ooo ~ c k 500

zr (w) o

-soo -~ u)H)~ ~ u).l,l I.II)~ )))ud ))I))l~ I I)))Ig I I)~),~ luI.~ )I)I~ -~oo ~0 -( 1.0 -3 10 -2 ~0 -~ ~.0 ~ iO ~ ~0 ~ I 0 ~ i 0 ( I 0 s

Frequency, Hz

800

600

40O

~ -200

- 4 0 0

Fig. | 5. Calculated low-frequency impedance values for a simulated nonstationary system in the frequency range 10 4 to 0.1 Hz. The elec- trical circuit of Fig. 2d was used with a variable capacitor C~ and nega- tive values of "inconsistency factor" f. The value of the capacitor de- creased during the course of the "experiment." Curve a: f = - 1 0 -e (AC1 = -15.5%), b: f = -5 x 10 -~ (ACI = -77.5%), c: f = -10 -7 (AC) = - 155%).

2500 ~1 lillll~ t lllllll t I III111~ I IIIIIll I I llllJl~ I lliJllJ] I IH(J~I~-[~_ eoo

2000 ~-.~. a / ' ~ a: =. . . - to k,z - .... .'.'.'.'.' :::::: soo -- b b: ula I = | kHz

A500

.;~ ~- 2oo N

N I 500

0

-~oo HllliJ ~ulll)l l llllll~ l lu,,I l lll,l~ luIlIl~ luI,d l ltll~ -2oo ~0 -4 ~,0 -3 ~,0 -2 iO -1 iO ~ ~01 iO 2 10 3 iO 4

F'requency, Hz Fig. 17. The effect of decreasing O)ma x on the algorithm attributable

to the problem of residuals at the high-frequency extreme. The data sets were from the equivalent circuit of Fig. 2d and co,~, = 0.1 Hz. Curve a: C%ax = 10 kHz; and b: O)ma x = 1 kHz. a, Calculated imped- ance values in the frequency range 10 -4 to 0.1 Hz when ~0=~ = 110 kHz; b, calculated impedance values in the frequency range 10 -4 to 0.1 Hz when (Oma x = 1 kHz.

tegrals of the KK relations. The proposed technique uses only the available impedance data, which is necessarily constrainted within a finite frequency domain. The only requirement for testing the consistency of the data using the method presented in this paper is that the highest measured frequency should be large enough to ensure that the high-frequency impedance data do not terminate in the capacitive region.

This KK algorithm should facilitate the analysis and in- terpretation of experimental data. This procedure comple- ments the method of fitting impedance data to an equiva- lent circuit; in part because a fitting procedure can be a tedious and t ime consuming exercise. By first appl~-ing the KK algorithm, an exper imenter will know whether the data are suitable for analysis by a fitting procedure. If the experimental data is found to satisfy the KK relations,

u C

L

2 q--

4~ C 0J U r 0J n

i01

10 o

l 0 -s

I i I I I I I l J I I I

o

&

I I

~0 ~ 2 3

B I/

A

W A

= ~ Z

~t SYNBOLS --

o R w �9 T N 0 FIt o C!

,~ R 2 ~ C 2

v R s

I I I I I I I I I I

4 5 6 7 eg ~.01 2 3 4 5

} Zr. expt - Zr. calc I /Zr. expt * :iO0

Fig. 16. The effect on inconsistency on the calculated parameters of the equivalent circuit of Fig. 2d. The abscissa represents the percent difference between the real component of the data set and the calcu- lated real impedance at con. The ordinate is the percent different be- tween the values of the circuit elements of the nonstationary and sta- tionary systems.

then an equivalent circuit could be found which will pro- vide a "good" and acceptable fit. Otherwise, the experi- menter will be aware of the inconsistency of the data set, and should interpret the results of the fitting procedure in light of a possible violation of one of the conditions of the KK relations.

A c k n o w l e d g m e n t s This material is based upon work supported by the Na-

tional Science Foundation under Grant no. EET-8617057 and on work supported by DARPA under the Optoelec- tronics program of the Florida Initiative in Advanced Mi- croelectronics and Materials. The writers also wish to ac- knowledge the contributions of Conrad B. Diem.

Manuscript submitted Jan. 15, 1990; revised manuscript received Aug. 2, 1990.

APPENDIX A Algorithm far Calculating Low-Frequency Impedance

The two forms of the KK relations (3, 7) employed in this algorithm were

Zi(r = - 10 x---- ~-_ w~ d x [A-l]

2 r| xZi(x) - coZi(~) z~(~o) : Z~(| + ~ ~o/ -~ - - ~ dx [A-2]

Zr(r and Zi(r are the real and imaginary components of the impedance, and r and x are angular frequencies. Equa- tion [A-l] represents the imaginary impedance that is inter- nally consistent with the real component of the experi- mental data. Equation [A-2] represents the real impedance that is internally consistent with the imaginary component of the experimental data.

Based on the assumptions presented in the section on Concept, the infinite integration region for the KK equa- tions is limited to r -< x -< r Experimental data were assumed to be available for the frequency range r --< X --< COmax; the real component reaches an asymptote Zr| and the imaginary component approaches zero at ~0max. The object of the algorithm was to determine the functions for the real and imaginary components

K

Zr(oJ1) = ~ a(k l) ( l o g ~o~) k [A-3] k=0

M Zi(00,) = • a~ ) (log to,) m [A-4]

m=O

74 J. Electrochem. Soc., Vol. 138, No. 1, January 1991 �9 The Electrochemical Society, Inc.

with in the low-f requency range r <- o)1 < com~ which, w h e n a p p e n d e d to the expe r imen ta l data set, force the data to satisfy the K K equat ions . These func t ions are po lynomia l express ions of order K and M, wi th coefficients a(0 ~) to a~ ) and a~ ~) to a~ ), respect ively The integrals of equa t ions [A-l] and [A-2] are evaluated over two regions; coo -< x < comin and O)mm--< X-< com~x. For a g iven f requency r Eq. [A-3] and [A-4] were subs t i tu ted into Eq. [A-l] and [A-2] to give

M a~)(log (1)1) TM

m=0 K K

E a~ )(l~ x ) ~ - E a~ )(l~ co,)k

X 2 2 0 -- "/.D~

K Z r ( X ) - E a(~ )(l~ co')~

X 2 2 dx min -- Xl

dx

[A-5]

and

K a~)(log ~ol) k = Z:,|

k=O M M

x E a(ma)(l~ x ) m - (91 E a(m 2)(lOg r176 = o = o

-t- X2 0 -- "L0f

dx

M am(lOg I) x Z i ( x ) - ~ol Y~ <~) co ~

+ X2 -- W 2 mm dx [A-6]

Z~(x) and Zi(x) in the range (Omi n <: X ~ (Oma x represen t the exper imenta l i m p e d a n c e data.

Equa t ions [A-5] and [A-6] were algebraical ly man ipu la t ed to collect l ike- terms for the po lynomia l coefficients a~ ) and a(2) such that m

~ ' a(m2'(1og col)m : -- x:- wy a x m =O rain

_ k~__ ~ a(kt ) ~min0 (log X) kx 2 ---- W~(1og ~O1) k dx

and

- (log 0)l)k min X 2 - w~ [A-7]

a(k')(log ~)l) k = Zr, | + - - - _ k=0 min x ~" - w~ dx

§ ( 2 1 ~ a(2m) { f: 'min :o o ~ --- W~ dx

f~ *max d x t -- col(lOg r rain X2 U -W~' [A-S]

Numer ica l in tegra t ion was employed due to the compli- cated form of the in tegrands . In the region coo -< x -< r the s ingular i ty at x = ~o] is avoided because the l imit of the in teg rand is zero u n d e r the same condi t ion (3). The recip- rocal of the d e n o m i n a t o r at this po in t can s imply be as- s igned a va lue of zero. A n al ternat ive me thod is to use Mac- laur in ' s fo rmula whi le accoun t ing for the par i ty of the po in t to avoid calcula t ing the d en o mi n a t o r w h e n x = co~ (15).

The t e chn ique suggested by Macdona ld et al. (3) for eval- ua t ing the integrals was employed. This involves subdi- v id ing the regions into several segments of evenly spaced f requencies as i l lustrated in Fig. 1. N~ represents the n u m - ber of segments in the low-f requency range, and N~ the n u m b e r of segments in the exper imen ta l f requency range. With in each s egm e n t in the f requency range

Xmi~ --< X <-- X . . . . the in terval is un i fo rmly spaced over 8i di- v is ions such that

Xmax -- Xmi n AX~ - [A-9]

8i

Po lynomia l express ions , as in Eq. [A-3] and [A-4], were ob ta ined for the discrete exper imenta l impedance data us ing least squares approximat ion . These equat ions were used to evaluate Z~(x) and Z~(x) with in the exper imenta l data segments at un i fo rmly spaced frequencies. Also

min x2 -- W~ 2(O1 I n [A-10] 0)max ~- COl comin -- COl

Since com~ > > r Eq. [ADO] was simplif ied to

/~,rnax dx . . . . 1 In (-co~i" + ~1 / [A-ell min X 2 -- W~ 2o)i (Omin -- COl /

Using Eq. [A:9] and [A-11], the integrals of Eq. [A-7] were approx imated by discrete s u m m a t i o n s wi th in each seg- m e n t to ob ta in a(1)f~(1) 4- ,~(1)r~(1) _~ rv(1)(~(1) _[_ + r,(1)f~(1)

0 ~0 - ~-~I '--1 - ~2 ~2 �9 �9 �9 ~K ~ K

(2) + a~ 2) + al 2) log o)l + a~2)(log r e + . . . + aM(log r M

( ~ ) jN__~I ~ Zr(Xn) = - : ~ - - - 2 AXj n=l X~ -- W l

where

C~) = ( - ~ ) [ ~ ~ (1OgXn)k--(logt01)k

0_<k_<K i=1 n=l X2n -- Wl 2 AXi

[A-12]

- - ( l ~ col)k2col In (~-~m~ -- comm + COl )]COl "

The same algebraic man ipu l a t i on was per formed on Eq. [A-8] to yield

a~ 1) + a~ ~) log co 1 + a~')(log o)1) 2 + . . . + a~)(log col) K

~(2)r,(2) + ~(2)f~(2) + ~(2)~(2) + . + (2)p(2) G0 k'0 t~l ~'~1 t~2 ~"2 ' " a M ~ M

= Zr(=) + -7-.~_2 AXj [A-13] 3=1 Xn -- W 1

where

C ~ ) : - ( ~ ) [ ~ ~ X n ( 1 O g X n ) m - - t01(1ogcol)m ~tXi

0<-rn-<M i=, n=l X2n -- W~

(l~ lntcomin+ (~ . 2 comin (11l

The K K relat ions of Eq. [A-l] and [A-2] have been re- duced to algebraic Eq. [A-12] and [A-13], where the polyno-

(1) (1) (2) (2) mial coefficients a 0 to a K and a 0 to a M comprise the ( K + M + 2) u n k n o w n s to be determined. Therefore, (K + M + 2) i n d e p e n d e n t equa t ions mus t be obta ined at f requencies wi th in the interval r 0 -<col <r where 1 - l -< (K + M + 2). This is conven ien t ly achieved by evenly d iv id ing coo <- x < r in logari thmic scale and as- s igning the (K + M + 2) values of f requency to o)i. The sum- mat ions in Eq. [A-12] and [A-13] were evaluated us ing S impson ' s rule. Final ly, a gauss ian e l imina t ion rout ine was employed to solve the l inear sys tem of equat ions.

Equa t ions [A-12] and [A-13] are algebraically coupled be- cause of the i n t e rdependence of the impedance com- ponents . This m eans that the po lynomia l coefficients cal- cula ted from each equa t ion m u s t be equal. The two equa t ions could be solved i ndependen t l y and m u s t yield equ iva len t values for the po lynomia l coefficients. How- ever, d iscrepancies m a y arise due to numer ica l errors asso- ciated wi th the regression analysis, the 81 in terpola t ions for the discrete data set, and the s u m m a t i o n rout ines involved

(1) (2) in ca lcula t ing the numer ica l coefficients C k and C m. An it- erative re f inement a lgor i thm mus t be im p lem en ted to

J. Electrochem. Soc., Vol. 138, No. 1, January 1991 �9 The Electrochemical Society, Inc. 75

match the set of polynomial coefficients for both equations where [for example, see algorithm 7.4, Ref. (16)].

This iterative routine may be avoided by using Eq. [A-12] ft and [A-13] simultaneously when constructing the linear h system of(K + M + 2) equations. For example, Eq. [A-12] is Z employed when l is odd, and Eq. [A-13] when I is even. This h direct method yields polynomial coefficients that are valid for both equations and which result in residuals of compa- P rable orders of magnitude. This was the approach taken in K the calculations presented in this paper. The computer program to implement the algorithm was written in Turbo Pascal (Boriand International, Version 5.0) and compiled ~2 in math mode. The gaussian elimination routine with ~1 scaled-column pivoting (16) and the Simpson's rule rou- tine were performed using 80 bit "extended" floating-point 0- numbers to minimize round-off errors.

A P P E N D I X B Theoretical Impedance Expressions

The analytic expressions employed for calculating the impedance response of the equivalent circuits depicted in Fig. 2 are listed in this section.

Circuit 2a.--The real and imaginary impedance com- ponents for the Randles circuit of Fig. 2a were calculated f r o m

ZT(tO ) = Z r (~ ) + jZi(tO)

= R~+ 2 s 2 - J " z 2 s 1+ tO R1C 1 1 + (o R1C 1 [B-l]

Circuit 2b.--The impedance response for the Randles circuit with a Warburg impedance shown in Fig. 2b is de- convoluted from the impedance response of the electrical circuit of Fig. 2d by reducing the value of the t ime con- stant R2C2. Consequently, the impedance data are calcu- lated from Eq. [B-6] presented in the following section on Circuit 2d by setting the values of R2 and Cs equal to zero.

Circuit 2c.--The electrical circuit of Fig. 2c was used to model the impedance behavior of mild steel in concen- trated hydrochloric acid (17). It has a constant phase ele- ment (CPE), whose impedance is given by

1 ZCPE -- - - [B-2]

Yo(JtOP

The rationale behind the use of this element is discussed in Ref. (18), Since

the analytic expression for this equivalent circuit is

( ctR1 ) ~R1 Zw(a) = Zr(cO) + jZi(~) = R~ + a s + B------T - j az + B2 [B-4]

where

~ = l + Y o R l t O n c o s ( ~ )

= Y o R l t o ~ s i n ( ~ )

Circuit 2d.--The equivalent circuit of Fig. 2d was em- ployed to model the impedance response of copper disso- lution in alkaline chloride solutions (19). The expression employed for the Warburg impedance Zw is given by (14)

tanh (X/jtOTw) Zw = Rw [B-5]

Vj~Tw

The analytic impedance response of the equivalent circuit shown in Fig. 2d is

Zr(tO) = Z~(tO) + jZ~(~)

= R ~ + A 2 + N 2 - j h s + Z ~ [B-6]

= pZ - ,ha = pA - , z

= coClp = K - ~ C 1 @

= Rw~l~ - d0eR2v2~ = R1K + RwX~ + 6~R2 = 6 ~

= 1 + ~2~ = 8 2 d- 0 -2 = RsC2 = ~/~ sin 6 - ~0- cos 4~ = ~ cos ~0 + ~/0- sin 6 = ~ sin O~ + ~/cos ~b = ~ cos 4~ - ~ sin = e* + e-* : e @ -- e -~

2V~Y~Tw 2

Simulation of inconsistent impedance data.--Inconsis- tent impedance data for a nonstationary system were sim- ulated by changing the value of a circuit element during the course of the "experiment ." A simple case is to assume that the value of the circuit element E varies proportion- ally with time. Therefore

i Ei = Eo + f N ~ ht~ [B-7]

n-1

where Eo is the value of the circuit e lement at start of the experiment, i is the number of data points collected, and Ei is the value of E as the experiment progresses. The "incon- sistency factor" f accounts for the direction of change of E; the value of E decreases with t ime when f < 0, and in- creases when f > 0. N represents the number of cycles used to make measurement at a given frequency. The time required to complete one cycle is ht = 1/~0, and NAt = N/to corresponds to the total t ime which has elapsed when the measurement was performed. The expression for Ei be- comes

z Ei = Eo + )~ E --1 [B-8]

n-1 tOn

where tO1 = 65 kHz since impedance experiments are usu- ally performed from 65 kHz down to low-frequency values. The percent difference between the initial value of the cir- cuit e lement and its value at any t ime during the "experi- ment" is given by

Ei - Eo fN ~ 1 hEro - - - • 100 = ~ - - • 100 [B-9]

Eo Eo ,,=1 tOn

Equation [B-8] was used to calculate the values of a vari- able resistor or variable capacitor, and was incorporated into the analytic expressions listed above to simulate a nonstationary system. The value of N was set to 6 when generating the inconsistent data sets.

REFERENCES 1. R. L. Van Meirhaeghe, E. C. Dutoit, F. Cardon, and

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The Effect of Tungsten on the Corrosion Behavior of Amorphous Fe-Cr-W-P-C Alloys in 1M HCI

Hiroki Habazaki, Asohi Kawashima, Katsuhiko Asami, and Koji Hashimoto Institute for Materials Research, Tohoku University, Sendai 980, Japan

ABSTRACT

The active current density of Fe-8Cr-7W-13P-7C alloy is almost three orders of magnitude lower than that of the Fe-8Cr-13P-7C alloy. The passive current density in the low potential region also decreases with increasing alloy tungsten content. The surface analysis by XPS reveals that chromium and tungsten ions are concentrated in the surface film on a tungsten containing alloy in the active region. The thickness of the surface film formed on the low tungsten alloy in the active region is x-ray photoelectron spectroscopically infinite, whereas the film thickness on the Fe-SCr-7W-13P-7C alloy in the active region is of the same order of magnitude as the thickness of the passive film. The chromium enrichment in the passive film becomes more significant with increasing tungsten content. Passivation of low tungsten alloys seems to occur by the formation of the passive film with the least enrichment of chromium ions necessary for passivation due to different dissolution rates of iron and chromium during a large amount of alloy dissolution. Passivation of tungsten containing al- loys takes place through transformation of the air-formed film to the passive film as a result of preferential dissolution of a small amount of iron without dissolution of chromium ions.

It is well known that amorphous metal-metalloid alloys with a certain amount of chromium have extremely high corrosion resistance in neutral and acidic solutions (1-8). Such a high corrosion resistance of amorphous alloys has been thought to be due to their chemical homogeneity and high passivating ability of amorphous alloys (3): amor- phous alloys are free of defects such as grain boundaries and dislocations, and of compositional fractuation formed by solid-state diffusion, Such a chemical homogeneity as- sists the formation of a uniform passive film on amorphous alloys. The high chemical reactivity of amorphous alloys is partly based on the existence of many coordinatively un- saturated atoms which enhances dissolution of elements unnecessary for the formation of the passive film. Rapid dissolution of such elements is responsible for the high passivating ability due to rapid surface concentration of ions, such as chromic ions, necessary for passive film for- mation.

The addition of molybdenum to amorphous iron-chro- mium-metalloid alloys further improves the corrosion resistance (4-8). When sufficient amounts of chromium and molybdenum are added, the amorphous iron-metalloid al- loys passivate spontaneously even in hot concentrated hy- drochloric acid (5-8). It is also well known that the addition of molybdenum to stainless steels increases the corrosion resistance, especially resistance to occluded cell corrosion (9-11).

Some of the present authors have carried out the analy- sis of surface films on amorphous and crystalline iron-base alloys containing chromium and molybdenum by x-ray photoelectron spectroscopy and attempted to clarify the role of molybdenum in increasing the corrosion resistance (7, 8, 12). It has been revealed from these investigations that in the active region, molybdenum ions are remarkably enriched in a surface film but not in the passive film and that the passive film is composed mostly of hydrated chro- mium oxyhydroxide. They proposed a possible role of mo- lybdenum in assisting the formation of the passive hy- drated chromium oxyhydroxide film by considering the analytical data as well as passivation kinetics.

The beneficial effect of tungsten has been known for many years. For example, tungsten improves resistance to pitting corrosion of stainless steels in chloride media (14, 15). Like molybdenum, the addition of tungsten im- proves the corrosion resistance of amorphous Fe-P-C al- loys with and without chromium in hydrochloric acid (4, 13). However, its effect in improving the corrosion resistance is not sufficiently understood yet.

This work has been performed to clarify the beneficial role of tungsten on the corrosion behavior of amorphous Fe-Cr-W-P-C alloys in hydrochloric acid by combination of electrochemical methods and x-ray photoelectron spec- troscopy.

Experimental Fe-xCr-yW-13 atom percent (a/o) P-7 a/o C (x = 4, 6, and

8 a/o; y = 0, 1, 2, 3, 5 and 7 a/o) alloy ingots were prepared by high-frequency induct ion melting of laboratory made iron phosphide and commercial iron carbide, tungsten, chromium, and iron under an argon atmosphere. Iron phosphide was prepared by sintering red phosphorus in iron. From these ingots, amorphous alloy ribbons of about 1 mm width and 10-30 ~m thickness were prepared by a single-roller method. X-ray diffraction patterns of the al- loys were measured with the x-ray diffractometer using Cu Ks radiation. All of the specimens used showed only halo patterns characteristic of amorphous structure.

Prior to electrochemical measurements, the amorphous alloy ribbons were polished in cyclohexane with silicon carbide paper up to no. 1000, degreased ultrasonically in acetone and dried in air. Electrochemical measurements were carried out at 303 K in deaerated 1M HC1 solution which was prepared from a reagent grade chemical and de- ionized water. Potentiodynamic polarization curves were measured with a potential sweep rate of 143 mV min-L Po- tentiostatic polarization was carried out for lh at various potentials by using different specimens to get different points in potentiostatic polarization curves. After the po- tentiostatic polarization or immersion for lh, these speci- mens were washed in water and dried in air, and then they