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Electrochemical noise analysis on multiple dissimilar electrodes: Theoretical analysis
Electrochimica Acta 56 (2011) 10270–10275
Contents lists available at SciVerse ScienceDirect
Electrochimica Acta journal homepage: www.elsevier.com/locate/electacta
Electrochemical noise analysis on multiple dissimilar electrodes: Theoretical analysis M. Curioni a,∗ , R.A. Cottis a , M. Di Natale b , G.E. Thompson a a b
Corrosion and Protection Centre, The University of Manchester, Manchester M13 9PL, United Kingdom Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, 20125 Milano, Italy
a r t i c l e
i n f o
Article history: Received 20 July 2011 Received in revised form 6 September 2011 Accepted 7 September 2011 Available online 16 September 2011 Keywords: Electrochemical noise Corrosion Multiple electrodes Galvanic coupling Dissimilar electrodes
a b s t r a c t The electrochemical potential and current noise generated by multiple electrodes are analyzed by considering the current flowing across each electrode and the electrochemical potential of the electrode array. By introducing the concept of a “virtual electrode”, the analysis of the electrochemical noise generated by an array of electrodes is reduced to the case of two dissimilar electrodes. For each electrode, an apparent impedance, Z *, can be determined as the square root of the power spectral density of potential divided by the power spectral density of the individual electrode current. When two dissimilar pairs of nominally identical electrodes are used and it is possible to assume that the pair of electrodes corroding more produces higher noise levels and displays lower impedance, the actual electrode impedance can be obtained with acceptable precision from the value of the nominal impedance. Further, the asymmetry between the two dissimilar pairs can also be quantitatively evaluated. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The measurement of electrochemical noise is well-established for corrosion studies both in the laboratory and in the field. On a single electrode, either the fluctuations in the open circuit potential or the current when the electrode is potentiostatically controlled can be measured. If two galvanically coupled electrodes are used, the fluctuations in the current flowing from one electrode to the other can be measured simultaneously with the electrochemical potential of the couple. From this combined measurement, information related to the type of corrosion [1,2], the frequency and magnitude of corrosion events [2–5] and the noise resistance [6–8] or the noise impedance [8–12] can be extracted. Noise resistance and impedance are of particular interest, since their physical meaning is similar to that of the polarization resistance [9] but, they can be estimated without introducing a potential or current perturbation to the freely corroding system. The process of electrochemical noise generation may be rationalized by representing the corroding surface by means of appropriate equivalent circuits. The equivalent circuits representing the corroding surface of one electrode generally fall within one of the following types: (i) current noise sources in parallel with constant impedances [8], (ii) potential noise sources in series with constant impedances [8], and (iii) constant potential sources
∗ Corresponding author. Tel.: +44 161 306 5971. E-mail address: [email protected] (M. Curioni). 0013-4686/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2011.09.013
(anodic and cathodic reaction equilibrium potentials) in series with fluctuating resistances (anodic and cathodic resistances) [13]. The first two equivalent circuits are equivalent from the electrical viewpoint and whichever is selected does not affect the numerical results of the noise analysis [8]; in contrast, the third type of equivalent circuit is intrinsically different and some differences are revealed in the analysis compared with the first two [13]. When the two electrodes are dissimilar, if the analysis is performed using type (i) and (ii) circuits and it is possible to assume a priori that one electrode produces the majority of the current noise, only the impedance of the less noisy electrode can be estimated [14]. This limitation is relatively severe in corrosion studies, since most of the noise is typically produced by the corroding electrode and information on its impedance is important. The use of type (iii) equivalent circuit has some advantage in interpreting the corrosion behaviour when the two electrodes are dissimilar, i.e. it allows estimation of the anodic and cathodic resistances on each of the two dissimilar electrodes, but requires a priori assumptions for the analysis to be valid. Specifically, it requires linearity of the anodic and cathodic reactions, the existence of a low-frequency impedance modulus plateau, and a priori knowledge of the equilibrium potentials of the anodic and cathodic reactions on each electrode. Further, to estimate the anodic and cathodic resistances, the two electrodes must be periodically coupled and decoupled [13]. A method based on the computation of the bispectrum or its integral to interpret the data on asymmetric electrodes has been proposed, but it does not allow information on the electrodes impedance or noise resistance to be extracted [15].
M. Curioni et al. / Electrochimica Acta 56 (2011) 10270–10275
For two electrodes, if it is not possible to assume a priori the relative magnitude of the noise sources for circuits (i–ii), or if the assumptions required to validate the analysis with circuit (iii), or the operation of coupling–decoupling must be avoided, the estimation of resistance or impedances on each of the two dissimilar electrodes becomes impossible. For two electrodes, this has been described in detail in Ref. [14] and arises from the existence in the measured system of 4 unknowns (two noise sources and two impedances for circuits (i–ii), and four resistances for circuit (iii)) with only two parameters (potential and current) being measured. Thus, if no a priori assumptions are made on the relative magnitude of the noise sources and of the impedances, the corroding system has two degrees of freedom, and cannot be fully characterized. Unfortunately, increasing the number of electrodes does not, of itself, facilitate the problem; for each additional electrode, one more current can be measured but two more unknowns are added. In this work, the two-electrode analysis is extended to an arbitrary number of galvanically coupled electrodes using type (i) circuit. Then, the special case of four electrodes is considered, in which pairs of electrodes are nominally identical. This approach enables approximated values of noise impedances on all the electrodes to be obtained, with the only required a priori assumption being that electrodes that corrode significantly more display higher noise levels and lower impedances compared with electrodes that corrode less.
2. Theory The potential and current noise produced by an array of electrodes can be measured using the experimental setup of Fig. 1a. Each zero-resistance ammeter (ZRA) must be connected with one terminal to one electrode and with the other to ground, enabling the acquisition of the individual electrode current. Further, a reference electrode and a voltmeter must be used to acquire the potential of the electrode array. Before describing the analysis of the noise associated with multiple electrodes, it may be helpful to state a few of the principles involved in electrical circuit analysis:
1. Kirchoff’s Current Law: the algebraic sum of currents in a network of conductors meeting at a point is zero. 2. Kirchoff’s Voltage Law: the sum of the emfs in any closed loop is equivalent to the sum of the potential drops in that loop 3. Norton’s Theorem: any collection of voltage sources, current sources, and resistors with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor, R. For single-frequency AC systems, the theorem can also be applied to general impedances, not just resistors. For AC systems over a wide frequency range, it is common to replace the simple resistor R with an equivalent circuit that may have a complex frequency dependence (an example being the equivalent circuit used in the interpretation of EIS data). 4. The Superposition Theorem: the response (voltage or current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, while all other independent sources are replaced by their internal impedances. a. To ascertain the contribution of each individual source, all of the other sources first must be “turned off” (set to zero) by: • Replacing all other independent voltage sources with a short circuit (thereby eliminating difference of potential, i.e. V = 0, the internal impedance of the ideal voltage source is zero (short circuit)).
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• Replacing all other independent current sources with an open circuit (thereby eliminating current, i.e. I = 0, the internal impedance of the ideal current source is infinite (open circuit)). 5. When adding noise voltages and currents, the power spectral densities add, not the amplitudes of voltage and current. Strictly this is a statistically ‘expected’ outcome; it results from the random phase relationship between the AC components at the various frequencies, which means that signals will sometimes reinforce (when the signals are in phase), sometimes cancel (when they are 180◦ out of phase), but will mostly be somewhere in between. 2.1. Properties of a virtual electrode Based on these principles, we assume that an array of k electrodes can be represented as a Norton equivalent circuit of current noise sources im (t) in parallel with their impedances Zm (where m = 1 to k). This implicitly assumes that the electrodes can be treated as linear, which is probably reasonable for the relatively low amplitude fluctuations associated with most electrochemical noise measurements. Before analysing specific cases, it is useful to derive the Norton equivalent for an assembly of several electrodes, which we shall term a virtual electrode. Considering a virtual electrode, , consisting of k parallel electrodes, we can sum the time varying current measured at each electrode, Im (t) (Fig. 1b), to obtain the time varying current flowing across the virtual electrode I (t). If the virtual electrode is disconnected, I (t) = 0 for all t and this gives: I (t) =
k
Im (t)
(1)
m=1
while for AC signals, the PSDs of the current noise source, add: i (t) =
k
im (t)
im , will
(2)
m=1
It is worth noting that in Eqs. (1) and (2) and in Fig. 1, Im (t) are the electrode currents that can be measured, while im (t) are the currents generated by the ideal noise sources and cannot be directly measured. In the following analysis, the time dependent Im (t) and im (t) are written Im and im m respectively. Considering that the parallel impedances add as inverses, the impedance of electrode is: Z =
k
1
m=1
(3)
(1/Zm )
As a result of the current i being applied to the impedance Z the PSD of potential, E , developed by an unconnected virtual electrode is given by
k 2 im m=1 W = i Z = 2 k m=1 (1/Zm )
(4)
2.2. Analysis of an array of electrodes Following the previous analysis, any array of k + 1 electrodes can be reduced to the two-electrode case: a real electrode , with current noise source i and impedance Z , connected with a virtual electrode , accounting for the remaining k electrodes and having a current noise source i and impedance Z , given by Eqs. (2) and (3) (see Fig. 2). Considering that the naming of the electrodes
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M. Curioni et al. / Electrochimica Acta 56 (2011) 10270–10275
Fig. 1. (a) Experimental setup required for the four-electrode analysis requiring four zero-resistance ammeters (ZRA) a voltmeter (V) and a reference electrode (REF). (b) Equivalent circuit representing electrode , and the construction of virtual electrode , comprising real electrodes 2–4.
is arbitrary, the analysis of the electrode is sufficient to analyze the complete array as it can be extended iteratively to any of the electrodes. Thus, following Bautista et al. [14], similar to two asymmetric electrodes, the power spectral density of the current flowing from the real electrode to the virtual electrode is
2 Z i + Z 2 i I = Z + Z 2
(5)
and the potential power spectral density is
Z Z 2 ( i + i ) E = Z + Z
(6)
Thus, the analysis of an array of k + 1 electrodes is reduced to the case of the analysis of two asymmetrical electrodes. The application of the above analysis to large electrode arrays will be examined in future publications but, for the purpose of the present work, it is interesting to examine the relationship between the nominal electrode impedance Z∗ and the actual electrode impedance Z . If all the electrodes are identical, replacing (2) and (3) in (7) gives the relationship between the apparent impedance Z∗ and the actual electrode impedance Z for an array of k + 1 electrodes:
∗ Z Z = √
(8)
k
of the apparent From the previous equations, the expression impedance modulus of the electrode , Z∗ , defined as the square root of the ratio between the power spectral density of the potential flowing across the electrode, and the power spectral density of the current of the electrode array, can be obtained:
If the electrodes are dissimilar, but it is possible to assume that most of the noise is generated by the virtual electrode ( i i ), Eq. (7) reduces to
∗ Z = Z Z
∗ Z ≈ Z Z
i +
i
2 Z i + Z 2 i
(7)
i = Z 2 Z i
(9)
M. Curioni et al. / Electrochimica Acta 56 (2011) 10270–10275
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Fig. 2. Equivalent circuit representing electrode that is galvanically coupled to the virtual electrode . The analysis of an array of multiple electrodes has been reduced to the two-electrode case.
Finally, if most of the noise is generated by electrode ( i i ), Eq. (7) reduces to:
∗ Z ≈ Z Z
i = Z 2 Z i
(10)
In practice, Eqs. (7)–(10) are an extension of the equations proposed by Bautista et al. for the case of an array of k + 1 electrodes, and state that for an array of k + 1 electrodes it is possible to estimate the impedances if the electrodes can be considered identical; otherwise, only the impedance of the less noisy electrode can be estimated, with a precision that increases with increasing difference in noise levels between electrodes. The latter statement introduces a severe limitation to the analysis if only two electrodes are used, because the noisier electrode is generally the one that is corroding and the information on its impedance is important. However, if larger electrodes arrays are used, it may be possible to overcome this limitation. This will now be examined for the particular case of four electrodes consisting of two pairs of nominally identical electrodes. 2.3. Dissimilar pairs of identical electrodes When two dissimilar pairs of identical electrodes are employed, only two conditions are possible: either one pair of identical electrodes produces more noise than the other, or all the four electrodes produce comparable levels of noise. Consequently, for each individual electrode coupled with the virtual electrode formed by the remaining three electrodes it is always the case that i ≤
i
(11)
In practice, the use of two dissimilar pairs of identical electrodes allows exclusion of the case that any single electrode produces more noise than the remaining three altogether, and the analysis can be progressed further.
Consider four electrodes, numbered 1–4, and assume that electrodes 1 and 3 are identical, ii = i3 and Z1 = Z3 and, in general, dissimilar from identical electrodes 2 and 4, i2 = i4 and Z2 = Z4 . The behaviour of electrode 1, and iteratively of all the other electrodes, must be analyzed under three alternative assumptions that cover all the possible cases: (i) electrode 1 belongs to the pair that is corroding less, (ii) electrode 1 belongs to the pair that is corroding more and (iii) all the electrodes behave quasi-identically. Thus in all cases, it is useful to consider electrode 1 as the individual electrode , and the remaining three electrodes as the virtual electrode . If electrode belongs to the pair that is corroding less, most of the noise is generated by the electrodes forming the pair that is corroding more, thus i i and, according to Eq. (9), the apparent impedance approaches the actual impedance of the electrode, Z∗ ≈ Z . In practice, the noise generated by the more corroding electrodes acts as the probing signal for the less corroding electrodes. Conversely, for the purpose of estimating the noise levels and impedance of the remaining electrodes forming the virtual electrode , if electrode belongs to the pair that is corroding more, the contribution to noise levels and decrease in impedance due to the less corroding electrodes can be neglected. Thus, noise levels and impedance of electrode are i =
Z =
i + 2 i2 ≈
i
i
1 ≈ Z (1/|Z |) + (2/|Z2 |)
i2
(12)
Z2 Z
(13)
Consequently, Eq. (7) reduces to the case of two identical electrodes and
∗ 2 Z ≈ Z
2 i = Z 2 2 Z i
(14)
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M. Curioni et al. / Electrochimica Acta 56 (2011) 10270–10275
Fig. 3. Pairs of nominally identical virtual electrodes that can be generated when electrode 1 is nominally identical to electrode 3 and electrode 2 is nominally identical to electrode 4 (C = D and E = F).
Finally, if all the four electrodes behave identically, Eq. (8) gives
√ ∗ Z = 3 Z
(15)
Considering that the previous analysis covers all the possible cases, it can be concluded that, using two dissimilar pairs of identical electrodes, the impedance of each electrode can be estimated √ with a maximum error of 3 ≈ 1.73 since it is always verified that ∗ √ ∗ Z < Z < 3 Z (16) It should be noted that the error in the impedance estimation is given by the product of the error on the estimation of Z∗ , dependent on the length of the dataset available and the number √ of averages, and the error introduced by the presented method, 3. The latter cannot be reduced because it is due to the use of dissimilar pair of identical electrodes and it is independent of the length of the dataset and the number of averages. 2.4. Symmetry factor Based on the previous method, a parameter accounting for the similarity between the pairs of dissimilar electrodes can be introduced. For four electrodes, when one of the two pairs is corroding more, the nominal impedances of each electrode approximate well the actual impedances. Conversely, when all the electrodes behave is obtained by multiidentically the actual electrodes impedance √ plying the apparent impedance by 3. Thus, a parameter ˛, varying from 1 to 3, accounting for the similarity between the two pairs of electrodes, can be introduced. The impedance of electrode K can be expressed as a function of the measured apparent impedance and of ˛: √ ∗ Zm = ˛ Zm m = 1, 2, 3, 4 (17)
˛ ≈ 1 is associated with high asymmetry between the pairs of electrodes and ˛ ≈ 3 is associated with close similarity between the four electrodes. The four-electrode configuration, with two dissimilar pairs of identical electrodes, enables the estimation of the value of ˛ by considering two nominally identical virtual electrodes, each comprising a pair of dissimilar electrodes (Fig. 3, electrodes C, D, E, F). In practice, taking as an example the two nominally identical virtual electrodes C and D of Fig. 3 (comprising electrodes 1–2 and 3–4), the impedance of each virtual electrode can be estimated by two independent methods: (a) by calculating the equivalent impedances of virtual electrodes C and D from the values of the nominal impedances corrected by an appropriate ˛
−1 −1 1 1 1 1 ZC ≈ ZD ≈ √ + √ ≈ √ +√ ˛ Z ∗ ˛ Z ∗ ˛ Z ∗ ˛ Z ∗ 1
2
3
4
(18)
(b) by the taking square root of the potential power spectral density divided by the square root of the power spectral densities of the virtual electrode current summed in the time domain prior to calculating the power spectral density.
ZC ≈ ZD ≈
(I1 + I2 ) E
(19)
This process can be repeated for the configuration of Fig. 3. If the nominally identical electrodes behave identically, the estimation of
M. Curioni et al. / Electrochimica Acta 56 (2011) 10270–10275
the impedance of the virtual electrodes by method (a) and method (b) should give similar results, provided that a suitable value of ˛ is selected or, equivalently, the difference between the estimations obtained by methods (a) and (b) can be minimized by adjusting the value of ˛. Thus, the ˛ produces the closest agreement between the two methods can provide an indication of the degree of asymmetry between the two dissimilar pairs of identical electrodes. It is likely, however, that the nominally identical electrodes do not behave perfectly identical and a residual error in the estimation of each electrode impedance is expected.
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estimation and providing information on the asymmetry between the pair of electrodes. Acknowledgements EPSRC is acknowledged for provision of financial support through the LATEST2 Programme Grant. Intertek-CAPCIS is acknowledged for providing a Concerto multichannel potentiostat free of charge. References
3. Conclusions A general method for the interpretation of electrochemical noise generated by an array of dissimilar electrodes has been presented. If two dissimilar pairs of identical electrodes are used, the impedance of each electrode can be estimated with acceptable approximation. The assumption required for the analysis is that, if a pair of identical electrodes is corroding less than the other, the electrodes of the non-corroding pair produce lower noise levels and have higher impedances compared with the electrodes of the corroding pair. If this can be accepted, the individual electrode impedances √ can be estimated with a maximum error of 3 ≈ 1.73 for a fourelectrode configuration. The four-electrode configuration enables an additional refinement of the analysis as two nominally identical virtual electrodes can be considered. This allows the estimation of two nominally identical virtual electrode impedances by two independent methods, reducing the error in the impedance
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
F. Mansfeld, Z. Sun, Corrosion 55 (1999) 915. R.A. Cottis, NACE – International Corrosion Conference Series, 2006, 064321. J.M. Sanchez-Amaya, R.A. Cottis, F.J. Botana, Corros. Sci. 47 (2005) 3280. J.M. Sanchez-Amaya, M. Bethencourt, L. Gonza¡lez-Rovira, F.J. Botana, Electrochim. Acta 52 (2007) 6569. A.M. Kuznetsov, Russ. J. Electrochem. 44 (2008) 1327. U. Bertocci, C. Gabrielli, F. Huet, M. Keddam, P. Rousseau, J. Electrochem. Soc. 144 (1997) 37. J.F. Chen, W.F. Bogaerts, Corros. Sci. 37 (1995) 1839. U. Bertocci, C. Gobrielli, F. Huet, M. Keddam, J. Electrochem. Soc. 144 (1997) 31. F. Mansfeld, C.C. Lee, J. Electrochem. Soc. 144 (1997) 2068. N.C. Rosero-Navarro, M. Curioni, R. Bingham, A. Durán, M. Aparicio, R.A. Cottis, G.E. Thompson, Corros. Sci. 52 (2010) 2489. F. Mansfeld, L.T. Han, C.C. Lee, C. Chen, G. Zhang, H. Xiao, Corros. Sci. 39 (1997) 255. F. Mansfeld, C.C. Lee, G. Zhang, Electrochim. Acta 43 (1998) 435. M. Curioni, R.A. Cottis, M. Di Natale, G.E. Thompson, Electrochim. Acta 56 (2011) 6318. A. Bautista, U. Bertocci, F. Huet, J. Electrochem. Soc. 148 (2001) B412. M. Kiwilszo, J. Smulko, J. Solid State Electrochem. 13 (2009) 1681.
Contents lists available at SciVerse ScienceDirect
Electrochimica Acta journal homepage: www.elsevier.com/locate/electacta
Electrochemical noise analysis on multiple dissimilar electrodes: Theoretical analysis M. Curioni a,∗ , R.A. Cottis a , M. Di Natale b , G.E. Thompson a a b
Corrosion and Protection Centre, The University of Manchester, Manchester M13 9PL, United Kingdom Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, 20125 Milano, Italy
a r t i c l e
i n f o
Article history: Received 20 July 2011 Received in revised form 6 September 2011 Accepted 7 September 2011 Available online 16 September 2011 Keywords: Electrochemical noise Corrosion Multiple electrodes Galvanic coupling Dissimilar electrodes
a b s t r a c t The electrochemical potential and current noise generated by multiple electrodes are analyzed by considering the current flowing across each electrode and the electrochemical potential of the electrode array. By introducing the concept of a “virtual electrode”, the analysis of the electrochemical noise generated by an array of electrodes is reduced to the case of two dissimilar electrodes. For each electrode, an apparent impedance, Z *, can be determined as the square root of the power spectral density of potential divided by the power spectral density of the individual electrode current. When two dissimilar pairs of nominally identical electrodes are used and it is possible to assume that the pair of electrodes corroding more produces higher noise levels and displays lower impedance, the actual electrode impedance can be obtained with acceptable precision from the value of the nominal impedance. Further, the asymmetry between the two dissimilar pairs can also be quantitatively evaluated. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The measurement of electrochemical noise is well-established for corrosion studies both in the laboratory and in the field. On a single electrode, either the fluctuations in the open circuit potential or the current when the electrode is potentiostatically controlled can be measured. If two galvanically coupled electrodes are used, the fluctuations in the current flowing from one electrode to the other can be measured simultaneously with the electrochemical potential of the couple. From this combined measurement, information related to the type of corrosion [1,2], the frequency and magnitude of corrosion events [2–5] and the noise resistance [6–8] or the noise impedance [8–12] can be extracted. Noise resistance and impedance are of particular interest, since their physical meaning is similar to that of the polarization resistance [9] but, they can be estimated without introducing a potential or current perturbation to the freely corroding system. The process of electrochemical noise generation may be rationalized by representing the corroding surface by means of appropriate equivalent circuits. The equivalent circuits representing the corroding surface of one electrode generally fall within one of the following types: (i) current noise sources in parallel with constant impedances [8], (ii) potential noise sources in series with constant impedances [8], and (iii) constant potential sources
∗ Corresponding author. Tel.: +44 161 306 5971. E-mail address: [email protected] (M. Curioni). 0013-4686/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2011.09.013
(anodic and cathodic reaction equilibrium potentials) in series with fluctuating resistances (anodic and cathodic resistances) [13]. The first two equivalent circuits are equivalent from the electrical viewpoint and whichever is selected does not affect the numerical results of the noise analysis [8]; in contrast, the third type of equivalent circuit is intrinsically different and some differences are revealed in the analysis compared with the first two [13]. When the two electrodes are dissimilar, if the analysis is performed using type (i) and (ii) circuits and it is possible to assume a priori that one electrode produces the majority of the current noise, only the impedance of the less noisy electrode can be estimated [14]. This limitation is relatively severe in corrosion studies, since most of the noise is typically produced by the corroding electrode and information on its impedance is important. The use of type (iii) equivalent circuit has some advantage in interpreting the corrosion behaviour when the two electrodes are dissimilar, i.e. it allows estimation of the anodic and cathodic resistances on each of the two dissimilar electrodes, but requires a priori assumptions for the analysis to be valid. Specifically, it requires linearity of the anodic and cathodic reactions, the existence of a low-frequency impedance modulus plateau, and a priori knowledge of the equilibrium potentials of the anodic and cathodic reactions on each electrode. Further, to estimate the anodic and cathodic resistances, the two electrodes must be periodically coupled and decoupled [13]. A method based on the computation of the bispectrum or its integral to interpret the data on asymmetric electrodes has been proposed, but it does not allow information on the electrodes impedance or noise resistance to be extracted [15].
M. Curioni et al. / Electrochimica Acta 56 (2011) 10270–10275
For two electrodes, if it is not possible to assume a priori the relative magnitude of the noise sources for circuits (i–ii), or if the assumptions required to validate the analysis with circuit (iii), or the operation of coupling–decoupling must be avoided, the estimation of resistance or impedances on each of the two dissimilar electrodes becomes impossible. For two electrodes, this has been described in detail in Ref. [14] and arises from the existence in the measured system of 4 unknowns (two noise sources and two impedances for circuits (i–ii), and four resistances for circuit (iii)) with only two parameters (potential and current) being measured. Thus, if no a priori assumptions are made on the relative magnitude of the noise sources and of the impedances, the corroding system has two degrees of freedom, and cannot be fully characterized. Unfortunately, increasing the number of electrodes does not, of itself, facilitate the problem; for each additional electrode, one more current can be measured but two more unknowns are added. In this work, the two-electrode analysis is extended to an arbitrary number of galvanically coupled electrodes using type (i) circuit. Then, the special case of four electrodes is considered, in which pairs of electrodes are nominally identical. This approach enables approximated values of noise impedances on all the electrodes to be obtained, with the only required a priori assumption being that electrodes that corrode significantly more display higher noise levels and lower impedances compared with electrodes that corrode less.
2. Theory The potential and current noise produced by an array of electrodes can be measured using the experimental setup of Fig. 1a. Each zero-resistance ammeter (ZRA) must be connected with one terminal to one electrode and with the other to ground, enabling the acquisition of the individual electrode current. Further, a reference electrode and a voltmeter must be used to acquire the potential of the electrode array. Before describing the analysis of the noise associated with multiple electrodes, it may be helpful to state a few of the principles involved in electrical circuit analysis:
1. Kirchoff’s Current Law: the algebraic sum of currents in a network of conductors meeting at a point is zero. 2. Kirchoff’s Voltage Law: the sum of the emfs in any closed loop is equivalent to the sum of the potential drops in that loop 3. Norton’s Theorem: any collection of voltage sources, current sources, and resistors with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor, R. For single-frequency AC systems, the theorem can also be applied to general impedances, not just resistors. For AC systems over a wide frequency range, it is common to replace the simple resistor R with an equivalent circuit that may have a complex frequency dependence (an example being the equivalent circuit used in the interpretation of EIS data). 4. The Superposition Theorem: the response (voltage or current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, while all other independent sources are replaced by their internal impedances. a. To ascertain the contribution of each individual source, all of the other sources first must be “turned off” (set to zero) by: • Replacing all other independent voltage sources with a short circuit (thereby eliminating difference of potential, i.e. V = 0, the internal impedance of the ideal voltage source is zero (short circuit)).
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• Replacing all other independent current sources with an open circuit (thereby eliminating current, i.e. I = 0, the internal impedance of the ideal current source is infinite (open circuit)). 5. When adding noise voltages and currents, the power spectral densities add, not the amplitudes of voltage and current. Strictly this is a statistically ‘expected’ outcome; it results from the random phase relationship between the AC components at the various frequencies, which means that signals will sometimes reinforce (when the signals are in phase), sometimes cancel (when they are 180◦ out of phase), but will mostly be somewhere in between. 2.1. Properties of a virtual electrode Based on these principles, we assume that an array of k electrodes can be represented as a Norton equivalent circuit of current noise sources im (t) in parallel with their impedances Zm (where m = 1 to k). This implicitly assumes that the electrodes can be treated as linear, which is probably reasonable for the relatively low amplitude fluctuations associated with most electrochemical noise measurements. Before analysing specific cases, it is useful to derive the Norton equivalent for an assembly of several electrodes, which we shall term a virtual electrode. Considering a virtual electrode, , consisting of k parallel electrodes, we can sum the time varying current measured at each electrode, Im (t) (Fig. 1b), to obtain the time varying current flowing across the virtual electrode I (t). If the virtual electrode is disconnected, I (t) = 0 for all t and this gives: I (t) =
k
Im (t)
(1)
m=1
while for AC signals, the PSDs of the current noise source, add: i (t) =
k
im (t)
im , will
(2)
m=1
It is worth noting that in Eqs. (1) and (2) and in Fig. 1, Im (t) are the electrode currents that can be measured, while im (t) are the currents generated by the ideal noise sources and cannot be directly measured. In the following analysis, the time dependent Im (t) and im (t) are written Im and im m respectively. Considering that the parallel impedances add as inverses, the impedance of electrode is: Z =
k
1
m=1
(3)
(1/Zm )
As a result of the current i being applied to the impedance Z the PSD of potential, E , developed by an unconnected virtual electrode is given by
k 2 im m=1 W = i Z = 2 k m=1 (1/Zm )
(4)
2.2. Analysis of an array of electrodes Following the previous analysis, any array of k + 1 electrodes can be reduced to the two-electrode case: a real electrode , with current noise source i and impedance Z , connected with a virtual electrode , accounting for the remaining k electrodes and having a current noise source i and impedance Z , given by Eqs. (2) and (3) (see Fig. 2). Considering that the naming of the electrodes
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M. Curioni et al. / Electrochimica Acta 56 (2011) 10270–10275
Fig. 1. (a) Experimental setup required for the four-electrode analysis requiring four zero-resistance ammeters (ZRA) a voltmeter (V) and a reference electrode (REF). (b) Equivalent circuit representing electrode , and the construction of virtual electrode , comprising real electrodes 2–4.
is arbitrary, the analysis of the electrode is sufficient to analyze the complete array as it can be extended iteratively to any of the electrodes. Thus, following Bautista et al. [14], similar to two asymmetric electrodes, the power spectral density of the current flowing from the real electrode to the virtual electrode is
2 Z i + Z 2 i I = Z + Z 2
(5)
and the potential power spectral density is
Z Z 2 ( i + i ) E = Z + Z
(6)
Thus, the analysis of an array of k + 1 electrodes is reduced to the case of the analysis of two asymmetrical electrodes. The application of the above analysis to large electrode arrays will be examined in future publications but, for the purpose of the present work, it is interesting to examine the relationship between the nominal electrode impedance Z∗ and the actual electrode impedance Z . If all the electrodes are identical, replacing (2) and (3) in (7) gives the relationship between the apparent impedance Z∗ and the actual electrode impedance Z for an array of k + 1 electrodes:
∗ Z Z = √
(8)
k
of the apparent From the previous equations, the expression impedance modulus of the electrode , Z∗ , defined as the square root of the ratio between the power spectral density of the potential flowing across the electrode, and the power spectral density of the current of the electrode array, can be obtained:
If the electrodes are dissimilar, but it is possible to assume that most of the noise is generated by the virtual electrode ( i i ), Eq. (7) reduces to
∗ Z = Z Z
∗ Z ≈ Z Z
i +
i
2 Z i + Z 2 i
(7)
i = Z 2 Z i
(9)
M. Curioni et al. / Electrochimica Acta 56 (2011) 10270–10275
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Fig. 2. Equivalent circuit representing electrode that is galvanically coupled to the virtual electrode . The analysis of an array of multiple electrodes has been reduced to the two-electrode case.
Finally, if most of the noise is generated by electrode ( i i ), Eq. (7) reduces to:
∗ Z ≈ Z Z
i = Z 2 Z i
(10)
In practice, Eqs. (7)–(10) are an extension of the equations proposed by Bautista et al. for the case of an array of k + 1 electrodes, and state that for an array of k + 1 electrodes it is possible to estimate the impedances if the electrodes can be considered identical; otherwise, only the impedance of the less noisy electrode can be estimated, with a precision that increases with increasing difference in noise levels between electrodes. The latter statement introduces a severe limitation to the analysis if only two electrodes are used, because the noisier electrode is generally the one that is corroding and the information on its impedance is important. However, if larger electrodes arrays are used, it may be possible to overcome this limitation. This will now be examined for the particular case of four electrodes consisting of two pairs of nominally identical electrodes. 2.3. Dissimilar pairs of identical electrodes When two dissimilar pairs of identical electrodes are employed, only two conditions are possible: either one pair of identical electrodes produces more noise than the other, or all the four electrodes produce comparable levels of noise. Consequently, for each individual electrode coupled with the virtual electrode formed by the remaining three electrodes it is always the case that i ≤
i
(11)
In practice, the use of two dissimilar pairs of identical electrodes allows exclusion of the case that any single electrode produces more noise than the remaining three altogether, and the analysis can be progressed further.
Consider four electrodes, numbered 1–4, and assume that electrodes 1 and 3 are identical, ii = i3 and Z1 = Z3 and, in general, dissimilar from identical electrodes 2 and 4, i2 = i4 and Z2 = Z4 . The behaviour of electrode 1, and iteratively of all the other electrodes, must be analyzed under three alternative assumptions that cover all the possible cases: (i) electrode 1 belongs to the pair that is corroding less, (ii) electrode 1 belongs to the pair that is corroding more and (iii) all the electrodes behave quasi-identically. Thus in all cases, it is useful to consider electrode 1 as the individual electrode , and the remaining three electrodes as the virtual electrode . If electrode belongs to the pair that is corroding less, most of the noise is generated by the electrodes forming the pair that is corroding more, thus i i and, according to Eq. (9), the apparent impedance approaches the actual impedance of the electrode, Z∗ ≈ Z . In practice, the noise generated by the more corroding electrodes acts as the probing signal for the less corroding electrodes. Conversely, for the purpose of estimating the noise levels and impedance of the remaining electrodes forming the virtual electrode , if electrode belongs to the pair that is corroding more, the contribution to noise levels and decrease in impedance due to the less corroding electrodes can be neglected. Thus, noise levels and impedance of electrode are i =
Z =
i + 2 i2 ≈
i
i
1 ≈ Z (1/|Z |) + (2/|Z2 |)
i2
(12)
Z2 Z
(13)
Consequently, Eq. (7) reduces to the case of two identical electrodes and
∗ 2 Z ≈ Z
2 i = Z 2 2 Z i
(14)
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Fig. 3. Pairs of nominally identical virtual electrodes that can be generated when electrode 1 is nominally identical to electrode 3 and electrode 2 is nominally identical to electrode 4 (C = D and E = F).
Finally, if all the four electrodes behave identically, Eq. (8) gives
√ ∗ Z = 3 Z
(15)
Considering that the previous analysis covers all the possible cases, it can be concluded that, using two dissimilar pairs of identical electrodes, the impedance of each electrode can be estimated √ with a maximum error of 3 ≈ 1.73 since it is always verified that ∗ √ ∗ Z < Z < 3 Z (16) It should be noted that the error in the impedance estimation is given by the product of the error on the estimation of Z∗ , dependent on the length of the dataset available and the number √ of averages, and the error introduced by the presented method, 3. The latter cannot be reduced because it is due to the use of dissimilar pair of identical electrodes and it is independent of the length of the dataset and the number of averages. 2.4. Symmetry factor Based on the previous method, a parameter accounting for the similarity between the pairs of dissimilar electrodes can be introduced. For four electrodes, when one of the two pairs is corroding more, the nominal impedances of each electrode approximate well the actual impedances. Conversely, when all the electrodes behave is obtained by multiidentically the actual electrodes impedance √ plying the apparent impedance by 3. Thus, a parameter ˛, varying from 1 to 3, accounting for the similarity between the two pairs of electrodes, can be introduced. The impedance of electrode K can be expressed as a function of the measured apparent impedance and of ˛: √ ∗ Zm = ˛ Zm m = 1, 2, 3, 4 (17)
˛ ≈ 1 is associated with high asymmetry between the pairs of electrodes and ˛ ≈ 3 is associated with close similarity between the four electrodes. The four-electrode configuration, with two dissimilar pairs of identical electrodes, enables the estimation of the value of ˛ by considering two nominally identical virtual electrodes, each comprising a pair of dissimilar electrodes (Fig. 3, electrodes C, D, E, F). In practice, taking as an example the two nominally identical virtual electrodes C and D of Fig. 3 (comprising electrodes 1–2 and 3–4), the impedance of each virtual electrode can be estimated by two independent methods: (a) by calculating the equivalent impedances of virtual electrodes C and D from the values of the nominal impedances corrected by an appropriate ˛
−1 −1 1 1 1 1 ZC ≈ ZD ≈ √ + √ ≈ √ +√ ˛ Z ∗ ˛ Z ∗ ˛ Z ∗ ˛ Z ∗ 1
2
3
4
(18)
(b) by the taking square root of the potential power spectral density divided by the square root of the power spectral densities of the virtual electrode current summed in the time domain prior to calculating the power spectral density.
ZC ≈ ZD ≈
(I1 + I2 ) E
(19)
This process can be repeated for the configuration of Fig. 3. If the nominally identical electrodes behave identically, the estimation of
M. Curioni et al. / Electrochimica Acta 56 (2011) 10270–10275
the impedance of the virtual electrodes by method (a) and method (b) should give similar results, provided that a suitable value of ˛ is selected or, equivalently, the difference between the estimations obtained by methods (a) and (b) can be minimized by adjusting the value of ˛. Thus, the ˛ produces the closest agreement between the two methods can provide an indication of the degree of asymmetry between the two dissimilar pairs of identical electrodes. It is likely, however, that the nominally identical electrodes do not behave perfectly identical and a residual error in the estimation of each electrode impedance is expected.
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estimation and providing information on the asymmetry between the pair of electrodes. Acknowledgements EPSRC is acknowledged for provision of financial support through the LATEST2 Programme Grant. Intertek-CAPCIS is acknowledged for providing a Concerto multichannel potentiostat free of charge. References
3. Conclusions A general method for the interpretation of electrochemical noise generated by an array of dissimilar electrodes has been presented. If two dissimilar pairs of identical electrodes are used, the impedance of each electrode can be estimated with acceptable approximation. The assumption required for the analysis is that, if a pair of identical electrodes is corroding less than the other, the electrodes of the non-corroding pair produce lower noise levels and have higher impedances compared with the electrodes of the corroding pair. If this can be accepted, the individual electrode impedances √ can be estimated with a maximum error of 3 ≈ 1.73 for a fourelectrode configuration. The four-electrode configuration enables an additional refinement of the analysis as two nominally identical virtual electrodes can be considered. This allows the estimation of two nominally identical virtual electrode impedances by two independent methods, reducing the error in the impedance
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
F. Mansfeld, Z. Sun, Corrosion 55 (1999) 915. R.A. Cottis, NACE – International Corrosion Conference Series, 2006, 064321. J.M. Sanchez-Amaya, R.A. Cottis, F.J. Botana, Corros. Sci. 47 (2005) 3280. J.M. Sanchez-Amaya, M. Bethencourt, L. Gonza¡lez-Rovira, F.J. Botana, Electrochim. Acta 52 (2007) 6569. A.M. Kuznetsov, Russ. J. Electrochem. 44 (2008) 1327. U. Bertocci, C. Gabrielli, F. Huet, M. Keddam, P. Rousseau, J. Electrochem. Soc. 144 (1997) 37. J.F. Chen, W.F. Bogaerts, Corros. Sci. 37 (1995) 1839. U. Bertocci, C. Gobrielli, F. Huet, M. Keddam, J. Electrochem. Soc. 144 (1997) 31. F. Mansfeld, C.C. Lee, J. Electrochem. Soc. 144 (1997) 2068. N.C. Rosero-Navarro, M. Curioni, R. Bingham, A. Durán, M. Aparicio, R.A. Cottis, G.E. Thompson, Corros. Sci. 52 (2010) 2489. F. Mansfeld, L.T. Han, C.C. Lee, C. Chen, G. Zhang, H. Xiao, Corros. Sci. 39 (1997) 255. F. Mansfeld, C.C. Lee, G. Zhang, Electrochim. Acta 43 (1998) 435. M. Curioni, R.A. Cottis, M. Di Natale, G.E. Thompson, Electrochim. Acta 56 (2011) 6318. A. Bautista, U. Bertocci, F. Huet, J. Electrochem. Soc. 148 (2001) B412. M. Kiwilszo, J. Smulko, J. Solid State Electrochem. 13 (2009) 1681.