# Review-Electrochemical Noise Applied in Corrosion Science: Theoretical and Mathematical Models towards Quantitative Analysis

Da-Hai Xia, Shizhe Song, Yashar Behnamian, Wenbin Hu, Y. Frank Frank Cheng, Jing-Li Luo, François Huet

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## To cite this version:

Da-Hai Xia, Shizhe Song, Yashar Behnamian, Wenbin Hu, Y. Frank Frank Cheng, et al., Review-Electrochemical Noise Applied in Corrosion Science: Theoretical and Mathematical Models towards Quantitative Analysis. Journal of The Electrochemical Society, 2020, 167 (8), pp.081507. 10.1149/1945-7111/ab8de3. hal-02864691

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Submitted on 11 Jun 2020

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1 Review — Electrochemical Noise Applied in Corrosion Science:

2 Theoretical and Mathematical Models towards Quantitative Analysis<fnref n="1" />

3

4 Da-Hai Xia<fnref n="1" />,², Shizhe Song<fnref n="1" />, Yashar Behnamian², Wenbin Hu<fnref n="1" />, Y. Frank. Cheng³,

5 Jing-Li Luo²,², François Huet⁴

6

7 <fnref n="1" /> Tianjin Key Laboratory of Composite and Functional Materials, School of Materials

8 Science and Engineering, Tianjin University, Tianjin 300354, People's Republic of

9 China

10 ²Department of Chemical and Materials Engineering, University of Alberta, Edmonton,

11 Alberta T6G 1H9, Canada

12 ³Department of Mechanical & Manufacturing Engineering, University of Calgary,

13 Calgary, Alberta, T2N 1N4, Canada

14 ⁴ Sorbonne Université, CNRS, Laboratoire Interfaces et Systèmes Electrochimiques,

15 F-75005 Paris, France

16

17

18 ²dahaixia@tju.edu.cn, jingli.luo@ualberta.ca

19

<footnote n="1">¹ Dedicated to Prof. Shizhe Song on the occasion of her eightieth birthday</footnote>
20 **Abstract**

21 Electrochemical noise (EN) can be used in situ to investigate
22 corrosion processes and to detect and monitor the corrosion of metallic
23 materials. EN data are largely influenced by the measurement mode, the
24 surface area of the working electrodes, the electrolyte resistance, and the
25 symmetry of the electrode system. Herein, the advantages and limitations
26 of electrochemical kinetics, equivalent circuit, and shot noise methods for
27 quantifying corrosion rates with EN are discussed.

28
Since Tyagai<sup>1-3</sup> and Iverson<sup>4</sup> documented the noise generated in electrochemical systems in the 1960s to 1970s, the use of electrochemical noise (EN) to detect metal dissolution has become widespread in corrosion science.<sup>5-37</sup> The development of electrochemical instrumentation,<sup>38-41</sup> advanced signal processing methods,<sup>34,42-44</sup> and inspired experimental design<sup>45-50</sup> in recent decades has promoted EN applications.

Early work used single measurements of electrochemical potential noise (EPN)<sup>51-55</sup> or electrochemical current noise (ECN)<sup>56,57</sup> to study the nature and rate of corrosion on a single working electrode (WE). The fact that the WE can be polarized at any potential under potentiostatic control or at any current under galvanostatic control allows the stochastic behavior of corroding electrodes to be investigated in various specific conditions, such as metastable and stable pitting corrosion,<sup>58-62</sup> gas evolution on stressed electrodes,<sup>63,64</sup> or corrosion protection by organic coatings.<sup>47,65-68</sup>

From the beginning of the nineties, most of the time EPN and ECN were measured synchronously on two WEs tested specifically at the corrosion potential, to get more information than single measurements of EPN or ECN, in particular to estimate the corrosion rate. Important information to be obtained prior to EN measurement includes (1) the electrode system, (2) the sensitivity limit of the measuring device, and (3) the magnitude of the solution resistance of the corrosion system.
Investigators employ mathematical methods and models for EN analysis to determine the mechanism, rate, nucleation, and propagation of corrosion over time. Mathematical methods and parameters used to identify corrosion forms and corrosion rates are summarized in Table I. They are classified into three groups, namely the time domain, the frequency domain, and the time-frequency domain. The details of each method are well-documented in the literature and will not be addressed in this paper.<sup>31,32,34</sup> Several parameters in Table I (e.g. noise resistance, noise impedance) are strongly related to the corrosion rate (see below). The most attractive aspect of detecting metal corrosion by measuring the EN is the absence of external excitation signal as opposed to standard electrochemical techniques, such as polarization techniques or electrochemical impedance spectroscopy, which average the current and potential signals over time and surface area of the electrodes. This allows the investigation of the early stages of corrosion processes in real time with high sensitivity. However, quantitative analyses can be difficult, as EN data are closely linked to the type of electrode system, the area of the electrodes, and the measurement mode. The theoretical aspects of quantitative EN analysis of corrosion are discussed in the following sections, including the electrode systems used in EN measurements, the various sources of EN, and the advantages and disadvantages of theoretical models for EN analysis.
Electrode systems suitable for EN measurement

73
74
75
76
77
78
79
80
EN measurements require a zero-resistance ammeter (ZRA) that can measure EPN and ECN synchronously, or a potentiostat that can measure ECN at an applied potential, or a galvanostat that can measure EPN at an applied current. The electrode systems numbered #1 to #9 can be arranged as shown in Table II. The corroding WE, reference electrode (RE) and counter electrode (CE) are identified in each schematic diagram of the electrode systems.

81
82
83
84
85
86
87
88
89
90
91
92
93
94
The ZRA is the most frequently adopted instrument in corrosion detection and monitoring, because EPN and ECN can be recorded synchronously. The electrodes usually used in this mode comprise one RE and two WEs, or one WE and one CE. EPN is measured between the WE and the RE, and ECN (coupling current) is recorded with an ammeter between the two WEs, or between the WE and the CE. In a “symmetrical” electrode system, WE1 and WE2 are made of nominally identical materials with the same surface area (electrode systems #1 and #2). In “asymmetrical” electrode systems, WEs have different surface areas (electrode systems #3 and #4), or a WE and a CE are used, each made of different materials (electrode system #5). Another mode of asymmetric electrode is similar to #1 but a bias potential is applied between the WEs to have a single corroding WE and a cathode (electrode system #6).⁷⁰

In the ZRA mode, an operational amplifier (Opamp) is often used to
measure the coupling current. The WE is directly connected to ground and the other WE or CE is linked to the inverting input of the Opamp while the non-inverting input is connected to ground. The current-measuring resistor between the output and the inverting input of the Opamp provides a voltage proportional to the coupling current at the output of the Opamp. Circuit components other than the operational amplifier can be used to de-noise and to increase the stability of the instrument. The current-measuring resistance of the ZRA plays a key role in the measurement of ECN.⁷¹

Typical current-measuring resistances in most commercial electrochemical instruments have values ranging from ohms to gigaohms, giving a current range from picoamps to amps. Instead of using a ZRA, sometimes a resistor is intentionally added between WE1 and WE2, partially decoupling the two WEs. This configuration was found to decrease the instrumentation noise,⁷² as simplifying the current-measuring circuit ensured that no instrumental noise due to active electronic components within the ZRA were fed back to the corroding electrodes. This may be important in some systems, but it is only one component of the instrumentation noise appearing in the final measurement, and the instrument noise associated with alternative measurement methods will depend strongly on the instrument design and cell properties as well as the measurement configuration.

Three other measurement modes are used less frequently nowadays. The open circuit potential (OCP) mode (electrode system #7) was the first
mode used to measure the fluctuations of the corrosion potential of an
electrode versus an RE over time, which can be regarded as EPN.⁴ Such
measurement enables an assessment of the transient dissolution of the
metallic surface with a simple voltmeter, but the information obtained is
purely qualitative. The last two modes concern the measurement of ECN
under potentiostatic control (electrode system #8) and the measurement of
EPN under galvanostatic control (electrode system #9). These modes were
widely used prior the introduction of the EN measurements in ZRA mode
in which the WEs are studied only at the corrosion potential (apart from
electrode system #6). Therefore, the potentiostatic/galvanostatic modes
can be used in conditions where the WEs are not at corrosion potential,
such as when studying (1) pitting corrosion of passive metals and
alloys,⁵,²⁰,²¹,⁴⁵,⁵¹,⁵³,⁵⁶,⁵⁸⁻⁶⁰ (2) gas evolution on WEs,⁶³,⁷³ (3) electroplating,⁷⁴
etc.

131

132 **Type of electrodes used in the ZRA mode.**—Symmetrical electrodes and
133 asymmetrical electrodes are successively considered.

134 **Symmetrical electrodes.**—Symmetrical electrodes incorporate two
135 nominally identical WEs with the same surface area plus one true, or
136 standard, RE, where “true” refers to its property of thermodynamic
137 equilibrium (electrode system #1), or three identical electrodes with the
138 same surface area (electrode system #2). In the latter case, the third
139 electrode acts as a pseudo-RE and its noise must be considered when the
140 EN results are analyzed.

141 Electrochemical noise can also be measured using two noisy REs
142 instead of one.⁷⁵ Such configuration allows measuring the cross-spectrum
143 of both signals issued from the REs to eliminate the noise generated by the
144 REs in the calculation of the noise impedance (this is efficient in some
145 cases, as shown in Fig. 8 of Ref. 75, in which the noise impedance is ten
146 times lower when using two REs instead of one). However, the
147 measurement scheme is obviously more complicated and seldom used,
148 except in some special situations.

149 Electrode system #1 is used for WEs undergoing either uniform
150 corrosion or localized corrosion. Since both WEs have similar corrosion
151 reactivity, the requirement of symmetry, that is, identical electrochemical
152 impedance for both WEs, can be easily met. After long-term exposure to
153 the environment, the appearance of corrosion products may or may not
154 affect significantly the impedance of each WE. Usually, a net current
155 flowing between the WEs appears with time without changing drastically
156 the impedance of each WE. In that case, the system is still symmetric
157 despite the existence of the net current. Most EN measurements have been
158 performed on symmetric systems, even when in some cases the asymmetry
159 between the WEs should have been considered in the quantitative
160 analysis⁷⁰.
161 *Asymmetrical electrodes*.—Asymmetrical systems include WEs of the
162 same material but different surface areas (electrode systems #3 and #4),
163 and systems with different materials in each of the WE, the CE, and the RE
164 (electrode system #5). In electrode systems #3 and #4, WE1 and WE2 have
165 different impedance values. The RE can either be a standard (true)
166 reference electrode or a pseudo-reference electrode (a corroding electrode),
167 but in the latter case, as for electrode system #2, the noise related to the
168 corrosion phenomena occurring on the pseudo RE cannot be ignored when
169 performing a quantitative analysis. In electrode system #5, the three
170 electrodes are constructed with different materials and might have different
171 surface areas and different impedance values. The CE is often made of an
172 inert material to diminish its noise level while the WE is a corroding
173 electrode studied at its corrosion potential. Moreover, the area of the CE
174 should be small enough, compared to the area of the WE, to lower the
175 galvanic current between the WE and the CE so that the WE remains at its
176 corrosion potential.⁸,¹⁰ However, a CE of small area has a higher impedance,
177 which increases the asymmetry between the two electrodes.⁷⁶ A Pt
178 electrode with a small area is often used as the CE. It draws the current
179 provided by the corrosion of the WE through a reduction reaction (for
180 example, oxygen reduction in neutral solution, or hydrogen evolution in
181 acidic medium) at the corrosion potential of the WE.

182 It is important to note that the type of electrodes employed when
measuring EN can affect the quantitative analysis of the EN signals.

Attempts to provide a symmetrical electrode system where the two coupled WEs have the same surface area, the same material composition, and the same corrosion potential and corrosion activity are quite never completely perfect, as these properties cannot be matched to perfection in different electrodes, and properties change with the passage of current. Therefore, the symmetry in the electrode systems listed in Table II is theoretical, even if often observed in practice.

**The impact of electrode area on EN measurement.**—Corrosion transients are assumed to be independent events; thus, the effects of electrode area on ECN and EPN data can be described using a single transient analysis.

*Analysis of EPN and ECN transients.*—Two important aspects are considered here, the shape of ECN and EPN transients, and how these transients superimpose or counteract as the WE area increases. EPN and ECN transients have been observed for a long time during chemically or mechanically triggered localized corrosion of various metals and alloys at OCP or at an applied anodic potential in the passive domain. Multiple factors influence the shape and amplitude of the transients, such as the nature and preparation of the metallic material, the electrolyte composition, the electrode potential, the surface area of the electrode, the measurement mode, and the instrumentation limitations as resolution, sampling
205 frequency, and filtering. The transients are the results of the competition
206 between the anodic process producing electrons and the cathodic process
207 consuming these electrons, in addition with the interaction of the
208 charge/discharge of the interfacial capacitance of the electrode.

209 Several types of ECN transients have been observed under
210 polarization control in the passive domain to study pitting corrosion of
211 various metals often in chloride-containing solutions, the most commonly
212 encountered consisting of a sudden rise followed by a slow decay (type 1)
213 and a slow rise followed by a sudden decay (type 2).<sup>56,59,77-80</sup> Type-1
214 transients correspond to the sudden breakdown of the passive film inducing
215 metal dissolution followed, after repassivation, by a current decay
216 corresponding to the recharge of the interfacial capacitance of the passive
217 area by the potentiostat.<sup>56</sup> Type-2 transients have been observed, for
218 example, on stainless steels on which metastable pits grow under a cover,
219 the current decaying suddenly when an opening appears in the pit cover.<sup>78</sup>
220 At OCP, using two identical electrodes connected through a ZRA, ECN
221 transients appear in both directions depending on which electrode supports
222 the localized corrosion event.<sup>81-84</sup> More complicated shapes than those of
223 type-1 and -2 transients have also been observed, which can be attributed
224 to the influence of the cathodic processes that contribute to consume
225 rapidly the electrons produced by the anodic event. In that case, the
226 amplitude of the measured current transient is lower than that of the anodic
227 metal dissolution current, which cannot be individually measured.⁸¹,⁸²
228 Other factors influence the shape of the current transients, such as the conductivity of the solution between the WEs,⁸⁵ or the cut-off frequencies of the filters used in the measuring instrumentation.⁸¹

231 Similar shapes have been observed for the EPN transients at OCP during metastable pitting corrosion of various metals,⁸²⁻⁸⁴,⁸⁶⁻⁸⁹ all potential transients starting in the negative direction. The shape and the amplitude of the potential transients depend as well on the kinetics of the anodic and cathodic reactions. If the cathodic process is fast, as for ferric ion reduction or for a high oxygen concentration in the solution, the charge produced by the anodic event can be consumed rapidly by the cathodic process, hence giving a fast return to the original potential value, and, therefore, a potential transient shape similar to that of the type-2 current transient. In contrast, for a slow cathodic reaction, the charge produced by the anodic corrosion event may be temporarily stored in the interfacial capacitance of the electrode before being consumed after repassivation by the slow cathodic reaction; the shape of the potential transient is then similar to that of the type-1 current transient with a slow recovery to the original potential value. This process was modelled by Pistorius who calculating the rate of change of the electrode potential as:⁹⁰

$$247 \qquad \frac{dE}{dt} = \frac{-1}{c_d(A_1+A_2)} [(A_1+A_2)i_{\text{cath}} + I_1 + I_2] \qquad [1]$$

248 where $C_d$ is the interface capacitance, $A_1$ and $A_2$ are the areas of WE1
and WE2, respectively, $i_{\text{cath}}(< 0)$ is the cathodic current density at the specific potential, and $I_1$ and $I_2$ ($> 0$) are the anodic currents on the two WEs.

*Impact of electrode area on ECN.*—As the electrode surface area increases, the number of current transients increases accordingly. Pistorius suggested that the standard deviation (SD) of the current, $S_I$, could be proportional to the area or the square root of the area depending on whether the ECN transients are separated in time or are superimposed.⁹⁰ However, the SD is calculated on long times during which it is very rare that transients superimpose exactly. Therefore, it is generally accepted that the SD of the current is proportional to the square root of the WE area,¹³,³¹,⁹⁰ and such a claim is expected to be valid for both uniform corrosion and localized corrosion. Therefore, in data analyses of EN, the $S_I$ should not be normalized versus the electrode area. This implies that the choice of electrode size is extremely important for EN measurements in general, and the size must be kept constant in order to compare results.

*Impact of electrode area on EPN.*—Increasing the electrode size will decrease the SD of the EPN, $S_V$, because of the increase of the interfacial capacitance $C_d$ and the area for the cathodic reaction.⁹⁰ According to Pistorius, the $S_V$ is expected to give no area dependence for superimposed transients or an inverse proportionality with respect to $\sqrt{\text{area}}$ for temporally separated transients.⁹⁰ However, transients superimposing
271 exactly are rare cases in real corrosion systems, therefore the $S_V$ is
272 believed, in most cases, to be inversely proportional to the square root of
273 the WE area.<sup>13,31</sup> Experimental data regarding SD dependence on WE area
274 in typical corrosion systems are limited, therefore it is unknown if this
275 relationship works for all corrosion systems, such as passive metals or
276 metals undergoing stress corrosion cracking or crevice corrosion.

277

278

## Noise sources during EN measurement

279 Noise sources during EN measurements include instrumentation noise,
280 aliased noise, thermal noise, and noise generated at the metal/electrolyte
281 interface. Aliasing occurs when the analog signal contains frequencies
282 higher than half the sampling frequency when it is sampled. Aliasing may
283 occur at all frequencies: for example, the frequency of 0.9 Hz is aliased at
284 0.1 Hz if the sampling frequency is 1 Hz. Aliased noise can be avoided by
285 using an analog anti-aliasing low-pass filter and by matching the cut-off
286 frequency of the filter to the sampling frequency. High levels of aliased
287 noise are often present in the absence of anti-aliasing filters in the
288 equipment or because of the way the filters are used. It is, therefore, very
289 important to use anti-aliasing filters prior to the analog-to-digital
290 conversion in EN measurements.<sup>91-93</sup> Electromagnetic interference may be
291 another noise source, but this can be reduced by using a DC power supply
292 and a Faraday cage.
**Instrumentation noise.**—Instrumentation noise cannot be avoided by experimental design. Because the noise levels of some corrosion systems are very low, e.g. passive metal electrodes and even sometimes electrodes undergoing uniform corrosion,⁹⁴,⁹⁵ it is critical to assess the measuring device limit of sensitivity to noise before conducting EN measurements. Instrumentation noise can be evaluated by using a dummy cell consisting of several resistors to validate the EN measurement equipment and determine its baseline noise performance. Measurements on dummy cells performed by the European Cooperative Group on Corrosion Monitoring of Nuclear Materials (www.ecg-comon.org) in the recent past have demonstrated that, because of the low level of thermal noise generated by resistors, few instruments are capable of measuring corrosion noises of such low level.⁹²,⁹³

**Thermal noise.**—Thermal noise in an electrochemical instrument is caused by the resistive elements, and, sometimes, an equivalent thermal noise resistance $R_e$ is used to represent the thermal noise level. $R_e$ is calculated from the power spectral density (PSD), $\Psi_V(f)$, of the voltage noise of the measuring instrument. The PSD shows which frequencies in the signal have a large amplitude and which ones have a small amplitude. The $R_e$ is calculated from the Nyquist equation:⁹⁶

$$R_e(f) = \frac{\Psi_V(f)}{4kT} \qquad [2]$$
315 where *k* is the Boltzmann constant and *T* is the temperature in Kelvin.

316 However, the above parameter is almost never used in the corrosion
317 field because it allows assessing the level of the voltage noise of the
318 measuring device, not its current noise, which is extremely dependent on
319 the value of the current-measuring resistor, as mentioned above, while the
320 voltage noise is not.⁷¹ Therefore it is preferable to use the PSDs of the
321 voltage and current noises of the measuring device to quantify the
322 instrumentation noise. In EN measurements for corrosion studies in ZRA
323 mode, the instrumentation noise can be evaluated by using dummy cells
324 composed of three identical pure resistors *R*, which gives thermal EPN and
325 ECN with a constant PSD, 6*k*TR for the EPN and 2*kT*/R for the ECN
326 (*R* is in ohms, *T* in Kelvin).⁹⁷ At low frequency, 1/*f* noise due to the
327 electronic components in the instrumentation may be observed. In
328 corrosion systems showing clear ECN and EPN transients, the PSD of the
329 instrumentation noise is much lower than that of the EN and, therefore, can
330 be neglected, at least at low frequencies. In other situations, especially for
331 passivation or slow uniform corrosion, the instrumentation noise may be
332 significant and EN is difficult to measure.

333

334 *Noise generated at the metal/electrolyte interface.*—Any changes in
335 state variables at the metal/electrolyte interface will lead to fluctuations of
336 potential and current. As detailed in several reviews,³¹,³⁴,³⁵,⁶⁴,⁹⁸,⁹⁹ EN is
337 mainly generated from the following corrosion and electrochemical events:

338 (1) Anodic dissolution during general corrosion. As the electrochemical activity at specific points in an electrode system fluctuates over time, the sites of anodic and cathodic reactions change with time during uniform corrosion. ECN during general corrosion is due to bursts of current of low amplitude related to the dissolution process that gives changes in the surface morphology. This noise due to the discrete nature of the charge carriers is called shot noise. When the charge carriers move independently of each other (Poisson process), according to the theory of shot noise, the mean corrosion current is equal to the product of the charge of each corrosion event and the emission frequency of the event (see below).<sup>44,98</sup>

349 (2) Fluctuations of species concentrations at the metal-electrolyte interface induced by elementary fluctuations in the flux of molecules or ions.<sup>100</sup> When the kinetics of the WE is controlled by mass transport, the electrolyte movement due to natural or forced convection enhances drastically the amplitude of EN.<sup>100,101</sup>

354 (3) Pitting corrosion. In addition to the typical potential and current transients induced by pitting described above, the electrochemical communication between the pits is the source of complex, meandering patterns in the EN signals.<sup>102</sup>

358 (4) Gas evolution. Hydrogen bubble evolution on the electrode surface
359 may occur in acidic medium and cause EPN and ECN due to the
360 fluctuations of dissolved hydrogen concentration at the interface and
361 to the fluctuations of active surface due to bubble screening.⁶⁴,¹⁰³

362 (5) Other noise sources include removal of surface layers by erosion or
cavitation,¹⁰⁴⁻¹⁰⁸ crevice corrosion,¹⁰⁹,¹¹⁰ intergranular corrosion,¹¹¹
363 stress corrosion cracking,¹¹,¹²,¹⁶,¹¹²,¹¹³ microbial-induced
corrosion,¹⁷,¹⁸ etc. Moreover, corrosion systems sometimes have
364
365 multiple sources of EN, leading to a complicated EN analysis.
366

367

368

# Theoretical models for EN analysis

369 EN contains fundamental information about the nature and rate of
370 metal corrosion. This information can be extracted from the EN signal
371 using theoretical and mathematical models.

372 *Electrochemical kinetics model.*—The electrochemical kinetics model
373 of potential and current fluctuations proposed by Chen and Bogaerts is
374 based on the Butler-Volmer equation.⁸,¹¹⁴ The authors made three
375 assumptions supporting the theoretical analysis:

376 (1) Both anodic and cathodic reactions occurring on the WE and the CE
377 are totally under activation polarization control; therefore, the
378 electrochemical kinetics of these reactions can be described by the
379 Butler-Volmer equation.¹¹⁵

380 (2) The corrosion potential of the WE, Ecorr, is far away from the
equilibrium potential of the anodic and cathodic reactions, $V_{e,a}$ and $V_{e,c}$ respectively ($V_{e,c} \ll E_{corr} \ll V_{e,a}$).

(3) Both anodic and cathodic reactions are under steady state condition, therefore, only the Faraday current is considered, the charge/discharge of the interface capacitance being neglected.

Theoretical derivation of the noise resistance.—In the symmetrical electrode system #1, schematically shown in Fig. 1 with the anodic and cathodic currents on each WE, the coupling current measured by the ZRA at time t, $I(t)$, flowing between WE1 and WE2 is due to a slight difference in the corrosion potentials of WE1 and WE2. The Butler-Volmer equation can be written:

$$I(t) = I_a(t) + I_c(t) = i_{0,a}A_a e^{\frac{2.303}{b_a}(V(t)-V_{e,a})} - i_{0,c}A_c e^{\frac{2.303}{b_c}(V(t)-V_{e,c})} \\ = I_{corr} \left( e^{\frac{2.303}{b_a}(V(t)-E_{corr})} - e^{\frac{2.303}{b_c}(V(t)-E_{corr})} \right) \quad [3]$$

where $V(t)$ is the potential of WE1, $b_a$ and $b_c$ the anodic and cathodic Tafel coefficients, $i_{0,a}$ and $i_{0,c}$ the exchange current densities, $A_a$ and $A_c$ the anodic and cathodic areas on WE1, and $I_{corr}$ the corrosion current. The introduction of the galvanic coupling current in Ref. 114 is obscure and the theoretical analysis is restricted to the case of coupling current lower than the corrosion current, so a new analysis is presented that does not have such restriction. Considering that $I(t)$ and $V(t)$ fluctuate slightly around the stable average values, $I_g$ and $V_g$, the linearization of
Eq. 3 gives:

$$404 \qquad I(t) = I_{\text{corr}} 2.303 (V(t) - E_{\text{corr}}) \left(\frac{1}{b_a} + \frac{1}{b_c}\right) \qquad [4]$$

or, after subtracting the mean value of Eq. 4:

$$406 \qquad I(t) - I_g = I_{\text{corr}} 2.303(V(t) - V_g) \left(\frac{1}{b_a} + \frac{1}{b_c}\right) \qquad [5]$$

The SD $S_I$ of the ECN, $I(t) - I_g$, is then related to the SD of the EPN, $V(t) - V_g$, through the equation:

$$409 \qquad S_I = I_{\text{corr}} 2.303 S_V \left(\frac{1}{b_a} + \frac{1}{b_c}\right) \qquad [6]$$

from which the noise resistance $R_n$, defined as the ratio $S_V/S_I$, can be derived:

$$412 \qquad R_n = \frac{S_V}{S_I} = \frac{b_a b_c}{2.303 I_{\text{corr}} (b_a + b_c)} \qquad [7]$$

The last term is the expression of the polarization resistance, $R_p$, derived by Stern and Geary.¹¹⁶ Thus, when the anodic and cathodic reactions occurring on the WEs are under activation polarization control, $R_n = R_p$. However, it should be noticed that in some cases the anodic or cathodic reaction does not strictly obey a Tafel behavior so that this relationship is sometimes not satisfied in practice.¹³,⁹⁰,¹¹⁷ Besides, the calculation of $R_n$ is slightly influenced by several factors such as the sampling frequency,¹¹⁷ by the DC removal method used,⁹,¹¹⁸,¹¹⁹ by the size of the WEs,¹³ or by the weak symmetry between the two WEs. All these factors can lead to conflicting values between $R_n$ and $R_p$. Although most works have claimed that a frequency range of 0.01 to 10 Hz enables satisfactory EN measurements of corrosion systems, the selection of
frequency range and, obviously, the corresponding sampling rate, depend considerably on the EN source and the intention of the measurements (fundamental research, testing, monitoring).<sup>14</sup> Also, there is no uniform standard for the DC removal method, the most frequently used methods appearing to be polynomial detrending,<sup>120</sup> wavelet analysis, and empirical mode decomposition.<sup>121</sup> Finally, the theoretically symmetric electrodes may become asymmetric over time, especially in cases of localized corrosion. Therefore, the noise resistance is not an accurate quantitative measure of corrosion rate for all corrosion systems. This is why, especially in the field, the corrosion current can be estimated from the ratio B/R<sub>n</sub>, where the Stern-Geary coefficient B is calibrated from weight-loss measurements. In contrast to the SD of the current, which should not be normalized vs the electrode area, R<sub>n</sub> can be normalized, as well as R<sub>p</sub>, so that the corrosion rate can be calculated in A cm<sup>-2</sup> whatever the size of the WEs.

Electrode systems #3 to #6 in Table II are typical asymmetrical systems for which there is no derived relationship between R<sub>n</sub> and R<sub>p</sub> in the electrochemical kinetics model. There may be some merit for these asymmetric electrode systems when quantitative analysis is not required, but useful aspects of asymmetric electrode systems are not considered in this review.

*Limitations of the electrochemical kinetic models.*—Though kinetics
models can be used to understand the correlation between noise resistance and polarization resistance, they have some drawbacks as listed below.

(1) The Butler-Volmer equation assumes Tafel kinetics on both electrodes, which is not the case in many corroding systems. For a single corroding electrode governed by active dissolution and concentration polarization, the relationship between the measured current $I$ and the electrode potential $V$ is written as:¹²²

$$I(V) = \frac{e^{\frac{2.303(V-V_{e,a})}{b_a}}}{\frac{1}{A_a i_{0,a}} + \left(\frac{1}{G_a}\right) e^{\frac{2.303(V-V_{e,a})}{b_a}}} - \frac{e^{\frac{2.303(V-V_{e,c})}{b_c}}}{\frac{1}{A_c i_{0,c}} + \left(\frac{1}{G_c}\right) e^{\frac{2.303(V-V_{e,c})}{b_c}}} \quad [8]$$

where $V_{e,a}$ and $V_{e,c}$ are the equilibrium potential of the anodic and cathodic reactions, $G_a$ and $G_c$ are the limiting diffusion currents of the anodic and cathodic reactions, $i_{0,a}$ and $i_{0,c}$ are the exchange current densities of the anodic and cathodic reactions, and $A_a$ and $A_c$ are the active anodic and cathodic areas. Curioni et al. proposed a numerical approach to calculate the potential and current noises.¹²³

They modified Eq. 8 in the simplified case of a cathodic reaction only under activation control, assuming that the anodic and cathodic areas, $A_a(t)$ and $A_c(t)$, vary with time during corrosion:

$$I(V,t) = \frac{e^{\frac{2.303(V-V_{e,a})}{b_a}}}{\frac{1}{A_a(t)i_{0,a}} + \left(\frac{1}{G_a}\right) e^{\frac{2.303(V-V_{e,a})}{b_{\text{diff}}} + \frac{2.303(V-V_{e,c})}{b_c}}} - A_c(t)i_{0,c}e^{\frac{2.303(V-V_{e,c})}{b_c}} - C_{\text{dl}} \frac{dV(t)}{dt} \quad [9]$$

where $b_{\text{diff}}$ is an equivalent Tafel coefficient accounting for the dependence of the limiting current diffusion on the potential and $C_{\text{dl}}$
467 is the double layer capacitance. The first two terms account for the
468 anodic and cathodic reactions, $I_a(V, t)$ and $I_c(V, t)$, and the last term
469 for the capacitive current, $I_{dl}(V, t)$, due to $C_{dl}$. According to the
470 authors, when WE1 and WE2, which are assumed to follow the
471 kinetics of Eq. 9, are galvanically coupled, the current $I_{1+2}(V, t)$
472 exchanged with an external circuit can be written:

$$473 \qquad I_{1+2}(V,t) = I_1(V,t) + I_2(V,t) = I_{a,1}(V,t) - I_{c,1}(V,t) - \\ 474 \qquad \qquad I_{dl,1}(V,t) + I_{a,2}(V,t) - I_{c,2}(V,t) - I_{dl,2}(V,t) \qquad [10]$$

475 with obvious notations. From a numerical point of view, the corrosion
476 potential is given by the solution of the previous equation in which
477 the first term $I_{1+2}(V, t)$ is equal to 0.¹²³ Then, to calculate the time
478 evolution of the potential, Eq. 9 was rearranged to obtain the
479 expression of the potential time-derivative $dV/dt$ from which the
480 potential at time $t+\Delta t$ could be approximated from its value at time $t$
481 by using Euler's method.

$$482 \qquad V(t + \Delta t) = V(t) + \Delta t \frac{dV}{dt} \qquad [11]$$

483 For given initial values of the anodic and cathodic areas, and of
484 the corrosion potential, the time evolution of the corrosion potential
485 could be calculated, from which the time evolution of the current
486 exchanged between the electrodes, the polarization resistance, and the
487 noise resistance could be derived with the numerical model. They
488 found that, in most cases, there was no unacceptable difference
between $R_n$ and $R_p$, even for severe asymmetry between the electrodes.

(2) The Butler-Volmer equation is sometimes valid for uniform corrosion, but it is not capable of interpreting localized corrosion. Although EN can be used to identify initiation and propagation stages of localized corrosion, the kinetic equations do not describe localized corrosion.

(3) For asymmetrical electrode systems, a quantitative analysis cannot be achieved, as mentioned above.

(4) The initially symmetric electrode systems listed in Table II may become asymmetric, especially those undergoing localized corrosion.

**Equivalent circuit approach with fluctuating potential and current sources.**—In the equivalent circuit (EC) approach introduced by Huet's group,²⁹,⁷⁰,⁸⁵,¹⁰⁰,¹²⁴⁻¹²⁶ the elements have clear physical significance. In the four circuits appearing in the first two rows of Table III, the EPN and ECN are modelled using a single circuit element, either a current source or a potential source, and it is assumed that the impedance of the electrodes does not vary during corrosion. ECs for electrode systems #1 to #6 in Table II are discussed in detail below and their theoretical predictions and experimental results are compared.

**Electrode systems #1, #3, #5 and #6**—The two ECs (EC1 and EC2) of
these electrode systems presented in the first row of Table III are
mathematically identical simplifications of real circuits. The circuit based
on current noise sources (EC1) is the Norton equivalent circuit and the
circuit based on potential noise sources (EC2) is the Thevenin equivalent
circuit¹²⁴. In EC1 and EC2, *i*₁ and *i*₂ represent the current noise sources of
WE1 and WE2 while *e*₁ and *e*₂ represent the potential noise sources of WE1
and WE2. Z₁, Z₂, and Z₃ denote the impedances of WE1, WE2, and RE. Δ*I*
is the current fluctuation flowing from WE1 to WE2 measured by the ZRA,
Δ*V* is the potential fluctuation of WE1, *R*<sub>s</sub> is the electrolyte resistance
between WE1 and WE2, and *x* is the fraction of *R*<sub>s</sub> between the RE and
WE1 (0 ≤ *x* ≤ 1). According to Ohm's law in the frequency domain (all
quantities, except *R*<sub>s</sub>, are frequency-dependent), the expressions of Δ*I*(*f*)
and Δ*V*(*f*) for these four electrode systems, are ¹²⁴:

$$
524 \qquad \Delta I = \frac{Z_1 i_1 - Z_2 i_2}{Z_1 + Z_2 + R_s} \quad ; \quad \Delta V = \frac{-Z_1 i_1 [Z_2 + R_s (1-x)] - Z_2 i_2 [Z_1 + R_s x]}{Z_1 + Z_2 + R_s} \quad [12]
$$

525 when using the current noise sources, and:

$$
526 \qquad \Delta I = \frac{e_2 - e_1}{Z_1 + Z_2 + R_s} \quad ; \quad \Delta V = \frac{e_1 [Z_2 + R_s (1-x)] + e_2 [Z_1 + R_s x]}{Z_1 + Z_2 + R_s} \quad [13]
$$

527 when using the potential noise sources. It is possible to convert one model
528 to the other with the following expressions of the potential noise sources:¹²⁴

$$
529 \qquad e_j(f) = Z_j(f) i_j(f) \qquad j = 1, 2 \qquad [14]
$$

From Eq. 12, the PSDs of the ECN and EPN can be derived as a
function of the PSDs of the current noise sources, $\Psi_{i_1}(f)$ and $\Psi_{i_2}(f)$,
which are assumed to be uncorrelated:¹²⁴
$$533 \qquad \Psi_I(f) = \left| \frac{Z_1}{Z_1+Z_2+R_s} \right|^2 \Psi_{i_1}(f) + \left| \frac{Z_2}{Z_1+Z_2+R_s} \right|^2 \Psi_{i_2}(f) \qquad [15]$$

$$534 \qquad \Psi_V(f) = \left| \frac{Z_1[Z_2+R_s(1-x)]}{Z_1+Z_2+R_s} \right|^2 \Psi_{i_1}(f) + \left| \frac{Z_2[Z_1+R_s x]}{Z_1+Z_2+R_s} \right|^2 \Psi_{i_2}(f) \qquad [16]$$

According to Eq. 14, the corresponding expressions of $\Psi_I(f)$ and $\Psi_V(f)$ as a function of the PSDs of the potential noise sources, $\Psi_{e_1}(f)$ and $\Psi_{e_2}(f)$, can be directly obtained by replacing $\Psi_{i_j}(f)$ with $\Psi_{e_j}(f)/|Z_j(f)|^2$ ($j=1,2$).

The noise impedance, $Z_n(f)$, initially called spectral noise resistance $R_{sn}(f)$, defined as the square root of the ratio of the PSD of EPN divided by the PSD of ECN, can be derived from Eqs. 15 and 16:29,70,124

$$543 \qquad Z_n(f) = \sqrt{\frac{\Psi_V(f)}{\Psi_I(f)}} = \sqrt{\frac{|Z_1[Z_2+R_s(1-x)]|^2 \Psi_{i_1}(f) + |Z_2(Z_1+R_s x)|^2 \Psi_{i_2}(f)}{|Z_1|^2 \Psi_{i_1}(f) + |Z_2|^2 \Psi_{i_2}(f)}} \qquad [17]$$

It is important to note that the noise impedance is a real number, not a complex number as a true impedance; in other words, it contains no phase information. If $R_s$ is much lower than the impedance of the electrodes, the expression of $Z_n$ becomes:

$$548 \qquad Z_n(f) = |Z_1 Z_2| \sqrt{\frac{\psi_{i_1}(f) + \psi_{i_2}(f)}{|Z_1|^2 \psi_{i_1}(f) + |Z_2|^2 \psi_{i_2}(f)}} \qquad [18]$$

Equation 18 indicates that, in general, $Z_n(f)$ is determined by four factors, the impedances of both electrodes and the PSDs of both current or potential noise sources. However, for electrode system #1 with identical electrodes immersed in the same electrolyte and working at the same corrosion potential, the impedances of both WEs are expected to be
554 identical $[Z_1 = Z_2$, denoted as $Z(f)]$. Equation 18 then becomes:

$$555 \qquad Z_n(f) = |Z(f)| \qquad [19]$$

556 Therefore, in the common use of two identical electrodes connected 557 through a ZRA, the noise impedance is equal to the modulus of the 558 impedance of the WEs whatever the origin of the noise sources (pits, 559 bubbles, cracks...) and whatever the noise level on each electrode.

560 For electrode system #3 with WEs of the same material but different 561 surface areas, the impedance of each WE is inversely proportional to its 562 surface area while the PSD of the current noise source is proportional to 563 the surface area.¹³,³¹ It can then be shown from Eq. 18 that:¹²⁵

$$564 \qquad Z_n(f) = \sqrt{|Z_1(f) Z_2(f)|} \qquad [20]$$

565 For electrode system #5 and #6, one electrode serves as the anode and 566 the other serves as the cathode, so $Z_1$ is now denoted as $Z_a$, and $Z_2$ as 567 $Z_c$. Using the current noise sources, Eq. 18 can be rewritten as:

$$568 \qquad Z_n(f) = |Z_a Z_c| \sqrt{\frac{\psi_{i_a}(f) + \psi_{i_c}(f)}{|Z_a|^2 \psi_{i_a}(f) + |Z_c|^2 \psi_{i_c}(f)}} \qquad [21]$$

569 The expression of the noise impedance can be further discussed 570 according to three different cases for these asymmetric electrode systems:

571 (1) The noise level of the cathode (WE2) is significantly higher than the 572 noise level of the anode (WE1) (for example, the cathode mainly 573 supports hydrogen bubble evolution, whereas the anode undergoes 574 uniform corrosion).
$$575 \qquad Z_n(f) = |Z_a(f)| \qquad [22]$$

The noise impedance is equal to the impedance of the anode, while the EPN and ECN time records provide information on the noiseier electrode (cathode).⁷⁰

579 (2) The noise level on the anode is significantly higher than the noise level on the cathode (for example, the cathode mainly supports oxygen reduction, whereas the anode undergoes pitting corrosion).

$$582 \qquad Z_n(f) = |Z_c(f)| \qquad [23]$$

The noise impedance is equal to the impedance of the cathode, while the EPN and ECN time records provide information on the noiseier electrode (anode).

586 (3) When the noise levels of the anode and cathode are comparable, $Z_n(f)$ ranges between the impedance moduli of the electrodes $|Z_a(f)|$ and $|Z_c(f)|$.⁷⁰

589 It is then important to underline the difficulty of the EN analysis in the case of asymmetric systems: EPN and ECN depend on four quantities (the impedance of each WE and the noise source generated by each WE) while EN measurements only give two quantities (EPN and ECN). 590
591
592 Therefore, the EN analysis can only be performed if the impedance of each WE has been measured.⁷⁰,⁷⁶

595 *Electrode systems #2 and #4*—Electrode systems #2 and #4 can be described by equivalent circuits EC3 or EC4.¹²⁴ If a pseudo RE is used, its
current noise source $i_3$ or potential noise source $e_3$ has to be taken into account. A new term, $Z_3 i_3$, must be added to the expression of $\Delta V$ in Eq. 597
598
599
12 while the expression of $\Delta I$ is unaffected. The noise impedance can then be expressed using the current noise sources of the three electrodes, which are assumed to be uncorrelated, as:¹²⁴

$$Z_n(f) = \sqrt{\frac{|Z_1[Z_2+R_s(1-x)]|^2\psi_{i_1}(f) + |Z_2(Z_1+R_s x)|^2\psi_{i_2}(f) + |Z_3(Z_1+Z_2+R_s)|^2\psi_{i_3}(f)}{|Z_1|^2\psi_{i_1}(f) + |Z_2|^2\psi_{i_2}(f)} \quad [24]$$

In electrode system #4, the impedances of the three electrodes are not identical and, therefore, the noise impedance cannot be related to the impedance of the electrodes. In contrast, in electrode system #2 with three identical electrodes, the electrodes have equal impedance ($Z_1 = Z_2 = Z_3$), denoted as $Z$). Therefore, when $R_s$ is neglected:

$$Z_n(f) = |Z(f)| \sqrt{1 + \frac{4\psi_{i_3}(f)}{\psi_{i_1}(f) + \psi_{i_2}(f)}} \quad [25]$$

When the three electrodes are identically noisy, which may be a reasonable assumption in some cases but cannot be experimentally checked since the current noise sources cannot be separately measured, then:

$$Z_n(f) = \sqrt{3} |Z(f)| \quad [26]$$

As for the electrode system #1, it is then still possible to determine the impedance modulus of the electrodes when using a third identical electrode as pseudo-RE.
618 Correlation between $R_n$ and $Z_n$.—For a random signal, the variance, 619 which is the square of the standard deviation, is equal to the integral of its 620 PSD, therefore, according to the definition of the noise resistance (Eq. 7):

$$R_n = \sqrt{\frac{\int_{0}^{+\infty} \psi_v(f) df}{\int_{0}^{+\infty} \psi_I(f) df}} = \sqrt{\frac{\int_{f_{\min}}^{f_{\max}} \psi_I(f) Z_n^2(f) df}{\int_{f_{\min}}^{f_{\max}} \psi_I(f) df}} \quad [27]$$

621 Actually, the PSD has a limited frequency bandwidth ($f_{min}$, $f_{max}$). 622 The lowest frequency analyzed of the spectrum of a discrete time record 623 sampled at frequency $f_s$ is given by $f_{min} = f_s/N = 1/N\Delta t$, where N is 624 the number of samples and $\Delta t = 1/f_s$ is the sampling interval. When the 625 PSD is calculated using the fast Fourier transform, the frequency resolution 626 is also given by $f_s/N$. As an example often encountered in the literature, 627 when $f_s = 2$ Hz and $N = 2,048$ points, the minimum frequency analyzed 628 and the frequency resolution are ~1 mHz. In practice, it is advised to 629 sample 10 times more points to obtain a sufficient accuracy of the spectrum 630 by PSD averaging. The $f_{max}$ being equal to one half of the sampling 631 frequency, the frequency range analyzed in the above example is (~1 mHz, 632 1 Hz). At higher frequencies, the noise is often due to the 633 instrumentation.126 634

635 For electrode system #1 with identical electrodes, the noise 636 impedance is equal to the impedance modulus of the electrodes, so Eq. 27 637 can be written as:
$$R_n = \sqrt{\frac{\int_{f_{\min}}^{f_{\max}} \psi_I(f) |Z(f)|^2 df}{\int_{f_{\min}}^{f_{\max}} \psi_I(f) df}} \quad [28]$$

Therefore, if $|Z(f)|$ does not depend on frequency in the range $(f_{\min}, f_{\max})$, that is, if $|Z(f)|$ is equal to the polarization resistance in this range, then $R_n = R_p$ regardless of the shape of $\psi_I(f)$. However, in the frequency range (~1 mHz, 1 Hz) often analyzed in practice, $|Z(f)|$ is frequency dependent in real corrosion systems and $R_n$ is in no simple way related to the electrode impedance according to Eq. 28. The conditions for which $R_n$ departs significantly from $R_p$ have been discussed in Ref 30. If $f_c$ denotes the critical frequency above which $|Z(f)|$ significantly decreases and if the slope of the current PSD is constant in the range $(f_{\min}, f_{\max})$, EN measurement times for which $f_c/f_{\min} > 10$ are sufficiently long in many cases to estimate $R_p$ from $R_n$.

Table IV lists some experimental values of $R_n$, $R_p$, $Z_n(f)$, and $|Z(f)|$. This table shows that $R_n$ is equal to $R_p$ only in some cases. Aballe et al.³⁰ claimed that the pre-condition for $R_n = R_p$ is that $|Z(f)|$ is equal to $R_p$ in the frequency bandwidth investigated, a reasonable assumption for sufficiently low values of the maximum frequency analyzed, with the notable exception of coated or stainless steel electrodes for which electrochemical impedance spectroscopy (EIS) measurements should be performed at extremely low frequencies (< 0.1 mHz), practically inaccessible, to get the value of $R_p$. In addition, Table IV shows that the
values of the noise impedance and of the modulus of the electrode impedance measured by EIS are in good accordance with the theoretical predictions (Eqs. 19 and 26).

**Equivalent circuit approach with fluctuating resistances.**—Curioni et al.¹²⁷ proposed a new EC approach to interpret noise generation on freely similar or dissimilar corroding electrodes. As shown in EC5 depicted in Table III, it is based on the assumption that the EN results from fluctuations of the electrical resistances, $R_{a,i}(t)$ and $R_{c,i}(t)$, of the surface layers (oxides, hydroxides or adsorbed species) covering the anodic and cathodic areas of each electrode ($i = 1,2$). In their model, the effects of the double layer capacitance of each electrode are neglected, so the analysis is limited to low frequencies. Moreover, the electrode impedances are assumed to be simple resistances, which limits the application of the model to corroding electrodes for which the EIS Bode plot displays a low-frequency plateau within practical measurement frequencies. Neglecting the solution resistance in conductive solutions and according to the direction of the currents in EC5, the currents flowing out of each electrode can be written from Ohm's law:

$$I_i(t) = I_{a,i}(t) + I_{c,i}(t) = \frac{V(t)-E_{a,i}}{R_{a,i}(t)} + \frac{V(t)-E_{c,i}}{R_{c,i}(t)} \quad [29]$$

where $V(t)$ is the potential of the coupled electrodes, and $E_{a,i}$, $E_{c,i}$ are the equilibrium potentials for the anodic and cathodic reactions on each electrode. To derive the time-varying resistances, $R_{a,i}(t)$ and $R_{c,i}(t)$, the
681 authors used the Thevenin equivalent potential source, $V_{eq,i}(t)$, in series
682 with the Thevenin equivalent resistance, $R_{eq,i}(t)$, for each electrode i:

$$683 \qquad V_{eq,i}(t) = E_{a,i} \frac{R_{c,i}(t)}{R_{a,i}(t)+R_{c,i}(t)} + E_{c,i} \frac{R_{a,i}(t)}{R_{a,i}(t)+R_{c,i}(t)} \qquad [30]$$

$$684 \qquad R_{eq,i}(t) = \frac{R_{a,i}(t)+R_{c,i}(t)}{R_{a,i}(t)R_{c,i}(t)} \qquad [31]$$

685 from which the potential measured versus the RE can be written:

$$686 \qquad V(t) = \frac{R_{eq,1}(t)}{R_{eq,1}(t)+R_{eq,2}(t)} V_{eq,2}(t) + \frac{R_{eq,2}(t)}{R_{eq,1}(t)+R_{eq,2}(t)} V_{eq,1}(t) \qquad [32]$$

687 Assuming that the potential $V_{eq,i}(t)$ of each electrode does not change
688 significantly with time during the coupling period, Eq. 32 was rewritten by
689 replacing $V_{eq,i}(t)$ with the value of the potential measured when the
690 electrode is decoupled and assumed constant, $\bar{V}_{eq,i}$. The Thevenin
691 equivalent resistances can then be estimated when the WEs are coupled
692 from the measurement of $V(t)$ and of the current $I(t)$ flowing from
693 WE1 to WE2 ($I = I_1$) with the expressions:

$$694 \qquad R_{eq,1}(t) = \frac{V(t)-\bar{V}_{eq,1}}{I(t)} \quad \text{and} \quad R_{eq,2}(t) = \frac{\bar{V}_{eq,2}-V(t)}{I(t)} \qquad [33]$$

695 from which the resistances, $R_{a,i}(t)$ and $R_{c,i}(t)$, can be derived with Eqs.
696 29 and 31:

$$697 \qquad R_{a,i}(t) = R_{eq,i}(t) \frac{E_{c,i}-E_{a,i}}{E_{c,i}-V(t)+R_{eq,i}(t)I_i(t)} \qquad [34]$$

$$698 \qquad R_{c,i}(t) = R_{eq,i}(t) \frac{E_{c,i}-E_{a,i}}{V(t)-E_{a,i}-R_{eq,i}(t)I_i(t)} \qquad [35]$$

699 By using a specific arrangement allowing coupling-decoupling EN
700 measurements on dissimilar electrodes, the authors could estimate the time
701 evolution of the anodic and cathodic resistances for AA2024T3 and
702 aluminum coupled electrodes in NaCl solution and validate their
theoretical approach by comparing the values of these resistances with the low-frequency limit of the impedance modulus measured by EIS.

**Practical applications of EC for EN analysis.**—Equivalent circuit approaches have been successfully applied in many corrosion systems with experimental results correlating well with theoretical predictions (see the results in Table IV).<sup>29,70,85,100,118,125,126,129</sup> One significant advantage for EC1 to EC4 in Table III, as proposed by Huet et al., is that a single EC can describe the experimental setup using two WEs connected through a ZRA.

Nothing is a priori assumed in the impedances $Z_1$ and $Z_2$, which are complex impedances including the double layer capacitance, and in the origin of the noise sources (pits, bubbles, cracks...). In contrast, EC5 needs further experiments to validate the EC design, in particular when the coupling of the WEs introduces a change in their corrosion potentials. This approach needs also to be extended to systems for which the impedance cannot be reduced to a simple resistance.

**Shot noise model.**—Shot noise refers to the fluctuations in current arising from the variations in time of the number of electrical charge carriers. It was applied to corrosion systems in recognition of the fact that the current can be considered as a sum of independent bursts of charge involving 1 to 4 electrons in an elementary electrochemical reaction, or a much larger number of electrons in transients, for example due to metastable pitting.<sup>15,31,44,77,98,100</sup>
The shot noise model derived by Cottis et al. for two identical electrodes connected through a ZRA and a noiseless RE is based on several assumptions:¹⁵,³¹,⁴⁴,⁹⁸

(1) The WEs undergo an anodic process with relatively large bursts of charge of short duration and independent of each other. In that case, the current on each WE follows a Poisson process, with an average value:

$$I_{\text{corr}} = qf_n \qquad [36]$$

and a PSD:

$$\Psi_{i_1} = \Psi_{i_2} = 2qI_{\text{corr}} = 2f_nq^2 \qquad [37]$$

where $q$ is the charge in each corrosion event (assumed to be the same for each event), $f_n$ is the mean emission frequency of corrosion events, and $I_{corr}$ is the corrosion current ($I_{corr}$ and $f_n$ are assumed to be the same for each WE). As a consequence of the short duration of the events, which in practice implies working at low sampling frequencies, the PSD is frequency independent in the measured frequency range. For events of longer duration, the PSD values considered in Eq. 37 are the low-frequency limit of the PSDs.

(2) The cathodic process on the WEs is noise-free, as for example oxygen reduction.

(3) Both WEs have the same impedance, which, at low frequency, is equal to the polarization resistance, $R_p$. The low-frequency limit of the
noise impedance, $Z_n(f)$, is then also equal to $R_p$ (Eq. 19).

Under these hypotheses, the PSDs of the ECN and EPN of the coupled electrodes can be expressed as, according to Eqs. 15 and 16 when neglecting the solution resistance:

$$ \Psi_I = qI_{\text{corr}} \quad \text{and} \quad \Psi_V = |Z|^2 q I_{\text{corr}} = \Psi_I R_p^2 \quad [38] $$

The three parameters of interest in the shot noise model, $I_{\text{corr}}$, $q$, and $f_n$-can then be estimated from Eqs. 36, 38 and from the Stern-Geary equation:

$$ I_{\text{corr}} = \frac{B}{R_p} = \frac{B}{Z_n} = B \sqrt{\frac{\psi_I}{\psi_V}} \quad [39] $$

$$ q = \frac{\sqrt{\psi_V \psi_I}}{B} \quad [40] $$

$$ f_n = \frac{I_{\text{corr}}}{q} = \frac{B^2}{\psi_V} \quad [41] $$

The three parameters are directly given as a function of the PSDs $\psi_V$ and $\psi_I$. It has also been proposed to use the SDs of EPN and ECN, which are directly calculated in the time domain, by replacing $\psi_V$ with $S_V^2/b$ and $\psi_I$ with $S_I^2/b$, where $b$ is the measurement frequency bandwidth. However, this is strictly valid under the assumption that the PSDs are frequency independent in the whole frequency bandwidth. This requires that the duration of each transient is short enough so that it can be considered as a Dirac delta function over the frequency range considered, which implies that the transient lasts for less than one period of the highest frequency included in the measurement.
The shot noise model is based on several assumptions that may not always be valid.⁹⁸ First, the PSDs are assumed to be flat at low frequency while the measured power spectra often show a 1/f behavior at the lowest analyzed frequencies. Second, $R_n$ is assumed to be equal to $R_p$, which is not true for some corrosion systems such as stainless steel or coated electrodes, as explained above.³⁰,¹²⁹ Third, the shot noise model cannot apply when the noise is produced by both anodic and cathodic reactions, such as when bubble evolution occurs in addition to the anodic process. Finally, all transients are expected to be uncorrelated and to have the same shape and amplitude, which is not observable on ECN time records showing transients as for metastable pitting.

## Conclusions and future work

The different electrode systems used in EN measurements in the corrosion field have been reviewed, most of them nowadays involving two symmetrical or asymmetrical electrodes connected through a ZRA. The impact of the electrode surface area on the ECN and EPN, which is a parameter of primary importance, has been considered. After a brief review of the various origins of the EN sources, the theoretical models used for analyzing the noise were presented.

Electrochemical kinetics models were introduced to explain the correlation between the noise resistance and the polarization resistance for
a symmetrical or asymmetrical corroding system. However, they have limitations, the most important being the assumption that the anodic and cathodic reactions are under activation control. Equivalent circuit approaches with potential or current noise sources or with fluctuating resistances have also been proposed to analyze EN data. While the latter approach is limited to corroding electrodes the impedance of which are reduced to simple resistance, the former approach does not presuppose anything on the impedance of the electrodes and on the origin of the noise sources. It is possible with this model to specify the conditions in which the noise resistance is or is not equal to the polarization resistance. Finally, the shot noise model was presented: under some assumptions (flat PSD at low frequency, $R_n$ equal to $R_p$, single anodic noise source), the mean occurrence frequency of corrosion events and the mean electrical charge involved in the transients can be calculated to discriminate between uniform corrosion and localized corrosion.

The existing theoretical models and mathematical methods need to be optimized as they suffer from serious limitations, as mentioned above, for studying important topics, such as the impact of electrode area on EN data, the shape and time duration of transients in metastable pitting, or the quantification of the extent of localized corrosion (crack length, pit depth, pit number). New developments are also necessary to better interpret the shape and amplitude of the EN power spectral densities, or to analyze EN
812 data produced by simultaneous processes occurring simultaneously on the
813 electrodes, such as pitting corrosion, stress corrosion cracking, bubble
814 formation, etc. Coupling EN measurements with other techniques giving
815 access to other quantities (i.e. video recordings, microscopy images,
816 acoustic emission) or simultaneous measurements of EN with other
817 fluctuating quantities (i.e. electrolyte resistance, double layer
818 capacitance...) would be of great help for improving the models.

819

820 Acknowledgements

821 This paper is dedicated to Prof. Shizhe Song on the occasion of her
822 eightieth birthday in 2020. The The authors acknowledge the National
823 Natural Science Foundation of China (No. 51701140). Thanks are
824 extended to Prof. Douglas Mills and Ruth Bingham from University of
825 Northampton (UK), Andreas Heyn from Otto-von-Guericke-Universität
826 Magdeburg (Germany), Jaber Neshati from Research Institute of
827 Petroleum Industry (Tehran, Iran), Ricardo Fabricio Escobar Jiménez from
828 Tecnológico Nacional de México (Mexico) and Tianshu Li from Ohio State
829 University (USA) for providing useful comments.

830
831 ORCID

832 Da-Hai Xia https://orcid.org/0000-0003-2336-8564

833 Yashar Behnamian https://orcid.org/0000-0002-7313-4021

834 Y. Frank Cheng https://orcid.org/0000-0002-3379-1828

835 Jing-Li Luo https://orcid.org/0000-0002-2465-7280

836 François Huet https://orcid.org/0000-0001-6176-6331

837
838 References

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840 2. V.A. Tyagai, "Noise in electrochemical systems." *Elektrokhimiya*, **10**, 3 (1974).

842 3. V. Tyagai and N.J.E. Luk'yanchikova, "Equilibrium fluctuations in electrochemical processes." *Elektrokhimiya*, **3**, 316 (1967).

844 4. W.P. Iverson, "Transient voltage changes produced in corroding metals and alloys." *J. Electrochem. Soc.*, **115**, 617 (1968).

846 5. H.S. Klapper, "Using electrochemical noise to elucidate the mechanisms involved in localized corrosion-A review." Paper 11236 presented at *Corrosion 2018*, NACE, 2018.

848 6. I. Obot, I.B. Onyeachu, A. Zeino, and S. Umoren, "Electrochemical noise (EN) technique: review of recent practical applications to corrosion electrochemistry research." *J. Adhes. Sci. Technol.*, **33**, 1453 (2019).

851 7. D. Eden, K. Hladky, D. John, and J. Dawson, "Simultaneous monitoring of potential and current noise signals from corroding electrodes." Paper 274 presented at Corrosion 1986, NACE, 1986.

853 8. J.F. Chen and W.F. Bogaerts, "Electrochemical emission spectroscopy for monitoring uniform and localized corrosion." *Corrosion*, **52**, 753 (1996).

855 9. Y. Tan, S. Bailey, and B. Kinsella, "The monitoring of the formation and destruction of corrosion inhibitor films using electrochemical noise analysis (ENA)." *Corros. Sci.*, **38**, 1681 (1996).

857 10. P.Q. Wu, Z.L. Quan, and J.P. Celis, "On-line corrosion and corrosion-wear monitoring using a modified electrochemical noise technique." *Mater. Corros.*, **56**, 379 (2005).

859 11. J. Kovac, C. Alaux, T.J. Marrow, E. Govekar, and A. Legat, "Correlations of electrochemical noise, acoustic emission and complementary monitoring techniques during intergranular stress-corrosion cracking of austenitic stainless steel." *Corros. Sci.*, **52**, 2015 (2010).

862 12. M. Breimesser, S. Ritter, H.-P. Seifert, T. Suter, and S. Virtanen, "Application of electrochemical noise to monitor stress corrosion cracking of stainless steel in tetrathionate solution under constant load." *Corros. Sci.*, **63**, 129 (2012).

865 13. X. Jiang, S. Nesic, F. Huet, B. Kinsella, B. Brown, and D. Young, "Selection of electrode area for electrochemical noise measurements to monitor localized CO₂ corrosion." *J. Electrochem. Soc.*, **159**, C283 (2012).

868 14. J. Goellner, A. Burkert, A. Heyn, and J. Hickling, "Using electrochemical noise to detect corrosion: Evaluation of a round-robin experiment." *Corrosion*, **55**, 476 (1999).

870 15. R.A. Cottis, M.A.A. Al-Awadhi, H. Al-Mazeedi, and S. Turgoose, "Measures for the detection of localized corrosion with electrochemical noise." *Electrochim. Acta*, **46**, 3665 (2001).

872 16. J. Kovač, M. Leban, and A. Legat, "Detection of SCC on prestressing steel wire by the simultaneous use of electrochemical noise and acoustic emission measurements." *Electrochim. Acta*, **52**, 7607 (2007).

875 17. C. Chandrasatheesh, J. Jayapriya, R.P. George, and U. Kamachi Mudali, "Detection and analysis of microbiologically influenced corrosion of 316L stainless steel with electrochemical noise technique." *Eng. Failure Anal.*, **42**, 133 (2014).

878 18. A.M. Homborg, C.F. Leon Morales, T. Tinga, J.H.W. de Wit, and J.M.C. Mol, "Detection of microbiologically influenced corrosion by electrochemical noise transients." *Electrochim. Acta*, **136**, 223 (2014).
<table>
  <tr>
    <td>881</td>
    <td>19.</td>
    <td>T. Haruna, Y. Morikawa, S. Fujimoto, and T. Shibata, "Electrochemical noise analysis for estimation of corrosion rate of carbon steel in bicarbonate solution." Corros. Sci., 45, 2093 (2003).</td>
  </tr>
  <tr>
    <td>882</td>
    <td>20.</td>
    <td>K. Tachibana, K. Miya, K. Furuya, and G. Okamoto, "Changes in the power spectral density of noise current on type 304 stainless steels during the long time passivation in sulfuric acid solutions." Corros. Sci., 31, 527 (1990).</td>
  </tr>
  <tr>
    <td>883</td>
    <td>21.</td>
    <td>T. Shibata, "Stochastic studies of passivity breakdown." Corros. Sci., 31, 413 (1990).</td>
  </tr>
  <tr>
    <td>884</td>
    <td>22.</td>
    <td>K. Sasaki, P. Levy, and H. Isaacs, "Electrochemical noise during pitting corrosion of aluminum in chloride environments." Electrochem. Solid-State Lett., 5, B25 (2002).</td>
  </tr>
  <tr>
    <td>885</td>
    <td>23.</td>
    <td>J. Wharton, B. Mellor, R. Wood, and C. Smith, "Crevice corrosion studies using electrochemical noise measurements and a scanning electrode technique." J. Electrochem. Soc., 147, 3294 (2000).</td>
  </tr>
  <tr>
    <td>886</td>
    <td>24.</td>
    <td>S. Matsch and H. Boehni, "Electrochemical investigations of pitting events at different temperatures by current transients analysis", in Pits and Pores II: Formation, Properties, and Significance for Advanced Materials, P. Schmuki, D.J. Lockwood, Y.H. Ogata, H.S. Isaacs, Editors, Proceedings Volume 2000-25, p. 70, The Electrochemical Society (2001).</td>
  </tr>
  <tr>
    <td>887</td>
    <td>25.</td>
    <td>J. Goellner, A. Burkert, A. Heyn, E. Boese, O. Ezers' ka, and J. Hickling, "State-of-the-art of corrosion testing by using electrochemical noise measurements." Mater. Sci., 37, 509 (2001).</td>
  </tr>
  <tr>
    <td>888</td>
    <td>26.</td>
    <td>T. Sourisseau, B. Baroux, and P.R. Roberge, "Electrochemical noise for the quantification of stainless steel resistances to pitting corrosion." in Corrosion and Corrosion Control in Saltwater Environments, Eds. D.A. Shifler, P.M. Natishan, T. Tsuru, S. Ito, Proceedings Volume 99-26, p. 286, The Electrochemical Society (2000).</td>
  </tr>
  <tr>
    <td>889</td>
    <td>27.</td>
    <td>A. Bautista, A. Vergara, J.V. Davila, L. Mariaca, and J.A. Gonzalez, "Comparison of electrochemical noise with impedance and polarization resistance techniques in the steel/concrete system." Rev. Metal., 33, 113 (1997).</td>
  </tr>
  <tr>
    <td>890</td>
    <td>28.</td>
    <td>L. Mariaca, A. Bautista, P. Rodriguez, and J.A. Gonzalez, "Use of electrochemical noise for studying the rate of corrosion of reinforcements embedded in concrete." Mater. Struct., 30, 613 (1997).</td>
  </tr>
  <tr>
    <td>891</td>
    <td>29.</td>
    <td>A. Bautista and F. Huet, "Noise resistance applied to corrosion measurements. IV - Asymmetric coated electrodes." J. Electrochem. Soc., 146, 1730 (1999).</td>
  </tr>
  <tr>
    <td>892</td>
    <td>30.</td>
    <td>A. Aballe, A. Bautista, U. Bertocci, and F. Huet, "Measurement of the noise resistance for corrosion applications." Corrosion, 57, 35 (2001).</td>
  </tr>
  <tr>
    <td>893</td>
    <td>31.</td>
    <td>R.A. Cottis, "Interpretation of electrochemical noise data." Corrosion, 57, 265 (2001).</td>
  </tr>
  <tr>
    <td>894</td>
    <td>32.</td>
    <td>A.M. Homborg, T. Tinga, E.P.M. van Westing, X. Zhang, G.M. Ferrari, J.H.W. de Wit, and J.M.C. Mol, "A critical appraisal of the interpretation of electrochemical noise for corrosion studies." Corrosion, 70, 971 (2014).</td>
  </tr>
  <tr>
    <td>895</td>
    <td>33.</td>
    <td>C. Ma, Z. Wang, Y. Behnamian, Z. Gao, Z. Wu, Z. Qin, and D.-H. Xia, "Measuring atmospheric corrosion with electrochemical noise: A review of contemporary methods." Measurement, 138, 54 (2019).</td>
  </tr>
  <tr>
    <td>896</td>
    <td>34.</td>
    <td>D.H. Xia, S.Z. Song, and Y. Behnamian, "Detection of corrosion degradation using electrochemical noise (EN): review of signal processing methods for identifying corrosion forms." Corros. Eng., Sci. Technol., 51, 527 (2016).</td>
  </tr>
  <tr>
    <td>897</td>
    <td>35.</td>
    <td>D.-H. Xia and Y. Behnamian, "Electrochemical noise: a review of experimental setup, instrumentation and DC removal." Russ. J. Electrochem., 51, 593 (2015).</td>
  </tr>
  <tr>
    <td>898</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>899</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>900</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>901</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>902</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>903</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>904</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>905</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>906</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>907</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>908</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>909</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>910</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>911</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>912</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>913</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>914</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>915</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>916</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>917</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>918</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>919</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>920</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>921</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>922</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>923</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>924</td>
    <td></td>
    <td></td>
  </tr>
</table>
<table>
  <tr>
    <td>925</td>
    <td>36.</td>
    <td>O.J. Ramos-Negrón, J.H. Arellano-Pérez, R.F. Escobar-Jiménez, J.F. Gómez-Aguilar, and D. Granados-Lieberman, "Electrochemical noise analysis to identify the corrosion type using the Stockwell transform and the Shannon energy." J. Electroanal. Chem., 836, 50 (2019).</td>
  </tr>
  <tr>
    <td>926</td>
    <td>37.</td>
    <td>J. Arellano-Pérez, R. Escobar-Jiménez, D. Granados-Lieberman, J. Gómez-Aguilar, J. Uruchurtu-Chavarín, and V. Alvarado-Martínez, "Electrochemical noise signals evaluation to classify the type of corrosion using synchrosqueezing transform." J. Electroanal. Chem., 848, 113249 (2019).</td>
  </tr>
  <tr>
    <td>927</td>
    <td>38.</td>
    <td>S.Z. Song, W.X. Zhao, J.H. Wang, J. Li, Z.M. Gao, and D.H. Xia, "Field corrosion detection of nuclear materials using electrochemical noise technique." Prot. Met. Phys. Chem. Surf., 54, 340 (2018).</td>
  </tr>
  <tr>
    <td>928</td>
    <td>39.</td>
    <td>E.A. Astafev, "Software and instrumentational methods of enhancing the resolution in electrochemical noise measurements." Russ. J. Electrochem., 54, 1031 (2018).</td>
  </tr>
  <tr>
    <td>929</td>
    <td>40.</td>
    <td>E. Astafev, "The instrument for electrochemical noise measurement of chemical power sources." Rev. Sci. Instrum., 90, 025104 (2019).</td>
  </tr>
  <tr>
    <td>930</td>
    <td>41.</td>
    <td>E. Astafev and A. Ukshe, "Peculiarities of hardware for electrochemical noise measurement in chemical power sources." IEEE Trans. Instrum. Meas., 68, 4412 (2019).</td>
  </tr>
  <tr>
    <td>931</td>
    <td>42.</td>
    <td>U. Bertocci, F. Huet, B. Jaoul, and P. Rousseau, "Frequency analysis of transients in electrochemical noise: mathematical relationships and computer simulations." Corrosion, 56, 675 (2000).</td>
  </tr>
  <tr>
    <td>932</td>
    <td>43.</td>
    <td>R. Cottis, A. Homborg, and J. Mol, "The relationship between spectral and wavelet techniques for noise analysis." Electrochim. Acta, 202, 277 (2016).</td>
  </tr>
  <tr>
    <td>933</td>
    <td>44.</td>
    <td>J.M. Sanchez-Amaya, R.A. Cottis, and F.J. Botana, "Shot noise and statistical parameters for the estimation of corrosion mechanisms." Corros. Sci., 47, 3280 (2005).</td>
  </tr>
  <tr>
    <td>934</td>
    <td>45.</td>
    <td>G. Monrrabal, F. Huet, and A. Bautista, "Electrochemical noise measurements on stainless steel using a gelled electrolyte." Corros. Sci., 148, 48 (2019).</td>
  </tr>
  <tr>
    <td>935</td>
    <td>46.</td>
    <td>J.M. Silva, R.P. Nogueira, L. de Miranda, and F. Huet, "Hydrogen absorption estimation on Pd electrodes from electrochemical noise measurements in single-compartment cells." J. Electrochem. Soc., 148, E241 (2001).</td>
  </tr>
  <tr>
    <td>936</td>
    <td>47.</td>
    <td>D.-H. Xia, Y. Song, S. Song, Y. Behnamian, L. Xu, Z. Wu, Z. Qin, Z. Gao, and W. Hu, "Identifying defect levels in organic coatings with electrochemical noise (EN) measured in Single Cell (SC) mode." Prog. Org. Coat., 126, 53 (2019).</td>
  </tr>
  <tr>
    <td>937</td>
    <td>48.</td>
    <td>D.-H. Xia, C. Ma, S. Song, L. Ma, J. Wang, Z. Gao, C. Zhong, and W. Hu, "Assessing atmospheric corrosion of metals by a novel electrochemical sensor combining with a thin insulating net using electrochemical noise technique." Sens. Actuators, B, 252, 353 (2017).</td>
  </tr>
  <tr>
    <td>938</td>
    <td>49.</td>
    <td>D. Xia, S. Song, J. Li, and W. Jin, "On-line monitoring atmospheric corrosion of metal materials by using a novel corrosion electrochemical sensor." Corros. Sci. Prot. Technol., 29, 581 (2017).</td>
  </tr>
  <tr>
    <td>939</td>
    <td>50.</td>
    <td>D.-H. Xia, J. Wang, Z. Wu, Z. Qin, L. Xu, W. Hu, Y. Behnamian, and J.-L. Luo, "Sensing corrosion within an artificial defect in organic coating using SECM." Sens. Actuators, B, 280, 235 (2019).</td>
  </tr>
  <tr>
    <td>940</td>
    <td>51.</td>
    <td>U. Bertocci, "Applications of a low-noise potentiostat in electrochemical measurements." J. Electrochem. Soc., 127, 1931 (1980).</td>
  </tr>
  <tr>
    <td>941</td>
    <td>52.</td>
    <td>K. Hladky and J.L. Dawson, "The measurement of corrosion using electrochemical 1/f noise." Corros. Sci., 22, 231 (1982).</td>
  </tr>
  <tr>
    <td>942</td>
    <td>53.</td>
    <td>J.C. Uruchurtu and J.L. Dawson, "Noise analysis of pure aluminum under different pitting</td>
  </tr>
  <tr>
    <td>943</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>944</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>945</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>946</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>947</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>948</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>949</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>950</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>951</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>952</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>953</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>954</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>955</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>956</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>957</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>958</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>959</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>960</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>961</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>962</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>963</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>964</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>965</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>966</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>967</td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>968</td>
    <td></td>
    <td></td>
  </tr>
</table>
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971 55. P. Roberge, “Analysis of spontaneous electrochemical noise for corrosion studies.” *J. Appl. Electrochem.*, **23**, 1223 (1993).

972 56. R. Oltra, C. Gabrielli, F. Huet, and M. Keddam, “Electrochemical investigation of locally depassivated iron. A comparison of various techniques.” *Electrochim. Acta*, **31**, 1505 (1986).

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976 60. H.S. Klapper, J. Goellner, A. Burkert, and A. Heyn, “Environmental factors affecting pitting corrosion of type 304 stainless steel investigated by electrochemical noise measurements under potentiostatic control.” *Corros. Sci.*, **75**, 239 (2013).

977 61. L. Guan, B. Zhang, X.P. Yong, J.Q. Wang, E.H. Han, and W. Ke, “Effects of cyclic stress on the metastable pitting characteristic for 304 stainless steel under potentiostatic polarization.” *Corros. Sci.*, **93**, 80 (2015).

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979 63. A. Benzaid, F. Huet, M. Jerome, F. Wenger, C. Gabrielli, and J. Galland, “Electrochemical noise analysis of cathodically polarised AISI 4140 steel. I. Characterisation of hydrogen evolution on vertical unstressed electrodes.” *Electrochim. Acta*, **47**, 4315 (2002).

980 64. A. Benzaid, F. Huet, M. Jerome, F. Wenger, C. Gabrielli, and J. Galland, “Electrochemical noise analysis of cathodically polarised AISI 4140 steel. II. Identification of potential fluctuation sources for unstressed electrodes.” *Electrochim. Acta*, **47**, 4325 (2002).

981 65. S.S. Jamali and D.J. Mills, “A critical review of electrochemical noise measurement as a tool for evaluation of organic coatings.” *Prog. Org. Coat.*, **95**, 26 (2016).

982 66. D. Mills, P. Picton, and L. Mularczyk, “Developments in the electrochemical noise method (ENM) to make it more practical for assessment of anti-corrosive coatings.” *Electrochim. Acta*, **124**, 199 (2014).

983 67. S.S. Jamali, D.J. Mills, and J.M. Sykes, “Measuring electrochemical noise of a single working electrode for assessing corrosion resistance of polymer coated metals.” *Prog. Org. Coat.*, **77**, 733 (2014).

984 68. H. Zheng, J. Liang, Z. Qin, S. Song, L. Xu, Z. Gao, W. Hu, and D.-H. Xia, “Identifying defect size in organic coatings by electrochemical noise, galvanostatic step and potentiostatic step techniques.” *J. Electroanal. Chem.*, **856**, 113596 (2020).

985 69. I. Jevremovic and A. Erbe, “The reassigned pseudo Wigner-Ville transform in electrochemical noise analysis.” *Phys. Chem. Chem. Phys.*, **21**, 24361 (2019).

986 70. A. Bautista, U. Bertocci, and F. Huet, “Noise resistance applied to corrosion measurements. V-Influence of electrode asymmetry.” *J. Electrochem. Soc.* **148**, B412 (2001).

987 71. F. Huet and K. Ngo, “Electrochemical noise—Guidance for improving measurements and data analysis.” *Corrosion*, **75**, 1065 (2019).
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1014
1015
1016 73. R.Y. Chen, V. Trieu, H. Natter, J. Kintrup, A. Bulan, and R. Hempelmann, "Wavelet analysis of chlorine bubble evolution on electrodes with different surface morphologies." *Electrochem. Commun.*, **22**, 16 (2012).

1017
1018
1019 74. X.Q. Huang, Y. Chen, T.W. Fu, Z. Zhang, and J.Q. Zhang, "Study of tin electroplating process using electrochemical impedance and noise techniques." *J. Electrochem. Soc.*, **160**, D530 (2013).

1020
1021
1022 75. U. Bertocci, F. Huet, and R.P. Nogueira, "Use of multiple reference electrodes in electrochemical noise measurements." *Corrosion*, **59**, 629 (2003).

1023
1024 76. R.A. Cottis, "The significance of electrochemical noise measurements on asymmetric electrodes." *Electrochim. Acta*, **52**, 7585 (2007).

1025
1026 77. C. Gabrielli, F. Huet, M. Keddam, and R. Oltra, "A review of the probabilistic aspects of localized corrosion." *Corrosion*, **46**, 266 (1990).

1028 78. G. Frankel, L. Stockert, F. Hunkeler, and H. Boehni, "Metastable pitting of stainless steel." *Corrosion*, **43**, 429 (1987).

1029
1030 79. R. Lillard, "Factors influencing the transition from metastable to stable pitting in single-crystal beryllium." *J. Electrochem. Soc.*, **148**, B1 (2001).

1031
1032 80. T. Li, J. Scully, and G. Frankel, "Localized corrosion: Passive film breakdown vs pit growth stability: Part II. A model for critical pitting temperature." *J. Electrochem. Soc.*, **165**, C484 (2018).

1033
1034
1035 81. M. Bierwirth and J. Goellner, "Modeling of electrochemical noise transients." *Mater. Corros.*, **58**, 992 (2007).

1036
1037 82. H.S. Klapper, J. Goellner, and A. Heyn, "The influence of the cathodic process on the interpretation of electrochemical noise signals arising from pitting corrosion of stainless steels." *Corros. Sci.*, **52**, 1362 (2010).

1038
1039
1040 83. H.S. Klapper, B. Zajec, A. Heyn, and A. Legat, "Elucidating nucleation stages of transgranular stress corrosion cracking in austenitic stainless steel by in situ electrochemical and optical methods." *J. Electrochem. Soc.*, **166**, C3326 (2019).

1041
1042
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1044
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1046
1047 86. H.S. Isaacs and Y. Ishikawa, "Current and potential transients during localized corrosion of stainless steel." *J. Electrochem. Soc.*, **132**, 1288 (1985).

1048
1049 87. H. Isaacs and A. Davenport, "The influence of surface capacitance on the measurements of localized corrosion transients." *J. Electrochem. Soc.*, **137**, 2196 (1990).

1050
1051 88. H.S. Isaacs, "Comments on "A stochastic analysis of potential fluctuations during passive film breakdown and repair of iron": by M. Hashimoto, S. Miyajima and T. Murata *Corros. Sci.*, **33**, 885 (1992)." *Corros. Sci.*, **34**, 525 (1993).

1052
1053
1054 89. M. Hashimoto, S. Miyajima and T. Murata, "A stochastic analysis of potential fluctuations during passive film breakdown and repair of iron." *Corros. Sci.*, **33**, 885 (1992).

1055
1056
1057 90. P.C. Pistorius, "Design Aspects of Electrochemical noise measurements for uncoated metals:
<table>
  <tr>
    <td>1057</td>
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  </tr>
  <tr>
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    <td>91. I.N. Bastos, F. Huet, R.P. Nogueira, and P. Rousseau, “Influence of aliasing in time and frequency electrochemical noise measurements.” J. Electrochem. Soc., 147, 671 (2000).</td>
  </tr>
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    <td>92. R.-W. Bosch, R.A. Cottis, K. Csecs, T. Dorsch, L. Dunbar, A. Heyn, F. Huet, O. Hyökyvirta, Z. Kerner, A. Kobzova, J. Macak, R. Novotny, J. Öijerholm, J. Piippo, R. Richner, S. Ritter, J.M. Sánchez-Amaya, A. Somogyi, S. Väisänen, and W. Zhang, “Reliability of electrochemical noise measurements: Results of round-robin testing on electrochemical noise.” Electrochim. Acta, 120, 379 (2014).</td>
  </tr>
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    <td>93. F. Huet and S. Ritter, “Electrochemical noise measurements with dummy cells: evaluation of a round-robin test series.” Corrosion, 74, 1457 (2018).</td>
  </tr>
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  </tr>
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    <td>95. F. Huet, “Comment on [“Laboratory-scale identification of corrosion mechanisms by a novel pattern recognition system based on electrochemical noise measurements.” J. Electrochem. Soc., 166, C284 (2019)] J. Electrochem. Soc., 166, Y31 (2019).</td>
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  </tr>
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    <td>98. R.A. Cottis, “Sources of electrochemical noise in corroding systems.” Russ. J. Electrochem., 42, 497 (2006).</td>
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    <td>101. A. Legat, “Influence of electrolyte movement on measured electrochemical noise.” Corrosion, 56, 1086 (2000).</td>
  </tr>
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    <td>102. K. Sasaki and H.S. Isaacs, “Origins of electrochemical noise during pitting corrosion of aluminum.” J. Electrochem. Soc., 151, B124 (2004).</td>
  </tr>
  <tr>
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    <td>103. C. Gabrielli, F. Huet, and R.P. Nogueira, “Electrochemical noise measurements of coalescence and gas-oscillator phenomena on gas-evolving electrodes.” J. Electrochem. Soc., 149, E71 (2002).</td>
  </tr>
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    <td>1071</td>
    <td>104. Y. Puget, K. Trethewey, and R.J.K. Wood, “Electrochemical noise analysis of polyurethane-coated steel subjected to erosion-corrosion.” Wear, 233-235, 552 (1999).</td>
  </tr>
  <tr>
    <td>1072</td>
    <td>105. R.J.K. Wood, J.A. Wharton, A.J. Speyer, and K.S. Tan, “Investigation of erosion-corrosion processes using electrochemical noise measurements.” Tribol. Int., 35, 631 (2002).</td>
  </tr>
  <tr>
    <td>1073</td>
    <td>106. A.K. Basak, P. Matteazzi, M. Vardavoulias, and J.P. Celis, “Corrosion-wear behaviour of thermal sprayed nanostructured FeCu/WC-Co coatings.” Wear, 261, 1042 (2006).</td>
  </tr>
  <tr>
    <td>1074</td>
    <td>107. M.R. Thakare, J.A. Wharton, R.J.K. Wood, and C. Menger, “Investigation of micro-scale abrasion-corrosion of WC-based sintered hardmetal and sprayed coating using in situ electrochemical current-noise measurements.” Wear, 267, 1967 (2009).</td>
  </tr>
  <tr>
    <td>1075</td>
    <td>108. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1076</td>
    <td>109. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1077</td>
    <td>110. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1078</td>
    <td>111. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1079</td>
    <td>112. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1080</td>
    <td>113. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1081</td>
    <td>114. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1082</td>
    <td>115. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1083</td>
    <td>116. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1084</td>
    <td>117. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1085</td>
    <td>118. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1086</td>
    <td>119. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1087</td>
    <td>120. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1088</td>
    <td>121. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1089</td>
    <td>122. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1090</td>
    <td>123. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1091</td>
    <td>124. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1092</td>
    <td>125. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1093</td>
    <td>126. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1094</td>
    <td>127. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1095</td>
    <td>128. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1096</td>
    <td>129. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1097</td>
    <td>130. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1098</td>
    <td>131. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1099</td>
    <td>132. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
  <tr>
    <td>1100</td>
    <td>133. R.J.K. Wood, D. Sun, M.R. Thakare, A.D. Rozas, and J.A. Wharton, “Interpretation of electrochemical measurements made during micro-scale abrasion-corrosion.” Tribol. Int., 43, 1100 (2009).</td>
  </tr>
</table>
1101 1218 (2010).

1102 109. Q. Hu, Y.B. Qiu, X.P. Guo, and J.Y. Huang, "Crevice corrosion of Q235 carbon steels in a solution of NaHCO3 and NaCl." *Corros. Sci.*, **52**, 1205 (2010).

1104 110. Q. Hu, G. Zhang, Y. Qiu, and X. Guo, "The crevice corrosion behaviour of stainless steel in sodium chloride solution." *Corros. Sci.*, **53**, 4065 (2011).

1106 111. M.G. Pujar, N. Parvathavarthini, R.K. Dayal, and S. Thirunavukkarasu, "Assessment of intergranular corrosion (IGC) in 316(N) stainless steel using electrochemical noise (EN) technique." *Corros. Sci.*, **51**, 1707 (2009).

1109 112. S. Ritter and H.P. Seifert, "Effect of corrosion potential on the corrosion fatigue crack growth behaviour of low-alloy steels in high-temperature water." *J. Nucl. Mater.*, **375**, 72 (2008).

1111 113. C.A. Loto and R.A. Cottis, "Electrochemical noise generation during stress-corrosion cracking of alpha-brass." *Corrosion*, **43**, 499 (1987).

1113 114. J.F. Chen and W.F. Bogaerts, "The physical meaning of noise resistance." *Corros. Sci.*, **37**, 1839 (1995).

1115 115. J. Bockris and A. Reddy, *Modern electrochemistry. 2nd ed.*, Kluwer Academic/Plenum Publishers, New York (2000).

1117 116. M. Stern and A.L. Geary, "Electrochemical polarization I. A theoretical analysis of the shape of polarization curves." *J. Electrochem. Soc.*, **104**, 56 (1957).

1119 117. G. Gusmano, G. Montesperelli, S. Pacetti, A. Petitti, and A. D'Amico, "Electrochemical noise resistance as a tool for corrosion rate prediction." *Corrosion*, **53**, 860 (1997).

1121 118. F. Mansfeld, Z. Sun, C.H. Hsu, and A. Nagiub, "Concerning trend removal in electrochemical noise measurements." *Corros. Sci.*, **43**, 341 (2001).

1123 119. S.Y. Arman, R. Naderi, and B.P. Markhali, "Effect of DC trend removal and window functioning methods on correlation between electrochemical noise parameters and EIS data of stainless steel in an inhibited acidic solution." *RSC Adv.*, **4**, 39045 (2014).

1126 120. U. Bertocci, F. Huet, R.P. Nogueira, and P. Rousseau, "Drift removal procedures in the analysis of electrochemical noise." *Corrosion*, **58**, 337 (2002).

1128 121. A.M. Homborg, T. Tinga, X. Zhang, E.P.M. van Westing, P.J. Oonincx, J.H.W. de Wit, and J.M.C. Mol, "Time-frequency methods for trend removal in electrochemical noise data." *Electrochim. Acta*, **70**, 199 (2012).

1131 122. W. Baeckmann and W. Schwenk, *Handbook of cathodic protection. The theory and practice of electrochemical corrosion protection techniques*, Portcullis Press, Ltd. Surrey, England (1975).

1133 123. M. Curioni, P. Skeldon, and G.E. Thompson, "Reliability of the estimation of polarization resistance of corroding electrodes by electrochemical noise analysis: Effects of electrode asymmetry." *Electrochim. Acta*, **105**, 642 (2013).

1136 124. U. Bertocci, C. Gabrielli, F. Huet, and M. Keddam, "Noise resistance applied to corrosion measurements. I-Theoretical analysis." *J. Electrochem. Soc.*, **144**, 31 (1997).

1138 125. U. Bertocci, C. Gabrielli, F. Huet, M. Keddam, and P. Rousseau, "Noise resistance applied to corrosion measurements. II-Experimental tests." *J. Electrochem. Soc.*, **144**, 37 (1997).

1140 126. U. Bertocci and F. Huet, "Noise resistance applied to corrosion measurements. III- Influence of the instrumental noise on the measurements." *J. Electrochem. Soc.*, **144**, 2786 (1997).

1142 127. M. Curioni, R. Cottis, M. Di Natale, and G.E. Thompson, "Corrosion of dissimilar alloys: Electrochemical noise." *Electrochim. Acta*, **56**, 6318 (2011).

1144 128. W. Qafsaoui, F. Huet, and H. Takenouti, "Analysis of the inhibitive effect of BTAH on localized corrosion." *J. Electrochem. Soc.*, **104**, 1747 (2007).
1145 corrosion of Al 2024 from electrochemical noise measurements." *J. Electrochem. Soc.*, **156**,

1146 C67 (2009).

1147 129. F. Mansfeld and C.C. Lee, "The frequency dependence of the noise resistance for polymer -

1148 coated metals." *J. Electrochem. Soc.*, **144**, 2068 (1997).

1149
1150 Table and figures

1151 Table I. Mathematical methods and parameters used to identify

1152 corrosion forms and corrosion rates.<sup>31,32,34</sup>

<table><thead><tr><th>Mathematical methods</th><th>Name of parameters</th><th>Relation to corrosion rate?</th><th>Relation to corrosion form?</th></tr></thead><tbody><tr><td rowspan="10">Time domain analysis</td><td>Standard deviation of the EPN or ECN</td><td>Related</td><td>-</td></tr><tr><td>Skewness</td><td>-</td><td>-</td></tr><tr><td>Kurtosis</td><td>-</td><td>-</td></tr><tr><td>Pitting index</td><td>-</td><td>Related</td></tr><tr><td>Noise resistance</td><td>Strong</td><td>-</td></tr><tr><td>Weibull distribution function and pit embryo formation rate</td><td>-</td><td>Related</td></tr><tr><td>Gumbel distribution</td><td>-</td><td>Related</td></tr><tr><td>Chaos analysis</td><td>Correlation dimension<br>Largest Lyapunov exponent</td><td>-</td><td>Related</td></tr><tr><td>Recurrence quantification analysis</td><td>Recurrence plot</td><td>-</td><td>Related</td></tr><tr><td rowspan="4">Frequency domain analysis</td><td>Fractal analysis</td><td>Hausdorff exponent<br>Hurst exponent<br>Spectral power exponents</td><td>-</td><td>Related</td></tr><tr><td rowspan="2">Shot noise</td><td>Charge in each corrosion event</td><td>Strong</td><td>Related</td></tr><tr><td>Frequency of occurrence of corrosion events</td><td>Strong</td><td>Related</td></tr><tr><td rowspan="4">Time-frequency domain analysis</td><td rowspan="4">Fast Fourier transform</td><td>Low frequency plateau</td><td>Related</td><td>-</td></tr><tr><td>Plateau at high frequencies</td><td>-</td><td>-</td></tr><tr><td>Roll-off slope</td><td>-</td><td>-</td></tr><tr><td>Knee frequency or critical frequency</td><td>-</td><td>-</td></tr><tr><td></td><td>Noise impedance</td><td>Strong</td><td>-</td></tr><tr><td rowspan="6">Time-frequency domain analysis</td><td>Hilbert-Huang transform</td><td>Hilbert spectrum</td><td>-</td><td>Related</td></tr><tr><td rowspan="3">Discrete Wavelet Transform</td><td>Wavelet entropy</td><td>-</td><td>Related</td></tr><tr><td>Wavelet based fractal dimension</td><td>-</td><td>Related</td></tr><tr><td>Wavelet energy distribution</td><td>-</td><td>Related</td></tr><tr><td>Stockwell transform<sup>36</sup></td><td>Frequency amplitude analysis</td><td>Related</td><td>Related</td></tr><tr><td>Synchrosqueezing transform<sup>37</sup></td><td>Frequency amplitude analysis</td><td>-</td><td>Related</td></tr><tr><td>Stockwell transform -Shannon energy<sup>36</sup></td><td>Shannon energy levels</td><td>-</td><td>Related</td></tr><tr><td>Synchrosqueezing transform-Shannon energy<sup>37</sup></td><td>Shannon energy levels</td><td>-</td><td>Related</td></tr></tbody></table>
<table><tr><td>Reassigned</td><td>Pseudo</td><td>Wigner-</td><td>Energy spectral densities</td><td>-</td><td>Related</td></tr><tr><td>Ville Transform<sup>69</sup></td><td></td><td></td><td></td><td></td><td></td></tr></table>

1153

1154
<table><caption>Table II. Electrode systems used in EN measurement.</caption><thead><tr><th>Measurement mode</th><th>Electrode system</th><th>Electrode system no.</th><th>Requirement of the area of the electrodes</th></tr></thead><tbody><tr><td rowspan="3">Symmetrical electrode system</td><td>#1</td><td>#1</td><td>S<sub>WE1</sub>=S<sub>WE2</sub></td></tr><tr><td>#2</td><td>#2</td><td>S<sub>WE1</sub>=S<sub>WE2</sub>=S<sub>RE</sub></td></tr><tr><td>#3</td><td>#3</td><td>S<sub>WE1</sub>≠S<sub>WE2</sub></td></tr><tr><td>ZRA mode</td><td>#4</td><td>#4</td><td>S<sub>WE1</sub>≠S<sub>WE2</sub></td></tr><tr><td rowspan="3">Asymmetrical electrode system</td><td>#5</td><td>#5</td><td>S<sub>WE1</sub>≠S<sub>CE</sub></td></tr><tr><td>#6</td><td>#6</td><td>S<sub>WE1</sub>=S<sub>WE2</sub></td></tr></tbody></table>
Open circuit potential

<figure><img src="image_3.png" /></figure>

#7

No requirement

Potentiostatic mode

<figure><img src="image_4.png" /></figure>

#8

No requirement

Galvanostatic mode

<figure><img src="image_5.png" /></figure>

#9

No requirement

1156

1157

1158
<table><caption>1159 Table III. Equivalent circuits proposed to analyze EN in ZRA mode.</caption><thead><tr><th>Electrode system</th><th>Equivalent circuit</th><th>Ref</th></tr></thead><tbody><tr><td></td><td>circuit based on current sources</td><td></td></tr><tr><td>#1</td><td rowspan="4">EC1</td><td rowspan="4">EC2</td></tr><tr><td>#3</td></tr><tr><td>#5</td></tr><tr><td>#6</td></tr><tr><td></td><td>circuit based on potential sources</td><td></td></tr><tr><td>#2</td><td rowspan="4">EC3</td><td rowspan="4">EC4</td></tr><tr><td>#4</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td><td>circuit based on current sources</td><td></td></tr><tr><td>#1</td><td rowspan="4">EC5</td><td rowspan="4">EC6</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td><td>circuit based on potential sources</td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr></tbody></table>
<table><caption>Table IV. Comparison of experimental values of $R_n$, $R_p$, $Z_n(f)$, and $|Z(f)|$.a)</caption><thead><tr><th>Electrode system No.</th><th>WE1 (surface area)</th><th>WE2 (surface area)</th><th>RE</th><th>Electrolyte</th><th>Sampling frequency</th><th>trend removal method</th><th>Correlation among R<sub>n</sub>, R<sub>p</sub>, Z<sub>n</sub>(f), or |Z(f)|</th><th>Ref.</th></tr></thead><tbody><tr><td>#1</td><td>Mild steel (3.02 cm<sup>2</sup>)</td><td>Mild steel (3.02 cm<sup>2</sup>)</td><td>Ag/AgCl electrode</td><td>640 ml of 3% NaCl brine</td><td>2Hz</td><td>moving average value</td><td>R<sub>n</sub> ≈ R<sub>p</sub></td><td>9</td></tr><tr><td>#1</td><td>Carbon steel</td><td>Carbon steel</td><td>SCE**</td><td>0.4 M Na<sub>2</sub>SO<sub>4</sub><br>0.1 M H<sub>2</sub>SO<sub>4</sub></td><td>50Hz<br>1 kHz</td><td>-</td><td>R<sub>n</sub> > R<sub>p</sub></td><td>90</td></tr><tr><td>#2</td><td>Carbon steel (3 cm<sup>2</sup>)</td><td>Carbon steel (3 cm<sup>2</sup>)</td><td>Carbon steel (3 cm<sup>2</sup>)</td><td>Na<sub>3</sub>PO<sub>4</sub> + NaCl, NaClO<sub>4</sub></td><td>2 Hz, 8 Hz, or 32 Hz</td><td>Linear method</td><td>R<sub>n</sub> is slightly higher than R<sub>p</sub> at a sampling frequency of 32 Hz; R<sub>n</sub> > R<sub>p</sub> at low sampling frequency (2 Hz and 8 Hz)</td><td>117</td></tr><tr><td>#1</td><td>Fe (0.20 cm<sup>2</sup>)</td><td>Fe (0.20 cm<sup>2</sup>)</td><td>SSE***</td><td>(i) in 1 M Na<sub>2</sub>SO<sub>4</sub>, pH 3;<br>(ii) in 1 M Na<sub>2</sub>SO<sub>4</sub>, pH 4</td><td>0.2 Hz to 200 Hz</td><td>polynomial fitting</td><td>Z<sub>n</sub>(f) ≈ |Z(f)|</td><td>125</td></tr><tr><td>#1</td><td>Al (5 cm<sup>2</sup>)</td><td>Al (5 cm<sup>2</sup>)</td><td>SCE**</td><td>1 M KCl.</td><td>0.2 Hz to 200 Hz</td><td>polynomial fitting</td><td>Z<sub>n</sub>(f) ≈ |Z(f)|</td><td>125</td></tr><tr><td>#3</td><td>Al (5 cm<sup>2</sup>)</td><td>Al (0.2 cm<sup>2</sup>)</td><td>SCE**</td><td>1 M KCl.</td><td>0.2 Hz to 200 Hz</td><td>polynomial fitting</td><td>|Z(f)|<sub>5cm<sup>2</sup></sub> < Z<sub>n</sub>(f) < |Z(f)|<sub>0.2cm<sup>2</sup></sub></td><td>125</td></tr><tr><td>#2</td><td>Fe (0.20 cm<sup>2</sup>)</td><td>Fe (0.20 cm<sup>2</sup>)</td><td>Fe (0.20 cm<sup>2</sup>)</td><td>(i) in 1 M Na<sub>2</sub>SO<sub>4</sub>, pH 3;<br>(ii) in 1 M Na<sub>2</sub>SO<sub>4</sub>, pH 4</td><td>0.2 Hz to 200 Hz</td><td>polynomial fitting</td><td>Z<sub>n</sub>(f) ≈ 2|Z(f)|</td><td>125</td></tr></tbody></table>

1163
<table><tr><td>#1</td><td>Mild steel<br>(4.5 cm²)</td><td>Mild steel<br>(4.5 cm²)</td><td>SCE</td><td>0.5 M NaCl (open to air)</td><td>2 Hz</td><td>Linear method</td><td>Z<sub>n</sub> ≈ |Z(f)|</td><td>118</td></tr><tr><td>#1</td><td>Pure aluminum<br>99.999% or<br>2024 T3<br>aluminum alloy<br>(0.2 cm²)</td><td>Pure aluminum<br>99.999% or<br>2024 T3<br>aluminum alloy<br>(0.2 cm²)</td><td>SCE</td><td>0.1 M Na₂SO₄ + 0.5 M<br>NaCl with and without<br>0.01 M BTAH.</td><td>100 and 10 Hz</td><td>polynomial<br>fitting</td><td>Z<sub>n</sub> ≈ |Z(f)|</td><td>128</td></tr><tr><td>#1</td><td>X-65 mild steel<br>(11.6 or 1 cm²)</td><td>X-65 mild steel<br>(11.6 or 1 cm²)</td><td>SCE</td><td>1% NaCl electrolyte<br>by bubbling CO₂, 80°C</td><td>1 Hz</td><td>polynomial<br>fitting</td><td>R<sub>n</sub> ≈ R<sub>p</sub> for small WEs,<br>R<sub>n</sub> > R<sub>p</sub> for large WEs</td><td>13</td></tr></table>

a) Note that an initial symmetric electrode system may become asymmetric, especially for electrodes undergoing localized corrosion. b) SCE: saturated calomel electrode. c) SSE: saturated mercury sulfate electrode.
1166

<figure><img src="image_6.png" /></figure>

1167

1168 **Figure 1.** Schematic diagram of electrode system #1 showing the anodic,

1169 $I_a(t)$ and $I'_a(t)$, and cathodic, $I_c(t)$ and $I'_c(t)$, currents on each WE.

1170