48 J. Electrochem. Soc., Vol. 144, No. 1, January 1997 The Electrochemical Society, Inc.

some interesting questions. What are the roles of the plas-ticizer aside from solvating the lithium salt and plasticiz-ing the polymer? Is the alternate movement along the seg-mental chain of PAN the only or the main form of Li ionictransport in the PAN-based electrolytes? Further investi-gation of these questions is in progress in our laboratory.

This work was financially supported by the Ford andNational Science Foundation of China, Contract No.09412304.

Manuscript submitted Feb. 21, 1996; revised manuscriptreceived Aug. 20, 1996.

REFERENCES1. K. M. Abraham and M. Alamgir, This Journal, 137,

1657 (1990).2. Z.-X. Wang, B.-Y. Huang, H. Huang, L.-Q. Chen, and

R.-J. Xue, Elect rochim. Acta, 41, 691 (1996).

3. Z.-X. Wang, B.-Y. Huang, H. Huang, R.-J. Xue, and L.-Q. Chen, This Journal, 143, 1510 (1996).

4. Z.-X. Wang, B.-Y. Huang, H. Huang, L.-Q. Chen, andR.-J. Xue, Solid State lonics, 86-88, 31 (1996).

5. T. C. Jao, I. Scott, and D. Steele, J. Mol. Spect rose., 92,1 (1982).

6. Z.-X. Wang, B.-Y. Huang, X.-J. Huang, L.-Q. Chen,and R.-J. Xue, J. Raman Spect rose., In press.

7. Z.-X. Wang, B.-Y. Huang, Z.-H. Lu, X.-J. Huang, L.-Q.Chen, and R.-J. Xue, Solid State lonics, In press.

8. W. H. Leong and D. W. James, Aust. J. Chem., 22, 499(1969).

9. D. W. James and R. E. Mayes, ibid., 35, 1775 (1982).10. R. L. Frost, D. W. James, R. Appleby, and R. E. Mayes,

J. Phys. Chem., 86, 3840 (1982).11. Z.-X. Wang, B.-Y. Huang, X.-J. Huang, L.-Q. Chen,

and R.-J. Xue, This Journal, In press.12. B.-Y. Huang, Z.-X. Wang, H. Huang, L.-Q. Chen,

and R.-J. Xue, Solid State lonics, 85, 79 (1996).

Spectroscopy Applications of the Kramers-Kronig Transforms:Implications for Error Structure Identification

Madhav Durbha* and Mark E. Orazem**

Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611, USA

Luis H. Garcia-Rubio

Department of Chemical Engineering, University of South Florida, Tampa, Florida 33620, USA

ABSTRACT

Stochastic errors were found to propagate through the Kramers-Kronig relations in such a manner that the respec-tive standard deivations of the real and imaginary components of complex quantities at any given frequency are equal.The only requirements were that the errors be stationary in the sense of replication at each measurement frequency, thaterrors be uncorrelated with respect to frequency, that the derivative of the variance with respect to frequency exists, andthat the Kramers-Kronig relations be satisfied. Experimental results for elecrochemical and rheological systems are pre-sented which support the conclusion that the real and imaginary components have the same variance. This result and theconclusions reported herein appear to be general and should apply for any physical system in which the real and imagi-nary components are obtained from the same measurement.

IntroductionWhile use of weighting strategies that account for the

stochastic error structure of measurements enhances theinformation that can be extracted from regression of spec-troscopic data,'1' independent assessment of the errorstructure is needed. The error structure for most radia-tion-based spectroscopic measurements such as absorp-tion spectroscopy and light scattering can be readily iden-tified.'2'13 The error analysis approach has been successfulfor some optical spectroscopy techniques because thesesystems lend themselves to replication and, therefore, tothe independent identification of the different errors thatcontribute to the total variance of the measurements. Incontrast, the stochastic contribution to the error structureof electrochemical impedance spectroscopy measurementsgenerally cannot be obtained from the standard deviationof repeated measurements because even a mild nonsta-tionary behavior introduces a nonnegligible time-varyingbias contribution to the error. Recent advances in the useof measurement models for filtering lack of replicacy havemade possible experimental determination of the stochas-tic and bias contributions to the error structure for imped-ance measurements. 14-18

The measurement model approach for identification oferror structures is widely used (see, for example, Ref. 12and 13 for applications to optical spectroscopies and Ref.

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

19 for general application to spectroscopy). Recently, ameasurement model approach has been applied to identi-fy the error structures of impedance spectra obtained for alarge variety of electrochemical systems.'4'8 One strikingresult of application of measurement models to impedancespectroscopy has been that the standard deviation of thereal and imaginary components of the impedance spectrawere found to be equal, even where the two componentsdiffered by several orders of magnitude. The only excep-tion was found when the data did not conform to theKramers-Kronig relations or when the precision of themeasurement did not allow calculation of the standarddeviation of one of the components (i.e., all significant dig-its reported by the instrumentation for the replicatedmeasurements were equal, see Ref. 20). The objective ofthe present work was to explore whether the equality ofthe noise levels in the real and imaginary parts of electro-chemical impedance spectra can be described as being adirect consequence of the manner in which errors propa-gate through the Kramers-Kronig relations when both realand imaginary components are obtained from the samemeasurement.

Experimental Motivation

Spectroscopic measurements which yield complex vari-ables are illustrated in hierarchical form in Fig. 1.Spectrophotometric techniques such as absorption spec-troscopy and light scattering record the light intensity as afunction of the wavelength of the incident radiation used

J. Electrochem. Soc., Vol. 144, No. 1, January 1997 The Electrochemical Society, Inc. 49

Mechanical

EHIQ RI00RMe2 To,qc.

,,c Eeds M. Lç'dM.y

SpectrcscoplcMeai,,emen01 Caniplex

Ouaha

ILt1 LJFig. 1. Hierarchical representation of spectroscopic measure-

ments. The shaded boxes represent measurement strategies forwhich the real and imaginary parts of Kramers-Kronig-trans-formable impedance had the same standard deviation. Followingcompletion of the analysis reported here, an experimental investi-gation was begun which showed that the real and imaginary partsof complex viscosity also have the same standard deviation if thespectra are consistent with the Kramers-Kronig relations.

to interrogate the sample. The frequency dependence aris-es from the wavelength of light employed. Electrochemicaland mechanical spectroscopic techniques employ a modu-lation of a system variable such as applied potential, andthe frequency dependence arises from the frequency of themodulation. In electrochemical, mechanical, and somespectrophotometric techniques, both real and imaginary(or modulus and phase angle) components are obtainedfrom a single measured variable (e.g., the impedance).

To date, the equality of the standard deviations for realand imaginary components was observed for several suchsystems which are highlighted in Fig. 1. The equality ofthe standard deviation for real and imaginary componentswas observed for

1. Electrohydrodynamic impedance spectroscopy(EHD), a coupled mechanical/electrochemical measure-ment in which the rotation rate of a disk electrode is mod-ulated about a preselected value. The impedance responseat a fixed potential is given by M/1l.'7

2. Electrochemical impedance spectroscopy (EIS). The database now includes measurements for electrochemical andsolid-state systems under both potentiostatic and galvanosta-tic modulation.'4'6 The equality of the standard deviation forreal and imaginary components was observed even for sys-tems with a very large solution resistance.2°

3. Optically-stimulated impedance spectroscopy. Imped-ance measurements were obtained for solid-state systemsunder monochromatic illumination.2'

The contention that the variances of the real and imagi-nary parts of Kramers-Kronig transformable complexvariables are equal is supported by experimental evidenceand has been validated statistically, as discussed in a sub-sequent section.

In contrast to results found for electrochemical andmechanical spectroscopies, the standard deviations for thereal and imaginary components of the complex refractiveindex from spectrophotometric measurements were corre-lated but not necessarily equal.22 However, for measure-ment of optical properties (e.g., the real and imaginarycomponents of the complex refractive index) over a suffi-ciently broad range of frequencies, different instrumentswith their particular error structures must be used; where-as in electrochemical and mechanical spectroscopies a sin-gle instrument is used to measure both the real and imag-inary components simultaneously.

For electrochemical and mechanical/electrochemicalspectroscopies, the experimental evidence for the equalityof the standard deviation of real and imaginary compo-nents is compelling and suggests that there may be a fun-damental explanation for the observed relationship be-tween the noise level of real and imaginary components ofthe impedance response.

Application of the Kramers-Kronig RelationsThe Kramers-Kronig relations are integral equations

which constrain the real and imaginary components ofcomplex quantities for systems that satisfy conditions ofcausality, linearity, and stability.2326 The Kramers-Kronigtransforms arise from the constitutive relations associatedwith the Maxwell equations for description of an electro-magnetic field at interior points in matter. Bode extendedthe concept to electrical impedance and tabulated variousforms of the Kramers-Kronig relations.26

An application of the Kramers-Kronig relations to vari-ables containing stochastic noise was presented byMacdonald, who showed, through a Monte Carlo analysiswith synthetic data and an assumed error structure, thatthe standard deviation of the impedance component pre-dicted by the Kramers-Kronig relations was equal to thatof the input component.27 The analysis was incompletebecause it did not identify correctly the conditions underwhich the variances of the real and imaginary componentsof experimental impedance data are equal, and the authorcontinued to use a weighting strategy in his regressionsbased on a modified proportional error structure for whichthe variances of the real and imaginary components ofexperimental impedance data are different.272°

The objective of the present work is to identify the errorstructure for frequency-dependent measurements. Hereinwe report an explicit relationship between the variances ofthe real and imaginary components of the error without apriori assumption of the error structure. The only require-ments are that the Kramers-Kronig relations be satisfied,that the errors be stationary in the sense of replication atthe measurement frequency, that the derivative of thevariance with respect to frequency exists, and that theerrors be uncorrelated with respect to frequency. In thesubsequent section, stochastic error terms are incorporat-ed into the derivation of Kramers-Kronig relations toexamine how the errors for the real and imaginary termsare propagated.

Absence of Stochastic ErrorsThe Kraniers-Kronig transforms can be derived under

the assumptions that the system is linear, stable, station-ary, and causal. The system is assumed to be stable in thesense that response to a perturbation to the system doesnot grow indefinitely and linear in the sense that theresponse is directly proportional to an input perturbationat each frequency. Thus, the response to an arbitrary per-turbation can be treated as being composed of a linearsuperposition of waves. The response is assumed to beanalytic at frequencies of zero to infinity. The statementthat the response must be analytic in the domain of inte-gration may be viewed as being a consequence of the con-dition of primitive causality, i.e., that the effect of a per-turbation to the system cannot precede the cause of theperturbation.3°

The starting point in the analysis is that the integralaround the closed loop (Fig. 2) must vanish by Cauchy'sintegral theorem3'

[Zr()(X) — Zr(cc)IdX = 0 [1]

where Z is the impedance response and x is the complexfrequency. Equation 1 yields (see, e.g., Ref. 26)

z () = Z(X) Zo) dx [2a]x —w

and

2w 1 —xZ.(x) ÷ wZ (w)Zr(W) — Zr(OO) = —2 2 dx [2b]'rio X (U

where only the principal value of the respective integralsis considered. The terms Z,(u) and uZ,(w) in Eq. 2a and b,respectively, facilitate numerical evaluation of the singu-

50 J. Electrochem. Soc., Vol. 144, No. 1, January 1997 The Electrochemical Society, Inc.

lar integrals, but do not contribute to the numerical valueof the integrals as

LX2_W2A similar development cannot be used to relate the real

and imaginary parts of stochastic quantities because Eq. 1is not satisfied except in an expectation sense.

Propagation of Stochastic ErrorsThe stochastic error can be defined by

ZOb(w) = Z(w) + €(w) = [Zr(w) + €r(w)] + j[Z(w) + €(w)] [31

where ZQb(w) is the observed value of the impedance at anygiven frequency w and Z(w), Zr(w), and Z(w) represent theerror-free values of the impedance which conform exactlyto the Kramers-Kronig relations. The measurement error€(w) is a complex stochastic variable such that e(w) = €r(w) +

je1(w). Clearly, at any frequency w

only if

E[Z(w)Ob] = Z(w)

E[€r(w)1 = 0

= 0

Equation 5 is satisfied for errors that are stochastic and donot include the effects of bias.

Transformation from real to imaginary—The Kramers-Kronig relations can be applied to obtain the imaginarypart from the real part of the impedance spectrum only inan expectation sense

=!.E[f

Zr(X) Zr(w) €r(x) — €r(w)

dx]It is evident from Eq. 6 that, for the expected value of theobserved imaginary component to approach its true valuein the Kramers-Kronig sense, Eq. 5a must be satisfied andthat

var€(w)2 Nr 1 2w 1 'ç' €k(x)

[61k=1

and

For the first condition to be met, the process must be sta-tionary in the sense of replication at every measurementfrequency. The second condition can be satisfied in twoways: in the hypothetical case where all frequencies can besampled, the expectation can be carried to the inside of theintegral, and Eq. 7 results directly from Eq. 5a. In the morepractical case where the impedance is sampled at a finitenumber of frequencies, €r(x) represents the error betweenan interpolated function and the "true" impedance valueat frequency x. This term is composed of contributionsfrom the quadrature and/or interpolation errors and fromthe stochastic noise at the measured frequencies w. In thelatter case, Eq. 7 represents a constraint on the integrationprocedure. In the limit that quadrature and interpolationerrors are negligible, the residual errors er(x) at a frequen-cy x = w should be of the same magnitude as the stochas-tic noise c,(w).

Under the conditions that Eq. Sa and 7 are satisfied andfor a given evaluation of Eq. 6

[4]Z(w) + e(w) =

E1fr — Zr(w) + er(x) — €r(w)d[Sal xl. LT0 x2—w2

[Sb] + LX2_W2} [8]

where €t(w) represents the error in the evaluation of theKramers-Kronig relations caused by the second integralon the right side. The variance of the transformed imagi-nary variable can be shown by the following developmentto be equal to the variance of the real variable. From Eq. 8

f €flj) 12

_tor2 — 2dxj [9]

where N is the number of replicate measurements which isassumed to be large. Under the assumption that Eq. 7 issatisfied

$ 1 N (2w 1 erk(x) \2var[€ (w)] = 1EIX ir tox2 w dxj [10]

As only principal values of the integrals are considered, itis appropriate to approach the point of singularity at x =w equally from both sides. In terms of the principal value,for specific values of k, the integral in Eq. 10 becomes

€fljX) dx =!5LJ

dxTrJox2 —w ii x —w

+ .2-5.J x2_w2 dx [11]

Under the transformations x = wy in the domain [0, wjand x =w/y in the domain [w, ]27

2w ( €rk(x) 2 1 (€(wy) — €rk(W/Y))

irj x2— 2dx=

—; • 1— 2 dy [12]

Equation 12 can be expressed in a summation sense as

2 C (€rk(WY) — €rk(w/y)) dyITO i—y2

= 2j [€r,(wYm)—

€r.k((O/Ym)w][13]'r1 [

where M is large, M — 1 represents the number of intervalsfor the domain of integration, and W(Ym) is the weighting

€r(x)E 2 2dxI=0[ 'rjox —w [7]

U)

C,4)

.4 I- ÷p-plane

(complex frequency plane)Imaginary axis

-0)

Fig. 2. Path of integration for the contour integral in the complex-

frequency plane.

J. Electrochem. Soc., Vol. 144, No. 1, January 1997 The Electrochemical Society, Inc. 51

1 6 11 16 21 26

factor which can be a function of the integration proce-1

dure chosen. From Eq. 10, 12, and 13

var[E(w)]

= I -Ym)W(yJl)2 [14]

j 01v2 N—i , t,,m=1 [ 1 — Ym

A general expression for the errors is given by €Zk(x) =Pk(O, 1)crr(x) where Pk(O, 1) is the kth observation of a sym- jj 0.01metrically distributed random number with a mean valueof zero and a standard deviation of 1, and ar(x) is the stan-dard deviation for the errors which is representative of the ii!error structure for the spectroscopy measurements and is Cassumed to be a continuous function of frequency. Under 0.001the assumption that the errors are uncorrelated withrespect to frequency

a 2(u) ={am[a(wym) + a(w/ym)}} [15] 0.0001

where the weighting factor is given by m

a = [W(ym) 12 [16] Fig. 3. Weighting factor for Eq. 160$ C function of m normalizedm [i — y, j to show relative confributions to the integral.

In the limit that M—*co, the trapezoidal rule yields

4 1 assumed to apply extends only 0.001 of the frequency u,am =x2(2m — 1)2

[17] e.g., 1 Hz at a frequency u of 1000 Hz.Substitution of Eq. 19 and 20 into Eq. 15 yields

and

am = 0.5 [18] a(w) = a(w) 2am [21]

As shown in Fig. 3, the am coefficients decay rapidly away where p c<M and accounts for intervals in the vicinity offrom y = 1. Beyond the first five terms, the individual con- y = 1. Following Eq. 18tribution of each term is less than 1% of the first term. Theseries approaches its limiting value to within 1% when 20 a*2(w) = a(u) [22]terms are used. The error associated with using a finitenumber of terms in Eq. 15 therefore can be made to be This result is consistent with the results of Monte Carlonegligibly small. calculations for assumed error structures22'27 but is

Under the assumption that 4(x) is continuous at x = w, obtained here without explicit assumption of an errora Taylor series expansion can be written in terms of fre- structure. The only requirements are that the Kramers-quency x which, when expressed in terms of the transfor- Kronig relations be satisfied, that the errors be stationarymation variable x = uym (valid for x < to), yields in the sense of replication at each measurement frequency,

r 2 1 that the errors be uncorrelated with respect to frequency,o(x) = a(w) + dtTr(x) [w(Ym — 1)] and that the derivative of the variance with respect to fre-

L dx quency exists.

+ —1)]2] [19] Transformation from imaginary to real.—The Kramers-

Kronig relations for obtaining the real part from the imag-Similarly, for the variable transformation x = W/Ym valid mary part of the spectrum can be expressed as Eq. 2b,for x> to which in terms of expectations becomes

2 2 do(x) 1crr(x) = ajw) + —

— 1) E[Zr(w) — Zr(oo)J

i 2' 2 xZ(x) + wZ3(w) — x€(x) + w3(w) 1+ O[w[_ - iJ] j [20a]= x - w2 dxj [23]

In the vicinity of y = 1, Eq. 20a can be expressed in a Following the discussion in the earlier section, the neces-form similar to Eq. 19 sary conditions for Kramers-Kronig transformability

r become Eq. Sb and2 2 Ida2(x)lar(x)=ar(w)—i r

I W(Ym —i[ dx j J2 1 xe3(x)

1tLFox2_u2' 0[24]

+ o([to(Y — 1)}2) [20b] .The variance of the error in the evaluation of Eq. 23 isgiven byEquation 20b is justified because the major part of the

contribution to the integral occurs within the range of y =1 to roughly y = 1 — 106. The assumption that higher 1 N r2 r —x€ x 12order terms in the expansion for a(x) can be neglected is var(€r) = —

2 dx I [25]justified because the region over which linearization is N — 1k=2 11 J o x — to2 j

52 J. Electrochem. Soc., Vol. 144, No. 1, January 1997 The Electrochemical Society, Inc.

10

4.0

(C0)I.-IL'

0aC0CO

>4)o 0.01

CO'0CCO

(00.001

0.01 0.1 1 10 100 1000 10000 100000

Frequency, Hz

Fig. 5. Error sfructure obtained for reduction of ferricyanide on aplatinum disk rotating at 120 rpm in a 1 M KCI solution underpotentiostatic modulation at 0.25 of the limiting current value; (0)standard deviation for real part of the impedance response, (Alstandard deviation for imaginary part of the impedance response,and (x) F test parameter F = o3,/4 Dashed lines correspond to sig-nificance levels for F for 25 replicates.

IC

IC ICIC 0.05 IC

C ICx IC

xICICIIC

IC

x "AIC x

ICIC IC ICxICICx oc

1/F005 05r0 .fw1

a 0A oo £000

Under the transformation used in the earlier section100

I XEJ,k(X)= —S'-

YJ,k(°Y)dy +

1l_

(1 / y)€5(w/y)dy s.

T0 X — U)2 0 1 — Y k2 1 — Y ,—

xEJk(x)________ 0)+2 3dx [26] _

— U)1

where a point of singularity at y = 0 (at w = cc) introduced aby the transformation x = w/y was avoided by further sub- c'

.9 0.1division of the integral. Since y can be chosen such thatthe third integral of Eq. 26 has negligible value and that 4)the first integral has a negligible contribution from the p 0.01range y = 0 to Y2, Eq. 26 can be solved by following a pro-

'0cedure similar to that adopted in the earlier section. A cform similar to Eq. 14 is obtained

001N 0.1 1 10 100 1000 10000 100000

var[eon4=

,I2N_1X Frequency, Hzk=1

2 Fig. 4. Error sfructure obtained for reduction of ferricvanide on a

( [ym€j,ym)—

(1/Ym)j,k(°/Ym)

w(Ym)]][27] platinum disk rotating at 3000 rpm in a 1 M KCI solution under

1 — y, potentiostatic modulation at one-half the limiting current value; (0)- standard deviation for real part of the impedance response, (A)In the region of interest (y—'l), y/(l — y,), and l/y(l standard deviation for imaginar1 part of the impedance response,

— y,,) tend toward 1/(1 — y). Hence and (x) F test parameter F = a ,/a. Dashed lines correspond to

N significance levels for F for three rep'icates.

var(€))— 4 1

k—i that the variance of the real part of the impedance is sta-2 tistically different from the variance of the imaginary part

over the measured frequency range can be rejected. The—

C1k(U)/Ym)W()]][28] dispersion of F about one as a function of frequency fur-

1 — ther supports the conclusion that the variance of the real

where pc<M and accounts for the intervals in the region ofpart of the impedance is equal to the variance of the imag-

y —* 1. Equation 28 is directly analogous to Eq. 14. mary part. The measurement model approach of Agarwalet at.'0'34'35'30 showed that the data were consistent with the

Following the discussion in the previous section Kramers-Kronig relations.o2(w) = on) [29] The value of F00, for rejection of the null hypothesis (i.e.,

= ci) approaches one when the number of replicatedThus, the variance of the real part of the impedance is measurements is increased. Results are presented in Fig. 5equal to the variance of the imaginary part of the imped- for the reduction of ferricyanide on a Pt disk rotating atance, independent of the direction of the transformation, 120 rpm where 25 replicate impedance scans were oh-if the Kramers—Kronig relations are satisfied in an expec- tamed. The test criterion, F = ff/oj, is compared in Fig. 5tation sense. For the Kramers-Kronig relations to be satis- with 1 and 5% limits for 25 replications. At frequenciesfied, the conditions stated through Eq. 5, 7, and 24 must be below 100 Hz, the hypothesis that a = cr1 is confirmed.satisfied. Equation 5a and b represent the usual constraints Above 100 Hz, the values for F and the trending as a func-on the experimental stochastic errors; whereas Eq. 7 and 24represent constraints on the integration procedure.

As summarized in the following section, the theoreticaldevelopment presented here is supported by experimental -.observations for various physical systems that satisfy theconditions of ICramers-Kronig relations. F001

Experimental ::.,<..F0 35

The equality of the standard deviations for real and A U x: : ..T°°1XX x x ,kCIC

imaginary parts of the impedance has been observed for :.:::::ac Q.J._4the impedance response of solid-state systems (GaAsSchottky diodes, ZnO varistors, ZnS electroluminescentpanels),15'16'33 corrosion of copper in seawater (under eitherpotentiostatic or galvanostatic modulation),16'33'34 electro-chemistry at metal hydride electrodes (LaNi, and mischmetal) the electrohydrodynamic impedance responsefor reduction of ferricyanide and oxidation of ferro-cyanide on Pt rotating disks,'7'36 the impedance response ofmembranes for which the solution resistance is large,3° andthe impedance response of electrical circuits with a largeleading resistance.2°

A standard statistical criterion can be used to confirmthe hypothesis that the standard deviations (or variances)of the real and imaginary parts of the impedance areequal.37 A typical result is presented in Fig. 4 for thereduction of ferricyanide on a Pt disk rotating at 3000 rpmwhere three replicate impedance scans were obtained. Thetest criterion, F = cr/cr, is compared in Fig. 4 with 1 and5% limits for three replications. On a frequency-by-fre-quency basis, 1/F00, .c F c F001; therefore, the hypothesis

0.1IC

2 )SCIC

0 —

0

A

J. Electrochem. Soc., Vol. 144, No. 1, January 1997 The Electrochemical Society, Inc. 53

tion of frequency suggest that a o. For this experiment,however, the measurement model approach of Agarwalet al."34'353' showed that the data collected at high fre-quency were inconsistent with the Kramers-Kronig rela-tions.

The derivation presented here suggests that, if a = crmust be true for systems that satisfy the Kramers-Kronigrelations, the experimental observation of the equality ofthe variance of the real and imaginary parts of the imped-ance cannot be attributed to unique features of electro-chemical systems or of the frequency-response analyzer(FRA). This conclusion was subsequently tested by meas-uring the frequency-dependent complex viscosity of vis-coelastic fluids using a Rheometrics RMS-800 parallel-plate viscometer.3' This system is nonelectrochemical anddoes not employ an FRA. For measurements that satisfiedthe Kramers-Kronig relations, the standard deviations forreal and imaginary parts of the viscosity were found to beequal.39 Results are presented in Fig. Ga for the complexviscosity of a concentrated polyethylene oxide solution.The F test, presented in Fig. 6b, confirms that ,r = cr'.

1 10 100 1000

Frequency, Hz

Fig. 6. (a) Complex viscosity and carresponding errar structureobtained for a concentrated solution of poyethyIene oxide (Mw300,000) in water; filled symbols represent the measured complexviscosity, and open symbols correspond to the standard deviation;(0) the real part of the viscosity and (E1) the imaginary part of theviscosity. (b) Test for inequality of the variance; (x) F test parameterF = a/oj, and dashed lines correspond to significance levels for Ffor three replicates.

The results presented here demonstrate theoreticallyand experimentally that r = a when the Kramers-Kronigrelations are satisfied. Manuscripts are in preparationwhich describe in greater detail the results for the experi-mental systems presented in Fig. 4 to 6.

DiscussionThe result of the development presented here is that, for

data that are consistent with the Kramers-Kronig rela-tions in an expectation sense, the standard deviation of thereal part of a complex spectrum at a given frequency mustbe equal to the standard deviation of the imaginary part.The development did not require any assumptions con-cerning analyticity of the stochastic noise with respect tofrequency; therefore, the result applies to spectra in whichdata are collected sequentially as well as to spectra inwhich a single observation is used to resolve a spectrum,as is done, for example, by Fourier transformation of tran-sient data.

The implications of this result are illustrated in Fig. 7,where the real and imaginary parts of an impedance spec-trum are presented as functions of frequency. The proba-bility distribution function for the data, corresponding toan equal standard deviation for the real and imaginaryparts, is shown at a frequency of 0.03 Hz. The real part ofthe impedance at this frequency is roughly 100 CI as com-pared to —3 Iii for the imaginary part, and the noise leveltherefore represents a much larger percentage of the imag-inary signal than the real. This result, which has been

-60

-40

-20

0.1

10000

I1000

100

jio

100

10

4bII

LL 1

0)413I-Li.

0.1

0.01

0.1

10

Frequency, Hz

100 1000

(a)

0.1 1

(b) F0,,,'C

'C

'C 'C 'C

'C 'C'C 'C

'C

xx 'C

'C 'C'C

K1F,,

10 100 1000 10000

Frequency, Hz

120

100

ag soCCu0aEj 40C

20

0

0.01

-120

a -100

-80•00.E

Cu

'CC)Cu

E

00.01

Frequency, Hz

Fig. 7. Real (a) and imaginary (b) parts of a typical electrochem-ical impedance spectrum as a function of frequency. The normalprobability distribution function, shown at a frequency of 0.03 Hz,shows that one consequence of the equality of the standard devia-tion for real and imaginary components is that the level of stochas-tic noise as a percentage of the signal can be much larger for onecomponent than the other.

(b)

0.1 1 10 100 1000 10000

54 J. Electrochem. Soc., Vol. 144, No. 1, January 1997 The Electrochemical Society, Inc.

observed in many experimental systems,'4'6'20'336 is nowshown to have a fundamental basis.

While the development presented here shows that thestandard deviations for the real and imaginary parts of aKramers-Kronig-transformable complex quantity areequal, the instantaneous realizations of the stochasticerrors for the respective components have been shownexperimentally to be uncorrelated.'6'34 The errors also werefound to be uncorrelated with respect to frequency,'6'34 anobservation that supports a key assumption made here.The assumption of the existence of the first derivative ofthe variance is supported by the identification of errorstructures presented in Fig. 4 to 6 and in Ref. 14-18, 20,and 33-36.

The need to identify an appropriate form for the imped-ance Z(x) in the presence of stochastic errors (e.g., Eq. 8and 23) supports the use of measurement models composedof line shapes that themselves satisfy the Kramers-Kronigrelations. The use of such measurement models is superiorto the use of polynomial fitting because fewer parametersare needed to model complex behavior. Experimental dataseldom contain a frequency range sufficient to approxi-mate the range of integration of 0 to required to evalu-ate the Kramers-Kronig integrals; therefore, extrapolationof the data set is required. Measurement models can beused to extrapolate the experimental data set, and theimplications of the extrapolation procedure are quite dif-ferent than from extrapolations with polynomials. Theextrapolations done with measurement models are basedon a common set of parameters for the real and imaginaryparts and on a model structure that has been shown to rep-resent the observations adequately. The confidence in theextrapolation using measurement models is, therefore,higher. In addition, as the line shapes used satisfy theKramers-Kronig relations, experimental data may bechecked for consistency with the Kramers-Kronig rela-tions without actually integrating the equations over fre-quency, avoiding the concomitant quadrature errors.18'39'3436,38,40 The correlation and bias errors introduced byquadrature errors are discussed further in Ref. 22. Theresults presented here also indicate clearly the importanceof identifying the nature of the error structure prior toapplication of the Kramers-Kronig transforms.

The analytic approach presented here establishes anexplicit relationship between r and cr with the onlyrequirement that the errors be stationary in the sense ofreplication at each measurement frequency, that errors beuncorrelated with respect to frequency, that the derivativeof the variance with respect to frequency exists, and thatthe Kramers-Kronig relations be satisfied. In addition, theconditions for the applicability of the Kramers-Kronigrelations to experimental data have been identified, i.e.,that Eq. 5, 7, and 24 be satisfied.

ConclusionKnowledge of the error structure plays a critical role in

interpreting spectroscopic measurements. An assessmentof the stochastic (or noise) component of the errors allowsrefinement of regression strategies and can guide design ofexperiments to improve signal-to-noise ratios. Assess-ment of consistency of data with the Kramers-Kronig rela-tions is also important because inconsistencies can beattributed to experimental bias errors which must beaccounted for during interpretation of measurements.

In this work, the experimental observation that real andimaginary parts of the impedance have the same standarddeviation was found to have a fundamental basis for sta-tionary error structures. The propagation of stochasticerrors through the Kramers-Kronig relations in bothdirections (real-to-imaginary and imaginary-to-real)yielded standard deviations that were equal. This resultappears to be general and should apply for all spectro-scopic measurements in which real and imaginary compo-nents are obtained simultaneously and with the sameinstrumentation. Thus, the observations concerning theerror structure of electrochemical, optical-electrochemi-

cal, and mechanical-electrochemical impedance spectrashould apply as well to purely mechanical spectroscopicmeasurements.'9

This work suggests also that evaluation of stochasticerrors could provide insight into the degree of consistencywith the Kramers-Kronig relations. The concept that thereexists a relationship between stochastic error structureand bias errors is supported by repeated experimentalobservation that the standard deviation of real and imag-inary parts of the impedance were equal except for spec-tra that were found to be inconsistent with the Kramers-Kronig relations.

AcknowledgmentThis work was supported in part by the U.S. Office of

Naval Research under Grant No. N00014-93-1-0056 (A. J.Sedriks, Program Monitor), and by the Engineering Re-search Center (ERC) for Particle Science and Technologyat the University of Florida, the National Science Foun-dation Grant No. EEC-94-02989, and the Industrial Part-ners of the ERC. Helpful conversations with Claude Des-louis and Bernard Tribollet (CNRS, Paris) and theassistance of Yudit Candocia are gratefully acknowledged.

Manuscript submitted March 7, 1996; revised manu-script received Sept. 27, 1996.

Univeristy of Florida assisted in meeting the publicationcosts of this article.

LIST OF SYMBOLSa weighting factor defined by Eq. 16f current, Aj the imaginary number, FiM number of nodes used for numerical evaluation of

integralsN number of observations for replicated experimentsW weighting factor for numerical evaluation of

integrals.x frequency, rad/sy integration variable defined above Eq. 12Z complex impedance, tl

complex stochastic errorcomplex viscosity

w frequency, rad/s(1 rotation speed, rad/sSubscripts

imaginary partk observation number for replicated experimentm index number used for numerical evaluation of

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Corrosion Protection of Copper by a Self-Assembled

Monolayer of Alkanethiol

Viqi Feng,° Wah-Koon Teo)' Kok-Siong Siow,a Zhiqiang Gao, Kuang-Lee Tan,c arid An-Kong Hsieha

"Department of Chemistry, 8Department of Chemical Engineering, and 'Department of Physics,National University of Singapore, Singapore 119260

ABSTRACT

A self-assembled monolayer of 1-dodecanethiol (DT) was formed on a copper surface pretreated using different meth-ods. The corrosion protection abilities of the monolayer were evaluated in an air-saturated 0.51 M NaC1 solutionusingvarious techniques including electrochemical impedance spectroscopy, polarization, coulometry; weight loss, andx-rayphotoelectron spectroscopy. It was found that the corrosion resistance of the monolayer was improved markedly by usinga nitric acid etching method. A minimum concentration of 10 M DT was needed to form a protective monolayer. The DT-monolayer retarded the reduction of dissolved oxygen and inhibited the growth of copper oxide in the NaC1 solution. Incomparison with other inhibitors, such as benzotriazole (BTA) and mercapto-benzothiazole (MBT), the DT-monolayershowed much better corrosion resistance in aqueous solution.

LnfroductionThe chemical-adsorption of alkanethiols (HS-C,,H2,1)

from solution onto the surface of gold and the formation ofdensely packed monolayer films (self-assembled monolay-ers) have been extensively studied.'-6 Laibinis and White-sides78 applied alkanethiol monolayers to freshly evapo-rated copper and studied the wettability of the monolayer.Although there were some early reports of the practicalapplication of alkanethiols as corrosion inhibitors for cop-per9" and iron,12 the corrosion resistance of such mono-layer-covered copper in atmosphere under ambient condi-tions was first investigated by Laibinis et al. using x-rayphotoelectron spectroscopy (XPS).'3 The monolayer-cov-ered copper substrate was oxidized to CuO after 391 hexposure in air.'3 Yamamoto et al. prepared an alkanethiol(n = 6 to 18) monolayer on copper by immersing galvano-statically reduced copper in a 5 mM alkanethiol ethanolsolution for 30 mm under the protection of N2, and the cor-rosion resistance of the monolayer were measured in 0.5 MNa2SO4 solution after 1 h immersion.'4 It was found, how-ever, that only moderate protection abilities of the filmagainst copper corrosion were observed.'4 A chemicallymodified layer was therefore made by reacting an 1 1-mer-

capto-1-undecanol H0(CH,),,SH (MUO) monolayer withalkyltrichiorosilance CH2,,,,SiC13 (C,TCS) and water toform a more protective alkylsiloxane film."6 The MUOmonolayer modified with C,,TCS has maximum protectionefficiency of 94.7% in 0.5 M Na2SO4 solution.16 Such amonolayer would be advantageous in protecting the sur-face of an ultrafine device against corrosion in which thestructure and thickness of the protective layers can becontrolled by molecular-level modification of the mono-layer with various reactants.

The alkanethiol monolayer could also be useful for thetemporary protection of copper equipment in industries,such as the protection of fresh surfaces after finishing,chemical cleaning, or during transportation, if the mono-layer is protective and can be formed and removed easily.The formation of the monolayer is a fast chemisorptionprocess, completed within 30 min,'4" but the corrosionprotection abilities of alkanethiol monolayer are still nothigh enough (the protection efficiencies of 1-dodecanethi-ol and 1-octadecanethiol monolayer are 65.3 and 80.3%,respectively, after 1 h immersion in 0.5 M NaSO4 solu-tion).'4 The formation of the monolayer is susceptible tooxidation of copper on air exposure and the adsorption of