To investigate the effects of galvanic coupling between dissimilar materials by electrochemical noise methods, an approach based on the use of an array of electrodes comprising two dissimilar pairs of identical electrodes is proposed. The individual currents flowing through each electrode are measured simultaneously with the potential of the array. For each electrode, the apparent noise resistance, , is determined from the square root of the potential variance divided by the individual current variance. For two dissimilar pairs of identical electrodes and under the assumption that nominally identical electrodes have identical resistance and are equally noisy, may approach closely to the true noise resistance of electrode n or be proportional to R_{n}. The analysis is facilitated by the introduction of “virtual electrodes,” corresponding to a convenient sub-array of real electrodes, and obtained by mathematical manipulation of the individual currents in the time or frequency domain. By using this approach, the validity of the assumptions on the relative magnitude of the noise sources and electrode resistances that are often necessary to evaluate R_{n} can be verified partially or completely. For example, the electrochemical noise data recorded from four identical electrodes and from two dissimilar couples of identical electrodes are discussed.

## INTRODUCTION

Electrochemical noise method is a mature technique that can be used in the laboratory and in the field for corrosion monitoring. Electrochemical potential or current noise arises from anodic and/or cathodic events on the surface of corroding electrodes, associated with the initiation or propagation of corrosion. A variety of methods are available for electrochemical noise analysis, ranging from techniques based on the statistical analysis of the current and/or potential signal, fast Fourier transform or wavelet transform-based analyses, and others.1–10 Such analysis techniques provide information on the corrosion rate, type of corrosion, and transition from metastable to stable pitting.9–10 Each method has specific advantages for the study or monitoring of a particular corroding system.1 Whatever the approach for the analysis, if both current and potential noises need to be acquired, generally two working electrodes and a third reference electrode are used.

Among the other analysis techniques, estimation of the noise resistance, defined as the square root of the ratio between the potential variance and current variance, generally is applied to a couple of nominally identical electrodes.2,4,11 Its wide use derives from the physical meaning of the noise resistance that is often similar to that of the polarization resistance, but it can be obtained without applying any probing signal to the corroding surface. With a conceptually similar approach, the noise impedance can be calculated by taking the square root of the ratio between the potential power spectral density and the current power spectral density, obtained by fast Fourier transform. Although the strength of this approach is to extract parameters that are directly related to the corrosion rate, a relatively severe limitation arises from the requirement to use two nominally identical electrodes. This limitation has been described in detail in various studies for experimental configurations with two working electrodes, and arises from the number of unknowns (one noise source and one noise resistance/impedance for each electrodes) and the number of measurable parameters (one current and one potential).11–13 Interestingly, when two dissimilar electrodes are used, it has been demonstrated in detail that if one of the electrodes produces the vast majority of the noise, it is possible to estimate the noise impedance/resistance of the other.11–12 This is unfortunate for corrosion studies because generally the electrode that produces most of the noise is the one that is corroding more significantly, and information on its noise resistance/impedance is important.

Using arrays off small (or micro) electrodes has some advantage over the two electrodes' configuration because it can highlight phenomena connected to corrosion localization with both spatial and temporal resolution.14–15 Concerning the calculation of noise resistance or impedance, recently, it has been shown theoretically that it is possible to extend the two-electrode analysis to an array of dissimilar electrodes, by using the concept of virtual electrodes.16 For a given array of electrodes, after the individual currents flowing across each electrode are acquired and the experiment is terminated, it is always possible to generate numerically a couple of virtual electrodes by adding in the time domain of the individual electrode currents for a subset of electrodes. Therefore, the analysis of any electrode array can be reduced to the analysis of a single pair of non-identical virtual electrodes.

In this work, the theoretical derivation of this result is summarized for the calculation of noise resistances. Subsequently, it is applied to the analysis of the effects as a result of galvanic coupling between dissimilar materials. In particular, the electrochemical noise signal is generated by a couple of AA2024-T3 (UNS A92024)^{(1)} electrodes galvanically coupled alternatively with the following:

—two AA2024-T3 electrodes

—two high-purity aluminum (99.99 wt%) electrodes

—two mild steel electrodes

—two AZ31 magnesium alloy specimens

For the first time, it has been possible to identify semi-quantitatively the effect of the galvanic coupling on the noise resistance of the AA2024-T3 electrodes.

## THEORY

Theoretical analysis for the general case of the estimation of impedances of an array of k electrodes has been reported in the literature in detail previously, and it is summarized here for the case of noise resistances. The following analysis is based on Kirchoff's current and voltage laws, the Norton and the superimposition theorems, and the concept of a virtual electrode.16

An array of k electrodes can be represented as a Norton equivalent circuit of current noise sources, i_{m}(t), in parallel with their impedances, Z_{m} (where m = 1 to k), under the assumption that the relatively small noise fluctuations enable the circuit to be treated as linear around its mean behavior. An array of dissimilar electrodes can be treated as a single virtual electrode, λ, formed by the parallel connection of k real electrodes (Figure 1).

The sum of the currents across each electrode gives the total current across the virtual electrode, λ. If the virtual electrode, λ, is not connected to any other electrode, I_{λ}(t) = 0 for all t.

Considering that the current sources from each electrode are independent, the variances (σ^{2}), and not the amplitude, will sum:

In Equations (1) and (2) and in Figure 1, I_{m}(t) are the electrode currents that can be measured, while i_{m}(t) represents the ideal current noise sources and are not measurable. For brevity, the time-dependent I_{m}(t) and i_{m}(t) are written I_{m} and i_{m}, respectively, in the following equations.

Considering that the real electrodes are connected in parallel, as indicated in Figure 1, the equivalent resistance of the electrode, λ, is given by:

As a result of all the noise currents being applied to the equivalent resistance, the variance of the potential of an isolated virtual electrode is given by:

The previous analysis allows any array of k+1 electrodes to be reduced to the usual two-electrode case: a real electrode, ρ, with current noise source, i_{ρ}, and noise resistance, R_{ρ}, connected with a virtual electrode, λ, accounting for the remaining k electrodes and having a current noise source, i_{λ}, and noise resistance, R_{λ}, as indicated by Equations (2) and (3) (Figure 2). Therefore, similar to two asymmetric electrodes, the variance of the current (I_{ρλ}) measured at the real electrode, ρ, connected to the virtual electrode, λ, is:

and the variance of the measured potential (E_{ρλ}) is given by:

Therefore, the apparent noise resistance of the electrode, ρ, , defined as the square root of the ratio between the potential variance and the current variance can be obtained:

Considering that the naming of the electrodes is arbitrary, the analysis of the electrode, ρ, can be extended iteratively to any of the electrodes forming the array. Therefore, Equations (6) and (7) and the concept of virtual electrode provide a general tool for the analysis of the noise generated by an electrode array, since any electrode array can be reduced to the usual two-electrode case. Further, if all the electrodes are identical and with the further assumption that identical electrodes are equally noisy:

Conversely, if the electrodes are dissimilar, but most of the noise is produced by the virtual electrode, λ (and it can be assumed that or σ^{2}[i_{ρ}]>>σ^{2}[i_{ρ}] if the resistances of the dissimilar electrodes are expected to be comparable), the apparent noise resistance approaches the actual noise resistance:

Otherwise, if the noise generated by electrode ρ dominates (σ^{2}[i_{ρ}]>>σ^{2}[i_{λ}]), the apparent resistance of electrode ρ approaches the resistance of the virtual electrode, λ:

In practice, as highlighted by Bautista, et al., for two real electrodes and extended here to two virtual electrodes, it is possible to estimate all the noise resistances if the electrodes can be considered identical; otherwise, only the impedance of the less noisy electrode can be estimated.11 Using two dissimilar pairs of nominally identical electrodes, it is impossible that a single electrode produces more noise than the remaining three (since it will be the same as one of the three) and the following is always true:

Therefore, only two cases are possible:

—one pair of identical electrodes produces more noise than the other

—all four electrodes produce comparable levels of noise

If it is possible to assume that the electrodes that corrode more generate higher levels of noise, then when electrode ρ (i.e., real electrode 1) is one of the two less corroding electrodes (i.e., real electrodes 1 and 3), most of the noise is generated by the electrodes forming the other pair (i.e., real electrodes 2 and 4); therefore, σ^{2}(i_{ρ}) << σ^{2}(i_{λ}) and, according to Equation (9), the apparent noise resistance approaches the actual noise resistance, ≈ R_{ρ}.

Conversely, if electrode ρ is one of the two electrodes that are corroding more, the contribution from electrodes that are corroding less (high resistance-low noise) can be neglected. Therefore, noise levels and resistance of electrode λ are:

and:

Consequently, Equation (7) reduces to the case of two identical electrodes and:

If all four electrodes behave identically (equally noisy and with identical resistances), Equation (8) gives:

Having covered all the possible cases with the previous analyses, using two dissimilar pairs of identical electrodes, it should be possible to estimate the actual resistance of each individual electrode with a maximum error of √3 ≈ 1.73, since it is always the case that:

The above approach was used to investigate the behavior of galvanically coupled materials. Specifically, the effect on noise resistance of an AA2024-T3 as a result of the galvanic coupling with selected materials was investigated.

## EXPERIMENTAL PROCEDURES

Electrochemical noise data were acquired simultaneously on four galvanically coupled electrodes (each having an exposed area of 2 cm^{2}) at a sampling frequency of 1 Hz, by using a Concerto^{†} multichannel potentiostat with a built-in anti-aliasing filter. The experimental setup and the electrical connections are presented in Figure 3. Prior to immersion in naturally aerated 3.5% sodium chloride (NaCl) solution, the electrodes were ground and subsequently mechanically polished to a 1 micron surface finishing using diamond paste. The electrode materials included high-purity aluminum (99.99 wt%), AZ31 magnesium alloy, mild steel, and AA2024-T3 aluminum alloys. Considering that the purpose of the study was to evaluate the effect of galvanic coupling on the corrosion behavior of the AA2024-T3, then for each experiment a pair of nominally identical electrodes of AA2024-T3 was coupled with a pair of nominally identical electrodes of one of the other materials. As a reference, one experiment with four nominally identical AA2024-T3 electrodes was also performed.

## NUMERICAL ANALYSIS

After acquisition, the electrochemical noise data were analyzed by a purpose-made program developed in-house. The plots of the time evolution of the individual apparent noise resistances, presented in Figures 4 through 7, were obtained with a procedure similar to that described previously and summarized as follows.17 From the original points data set (acquired at 1 Hz), comprising one potential record and four individual current records, a segment 512 points long is extracted starting at time t = 0 s. From this segment, one potential and four current variances are calculated. The square roots of the ratio between the potential variance and each of the current variances provide four individual values of apparent noise resistance, each being associated with one real electrode.17 For each electrode the obtained value of apparent noise resistance is associated (arbitrarily) to the time at which the first point of the segment used for the calculation was acquired, e.g., for the first segment t = 0. This procedure is repeated for the following segment, displaced 10 s forward on the time axis, thereby generating four new values of apparent noise resistances related to t = 10 s. As a result of the subsequent similar iterations, with an approach similar to that described previously for low-frequency noise impedance, plots of the time evolution of the apparent noise resistances are obtained.17

## RESULTS AND DISCUSSION

The effects on the noise resistance of an AA2024-T3 as a result of galvanic coupling with dissimilar materials were examined by coupling two AA2024-T3 electrodes with the following:

—(1) two AA2024-T3 electrodes (4 nominally identical electrodes configuration)

—(2) two high-purity aluminum (99.99 wt%) electrodes

—(3) two mild steel electrodes

—(4) two AZ31 magnesium alloy specimens (2 through 4, two dissimilar pairs of identical electrodes configuration).

When the two AA2024-T3 electrodes were coupled with two nominally identical AA2024-T3 electrodes, a potential of about −0.8 V vs. saturated calomel electrode (SCE) was measured initially (Figure 4). After immersion, the potential progressively increased to approach −0.6 V_{SCE} after 2,000 s and, subsequently, remained unchanged. The coupling current was initially low, but increased significantly after 1,000 s to values in the region of 10^{−5} A. One of the electrodes became a preferential anode after about 3,500 s, while the remaining electrodes all become net cathodes. The apparent noise resistances were initially in excess of 30 kΩ, but progressively decreased to about 1 kΩ after 2,000 s, showing some fluctuation.

Significant differences were observed during a similar experiment performed on two AA2024-T3 and high-purity aluminum (Figure 5). The corrosion potential did not show a significant initial drift, but displayed fluctuations for the duration of the experiment. Whereas the potential of the electrodes' array formed only by AA2024-T3 specimens was in the region of −0.6 V_{SCE}, the array containing both high-purity aluminum and AA2024-T3 displayed a more negative potential (−0.74 V_{SCE}). As in the case of the AA2024-T3 array, the apparent resistance of three of the four electrodes progressively decreased with time, whereas one AA2024-T3 electrode displayed a slightly higher apparent resistance that was approximately constant for the duration of the experiment and close to 10 kΩ.

For the experiment with AA2024-T3 alloy coupled with mild steel (Figure 6), an initial potential drift was observed from about −0.7 V_{SCE} to about −0.6 V_{SCE}, which continued for the first 1,500 s. Subsequently, the potential steadily approached −0.6 V_{SCE}. The mild steel electrodes behaved as net anodes, while the AA2024-T3 electrodes were always net cathodes. Initially, the AA2024-T3 electrodes displayed apparent resistances higher compared to the mild steel electrodes; subsequently, both the apparent noise resistances of both the mild steel and the aluminum alloy approached 1 kΩ.

Markedly different behavior was observed for galvanic coupling between AA2024-T3 and the AZ31 magnesium alloy (Figure 7). The potential was initially close to −1.55 V_{SCE}, and progressively increased to about −1.37 V_{SCE} during the first 1,500 s. Subsequently, the potential slowly drifted toward more positive values to achieve about −1.33 V_{SCE} at 10,000 s. A similar effect was observed for the galvanic currents, which progressively increased from 0 to about 10 mA during the first 2,000 s and subsequently were steady. The apparent resistance of all four specimens was in the region of 20 Ω for the duration of the experiment, both for the AA2024-T3 and for the AZ31 electrodes.

The time evolution of the apparent noise resistances for the four nominally identical AA2024-T3 electrodes shows that their behavior is close to that expected from identical electrodes. Therefore, according to the presented analysis, the apparent resistance should be multiplied by √3 to obtain the real electrode's resistance, according to Equation (15). In the case of AA2024-T3 coupled with high-purity aluminum, the AA2024-T3 specimens display a higher resistance compared to pure aluminum for most of the duration of the experiment, suggesting that the apparent noise resistance provides a reasonable estimation of the noise resistance. For the coupling with mild steel, except during the early stages, the values of apparent resistances were similar for all the specimens. This suggests that the corrosion behavior is similar and therefore an approximation of the real noise resistance after the initial transient is obtained by multiplying the apparent noise resistance by √3, according to Equation (15). A similar argument applies to the couple AA2024 with AZ31. These considerations have been applied to create Figure 8. In this case, the average of the apparent noise resistance of the two AA2024-T3 was taken and multiplied by √3 when appropriate. Based on the argument leading to Equation (16), the plotted values for the noise resistance must be read as maximum values for AA2024-T3 alone and coupled with AZ31 and mild steel (with the assumption of identical behavior) and minimum values for the AA2024-T3 coupled with high-purity aluminum (with the assumption of strongly different behavior).

From the resulting plots of estimated noise resistances with time (Figure 8), the effect of the galvanic coupling can be investigated. To compare the noise resistance of the AA2024-T3 alone with the noise resistance of the AA2024-T3 coupled with a dissimilar material, the values of noise resistance obtained for the experiment with four AA2024-T3 electrodes are taken as a reference. In practice, when the AA2024-T3 alloy was coupled to the high-purity aluminum, a small increase of the noise resistance was observed on the AA2024-T3 electrodes. This can be rationalized by considering that the number of active cathodic sites on the high-purity aluminum is relatively low. So, to sustain some anodic reactions on the high-purity aluminum, some of the anodic reactions on the AA2024-T3 must be deactivated, resulting in a moderate increase in noise resistance. A similar situation is generated when the AA2024-T3 alloy is coupled with mild steel. In the naturally aerated 3.5% NaCl solution, the steel behaves as an active material, with active anodic and cathodic sites. However, the presence of copper-rich intermetallics on the AA2024-T3 alloy provided additional cathodic sites for the anodic reaction of steel. Initially, the apparent resistance of the AA2024-T3 was not affected significantly by the increased cathodic activity. However, as corrosion initiated also on the aluminum alloy, the apparent noise resistance decreased to values comparable to those estimated for the mild steel electrodes. A very contrasting behavior is observed when coupling between AA2024-T3 and AZ31 magnesium alloy is undertaken. In the present study, because of the much lower corrosion potential of magnesium, hydrogen evolution becomes extensive also on the aluminum surface and specifically on the copper-rich intermetallics. Consequently, extensive alkalinization can take place on the aluminum surface that becomes very active, because of the increased solubility of alumina with pH. As a result of this, widely extensive corrosion takes place on the aluminum alloy surface, with a very significant increase of the anodic and cathodic reaction rates. This is reflected by the much lower values of the noise resistances.

## CONCLUSIONS

❖A method for the analysis of the electrochemical noise generated by arrays of multiple dissimilar electrodes has been presented in the paper. The method that was studied is based on the measurement of the individual electrode currents and subsequent analysis using the concept of virtual electrode. A virtual electrode is obtained by summing in the time domain the individual currents of a given number of real electrodes. As a result, it was found that any array of dissimilar electrodes can be reduced to the case of two dissimilar electrodes. When two dissimilar pairs of identical electrodes are used, the analysis can be progressed further and the estimation of the electrode noise resistance is possible. The practical case of AA2024-T3 coupled with high-purity aluminum, steel, and magnesium has been examined. It was found that coupling with high-purity aluminum increases the noise resistance and, therefore, has a beneficial effect on corrosion resistance. Conversely, coupling with steel produces a slight decrease in noise resistance. Finally, coupling with the magnesium alloy is detrimental since it results in activation of the aluminum alloy and a three order of magnitude decrease in noise resistance.

## ACKNOWLEDGMENTS

EPSRC is acknowledged for provision of financial support through the LATEST2 Programme Grant. Intertek-CAPCIS^{†} is acknowledged for providing a Concerto^{†} multichannel potentiostat.

## REFERENCES

^{(1)} UNS numbers are listed in *Metals and Alloys in the Unified Numbering System*, published by the Society of Automotive Engineers (SAE International) and cosponsored by ASTM International.

^{†}Trade name.

## Author notes

^{*}Corrosion and Protection Centre, The University of Manchester, Manchester, M13 9PL, United Kingdom.