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Volume 3 Paper 4


Simulation of Electrochemical Noise Due to Metastable Pitting

R.A. Cottis
Corrosion and Protection Centre, UMIST, P.O. Box 88, Manchester M60 1QD, UK

Abstract

This paper presents a program that simulates electrochemical noise time records for a process that produces pulses of charge. The output from the simulation is discussed in relation to a range of criteria that have been proposed for the interpretation of electrochemical noise data, and, in particular, for the identification of localized corrosion. The nature of the simulation introduces some artificial limitations, and the results do not adequately test parameters relating to the shape of power spectra. Of the other parameters, the characteristic charge and characteristic frequency are proposed as useful general purpose parameters for the interpretation of electrochemical noise data, while the skew and kurtosis may be useful single parameters for the identification of localized corrosion in well-characterized situations.

Keywords: Electrochemical noise, pitting corrosion, simulation.

Introduction

The measurement of electrochemical noise has received considerable attention for corroding systems since the original publication of Iverson [] (Tyagai published earlier work on a basic shot noise analysis [], but this has rarely been read or referenced by corrosion workers, probably because it is mostly in Russian). The theoretical basis of the analysis of electrochemical noise is relatively well-established for general corrosion, where the electrochemical noise resistance and impedance techniques appear valid under most circumstances []. However, electrochemical noise measurements for general corrosion offer little that cannot be achieved by conventional dc or ac methods, and the most interesting application is to the identification and quantification of localized corrosion. Several parameters have been suggested as being indicators of localized corrosion, but relatively little agreement has been reached about their validity. One factor that has contributed to this has been the lack of good quality electrochemical noise data with known corrosion properties. Thus many workers who have nominally been studying the application of EN techniques to the identification of localized corrosion have compared the analytical parameters to a visual interpretation of the EN time record, rather than testing them against independent measures of the type of corrosion. The objective of this paper is to present a simple simulation of corrosion occurring according to a well-defined physical process, thereby providing artificial data that can be used to investigate the behaviour of the range of possible parameters. 

The Model

The simulation is based on the conventional three-electrode method in which potential and current noise are measured simultaneously. The current noise is measured as the current between two nominally identical working electrodes (using a zero-resistance ammeter to eliminate potential differences), and the potential noise is measured relative to a quiet reference electrode (in the case of this simulation the reference electrode is, of course, ideally quiet!).

A shot noise model is assumed for the corrosion process. The anodic reaction is assumed to consist of instantaneous pulses of charge (mathematically they are Dirac delta functions, current pulses with zero width and infinite height, but finite charge). The amplitude of these pulses may be constant, or it may have an exponential distribution (i.e. there will be many small pulses and few large pulses). The pulses are assumed to be independent of one another (i.e. this is a Poisson process), such that the interval between pulses will have an exponential distribution. The probability of a pulse being emitted, expressed as the mean inter-pulse interval, l (s), is assumed to depend on the electrode potential according to a Tafel type law. In order to permit the examination of the effects of asymmetry of the two working electrodes, the mean inter-pulse interval on one electrode may differ from that for the other electrode (this is expressed as the relative probability of a pulse occurring on working electrode 1). The cathodic reaction is assumed to occur at a constant current, as would be the case for mass-transport limited oxygen reduction. This is probably not very realistic, but it allows for a simple analytical solution for the integration of the potential and current, and thereby provides faster computation. The immediate effect of the anodic pulses of charge is to change the potential of the electrode on which they occur by adding charge to the double-layer capacitance. This leads to a difference in potential of the two working electrodes and the current between them will be a result of the potential difference applied across the solution resistance.

In order to avoid artefacts associated with aliasing (due to sampling at a single time) and quantization (due to always starting a pulse at a fixed time relative to the sample time), a relatively complex calculation procedure is used. From a given starting point the time to the next pulse on each of the two working electrodes is computed as a sample from an exponential distribution with the mean inter-pulse time being determined by the electrode potential (the two electrodes are treated independently to allow for the possibility that they may be at significantly different potentials). The potential and current are then integrated analytically up to the next pulse or the next sample time. This minimizes the extent of aliasing of frequencies above the Nyquist frequency into the sample.

The estimate of the time to the next pulse is based on the assumption that the potential will not change significantly (and thereby change the probability of emitting a pulse) before the next pulse occurs. In many circumstances (particularly with a large cathodic current or a long mean inter-pulse interval) this assumption will not be valid. It is difficult to solve this problem in a general way. The ideal approach would be to modify the distribution used to account for the change in potential; this may be possible for a single electrode, but it is inherently not possible when the potential of the other electrode may change and thereby change the potential of the coupled working electrode pair. As a simple, though imperfect, solution the time to the next pulse is recalculated at every sample interval. This is valid because the probability of the time to the next pulse is independent of the history of the electrode (given the assumptions made in this model); it may not be valid for other distributions.

Implementation

The simulation has been programmed in Object Pascal (Delphi V5.0 - Inprise Corporation). The source code for the main module and two supplementary modules forms a part of this paper, as does the complete program.

Performance

The simulation can potentially process very many pulses in a time record, with the worst case being very frequent pulses in a very long time record. For large numbers of pulses relative to the number of samples the algorithm is first order in the number of pulses processed, and Figure 1 shows the processing time as a function of the expected number of pulses for a fixed time record duration of 65536 seconds. These results were obtained using a computer based on a 350 MHz Pentium II running Windows NT.

Figure 1 Processing time for a simulation over 65536 seconds as a function of the expected number of pulses for a constant pulse amplitude. (click on the image for an enlarged view)

Results

The simulation has been used to investigate the applicability of a range of parameters that may be able to identify whether a given corrosion process is localized or not. These results have been presented in greater detail elsewhere [], but some of the main conclusions are summarized here.

Coefficient of Variation

The coefficient of variation suffers from its dependence on the mean current, which is a largely arbitrary result of the asymmetry of the two electrodes. Thus Figure 2 shows the effect of electrode asymmetry on the coefficient of variation. This result is to be expected, since the coefficient of variation is only valid for a variable that can only have values that are one side of zero. One solution to this problem is to use the corrosion current, Icorr, rather than the mean current. This can be done by estimating Icorr from Rn, which leads to the conclusion that the standard deviation of potential should be a good indicator of localized corrosion (see also the characteristic frequency below).

Figure 2a Effect of electrode asymmetry on coefficient of variation 

Figure 2b Effect of electrode asymmetry on coefficient of variation (enlargement of y axis)

Localization Index

The localization index is a simple transformation of the coefficient of variation [], and suffers from the same limitations.

Characteristic Charge

The characteristic charge may be deduced from a shot-noise analysis [ - but see the addendum], and is identical to the charge used in this simulation. It is hardly surprising, therefore, that the calculation provides an accurate estimate. Note, however, that the analysis is based on the low frequency limit of the potential and current noise, and an erroneous value will be obtained if data containing too high a frequency are used (figure 3). The term 'characteristic charge' is suggested to cover the general situation where the data may not be derived from a shot noise process, and where it is not strictly valid to talk in terms of individual events that have a particular charge. In the more general case the large amplitude of both potential and current noise that is required to obtain a large characteristic charge are still expected to be an indicator of the severity of the corrosion process.

v3p4_fig3.gif (9427 bytes) 

Figure 3 Effect on sampling frequency (and hence measurement bandwidth) on estimated charge and pulse frequency

Note that a large characteristic charge does not, of itself, indicate that localized corrosion is occurring as it may equally well indicate a very high rate of general corrosion. It should therefore be taken as an indicator of the severity or intensity of the corrosion, and it is suggested that it may correlate with the maximum penetration rate.

Characteristic Frequency

As it is possible to calculate the characteristic charge (from the shot noise analysis) and the corrosion rate (from Rn), it is also possible to calculate a characteristic frequency - the number of characteristic charges per second that are equal to the corrosion current (i.e. Icorr/q). It can be shown [] that this is inversely proportional to the variance of potential. Then a large characteristic frequency (of the order of 1000 Hz/cm2 or above) implies a relatively uniform corrosion process, while a lower frequency implies a more localized corrosion process. Note that the frequency is expected to be proportional to the specimen area, as implied in the units above (see [] and [] for a general discussion of the area dependence of electrochemical noise parameters).

When coupled with the estimate of the characteristic charge, the characteristic frequency provides the ability to discriminate a range of corrosion types. Thus experimental work to be reported elsewhere [] has shown that uniform corrosion, pitting corrosion and passivity can be distinguished for a specific system by using these parameters.

Roll-off Slope

The roll-off slope has been found in relatively narrow sets of experiments to be an indicator of localized corrosion, with a larger slope corresponding to localized corrosion. However, more general comparisons tend to show considerable overlap for the various corrosion types []. This simulation effectively fixes the roll-off slope by the assumptions made, and does not provide a valid test of this parameter.

Skew

Providing the pitting process produces distinct uni-directional transients (as it does for the potential noise in some situations and for the current noise when the electrodes are asymmetrical), then the resultant distribution of values will be skewed, and the skew may be a useful parameter for the identification of localized corrosion processes. However, when bi-directional transients occur, or when there are many overlapping transients, then the skew will tend to zero, and will not be a reliable indicator. This is illustrated in Figures 4 and 5. The measured skew is also strongly sensitive to the nature of any drift in the signal []. 

Figure 4 Effect of electrode asymmetry on skew 

Figure 5 Effect of mean inter-pulse interval on skew 

Where the skew does provide information about a localized corrosion process (e.g. in the case of potential noise), it does have interesting characteristics. By optimizing the sampling frequency, and hence the measurement bandwidth (or by taking the PSD at a specific frequency) skew due to high frequency transients that may be attributable to relatively uniform processes may be lost (in effect this is the central limit theorem at work), while lower frequency transients will show up as a significant skew.

Kurtosis

The normalized kurtosis (the kurtosis with 3 subtracted from it to make it zero for a normal distribution) will typically be positive for signals showing distinct transients (whether uni- or bi-directional), and it has been used as an indicator of localized corrosion []. It can be seen from figures 7 that the kurtosis is not much affected by the electrode asymmetry, and in Figure 8 it can be seen that the kurtosis is strongly affected by the frequency of transients. This is particularly true for the current kurtosis in this simulation (since the potential wave transient shape tends not to lead to a large effect on the kurtosis, even when there are few large transients). The current kurtosis (if measured over the optimal frequency range) is effectively an indicator of few, large transients, and as such appears to be a good indicator of the occurrence of localized corrosion. 

Figure 6 Effect of electrode asymmetry on normalized kurtosis

Figure 7a Effect of mean inter-pulse interval on kurtosis

Figure 7b Effect of mean inter-pulse interval on kurtosis (enlarged y axis)

Noise Resistance and Impedance

The electrochemical noise resistance and impedance are not expected to reveal useful information about localized corrosion, but it is interesting to see how the results obtained for this simulation compare with that predicted from simple theory. Note that the model does not incorporate an explicit charge transfer resistance, this being provided indirectly by the response of the pulse emission process to potential. This model therefore provides a slightly more complete test of the relationship between Rn and Rp than is provided by most previous theoretical analyses, which assume either explicitly or implicitly that the current noise is described by the effect of the potential noise on the metal-solution impedance (or vice versa). 

Figure 8 shows a comparison of the measured electrochemical noise impedance, the set value of the double-layer capacitance, Cdl, and the expected value of Rct.

Figure 8 Comparison of amplitude of impedance (average of 6 MEM spectra with order 50) derived from electrochemical noise analysis and from equivalent circuit analysis

Conclusions

References

"Transient Voltage Changes Produced in Corroding Metals",  W.P. Iverson, J. Electrochem. Soc., vol. 115, p.617, 1968.

"Faradaic Noise of Complex Electrochemical Reactions", V.A. Tyagai, Electrochimica Acta, vol. 16, 1647-1654 (1971).

"Corrosion Testing Made Easy : Electrochemical Impedance and Electrochemical Noise", S. Turgoose and R.A. Cottis, NACE International, February 2000. (see also the Addendum Web page)

"Electrochemical Noise Analysis of Carbon Steel in Sodium Chloride Solution with Sodium Nitrite as an Inhibitor", H.A. Al-Mazeedi, R.A. Cottis and S.Turgoose, to be presented to EuroCorr 2000, London, September 2000

G. Bagley, PhD Thesis, UMIST, 1999

"Higher order measures for the analysis of electrochemical noise", G. Bagley, R.A. Cottis and P.J. Laycock, Paper 191 Corrosion 1999, NACE International, 1999.

"Noise monitoring at Canada's Simonette Sour Oil Processing Facility", E.E. Barr, R. Goodfellow and L.M. Rosenthal, Paper 414, Corrosion 2000, NACE International, 2000.