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

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


Introduction 29
Since Tyagai 1-3 and Iverson 4 documented the noise generated in 30 electrochemical systems in the 1960s to 1970s, the use of electrochemical 31 noise (EN) to detect metal dissolution has become widespread in corrosion 32 science.  The development of electrochemical instrumentation, 38-41 33 advanced signal processing methods, 34,[42][43][44] and inspired experimental 34 design [45][46][47][48][49][50] in recent decades has promoted EN applications. 35 Early work used single measurements of electrochemical potential 36 noise (EPN) 51-55 or electrochemical current noise (ECN) 56,57 to study the 37 nature and rate of corrosion on a single working electrode (WE). The fact 38 that the WE can be polarized at any potential under potentiostatic control 39 or at any current under galvanostatic control allows the stochastic behavior 40 of corroding electrodes to be investigated in various specific conditions, 41 such as metastable and stable pitting corrosion, 58-62 gas evolution on 42 stressed electrodes, 63,64 or corrosion protection by organic coatings. 47,65-68 . 43 From the beginning of the nineties, most of the time EPN and ECN 44 were measured synchronously on two WEs tested specifically at the 45 corrosion potential, to get more information than single measurements of 46 EPN or ECN, in particular to estimate the corrosion rate. Important 47 information to be obtained prior to EN measurement includes (1) the 48 electrode system, (2) the sensitivity limit of the measuring device, and (3)  49 the magnitude of the solution resistance of the corrosion system. 50 the other WE or CE is linked to the inverting input of the Opamp while the 96 non-inverting input is connected to ground. The current-measuring resistor 97 between the output and the inverting input of the Opamp provides a voltage 98 proportional to the coupling current at the output of the Opamp. Circuit 99 components other than the operational amplifier can be used to de-noise 100 and to increase the stability of the instrument. The current-measuring 101 resistance of the ZRA plays a key role in the measurement of ECN. 71 102 Typical current-measuring resistances in most commercial electrochemical 103 instruments have values ranging from ohms to gigaohms, giving a current 104 range from picoamps to amps. Instead of using a ZRA, sometimes a resistor 105 is intentionally added between WE1 and WE2, partially decoupling the two 106 WEs. This configuration was found to decrease the instrumentation noise, 72 107 as simplifying the current-measuring circuit ensured that no instrumental 108 noise due to active electronic components within the ZRA were fed back 109 to the corroding electrodes. This may be important in some systems, but it 110 is only one component of the instrumentation noise appearing in the final 111 measurement, and the instrument noise associated with alternative 112 measurement methods will depend strongly on the instrument design and 113 cell properties as well as the measurement configuration. 114 Three other measurement modes are used less frequently nowadays. 115 The open circuit potential (OCP) mode (electrode system #7) was the first 116 same material but different surface areas (electrode systems #3 and #4), 162 and systems with different materials in each of the WE, the CE, and the RE 163 (electrode system #5). In electrode systems #3 and #4, WE1 and WE2 have 164 different impedance values. The RE can either be a standard (true) 165 reference electrode or a pseudo-reference electrode (a corroding electrode), 166 but in the latter case, as for electrode system #2, the noise related to the 167 corrosion phenomena occurring on the pseudo RE cannot be ignored when 168 performing a quantitative analysis. In electrode system #5, the three 169 electrodes are constructed with different materials and might have different 170 surface areas and different impedance values. The CE is often made of an 171 inert material to diminish its noise level while the WE is a corroding 172 electrode studied at its corrosion potential. Moreover, the area of the CE 173 should be small enough, compared to the area of the WE, to lower the 174 galvanic current between the WE and the CE so that the WE remains at its 175 corrosion potential. 8,10 However, a CE of small area has a higher impedance, 176 which increases the asymmetry between the two electrodes. 76 A Pt 177 electrode with a small area is often used as the CE. It draws the current 178 provided by the corrosion of the WE through a reduction reaction (for 179 example, oxygen reduction in neutral solution, or hydrogen evolution in 180 acidic medium) at the corrosion potential of the WE. 181 It is important to note that the type of electrodes employed when 182 measuring EN can affect the quantitative analysis of the EN signals. 183 Attempts to provide a symmetrical electrode system where the two coupled 184 WEs have the same surface area, the same material composition, and the 185 same corrosion potential and corrosion activity are quite never completely 186 perfect, as these properties cannot be matched to perfection in different 187 electrodes, and properties change with the passage of current. Therefore, 188 the symmetry in the electrode systems listed in Table II  state variables at the metal/electrolyte interface will lead to fluctuations of 335 potential and current. As detailed in several reviews, 31,34,35,64,98,99 EN is 336 mainly generated from the following corrosion and electrochemical events: 337 (1) Anodic dissolution during general corrosion. As the electrochemical 338 activity at specific points in an electrode system fluctuates over time, [3] 394 where is the potential of WE1, and the anodic and cathodic 395 Tafel coefficients, , and , the exchange current densities, and 396 the anodic and cathodic areas on WE1, and the corrosion current. 397 The introduction of the galvanic coupling current in Ref. 114  [7] 412 The last term is the expression of the polarization resistance, , 413 derived by Stern and Geary. 116 Thus, when the anodic and cathodic 414 reactions occurring on the WEs are under activation polarization control, 415 = . However, it should be noticed that in some cases the anodic or 416 cathodic reaction does not strictly obey a Tafel behavior so that this 417 relationship is sometimes not satisfied in practice. 13,90,117 Besides, the 418 calculation of is slightly influenced by several factors such as the 419 sampling frequency, 117 by the DC removal method used, 9,118,119 by the size 420 of the WEs, 13 or by the weak symmetry between the two WEs. All these 421 factors can lead to conflicting values between and . Although most 422 works have claimed that a frequency range of 0.01 to 10 Hz enables 423 satisfactory EN measurements of corrosion systems, the selection of 424 considerably on the EN source and the intention of the measurements 426 (fundamental research, testing, monitoring). 14 Also, there is no uniform 427 standard for the DC removal method, the most frequently used methods 428 appearing to be polynomial detrending, 120 wavelet analysis, and empirical 429 mode decomposition. 121 Finally, the theoretically symmetric electrodes 430 may become asymmetric over time, especially in cases of localized 431 corrosion. Therefore, the noise resistance is not an accurate quantitative 432 measure of corrosion rate for all corrosion systems. This is why, especially 433 in the field, the corrosion current can be estimated from the ratio B/R n , 434 where the Stern-Geary coefficient B is calibrated from weight-loss 435 measurements. In contrast to the SD of the current, which should not be 436 normalized vs the electrode area, R n can be normalized, as well as R p , so 437 that the corrosion rate can be calculated in A cm -2 whatever the size of the 438

WEs. 439
Electrode systems #3 to #6 in Table II   can be used to identify initiation and propagation stages of localized 493 corrosion, the kinetic equations do not describe localized corrosion. 494 (3) For asymmetrical electrode systems, a quantitative analysis cannot be 495 achieved, as mentioned above. 496 (4) The initially symmetric electrode systems listed in Table II Table III, the EPN and ECN  503 are modelled using a single circuit element, either a current source or a 504 potential source, and it is assumed that the impedance of the electrodes 505 does not vary during corrosion. ECs for electrode systems #1 to #6 in Table  506 II are discussed in detail below and their theoretical predictions and 507 experimental results are compared. 508 509 Electrode systems #1, #3, #5 and #6-The two ECs (EC1 and EC2) of 510 these electrode systems presented in the first row of Table III are  511 mathematically identical simplifications of real circuits. The circuit based 512 on current noise sources (EC1) is the Norton equivalent circuit and the 513 circuit based on potential noise sources (EC2) is the Thevenin equivalent 514 circuit 124 . In EC1 and EC2, i 1 and i 2 represent the current noise sources of 515 WE1 and WE2 while e 1 and e 2 represent the potential noise sources of WE1 516 and WE2. Z 1 , Z 2 , and Z 3 denote the impedances of WE1, WE2, and RE. ∆ 517 is the current fluctuation flowing from WE1 to WE2 measured by the ZRA, 518 ∆ is the potential fluctuation of WE1, is the electrolyte resistance 519 between WE1 and WE2, and is the fraction of between the RE and 520 Therefore, in the common use of two identical electrodes connected 556 through a ZRA, the noise impedance is equal to the modulus of the 557 impedance of the WEs whatever the origin of the noise sources (pits, 558 bubbles, cracks…) and whatever the noise level on each electrode. 559 For electrode system #3 with WEs of the same material but different 560 surface areas, the impedance of each WE is inversely proportional to its 561 surface area while the PSD of the current noise source is proportional to 562 the surface area. 13,31 It can then be shown from Eq. 18 that: 125 563 | | [20] 564 For electrode system #5 and #6, one electrode serves as the anode and 565 the other serves as the cathode, so is now denoted as , and as 566 . Using the current noise sources, Eq. 18 can be rewritten as: 567 The expression of the noise impedance can be further discussed 569 according to three different cases for these asymmetric electrode systems: 570 (1) The noise level of the cathode (WE2) is significantly higher than the 571 noise level of the anode (WE1) (for example, the cathode mainly 572 supports hydrogen bubble evolution, whereas the anode undergoes 573 uniform corrosion). 574

| | [22] 575
The noise impedance is equal to the impedance of the anode, 576 while the EPN and ECN time records provide information on the 577 noisier electrode (cathode). 70 578 (2) The noise level on the anode is significantly higher than the noise 579 level on the cathode (for example, the cathode mainly supports 580 oxygen reduction, whereas the anode undergoes pitting corrosion). 581 The noise impedance is equal to the impedance of the cathode, 583 while the EPN and ECN time records provide information on the 584 noisier electrode (anode). 585 (3) When the noise levels of the anode and cathode are comparable, 586 ranges between the impedance moduli of the electrodes 587 As for the electrode system #1, it is then still possible to determine 615 the impedance modulus of the electrodes when using a third identical 616 electrode as pseudo-RE. 617

Correlation between
and .-For a random signal, the variance, 618 which is the square of the standard deviation, is equal to the integral of its 619 PSD, therefore, according to the definition of the noise resistance (Eq. 7): 620 [27] 621 Actually, the PSD has a limited frequency bandwidth ( , ). 622 The lowest frequency analyzed of the spectrum of a discrete time record 623 sampled at frequency f s is given by ⁄ 1/ ∆ , where is 624 the number of samples and ∆ 1 ⁄ is the sampling interval. When the 625 PSD is calculated using the fast Fourier transform, the frequency resolution 626 is also given by ⁄ . As an example often encountered in the literature, 627 when = 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 being equal to one half of the sampling 631 frequency, the frequency range analyzed in the above example is (~1 mHz, 632    Table IV). 29,70,85,100,118,125,126,129 One significant advantage for EC1 708 to EC4 in Table III where is the charge in each corrosion event (assumed to be the 735 same for each event), is the mean emission frequency of corrosion 736 events, and is the corrosion current ( and are assumed 737 to be the same for each WE). As a consequence of the short duration 738 of the events, which in practice implies working at low sampling 739 frequencies, the PSD is frequency independent in the measured 740 frequency range. For events of longer duration, the PSD values 741 considered in Eq. 37 are the low-frequency limit of the PSDs. 742 (2) The cathodic process on the WEs is noise-free, as for example oxygen 743 reduction. 744 (3) Both WEs have the same impedance, which, at low frequency, is equal 745 to the polarization resistance, . The low-frequency limit of the 746 noise impedance, , is then also equal to (Eq. 19 However, this is strictly valid under the assumption that the PSDs are 762 frequency independent in the whole frequency bandwidth. This requires 763 that the duration of each transient is short enough so that it can be 764 considered as a Dirac delta function over the frequency range considered, 765 which implies that the transient lasts for less than one period of the highest 766 frequency included in the measurement. 767 always be valid. 98 First, the PSDs are assumed to be flat at low frequency 769 while the measured power spectra often show a 1/f behavior at the lowest 770 analyzed frequencies. Second, is assumed to be equal to , which is 771 not true for some corrosion systems such as stainless steel or coated 772 electrodes, as explained above. 30,129 . Third, the shot noise model cannot 773 apply when the noise is produced by both anodic and cathodic reactions, 774 such as when bubble evolution occurs in addition to the anodic process. 775 Finally, all transients are expected to be uncorrelated and to have the same 776 shape and amplitude, which is not observable on ECN time records 777 showing transients as for metastable pitting. 778 779

Conclusions and future work 780
The different electrode systems used in EN measurements in the 781 corrosion field have been reviewed, most of them nowadays involving two 782 symmetrical or asymmetrical electrodes connected through a ZRA. The 783 impact of the electrode surface area on the ECN and EPN, which is a 784 parameter of primary importance, has been considered. After a brief review 785 of the various origins of the EN sources, the theoretical models used for 786 analyzing the noise were presented. 787 Electrochemical kinetics models were introduced to explain the 788 correlation between the noise resistance and the polarization resistance for 789 a symmetrical or asymmetrical corroding system. However, they have 790 limitations, the most important being the assumption that the anodic and 791 cathodic reactions are under activation control. Equivalent circuit 792 approaches with potential or current noise sources or with fluctuating 793 resistances have also been proposed to analyze EN data. While the latter 794 approach is limited to corroding electrodes the impedance of which are 795 reduced to simple resistance, the former approach does not presuppose 796 anything on the impedance of the electrodes and on the origin of the noise 797 sources. It is possible with this model to specify the conditions in which 798 the noise resistance is or is not equal to the polarization resistance. Finally, 799 the shot noise model was presented: under some assumptions (flat PSD at 800 low frequency, equal to , single anodic noise source), the mean 801 occurrence frequency of corrosion events and the mean electrical charge 802 involved in the transients can be calculated to discriminate between 803 uniform corrosion and localized corrosion. 804 The existing theoretical models and mathematical methods need to 805 be optimized as they suffer from serious limitations, as mentioned above, 806 for studying important topics, such as the impact of electrode area on EN 807 data, the shape and time duration of transients in metastable pitting, or the 808 quantification of the extent of localized corrosion (crack length, pit depth, 809 pit number). New developments are also necessary to better interpret the 810 shape and amplitude of the EN power spectral densities, or to analyze 844 Electrochem. Soc., 115, 617 (1968  inhibitor films using electrochemical noise analysis (ENA)." Corros. Sci., 38, 1681Sci., 38, (1996.   and evaluation of electrode asymmetry." Corros. Sci., 77, 281 (2013).