interpretation of electrochemical noise data · potential noise of the working electrode pair can...

CRITICAL REVIEW OF CORROSION SCIENCE AND ENGINEERING

265CORROSION–Vol. 57, No. 30010-9312/01/000051/$5.00+$0.50/0

© 2001, NACE International

Submitted for publication September 2000; in revised form,December 2000.

* Corrosion and Protection Centre, UMIST, PO Box 88, ManchesterM60 1QD UK.

Interpretation of Electrochemical Noise Data

R.A. Cottis*

ABSTRACT

The measurement of the electrochemical noise associatedwith corroding metals places a number of requirements onthe measurement system if good quality data are to be ob-tained. In particular, aliasing and quantization noise shouldbe avoided. Analysis methods may ignore the ordering of themeasured potential or current values by using sequence-independent parameters such as the mean, standard devia-tion, skew, and kurtosis, or they may take the sequencinginto account by computing the autocorrelation function,power spectra, or higher-order spectra. When potential andcurrent noise are measured simultaneously, additional meth-ods are available, including the calculation of electrochemicalnoise resistance, electrochemical noise impedance, character-istic charge, characteristic frequency, and various cross-correlation methods. Newer, somewhat more speculativemethods include wavelet and chaos analysis. One of themost attractive prospects of electrochemical noise measure-ment methods is the ability to identify the type of corrosion,something that is not possible with alternative electrochemi-cal methods (except for methods specific to particular metal-environment systems). The characteristic frequency andcharacteristic charge appear to offer simple but usefulparameters for this purpose.

KEY WORDS: chaos, characteristic charge, characteristicfrequency, coefficient of variation, corrosion fatigue, crevicecorrosion, noise, intergranular corrosion, localization index,pitting, power spectra, stress corrosion cracking, wavelet

INTRODUCTION

The measurement of electrochemical noise (EN) forcorrosion studies was first described by Iverson in1968,1 and Tyagai examined EN in electrochemicalengineering at about the same time.2-3 Initial worklooked at the fluctuations in electrochemical poten-tial;4-7 subsequently, it was realized that fluctuationsin current could also be studied.8-10 At first the latterworkers were concerned with a detailed investigationof individual transient events, notably those associ-ated with metastable pitting.11 Later, it was appreci-ated that the combination of electrochemicalpotential and current noise is more powerful thanthe individual measurements, and several workersexamined the electrochemical noise resistancemethod (the acronym ENRM is used here for themethod, while the coventional Rn is used for themeasured value) and then the electrochemical noiseimpedance. These techniques were probably firstreported in a patent obtained by Eden, et al.,12

although they may have been independently rein-vented by other authors. A brief summary of themeasurement methods for EN data is presentedhere, but the primary objective of this review is toexamine the methods that have been used for theanalysis and interpretation of EN data. For a reviewof other aspects of EN, readers are referred to thepaper by Eden,13 the proceedings of an ASTM sympo-sium held in Montreal,14 and papers presented tothe EN Symposia at the Annual NACE CorrosionConference.


266 CORROSION–MARCH 2001

MEASUREMENT OF EN

Electrochemical potential noise is measured asthe fluctuation in potential of a working electrodewith respect to a reference electrode or as the fluc-tuation in potential difference between two nominallyidentical working electrodes. The measurement isreasonably straightforward, although care is neededto avoid a number of problems, such as instrumentnoise (summarized in the Appendix) and extraneousnoise, aliasing, and quantization.15

Electrochemical current noise is generally meas-ured as the galvanic coupling current between twonominally identical working electrodes, with the current being measured using a zero resistance am-meter (ZRA) to ensure that the two electrodes are atthe same electronic potential.(1) Some workers16 haveused a low-value resistor and measured the currentnoise as the fluctuation in voltage across the resistor,on the grounds that this is subject to less interfer-ence from the electronic circuitry, although littleevidence justifies this fear. Electrochemical currentnoise may also be measured as the current to asingle working electrode that is held at a fixed poten-tial. This was used by earlier workers who wereanalyzing individual transients, often at artificiallyelevated potentials, but it is less commonly used ingeneric EN work since it does not permit the simulta-neous measurement of a potential noise. Both poten-tial and current noise may be low in amplitude anddifficult to measure. The measurement of potentialnoise is expected to be particularly difficult for uni-form corrosion of large electrodes because the powerspectral density (PSD) of potential noise is expectedto be inversely proportional to specimen area, and aresolution of 1 or 0.1 µV will typically be required toavoid excessive quantization. Somewhat counter-intuitively, the measurement of current noise is alsoexpected to be more difficult for larger electrodesbecause the lower impedance of the larger electrodeleads to a poorer signal-to-noise ratio. Consequently,EN measurements should generally use relativelysmall electrodes. They should not be so small, how-ever, that any corrosion “events” become excessivelyrare. Consideration should also be given to statisticalaspects; if small electrodes are used to monitor forthe possibility of localized corrosion in a large struc-ture, then it is likely that the structure will sufferfrom localized corrosion well before the small moni-toring electrodes.

General-purpose data acquisition and electro-chemical instruments are typically rather poor atmeasuring EN, especially current noise. In particular,

the use of autoranging for the current-sensing resis-tor frequently seems to create problems.

If the current noise is monitored as the currentbetween two nominally identical electrodes, then thepotential noise of the working electrode pair can bemeasured with respect to a reference electrode or athird working electrode. This is now effectively thestandard method used to measure EN. It has theadvantage of measuring the current and potentialnoise essentially simultaneously, and both measure-ments relate to the same working electrode pair. Thelatter point is only partially correct if a third workingelectrode is used as the reference electrode since thepotential noise from the latter may dominate themeasurement (because it will usually have a smallerarea than the combined area of the two current-measuring electrodes).

MEASUREMENT METHODS

While it is feasible to derive summary EN param-eters directly through analogue electronic methods,it is more common to record EN data as digital timerecords. The process of converting the real (continu-ous-time analogue) potential or current signal to adigital (discrete-time digitized) time record introducesa number of errors or limitations:

— The sampling process limits the maximumfrequency that can be represented in the time record.In simple terms, at least two samples are requiredwithin each cycle of a given frequency in order to reg-ister the amplitude at that frequency. Consequently,there is a frequency limit of half the sampling fre-quency, known as the Nyquist limit or the Nyquistfrequency (ƒN).

— Frequencies above the Nyquist limit that arepresent in the signal immediately before it is sampledwill reappear at a lower frequency by the processknown as aliasing. There is no way to distinguishbetween real and aliased signals, and it is essentialto avoid aliasing by filtering out frequencies abovethe Nyquist limit before sampling.17-18

— Converting the continuous analogue voltage orcurrent to a digital form will introduce a limit on theresolution of the digital signal, which will only beable to represent a discrete set of values. This effectis known as quantization and introduces a noise intothe signal known as quantization noise. Providing thatthe fluctuations in the analogue signal are larger thanthe quantization step size, and assuming that theanalogue-to-digital conversion process is ideal, theerror introduced by quantization will be uniformlydistributed between –0.5Eq and 0.5Eq, where Eq is thequantization step size. Quantization noise is white—that is, it has the same PSD at all frequencies—andthe power spectrum is thus flat. Quantization noiseis therefore typically more important at higher fre-quencies, where the EN amplitude is normally lower.

(1) The term “electronic potential” is used here to indicate thepotential of one piece of metal relative to another, measured in themetallic, electron-conducting circuit. The electrochemicalpotentials of the two electrodes may, of course, differ as a result ofIR drops in the solution between them.


267CORROSION–Vol. 57, No. 3

Quantization noise is often visible in the timerecord as a “banding” of the data (Figure 1). If suchfeatures are observable, then quantization noise isprobably significant at higher frequencies.

The electronics of the signal-conditioning circuit-ing will introduce noise into the measurement. Thefull noise analysis of a typical amplifier has beenperformed by Bertocci and Huet.19 The analysis isreproduced in the Appendix in a simplified form(and, as such, it ignores some sources of error, suchas the effect of the ZRA input current noise on themeasured potential noise). EN, at least noise that ismeasurable, is generally experienced at low frequen-cies, typically in the range of 10–3 Hz to 1 Hz. Thereis some disagreement over the optimum samplingfrequency (and hence over the range of frequenciesincluded in the measurement). Most work has beenperformed with a sampling frequency in the regionof 1 Hz. The selection of this frequency is largelyaccidental; it is a “nice round number,” it is easilyachieved with standard digital voltmeters, and it iswell clear of power line frequencies, so interferencecan be avoided easily. Much of the early work (andsome current work) using digital voltmeters tosample the data did not use anti-aliasing filters,relying instead on the voltmeter to exclude power lineinterference. While they do this effectively, there isfrequently evidence of aliasing of frequencies betweenthe Nyquist limit and the frequencies rejected by thevoltmeter. This aliasing typically leads to a high-fre-quency plateau.17 Some workers may have interpretedthis plateau as evidence of having attained a high-frequency limit, thereby justifying the sampling fre-quency used. However, little evidence validates this.

Some other workers, notably a consortium inEurope,20 have used higher sampling frequencies, upto 30 Hz. This is much more difficult because there islittle clearance between the sampling frequency andthe power line frequency. Unfortunately, Reference20 is not specific about the differences observed bythe various workers, and it is hard to draw clear con-clusions about the value of higher frequency infor-mation. Nevertheless, this area does seem to meritfurther work.

SYMBOLS

A wide variety of methods can be used to de-scribe electrochemical potential and current noisetime records and parameters derived from them. Inthis paper the same symbols are used as in Refer-ence 15. The basic potential-time signal is given thesymbol E(t), with t indicating a time-varying quantity.

The same signal may be represented in the frequencydomain as E(ƒ), with ƒ representing frequency.The parenthetical t or ƒ may be omitted where themeaning is obvious or irrelevant. If the mean poten-tial is subtracted from the data, such that the result-ant signal specifically has zero mean, then En isused—En only has real significance for data in thetime domain, so the (t) is omitted. When the signalhas been sampled, the kth sample is indicated asEn[k]. As far as possible, derived parameters areexpressed by the mathematical operations used toobtain them. Thus, the variance of potential is giventhe symbol En

2, and the standard deviation is giventhe symbol En

2 (the square root of the mean value ofthe square of the potential noise).

AREA EFFECTS

The effect of specimen area on EN measurementshas not been fully established. However, it seemsprobable that there is no unique relationship and thatdifferent area effects are possible. The range of rea-sonable relationships becomes evident by consideringtwo situations that probably represent limiting cases:

— Current noise is considered to be produced bya large number of independent (and henceuncorrelated) current sources. The potential noiseresults from the application of the current noise tothe polarization resistance, Rp, of the electrodes(more precisely, the metal-solution impedance, ratherthan just Rp, should be considered). In this case,since the power of the uncorrelated current noisefrom the various regions of the electrode will add, thevariance of the current noise is proportional to area.Hence, the amplitude of the current noise (measuredas the standard deviation) is proportional to √Area,while the amplitude of the potential noise is propor-tional to 1/√Area.

— The current noise from different areas of agiven electrode is considered to be correlated (whilethis seems rather unlikely, it is not completely im-plausible).(2) In this case, the amplitude of the current

(2) Note that it is also necessary for the current noise of the twoworking electrodes to be uncorrelated—if the two noise currentsare correlated, then their sum will be the difference between twocorrelated signals, and will have an expected value of zero,assuming the two current powers are equal.

FIGURE 1. Time record showing quantization as “banding” of thecurrent data (the quantization step size in this case is 10–6 A).



noise will be proportional to specimen area, and, as-suming that the potential noise is again produced bythe action of the current noise on Rp, it will be inde-pendent of area.

In the more general case, if the potential noise isassumed to be produced by the action of the currentnoise on Rp, then amplitude of current noise ∝ An andamplitude of potential noise ∝ An–1, where n is ex-pected to range from 0.5 (for uncorrelated noise) to 1(for correlated noise). Relatively little work has beendone to test this relationship experimentally. Whatevidence there is tends to support a value of n in theregion of 0.5.21-27

One important implication of these results isthat it is not appropriate to normalize electrochemi-cal current noise data by dividing the amplitude ofthe current by the area. Equally important, it is notappropriate to report potential noise without indicat-ing the specimen area. Thus, the only logical ap-proach is to report current and potential noise dataas recorded, being sure to quote the specimen areaused for the measurements.

ANALYSIS METHODS

Having obtained potential and/or current noisetime records, many methods can be used to analyzethe data. These are summarized below.

Analysis methods may be divided into those thattreat the collection of voltage or current values with-out regard to their position in the sequence of read-ings and those that take the sequence into account.

SEQUENCE-INDEPENDENT METHODS

The sequence-independent methods mightalso be described as statistical, although this isslightly misleading because many of the sequence-dependent methods are also based on soundstatistical procedures.

MomentsMoments are a set of generalized statistical

parameters derived from a sample or population.The nth moment of a sample is given by

n momentx k

Nth

n

k

N

= =∑ [ ]

1 (1)

where x[k] is the kth measurement of x, and N is thenumber of measurements in the sample.

The first moment is clearly the mean. Highermoments, when they are defined as above, are re-lated to other statistical parameters such as the vari-ance (for the second moment). However, the highermoments are heavily influenced by the mean, and itis common practice to use the central moment. Thecentral moment is obtained by subtracting the meanfrom the sample before calculating the moment, asin the following:

n momentx k x

Nth

n

k

N

central =( )

=∑ [ ] –

–1

1 (2)

Then the second central moment is equivalent to thevariance. The power n used in calculating the mo-ment or related parameters is known as the order.The denominator in Equation (2) is N–1 because theuse of the mean for the sample (rather than for thepopulation) reduces the number of degrees of free-dom by one.

Mean, Variance, and Standard DeviationThese standard statistical parameters are widely

used in EN analysis. In particular, the standard de-viation (the square root of the variance) is the mostnatural parameter to describe the “amplitude” of anoise signal. It is important to appreciate that thestandard deviation obtained for a given measurementdepends on the sampling frequency. In general, in-cluding a wider range of frequencies in the measure-ment will lead to increased standard deviation. Thisdirectly results from the relationship between thevariance and the PSD.

SkewSkew (or skewness) is a third-order statistic, and

it is derived as the third central moment divided bythe cube of the standard deviation:(3)

Skew

x k x

N x k

k

N

=( )

( )=

∑ [ ] –

( – ) [ ]

3

1

2321

(3)

The above operation normalizes the value obtainedsuch that the skew is a dimensionless description ofthe extent to which the distribution of values areskewed about the mean. The standard error of theestimate of the skew is relatively large, 6 / N for anormal distribution,28 where N is the number of sam-ples. In the case of a typical noise time record con-sisting of 1,024 samples, the standard error in theskew will therefore be 0.077, and this should betaken into consideration when the skew is being usedin analysis of EN.

(3) Note that Reference 15 uses N in place of N–1. The latter is strictlycorrect when dealing with a sample from a larger population,although the difference is negligible for normal time recordlengths.



KurtosisKurtosis is a fourth-order statistic, derived by

dividing the fourth central moment by the fourthpower of the standard deviation:

Kurtosisx k x

N x k

nk

N

n

=( )

( )=

∑ [ ] –

( – ) [ ]

4

1

22

1 (4)

The kurtosis for a normal distribution is 3, and it iscommon to report the value obtained after subtract-ing 3:

Normalized kurtosisx k x

N x k

nk

N

n

=( )

( )=

∑ [ ] –

( – ) [ ]–

4

1

22

13 (5)

This can lead to some confusion, and the term “nor-malized kurtosis” is suggested to indicate (kurtosis-3). The standard error of the estimate of the kurtosisis large, 24 / N for a normal distribution.28

Higher-Order StatisticsIn principle, statistics of even higher order than

the kurtosis can be used, but in practice the stan-dard error of the estimated value becomes so largethat it is unusual for differences obtained to besignificant.

Root Mean Square, Mean,and Standard Deviation

The root mean square (rms) is similar to thestandard deviation, but calculated without removingthe mean from the data. Consequently, the rms,mean, and variance are related:(4)

x x xrms n2 2 2= + (6)

SEQUENCE-DEPENDENT METHODS

Methods that take account of the data sequenc-ing within the time record are inevitably more com-plex than the sequence-independent methods, andthey produce more complex outputs.

Autocorrelation FunctionProbably the simplest sequence-dependent

analysis method is the calculation of the autocorrela-tion function (ACF). In essence, the ACF is theexpected value of the product of the time series at

one time and at certain later times. Consequently,the ACF is a function of the lag time, the time differ-ence between the two samples. For Gaussian whitenoise, where each sample in the time record is anindependent sample from a normal distribution, theACF is zero for all lags except zero. Kriston andLakatos-Varsányi have recently used the ACF to in-terpret EN data,29 although this method has generallybeen superseded by power spectra, which present thesame information in a slightly more intuitive way.

Power SpectraAnother sequence-dependent analysis estimates

the power present at various frequencies included inthe signal. This process is known as spectral estima-tion, on the basis that the objective is to estimate thepower present, treating the signal being analyzed asa sample from a larger population of signals extend-ing backwards and forwards in time. The power spec-trum obtained plots the PSD, the power per unit offrequency, as a function of frequency. The PSD iswritten here as ψE for potential and ψI for current,following Bertocci, et al.,30 and has units of V2/Hz orA2/Hz, respectively. There is no correct way to esti-mate a power spectrum, and many methods can beused, of which the following two are the most com-mon in the corrosion field:

Fourier Transform Methods — The most directmethod of power spectrum estimation computes theset of sine waves that would need to be combined toobtain the observed signal. For sampled data this isdone using the discrete Fourier transform (DFT),usually in the form of the fast Fourier transform(FFT) algorithm. The PSD is then determined as theamplitude squared of the sine waves, divided by thefrequency separation. In order to minimize artifactsin the spectrum, trend removal and windowing(a process that reduces the amplitude of the timerecord towards the start and finish in order to mini-mize errors associated with the sharp changes invalue)28 are generally applied to the time recordbefore computing the spectrum. It is important toconduct trend removal before windowing; if thewindowing is performed first, the subsequent trendremoval will destroy the benefits of the windowingprocess.

Because of the reasonably obvious relationshipbetween the time record and the resultant powerspectrum, the FFT method is often implied to havesome form of fundamental “correctness.” However,this is misleading. The start time for the measure-ment and the sample interval is an accident of thesampling process—sampling at a different frequencyor starting at a different time would give a differentset of values and a different power spectrum. Bothare equally valid, and neither is, in any fundamentalway, more correct when referring to a single spec-trum. This is evidenced by the frequent use of the

(4) This (as with most of the other results in this paper) is an“expected” result (i.e., it is the best estimate of what the result willbe), and it will not necessarily be exact for a particular data set.



averaging of multiple spectra to reduce scatter, whichcan be interpreted as an error in the estimate of the“correct,” smooth spectrum that would be obtainedwith an infinite number of samples.

In the FFT method of power spectrum estima-tion, N values in the time record are used to estimateN/2 values of PSD in the power spectrum. The esti-mation is thus comparable to estimating the mean ofa population on the basis of two samples, and it isnot surprising that the error in the estimate is largeand the power spectra appear noisy. Indeed, a powerspectrum estimated using the FFT method withoutany averaging is suspect when it is not noisy.

As with “normal” statistics, averaging severalvalues can reduce the error of the estimate. In thecase of power spectrum estimation, points can be av-eraged at the same frequency from several equivalentspectra, or values can be averaged for several adja-cent frequencies within one spectrum. The twoapproaches are mathematically equivalent,28 and

either way the standard error is reduced by a factorof 1/√M, where M is the number of samples used incalculating the average.

As an alternative to averaging groups of points inthe power spectrum, the spectrum can be smoothedor have functions fit to it. This approach has beenrather little used (except that the maximum entropymethod [MEM] is effectively a method of fitting afunction to the power spectrum) and probably meritsfurther investigation.

The Maximum Entropy Method — Several powerspectrum estimation methods take advantage of thefact that the power spectrum may be calculated asthe Fourier transform of the ACF. If the “raw” ACF isused, the result is exactly the same as that obtainedfrom the Fourier transform of the original timerecord. However, functions may be fitted to the ACF,and these functions transformed into power spectra.This has the effect of smoothing the power spectrum.

The MEM is one such method. It has a numberof interesting characteristics, but it is also widelymisunderstood, at least in the corrosion literature.The MEM computes a number of coefficients that de-scribe the ACF and hence, indirectly, the power spec-trum. The number of coefficients is known as theorder of the MEM and is a user-defined variable.(5)

The order used has an important effect on the powerspectrum obtained. Each coefficient in the MEM maybe thought of as defining a control point for the spec-trum. If the order is small, the spectrum is necessar-ily smooth and simple. If the order is large, thespectrum may appear much noisier. This has beeninvestigated experimentally31 and is illustrated for anexample noise time record in Figure 2.

Previous comparisons between FFT and MEMpower spectra in the corrosion literature have im-plied that the MEM and FFT are somehow fundamen-tally different. Some workers have also concludedthat the MEM is fundamentally unsuited to the esti-mation of power spectra.32 However, such conclu-sions are usually based on the comparison of theunsmoothed FFT and the MEM with a relatively loworder. If the FFT is smoothed by averaging, or theorder of the MEM is increased, such that the numberof points defining the spectrum is comparable, thespectra are found to look rather similar, and a ratio-nal approach would use the FFT as a guide to theoptimization of the MEM order, on the basis that theMEM provides a good general purpose method for thenear-optimal smoothing of the power spectrum. It isimportant not to use the MEM alone with a fixed or-der because this risks failing to detect low-frequencyfeatures, such as are present in Figure 3.

A second property that has been claimed of theMEM is the ability to extrapolate to frequencies lowerthan 1/(time record duration). This is probably basedon a misunderstanding of MEM features. In particu-lar, the MEM makes the minimum possible assump-

(5) There are methods for optimizing the order automatically, andthese have been used for EN analyses. This may explain theunfortunate observation that the order is rarely given when MEMresults are printed.

(a)

(b)

FIGURE 2. Illustration of the influence of MEM order on powerspectrum characteristics (F = FFT, M1 = MEM with order 1, M10 =MEM with order 10, etc.). The MEM spectra are displaced upwardsto allow them to be seen clearly. The time record has 1,024 points,so the MEM with order 500 has similar degrees of freedom to theFFT, as well as a similar amount of noise. This particular powerspectrum is modeled naturally by the MEM, so even an order of only1 gives a reasonable fit: (a) time record and (b) power spectra.



tions about what happens at frequencies outside therange that can be analyzed, but this is not the sameas producing a valid extrapolation. In fact, if the con-trol points associated with the MEM coefficients areconsidered, as well as the fitting process being essen-tially linear with frequency, then the control pointscan be expected to be reasonably uniformly distrib-uted along the linear frequency axis. Thus, the lowestfrequency that can be estimated reliably is in the re-gion of ƒN/M, where M is the MEM order. In practice,the fit may be acceptable down to a somewhat lowerfrequency because of the way the coefficients operateon the spectrum, combined with the tendency formost of the power in EN signals to be concentratedat lower frequencies. As can be seen by comparisonof Figure 2 and Figure 3, there is no general ruleabout the order required to obtain a good descriptionof the spectrum.

The assumptions made about frequencies out-side the valid range actually translate into anassumption that the PSD attains constant low- andhigh-frequency limits outside the range that can bemodeled. The almost invariable observation of a low-frequency plateau may be as much a function of theuse of the MEM as of a physical reality! Conse-quently, a large MEM order should be used to obtaina good estimate of the low-frequency end of the spec-trum, with 10% of the points in the time record beinga reasonable starting point.28 This is a much largervalue than is commonly used. It also leads to a rela-tively noisy behavior in the high-frequency partsof the spectrum. This is illustrated in Figure 3, al-though it should be noted that having an increasingPSD at very low frequencies is somewhat unusualand therefore requires a very large order to describethe spectrum correctly. It is more common for ENsignals to have a spectrum closer to that shown inFigure 2, which reveals that even an order of onlyone gives a reasonable description of the spectrum.(6)

Relationships Between PSD and VarianceThere is a direct link between PSD and variance.

If a narrow range of frequencies is selected by suit-able filtering, the variance observed will be the PSDmultiplied by the bandwidth of the filter. More gener-ally, the variance observed for a given signal is theintegral divided by frequency of the PSD.

PSD and variance divided by bandwidth may beinterchanged in many of the theoretical results, suchas the shot noise formula, providing that the fre-quency range included in the measurement of thevariance is taken into account.

Some parameters, such as the Rn, are ideallymeasured at a low frequency. These will typically beestimated better by taking the PSD at a suitable fre-quency than by using the variance or standard devia-tion measured over an arbitrary range of frequencies.If the PSD attains a valid low-frequency limit, thenthis is usually an appropriate value to use. If thepower spectrum exhibits 1/ƒ noise, as shown inFigure 3 for example, then it is more difficult to de-termine an appropriate value to use; as an arbitrarysuggestion, the PSD at 10–3 Hz is probably as good asany, and will be included in most measurements.

Higher-Order SpectraJust as higher-order statistical parameters than

the mean and variance may be considered, so mayhigher-order spectra. Unfortunately, the results areeven more difficult to interpret, and while there doesappear to be relevant information in higher-orderspectra, it is not known, as yet, how to extract it.

It can be seen how higher-order spectra areobtained by considering the derivation of the powerspectrum by way of the ACF. The ACF is derived asthe expected value of the product of two samplesfrom the signal that are separated in time by a giventime lag.

(a)

(b)

FIGURE 3. Effect of MEM order on low-frequency behavior. Thepower spectra are plotted in the same way as in Figure 2. Note thatalthough all MEM spectra indicate an increase in PSD at lowerfrequencies, the frequency range indicated falls as the orderincreases and is only correctly identified with an order of 300.Too low an order gives a false indication of a low-frequency plateau:(a) time record and (b) power spectra.

(6) The time records in Figures 2 and 3 were actually measured asthe instrument noise for a general-purpose commercialelectrochemical system. The initial transient in potential is clearlyan instrumental artifact (it is not cause by the instrumentwarming up because it happens even after extended periods ofoperation), and these results demonstrate the need to check theinstrumentation very carefully.



The next-higher-order equivalent to the ACF(known as the third-order cumulant) is obtained asthe expected value of the product of three samplesfrom the signal, separated by two independent timelags. As a result of the two independent time lags,the cumulant is two-dimensional, and thebispectrum is obtained as the two-dimensionalFourier transform of the cumulant.

Just as the skew is normalized by dividing thethird central moment by the cube of the standarddeviation, so it facilitates interpretation of thebispectrum to normalize it so that a Gaussian signalwill produce a bispectral content of zero over thewhole plane.

Figures 4 and 5 show sample normalizedbispectrum and the time records from which theywere obtained.33 In the case of steel corrosion in acid(Figure 4), the normalized bispectrum is essentiallyzero over the whole plane, which is consistent withthe observed lack of structure in the time record.In contrast, for steel in alkaline chloride (Figure 5),

the structure in the normalized bispectrum appearsto indicate the structure in the original time record.The general form of this spectrum is relativelyreproducible.

Wavelet MethodsThe technique of wavelet analysis may be re-

garded as a variant of Fourier analysis in which thecontinuous sine waves used in the Fourier transformare replaced by transients with a finite duration,known as wavelets. A set of wavelets of varyingamplitude, duration, and location can then be con-structed such that their sum reproduces the signal ofinterest. This is an effective approach to analyzingsignals that have a nonstationary character.(7) SinceEN signals are frequently nonstationary, this offersan attractive prospect.

Unfortunately, the “raw” wavelet transform haslimitations as a data-reduction technique. The formof the wavelet transform is potentially very fluid: anessentially infinite variety of wavelet functions can beused, and the distribution of wavelet durations andlocations used is not fixed. It is common practice tofix on a specific set of wavelet locations and dura-

(a)

(b)

FIGURE 4. Normalized bispectrum for steel in HCl (after Bagley):33

(a) time record and (b) normalized bispectrum.

FIGURE 5. Normalized bispectrum for steel in 0.05 M Ca(OH)2 +0.025 M NaCl (after Bagley32): (a) Time record (current is lower tracewith shorter transients) and (b) normalized bispectrum for current.

(a)

(b)

(7) The term “stationary” is used in this paper in its strict statisticalsense, and a signal is therefore stationary if all of its statisticalproperties are constant over time.



tions, but the wavelet function is often treated as auser-adjustable parameter.

A second problem with the wavelet transform isthat it produces a complex output (the amplitudesfor each of the combinations of duration and locationused). This can be displayed graphically,34 but it re-mains rather difficult to interpret. Some form of datareduction is therefore required in order to derivesimple parameters that bear some relation to the cor-rosion process. The simplest approach is to averagethe amplitudes for wavelets of a given duration.35

This produces results that appear similar to a powerspectrum, and it is unclear whether there is any realadvantage in using wavelet methods rather thanmore conventional power spectrum estimators.More recently, a number of investigators have beenattempting to derive alternative indicators of corro-sion type and severity from wavelet analysis.36-37

These approaches appear interesting, but it remainsto be seen whether the methods proposed havegeneral applicability.

Chaos MethodsChaos analysis attempts to characterize behavior

that is deterministic but highly unstable. It is plausi-ble that some corrosion processes come into this cat-egory, and a number of workers have applied chaosanalysis methods to the analysis of EN data.38-40

Reported analyses have typically found indicationsof chaotic behavior (although there is presumablyan element of self-selection, in that experiments thathave not found chaotic behavior have not been re-ported). The results are of theoretical interest, but itis not clear that they yet provide a general approachto interpreting EN data.

DriftDrift may be defined as a change of the mean

potential or current divided by time. Note that thisdefinition is much like that for EN, and there are un-resolved fundamental questions about when some-thing is drift and when it is noise. Another problemwith drift is that its existence implies that the signalis nonstationary, and, consequently, virtually all ofthe standard analysis procedures become invalid. Ifthe drift consists of a linear change in the mean di-vided by time, it can be removed simply by subtract-ing the linear regression line from the data, acommon method of treating drift, especially prior tospectral estimation. A similar result may be obtainedby applying an analog high-pass filter to the signalprior to sampling, although very low-filter frequen-cies pose some problems, including the difficulty ofobtaining the very large capacitors that are neededand the very long times required for the filter to sta-bilize. More complex drift functions, such as expo-nential decays, can potentially be removed aftermeasurement in a similar way (i.e., by using regres-

sion procedures to fit the drift function to the dataand then subtracting it from the data to leave thenoise). However, as the drift function becomes morecomplex, concerns about the possibility of subtract-ing real low-frequency noise increase, and it is prob-ably best to limit drift removal to the subtraction of astraight line unless there is sound evidence for theexistence of an alternative drift function.

If it is not removed, a straight line drift will su-perimpose a “boxcar” distribution (i.e., a distributionwith constant probability density inside a windowand zero outside) on the noise distribution. This willdirectly increase even-order statistics, such as thevariance and the kurtosis, and will indirectly de-crease normalized odd-order statistics, such as theskew, by virtue of the increase in standard deviation.Nonlinear drift may also directly affect odd-orderstatistics, such as the skew.

If spectral estimation procedures using the FFTare applied to a signal with linear drift, this willeffectively appear as a sawtooth signal, which willadd power with a 1/f2 characteristic. The MEM isless affected by drift, but the effects are also seen atlow frequencies. In order to minimize these effects,linear drift removal (also known as trend removal)is generally applied to noise signals prior to spectralestimation.

METHODS USING POTENTIALAND CURRENT NOISE

So far, parameters obtained from an individualpotential or current time record have been consid-ered. However, it is feasible to measure potential andcurrent noise simultaneously, and additional param-eters can be derived by using the two time records.

The simplest parameters are obtained by simplecombinations of the standard deviation (or other sta-tistical parameters) of potential and current. Becausethese do not take account of the sample-to-samplecorrelation of the potential and current, it is notnecessary for the two time records to be obtainedsimultaneously; however, they should obviously cor-respond to similar corrosion situations, and are bestmeasured simultaneously.

Noise ResistanceThe Rn is obtained by dividing the standard de-

viation of potential by the standard deviation of cur-rent. Several workers have analyzed the relationshipbetween the Rn and the Rp, and some claim to haveproved that the two are equivalent. However, all ofthese analyses assume, either explicitly or implicitly,that the potential noise can be modeled as the actionof the current noise on the metal-solution imped-ance, with the latter usually treated as the Rp, andthey therefore effectively assume that the Rn and Rp

are equivalent. The most comprehensive such analy-



sis is arguably that of Bierwagen, et. al.,41 which pre-sents a statistical description based on the derivationof thermal noise. However, this still embeds anassumption that the potential noise depends on themetal-solution impedance. Furthermore, as Bertocci,et al.,30,42 and Mansfeld, et al.,43 have shown, theanalyses tend to ignore the difference in the effectivefrequency at which the Rp and noise resistance Rn aremeasured. Whereas Rp is typically measured at aneffective frequency of ~ 10–2 Hz (Mansfeld generallydefines Rp as the limit of the impedance as the fre-quency tends to zero; this is a good fundamentaldefinition, but this is not the normal way that Rp ismeasured), Rn is normally measured using a range offrequencies up to ~ 1 Hz; consequently, the two val-ues may be quite different. Note that this is not afundamental limitation of the ENRM—the frequenciesincluded in the measurement of Rn can be modifiedby sampling and/or analysis procedures, and Rn canbe estimated with a reasonable error at any fre-quency down to the inverse of the time record dura-tion. Lee and Mansfeld have argued that Rp (usinghis definition) can be estimated by extrapolation tofrequencies below the lowest measurement frequencyfor an electrochemical impedance spectroscopy (EIS)measurement,44 and it is probably generally true thatEIS provides a better estimate of the impedance thanEN does, not least because measurements can bemade on a single electrode, without the uncertaintyassociated with electrode asymmetry.

Confusion remains with respect to the terminol-ogy used, and some workers talk about resistancenoise rather than noise resistance. While the differ-ence between the two terms may appear slight, theformer term implies a fluctuation in a resistance (ashas been measured by Gabrielli, et al.,45 for example),and should not be used to mean a resistance com-puted from noise measurements.

Electrochemical Noise ImpedanceElectrochemical noise impedance can be esti-

mated in essentially the same way as the Rn, bydividing the PSD of the potential noise by the PSD ofthe current noise:30,43,46

Noise impedance E

I

= ψψ (7)

where the calculation is performed at each validfrequency.

If the potential and current noise are correlated,it may be possible to get a reasonable estimate of thenoise impedance using an unsmoothed FFT. How-ever, it is generally expected that potential and cur-rent time records will be poorly correlated since thepotential is proportional to the sum of the currentsfrom the two working electrodes, while the measured

current is proportional to the difference between thetwo currents. If the variances of the currents fromeach of the two electrodes are the same (which is theexpected situation, and the situation assumed in thesimpler analyses) the sum and difference operation isexpected to eliminate the correlation.47 In this case,the best results (those that are least scattered) areprobably obtained with potential and current spectraderived from the MEM (with due care taken to use asufficiently large order), although heavily smoothedFFT spectra will give comparable results.

Note that the noise impedance calculation actu-ally estimates the modulus of the impedance. It isnot generally possible to estimate the phase of theimpedance. There are two basic reasons for this:

— The computation of the power spectrum withthe MEM loses the phase information about the indi-vidual potential and current spectra, while the FFT,which nominally retains the phase information, isusually too noisy.

— As indicated above, the potential noise andcurrent noise are expected to be uncorrelated. Whilethe noise powers are expected to be related via themodulus of metal-solution impedance in this situa-tion, the phases do not behave predictably.

The terminology for noise impedance is some-what varied, with several terms being used. Never-theless, “electrochemical noise impedance” clarifieswhat is being measured, and there is a direct analogywith “electrochemical noise resistance.” One slightinaccuracy of this term, however, is that the imped-ance is not actually measured, only the modulus ofthe impedance. This justifies the alternative termi-nology of “spectral noise resistance.” “Spectral noiseimpedance” has also been used, but it is tautological(impedance is necessarily a function of frequency, so“spectral” is redundant), as well as implying a fullimpedance measurement.

Note that noise methods can also be used to esti-mate the impedance of an electrode where the noiseis applied by the measuring instrument, either delib-erately or as a result of the noise characteristics ofthe potentiostat.9 This potentially offers advantagesin terms of measurement speed when compared toconventional one-frequency-at-a-time measurements,since all frequencies are measured simultaneously.However, it is subject to a range of problems associ-ated with factors such as the non-linearity of thesystem, and practical experience is that the expectedspeed advantages are difficult to achieve.

Area Effects on ENRM and ImpedanceIt has been suggested above that the relationship

between potential and current noise and specimenarea is uncertain, and it might be expected that simi-lar problems would arise for Rn and impedance mea-surements. However, present indications are that thepotential and current noise have a constant relation-



ship with one another, such that the measured Rn orimpedance are expected to be inversely proportionalto specimen area. Note that this conclusion is basedlargely on theoretical analyses (which generallyassume the area dependence in the first place),and the author is not aware of any systematic experi-mental studies in this area. As with potential andcurrent noise measurements, the specimen areashould always be reported along with Rn and imped-ance data, although it is probably appropriate to useΩ-cm2 units when reporting the actual data.

Effects of Electrode AsymmetryMuch of the theoretical work on EN has assumed

that the two electrodes between which the currentnoise is measured are similar in that the averagecurrent noise level and metal-solution impedance areessentially the same. However, this is frequentlyfound not to be the case. For example, the timerecord of Figure 4 exhibits pitting transients on onlyone of the two current-measuring electrodes. Thisobservation is not a mere coincidence; this behavioris often observed for this system and can be ex-plained by the small pH changes that occur as aresult of the net current between the two electrodes.The nonpitting electrode will be a net cathode, andthe neighboring solution will become more alkaline,which will inhibit pit initiation; the pitting electrodewill be a net anode, which will cause the solution tobecome slightly more acidic, and thus facilitate pitinitiation. Similarly, when EN is used with paintedmetal samples, it is almost inevitable that there willbe a significant difference between the two samples.This system has been analyzed in detail by Bertocci,et al.,30 but the results of the analysis are rather un-satisfying because they consist of functions of fourunknowns (the metal-solution impedance and theelectrochemical current noise for each of the two cur-rent-measuring electrodes), with only two measur-able parameters (the potential and current noise ofthe coupled electrode pair). Consequently, the corro-sion behavior can be interpreted only if the relation-ship between the metal-solution impedance and thecurrent noise is known (or assumed). This may befeasible in specific cases, but in general it is not.

CROSS SPECTRA

ENRM and impedance measurement regard thepotential and current as being linked through a con-stant of proportionality, but in this case the link istreated in terms of an average over the time record.Clearly, the potential and current noise may be ex-pected to be linked to each other in time, althoughthe linkage may be somewhat complex. For example,Figure 5 shows the potential and current timerecords for steel samples undergoing metastablepitting in hydroxide/chloride solution. This measure-

ment was made using a true reference electrode, sometastable pits only occurred on one or another of theworking electrode pair, and always resulted in nega-tive-going potential transients. However, an individualmetastable pit may occur on either working electrode,and the current transient may be either negative- orpositive-going. This particular measurement exhibitspitting on only one of the two working electrodes, sothe current transients are also unidirectional.

Surprisingly little attention has been given tothis aspect of EN analysis, especially since one mightexpect the correlation between events in potentialand current to be an important aspect of analysis.Indeed, experienced EN analysts will generally lookat the correlation in time between potential and cur-rent events as a key indicator of whether the eventsare “real” EN or experimental artifacts.

Correlation is typically detected by taking theproduct of the two parameters; cross correlationoccurs when the two parameters are different. Thesimplest cross correlation is obtained by taking theproduct of the potential and the current on a sample-by-sample basis, and this might be expected to revealcorrelated potential and current events. While theresult is complicated by drift and similar features,most of these can be removed by a variety of tech-niques, and this simple plot is potentially useful.

More sophisticated cross-correlation methodsallow for shifts in time between the potential andcurrent and compute the cross-correlation function(CCF), which is analogous to the ACF. Just as theFourier transform of the ACF produces the powerspectrum, so the Fourier transform of the CCF leadsto the cross spectrum, and this has been applied toEN data by Alawadhi and Cottis.48 It appears that thelow-frequency limit of the cross spectral density maybe related to the charge involved in the transients,and this may be a useful indicator of localized corro-sion. The cross spectrum probably contains otheruseful information about corrosion processes, butthis has not, as yet, received much attention. Thereare questions about the loss of correlation betweenthe current and potential as a result of the bidirec-tional nature of the current transients, but the com-monly observed tendency for only one electrode to pit(as demonstrated in the time record of Figure 4) mayallow the method to be applied in many situations.

NOISE PROCESSES

A number of noise-generating processes havebeen identified in the general scientific literature.Many of the analyses have been developed in thecontext of electronic devices, where noise has impor-tant technical consequences. However, the theoreti-cal analyses are found to apply to a wide range ofphysical phenomena, and they are almost certainlyapplicable to EN.



Thermal NoiseThermal noise results from the thermally-acti-

vated motion of charge carriers. The motion causes arandom separation of charge across resistors andthereby generates a voltage noise given by:

ψE kTR= 4 (8)

where k = Boltzmann constant, T = absolute tem-perature, and R = resistance. Note that ψE is fre-quency-independent; hence, thermal noise is white.

Because of the relationship between noise powerand PSD, this relationship may also be written as:

Potential noise power = 4 kTRb (9)

or:

Potential noise standard deviation = 4 kTRb (10)

where b = bandwidth of measurement.It can be seen that thermal noise will be most

significant when large value resistances are involvedand where the measurement bandwidth is large. Notethat the resistance need not be that of a conventionalelectronic device; thermal noise will also be gener-ated across the impedance of a metal-solutioninterface, and this may be a significant effect forhigh-impedance systems such as passive metals andpainted samples.

Shot NoiseShot noise results from the fact that charge is

quantized, and, consequently, the number of chargecarriers passing a given point will be a random vari-able. If the charge carriers move independently ofeach other, the statistics are relatively simple22 andresult in current noise given by:

ψ I nqI q= = ƒ2 2 2 (11)

where q = charge on charge carrier, I = mean current,and ƒn = mean frequency of charge emission. Notethat ψI does not vary with measurement frequency;hence, shot noise is white.

This can also be written in terms of currentnoise power (or variance):

Current noise power = I n2 = 2qI b (12)

where b = bandwidth of measurement.

In electronic circuits the charge, q, is normallythe charge on the electron, but in electrochemicalsituations the charge may be greater than this. Foractivation-controlled processes the charge will bethat involved in the elementary electrochemical reac-tion and will typically correspond to the charge on1 to 4 electrons. However, metastable pitting andsimilar processes may involve much larger quantitiesof charge. Providing the frequency is low enough thatthe events may be treated as instantaneous, andproviding individual events are independent of otherevents, the shot noise analysis is applicable;(8) in thefollowing analysis, the PSD referred to is that at lowfrequencies, where the noise is white (hence, the PSDis frequency-independent). If it is also assumed thatthe potential noise may be analyzed using the Rn,then it can be shown that it is possible to determinethe corrosion current, Icorr; the average charge, q, ineach metastable pitting event; and the frequency ofthese events, ƒn.22 Earlier calculations of q and ƒn aresomewhat unclear,15 and the derivation is reproducedhere in a more thorough (and hopefully more accu-rate) fashion.

It is assumed that the corrosion process consistsof an anodic process producing relatively large pulsesof charge of relatively short duration (such as meta-stable pitting), while the cathodic process is essen-tially noise-free. Consequently, the process may betreated as a shot noise process with a current of Icorr

and a charge of q in each pulse. It is also assumedthat the two working electrodes between which thecurrent is measured have the same corrosion rate,and that all anodic pulses are uncorrelated. For sim-plicity, the solution resistance is assumed to be zero,so half of the anodic current produced by each work-ing electrode travels to the other working electrodeand is therefore measured.

Thus, the current noise PSD associated witheach electrode is given by the shot noise formula:

ψ I corrqI= 2 (13)

Half of the current noise passes through the measur-ing circuit and the other half stays on the electrodethat generated it and is not measured. Halving thecurrent causes the PSD to decrease by a factor of 4,so the measured PSD attributable to the currentnoise produced by one electrode is qIcorr/2.

The PSD associated with the two electrodesadd (since the two currents are uncorrelated), sothe measured PSD is twice that produced by oneelectrode:

ψ I meas corrqI, = (14)

The potential noise results from the effect of the cur-rent noise on the impedance of the metal solutioninterface. At the low-frequency limit the impedence is

(8) More generally, the anticipated spectrum of the sum of acollection of individual transients is the sum of the power spectraof the individual transients. Typical transients will have a whitespectrum at low frequencies, with a PSD given by the shot noiseformula.



given by Rp. The current noise associated with oneelectrode, ψI, will be applied to both electrodes, so theresultant potential noise PSD is given by:

ψ ψE IpR

= ×

2

2

(15)

The potential noise PSD produced by the currentsfrom each of the two electrodes will add, so the mea-sured potential PSD will be given by:

ψ ψ

ψ

E meas Ip

I meas p

R

R

,

,

= ×

= ×

2

2

2 (16)

(Note that Rn has been “proven” equivalent to Rp—this is essentially the same as most other suchproofs, in that it is based on the assumption than thetwo are equivalent).

From the Stern-Geary equation:

IBR

B

corrp

I meas

E meas

=

=ψψ

,

,

(17)

Substituting in (14):

ψψψ

ψ ψ

I measI meas

E meas

E meas I meas

qB

therefore qB

,,

,

, ,

=

= (18)

Then Icorr is produced by ƒn pulses of charge q everysecond, so:

ƒ =

=

ncorr

E meas

Iq

B2

ψ ,

(19)

With a shot noise process, ƒn is proportional to thespecimen area, and it may be appropriate to normal-ize it by dividing by specimen area.

These estimates of Icorr, q, and ƒn are derived froma statistical analysis of the expected result of addingmany independent events. They also assume that theeffect of the double-layer capacitance can be ignoredand that the individual events produce a white powerspectrum at the frequencies considered. For all thesereasons these parameters should be estimated at low

frequencies. It is not necessary to be able to identifyindividual events in the time record for the analysisto be applicable; indeed, if individual events can bedistinguished, the values will probably not be cor-rectly estimated from the standard deviation. In gen-eral, therefore, it is best to estimate these parametersfrom the low-frequency limit of the PSD. The latter isbest obtained using the MEM, or by averaging severalof the low-frequency points obtained using the FFT.In either case, the objective is to estimate the low-frequency PSD with a reasonably low standard error.

Electrochemical processes that do not occur bya “pure” shot noise process may not exhibit the ex-pected low-frequency plateau in the current and/orpotential power spectra. In this case, it is clearlyquestionable to use the shot noise analysis, and theestimation of Icorr, q, and ƒn is not strictly feasible.However, it is plausible that computing these param-eters at an arbitrary frequency, such as 10–3 Hz or10–2 Hz, may provide a useful practical value (just asuseful Rp measurements for steel in concrete can bemade in ~ 1 min, which is far too short a time tomeasure the true low-frequency limit of Rp).49

Providing that a true low-frequency plateau ex-ists in both the potential and current power spectra,these results can also be expressed in terms of thestandard deviation or variance of the potential andcurrent noise (providing these are measured over afrequency range that lies in the plateau region),replacing the PSD with the variance divided by themeasurement bandwidth. Where there is no low-frequency plateau, or where frequencies above theplateau are included in the measurement, theparameters derived in this way may still be of quali-tative use (although there seems little point in usingthese results instead of more valid results derivedfrom spectral analysis).

Note that the ψE,meas ψI,meas term in Equation (18)is similar to a correlation, and it appears that it mayalso be possible to estimate q from the low-frequencylimit of the cross-spectral density,48 although thereare unresolved questions about the lack of correla-tion between the potential and the current noise.

1/f or Flicker NoiseThe origin of this form of noise is somewhat less

clear than thermal or shot noise, but it is widelyobserved in a range of physical systems. The namederives from the dependence of noise PSD onfrequency, and “true” 1/f noise has PSD ∝ 1/f. Thereis a tendency, however, to describe any noise as be-ing 1/f, providing the amplitude falls with increasingfrequency, typically represented as PSD ∝ ƒn, wheren < 0. Theoretical descriptions of the source of 1/fnoise are available,13 although it tends to be difficultto relate these to real electrochemical processes. It isalso important to be aware of other effects that tendto give a PSD that falls with increasing frequency. In



particular, white current noise will produce 1/ƒ2 po-tential noise if it is applied to a capacitance, such asthe capacitance of the electrochemical double layer.If white current noise is applied to a metal-solutionimpedance exhibiting a diffusional impedance, then1/ƒ potential noise will result. A general observationin this respect is that it is important to note whethercurrent or potential noise is being considered, as itwould normally be expected that the two will havedifferent roll-off slopes.

NOISE-GENERATION PROCESSESFOR CORRODING SYSTEMS

In addition to the fundamental sources of noisediscussed above, corroding systems may generate ENby a number of mechanisms:

Metastable PittingThe nucleation, growth, and death of metastable

pits produces current transients with a duration ofthe order of a few seconds. The expected power spec-trum for the sum of a series of transients is the sumof the power spectra for the individual transients.Consequently, if the transients are similar in ampli-tude and duration, the resultant power spectrum willhave the same shape as that of the underlying tran-sients, but with the PSD multiplied by the numberof transients in the time record. At low frequencies(corresponding to periods much longer than theduration of the transient), the PSD associated with atransient will be white and proportional to the squareof the area under the transient. Bertocci, et al., hasshown that, at higher frequencies, the power spec-trum will have a roll-off slope that depends on thenumber of times the function describing the tran-sient can be differentiated.50

Metastable pits typically consist of a slow cur-rent rise followed by a sharp current fall (for stain-less steel) or a moderate current rise followed by aslower fall (for carbon steel and aluminium alloys). If

the current exactly follows these simplified growthlaws, then the expected power should be as shown inFigure 6, since the sharp transition from the rising tothe falling part of the transient leads to a discontinu-ity in the first differential. However, if there is some“rounding” of the transition, this will extend the dif-ferentiability and thereby produce a steeper slope athigher frequencies.

The potential noise time record will be producedby the action of the current noise on the double layercapacitance, and it will therefore have a steeper roll-off slope.

At low frequencies, such that individual meta-stable pitting events may be regarded as instanta-neous, the shot noise analysis can be applied. Thisresult has apparently provided a successful estimateof the pit dimensions for the pitting corrosion of Al,22

although further validation is required. Note that thisanalysis assumes that essentially all of the corrosioncurrent is associated with the metastable pitting.More recently, the use of q and ƒn has been success-ful in describing the behavior of steel in chloride/nitrite mixtures.51 Both of these estimates weremade using the variance divided by the bandwidth,rather than the preferable method of using the low-frequency limit of the PSD.

Turbulent Mass TransportTurbulent flow is known to have noise-like

characteristics,52 and one might expect EN resultingfrom mass transport fluctuations to be amenableto similar analysis. The authors are not aware ofdetailed studies in this area, but EN measurementshave been used to study corrosion in multiphaseflow conditions.53-54

Particle ImpactThe impact of solid particles or gas bubbles on

the surface may transiently increase the corrosionrate through removal of the corrosion product film, itmay reduce it for the period of the impact (throughshielding of the surface), or a combination of botheffects may be observed. The EN produced may beanalyzed by essentially the same method as for pit-ting corrosion.55 Since the steady-state corrosioncurrent without the particle impacts is typically non-zero, the calculation of charge using the shot noiseanalysis will not be valid. A similar system has beenstudied by Gabrielli, et al., using a high-frequencyresistance measurement to determine the changes insolution resistance associated with the passage ofparticles across the surface of the electrode, and thecombination of EN measurements with this methodmay prove valuable.45

Bubble Nucleation and SeparationCorrosion in acids with hydrogen evolution will

also give rise to transient currents associated with

FIGURE 6. Shape of power spectra for typical metastable pittingtransients: t = theoretical and s = simulated (after Bertocci, et al.).50



the growth of bubbles followed by their subsequentseparation. Figure 4 shows the data for steel inhydrochloric acid (HCl). Again, this may be treatedby methods similar to those for metastable pitting,although the estimation of charge is clearly not valid.

Activation-Controlled DissolutionActivation-controlled dissolution may be ex-

pected to produce bursts of charge as the dissolutionof ledges is “unlocked” by the loss of the first atomwithin the ledge.56 This will give rise to a shot noiseprocess, with a charge corresponding to 100 elec-trons to 300,000 electrons, depending on the disloca-tion density. This will result in a noise PSD in theregion of 2neIcorr A2/Hz, where n is the number ofelectrons involved in each burst of charge and e isthe charge on the electron (1.6 × 10–16 C). Assumingthe reaction proceeds at a reasonable rate (so thesource impedance is low compared to the current-measuring resistor in the current amplifier), theinstrument noise will be dominated by the input volt-age noise of the current amplifier (Appendix),57 andthe effective current noise PSD will be ≈ e2

inI2corr/B2,

where e2in is the amplifier input voltage noise PSD.

This leads to a signal-to-noise ratio (expressed as thesquare root of the ratio of the real PSD to the instru-ment noise PSD) of 2 2 2neB e Iin corr/ , and this form ofnoise can therefore be measured most sensitively bysmall electrodes with a low corrosion current. How-ever, only a limited benefit can be achieved by reduc-ing the electrode size since the amplifier input noisecurrent will start to become significant as Icorr de-creases. If a typical activation-controlled systemis considered, plausible or known values for the vari-ous parameters are n = 1,000, e = 1.6 × 10–19 C, B =0.026 V, e2

in = 1.2 × 10–17 V2/Hz (for the OPA37 low-noise amplifier at frequencies > 10 Hz),58 Icorr = 10–6 A(corresponding to a penetration rate of ~ 1 mm/y ona 1-mm2 electrode). These values will give a currentnoise PSD of 3.2 × 10–22 A2/Hz and a reasonable sig-nal-to-noise ratio (~ 130), and this noise should bemeasurable with care. At frequencies below 1 Hz to10 Hz, the input noise voltage of typical amplifiersexhibits a 1/f characteristic, and measurement willbecome more difficult (chopper-stabilized amplifiers59

are claimed to be free of this 1/f region and may helpto reduce this problem, but published specificationsdo not generally cover frequencies < 1 Hz).

The measurement of the shot noise associatedwith the elementary dissolution event (where n = 1 to4, depending on the electrochemical process) is moredifficult, but it is being attempted by Mézáros, et al.60

Crevice CorrosionThe initiation and aspects of the propagation of

crevice corrosion are similar to metastable pittingand will give similar characteristics. However, the on-set of crevice corrosion is typically accompanied by a

large drop in the potential as the active crevice polar-izes the passive external surface. Under some cir-cumstances, the passive surface may be unable tosustain the current needed to maintain the crevice inan active state, and potential oscillations may beobserved (the oscillations will typically have chaoticrather than completely deterministic oscillatory char-acter, and similar oscillations may also be observedfor pitting corrosion).61 In other cases, the potentialwill fall and remain low (Figure 7). In the latter case,it is clear that a simple visual analysis of the stepchange in potential (i.e., apply the rule “if there is apermanent large drop in potential, then crevice cor-rosion has probably initiated”) is more appropriatethan any of the more sophisticated procedures, andthis may also be true for oscillatory behavior.

Stress Corrosion CrackingStress corrosion cracks may propagate by dis-

continuous or continuous processes, and these maybe expected to have quite different EN characteris-tics. Discontinuous processes will produce currenttransients as the new surface at the crack tip is ex-posed by a crack-advance event.62 This will be similarto the transients for metastable pitting and may beanalyzed by a similar approach. Continuous proc-esses will presumably give lower noise levels as aresult of “random” fluctuations in the crack-growthrate. As the crack propagates, the crack-advanceevents or fluctuations in crack-growth rate may beexpected to grow as a result of the increasing stress-intensity factor and crack-growth rate. However, thecrack will also become more effective at “shielding”the crack tip from the external measuring circuits,and the EN associated with the crack tip will prob-ably fall with increasing crack length. This presentssomething of a limitation on using EN to detectstress corrosion cracking.

Corrosion FatigueEN measured during corrosion fatigue tests may

be expected to be dominated by the applied load

FIGURE 7. Crevice corrosion time record.



cycle, and this was observed many years ago (beforethe concept of EN had been invented) by the authorin an unpublished work on the fluctuation of ca-thodic protection current during the corrosion fatigueof structural steels in seawater. Variations in corro-sion current and potential will arise from the effectsof drawing fresh solution into the crack, from theexposure of bare metal by yielding at the crack tip, orfrom other crack advance processes. There may alsobe a cyclic EN component associated with metalaway from the crack or with solution-stirring effectsrelated to fluid agitation by the deflection or vibrationof the specimen. It is possible that the cyclic EN as-sociated with a propagating crack may be signifi-cantly greater than that associated with a cyclicallystressed, but uncracked surface, and this may beapplicable as a method of detecting corrosion fatiguecracks. However, it will probably be subject to shield-ing effects, as with stress corrosion cracking.

Intergranular CorrosionIntergranular corrosion may be expected to give

EN characteristics similar to stress corrosion occur-ring by a continuous process63 (indeed, the two pro-cesses may coincide for some systems). As withstress corrosion cracking, the shielding of the grow-ing tip of the corrosion penetration by the passivewalls may be expected to cause a reduction in themeasured EN as the depth of penetration increases,and this has been observed recently by Stephenson.64

However, the technique has been used to monitorfor intergranular corrosion of lead-acid batteryelectrodes.65

Galvanic CorrosionSome aspects of galvanic corrosion have been

studied through monitoring the coupling current be-tween electrodes of different metals, and this currentcan be treated as a form of electrochemical currentnoise. Similarly, the associated fluctuations inelectrode potential can be treated as electrochemicalpotential noise. However, the authors are not awareof any systematic studies in this area that use ENanalysis methods.

Uniform CorrosionWhen uniform corrosion is obtained, ENRM has

both theoretical and practical support, and can beused to estimate corrosion rate. However, some cave-ats are in order:

— The frequency range included in the measure-ment should be appropriate (and typically lower thanthat normally used).

— Any filters used for preventing aliasing orrestricting the frequency range must be well-matchedbetween the potential and current channels.

— The measured noise (both potential and cur-rent) must be dominated by EN rather than instru-

ment noise (there are circumstances where the in-strument noise can be used as a source of perturba-tion for the measurement of the metal-solutionimpedance,66 but this is not recommended becauseit has significant disadvantages compared to conven-tional methods).

Coated MetalsEN methods have been used by several groups,

notably those led by Mansfeld43,67-68 and byBierwagen,69-71 to study painted metal samples.Because they do not explicitly apply a current to thesystem under study, some workers expected (beforethe theoretical limitations of the method were fullyunderstood) that EN methods would not be affectedby the high resistance of the paint coatings. This isnow known to be incorrect, except for potential noisemeasurements on a single electrode, because thecurrent must pass through the solution and coatingresistance in just the same way for a linear polariza-tion or impedance measurement. Furthermore,coated samples are particularly likely to exhibitasymmetry between the two current-measuring elec-trodes, with the concomitant uncertainty in what isactually being measured. Some of the earlier work oncoated samples (and, quite possibly, on other sys-tems) did not fully appreciate the levels of instrumentnoise in the measurements, and the data obtainedshould be treated with a degree of caution. Reporteddata may be checked by comparison with the instru-ment noise estimates given in the Appendix, whichrepresent a minimum plausible level of instrumentnoise (in principle the amplifier noise levels could beimproved, but they are unlikely to have been betteredin the past).

PARAMETERS FOR IDENTIFYINGLOCALIZED CORROSION

A particularly interesting application of EN is foridentifying the type of corrosion, specifically localizedcorrosion. The characteristics of the noise producedby the various corrosion processes have been intro-duced above; here, the applicability of the variousparameters that have been proposed are examined.

Many localized corrosion processes give rise totransient events, to which the shot noise analysiscan be applied. Providing that Icorr is dominated bythe current associated with the transient events,then Icorr, q, and ƒn can be estimated using the shotnoise analysis. While these three parameters are in-terrelated through Icorr = q ƒn, there are significantadvantages to considering q and ƒn when attemptingto identify the type of corrosion. A large value of ƒn

(typically > 1 kHz/cm2) indicates that many eventsare occurring, and this may be taken to indicate thatthe corrosion will be relatively uniform. Conversely, asmaller value of ƒn indicates a localized process. The



charge, q, indicates the magnitude of the individualevents and may be thought of as evidence of thecorrosion process “intensity” since it signifies theamount of material removed in an individual event.In previous work, it has been suggested that a largevalue of q indicates localized corrosion. However, ithas recently been shown that “uniform” corrosion ofsteel in sodium chloride (NaCl) solution exhibits arelatively large value of q, but coupled with a largevalue of ƒn.51 Corrosion of the same system withadded sodium nitrite (NaNO2), which leads to pittingcorrosion, exhibits a similar value of q, but a signifi-cant reduction of ƒn, suggesting that the process re-mains essentially the same, with the NaNO2 justreducing the area reacting. Passive systems willtypically exhibit a very small value of q, and a highor low value of ƒn (however, EN measurements forpassive systems will often be contaminated byextraneous noise sources, leading to false estimatesof q and ƒn).

The shot noise model is strictly only applicable toa process that produces pulses of charge. However, itseems plausible that the q and ƒn parameters aremore generally applicable, and the terms “character-istic charge” and “characteristic frequency” are pro-posed to take account of situations lacking directevidence of the strict applicability of a shot noiseanalysis. As noted above, questions may arise aboutcomputing these parameters in cases without a validlow-frequency plateau in the potential and currentnoise power spectra, and this requires further work.

A number of other parameters have been pro-posed as indicators of the extent of corrosion local-ization, but few have as good a theoretical basis(though they have generally been tested by morepractical experiments).

Two of the parameters compare the amplitude ofthe current noise with the mean current:

— The coefficient of variation of current (stan-dard deviation divided by the mean) is expected to belarge if localized corrosion occurs.72 It suffers thefundamental problem that the expected value of themean current between two identical samples is zero,and it is possible for a large coefficient of variation tobe obtained even if the standard deviation is small.

— Nominally as a means of avoiding problemsassociated with the expected value of zero for themean current, the localization parameter or localiza-tion index (standard deviation of current divided byrms current) has been used.13 However, it can beshown that this is a simple transformation of the co-efficient of variation,15 and it thus suffers equivalentlimitations.

While the coefficient of variation and the localiza-tion index may be viable indicators of the nature ofthe corrosion process for a specific metal-environ-ment system, their sensitivity to the mean current,and hence to any asymmetry between the two cur-rent-measuring electrodes, can lead to inconsistentresults. Thus, Sun and Mansfeld have found for mildsteel in NaCl solution and Ti in Ringer’s solution thatthe localization index was dominated by the meancurrent (and hence the asymmetry of the two currentmeasuring electrodes), rather than any real tendencytoward localized corrosion.73

The skew and kurtosis of the current and/orpotential time record have also been used to identifythe type of corrosion, and have proved effective inpractical applications.74-75 Considering a processproducing relatively short current or potential tran-sients, such as that shown in the time record ofFigure 5, the unidirectional transients will producea skewed distribution, together with a positive kurto-sis. If the transients are bidirectional, this will tendto give zero skew, but the kurtosis will remain posi-tive. However, these parameters have a number oflimitations:

— As the number of transients increases, thedistribution of current and potential values will tendtoward a normal distribution (as a consequence ofthe central limit theorem), and the skew and normal-ized kurtosis will tend toward zero.

— Both parameters have a rather large standarderror (see above), which needs to be taken into ac-count.

— Both parameters are sensitive to drift andlong-term transient processes. In some systems itmay be possible to limit the effects of drift by suitablefiltering or trend removal.

The roll-off slope of the potential and/or currentpower spectra has also been proposed as an indicatorof the type of corrosion.76-77 However, such proposalsare typically based on a fairly restricted data set(9)

obtained solely by the group performing the analysis.When the roll-off slopes for a reasonably broad dataset are compared, it is typically found that there issufficient overlap to render the roll-off slope of littlevalue as a general identifier of type of corrosion.Thus, Figure 8 shows data obtained by Bagley fromher own work and from the literature, where it isclear that the different forms of corrosion cannot bedistinguished reliably on the basis of the roll-offslope alone.78 However, with a narrow, well-charac-terized set of corrosion conditions, it is feasible thatthe roll-off slope will correlate reliably with the typeof corrosion.

Neural network methods have been proposed foridentifying localized corrosion.74 This approach relieson the availability of a large collection of EN timerecords for which the type of corrosion is known. Arange of parameters derived from each time record

(9) A fundamental problem with developing EN analysis techniqueshas been the difficulty of publishing “raw” EN data such that arange of analysis procedures can be tried on a selection of datafrom different sources. The author therefore proposes to establishan electronic data collection for EN and other corrosion data.



are used as inputs to a neural network, which istrained to indicate the type of corrosion as an output.This is an attractive approach since it permits analy-sis without requiring one to solve the problems offully understanding the features characterizing aparticular type of corrosion. However, the develop-ment of the training data set presents a major chal-lenge. When training neural networks, it is importantto cover the “problem domain” (the N-dimensionalspace defined by the N input variables) as thoroughlyas possible. Unfortunately, the input variables ob-tained from EN time records will be outside the in-vestigator’s control and are liable to be quite stronglyclustered. Perhaps a more fundamental problem is todevelop data sets that represent practical exposureconditions for which the type of corrosion is known.Reference 74 does not state how this has beenachieved, but the evaluation of corrosion type isbelieved to have been obtained from an “expert”

analysis of the EN time records since it is difficult tosee how else the corrosion type can be determinedunambiguously for an individual time record (pittingcorrosion observed when a coupon is removed fromtest does not necessarily indicate that pitting oc-curred during all time records associated with thattest). If this is indeed the case, then the neural net-work has been trained to emulate the expert, not toproduce any fundamentally correct identification ofthe type of corrosion.

Several of the wavelet methods appear to offerthe prospect of identifying transients and othercharacteristics of the time record associated withlocalized corrosion. These methods may be betterthan those that depend on the overall statisticalproperties of the signal (such as q, ƒn, and thecoefficient of variation) when there are only a fewtransients in each time record, but further work isrequired to determine the general applicability ofthese techniques.

The higher-order spectral methods also offerinteresting possibilities for determining the type ofcorrosion, but the complexity of the output producedcurrently makes them very difficult to use. Furtherwork is needed to clarify the relationship between thefeatures of the higher-order spectra and to producesimple parameters that characterize corrosion type.

CONCLUSIONS

EN measurement and analysis is starting tomature, in that there are reasonably sound analysesof some of the simpler noise-generation processes.The measurement of Rn generally gives a reliable in-dication of corrosion rate, and those circumstanceswhere the method fails are generally explicable. It appears that the characteristic charge and char-acteristic frequency provide reasonably general toolsfor identifying the type of corrosion. However, furtherwork is required to establish their applicability be-yond “pure” shot noise systems. Asymmetries between the two current-measuringelectrodes present a major practical and theoreticalchallenge. It seems likely that an understanding of the con-text of the measurements (i.e., the metal, the envi-ronment, and similar factors) will always benecessary to obtain reliable results for anythingother than uniform corrosion.

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APPENDIX

Instrument NoiseIn this appendix the expected noise levels from

conventional instruments are described. Some noisesources are omitted on the basis that they are rarelysignificant. In particular, it is assumed that both thevoltage and current are conditioned using an opera-tional amplifier circuit such that the noise fromsubsequent processing is minimized; in the case ofthe voltage amplifier, this implies that there is a sig-nificant voltage gain while, for the current amplifier,this implies that the feedback resistor produces areasonably large output voltage for the full-scale in-put current. Operational amplifiers are specified interms of two sources of noise: the input voltage noise(a conceptual voltage noise source between each ofthe input terminals and a hypothetical ideal ampli-fier) and the input current noise (a real current noisesource at each of the input terminals). These noisesources interact with the impedance of the circuitcomponents, leading to contributions to the noise inthe measured signal, as indicated in Table A1 andTable A2. Noise from the various sources will normallyadd as PSD. Typical values of the various parameters(based on the OPA37 amplifier, which exhibits whiteinput voltage noise, e2

in, of 9 × 10–18 V2/Hz at highfrequencies, with 1/f noise below about 10 Hz, andwhite input current noise, i2in, of 1.6 × 10–25 A2/Hz,at high frequencies, with 1/f noise below about100 Hz) result in the voltage amplifier noise beingdominated by the amplifier potential noise at low

TABLE A1Noise of Voltage Amplifier

Contribution to Effective InputNoise PSD (V2/Hz) Notes

Amplifier input current noise i2inZ2s (a)

Amplifier input voltage noise e2in (b)

Thermal noise on feedback resistor 4kTRƒvG2 (c) and (d)Thermal noise on Zs 4kTZs

Notes:(a) i2in is the PSD of the input noise current; Zs is the impedance of the voltage source.(b) e2

in is the PSD of the input noise voltage.(c) k is the Boltzmann constant, T is the absolute temperature, Rƒv is the resistance of the gain-determining

resistor between the amplifier output and the inverting input, and G is the voltage gain of the amplifier[=(Rƒv + Rgv)/Rgv].

(d) The other gain-determining resistor, Rgv, will not generally produce a significant noise compared toRƒv.



frequencies and the thermal noise of the feedbackresistor or the source impedance at high frequencies,except for systems with a very high value of thesource impedence, Zs, when the input current noisemay become important. The current amplifier noiseis usually also dominated by the input voltage noiseterm (dependent on 1/Zs, which leads to particularproblems for low-impedance systems such as activelycorroding metals), except for very high values of Zs,

TABLE A2Noise of Current Amplifier

Effective Input Noise PSD (A2/Hz) Notes

Amplifier input current noise iin2

Amplifier input voltage noiseeZ

in

s

2

2 (a)

Thermal noise on Rƒ

4kTRƒ

(b)

Thermal noise on Zs

4kTZs

Notes:(a) Bertocci and Huet argue that the input voltage noise of the current amplifier is not affected by the

input voltage noise of the operational amplifier when using an instrumentation amplifier to detect thevoltage across the current measuring resistor.19 However, their analysis ignores the noise at the invertinginput of the operational amplifier, and the analysis is thus believed to be incorrect. The results quotedhere assume a conventional current amplifier based on a single operational amplifier.58

(b) Rƒ is the resistance of the current-measuring resistor.

when the input current noise becomes significant.Note that the OPA37 is a bipolar amplifier with arelatively low-input impedance and voltage noiseand a relatively high-input current noise. A field-effect transistor (FET) and similar high-input imped-ance amplifiers will provide a much lower input cur-rent noise (at the expense of a higher voltage noise),and these may be better for very high-impedancesystems.

interpretation of electrochemical noise data · potential noise of the working electrode pair can...

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