YongJun Tan
Western Australian Corrosion Research Group, School of Applied Chemistry,
Curtin University of Technology, GPO Box U1987, Perth 6001, Western Australia
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A theoretical analysis on the electrochemical noise resistance is carried out based on basic electrochemical corrosion theory. It is proven that the noise resistance is equivalent to the polarisation resistance and is in fact a special form of linear polarisation resistance, namely statistical linear polarisation resistance.
Keywords:
The electrochemical noise resistance (Rn) [1], is defined as the ratio of the standard deviations of potential noise and of the current noise (Rn = s v/s i) between two identical working electrodes which are linked by a zero resistance ammeter. In 1986, Eden et al found that the noise resistance is comparable to polarisation resistance (Rp), i.e. Rn � Rp [1] and thus it can be used as a method of corrosion rate measurement. This relationship has received considerable interest and has been experimentally confirmed in some laboratories [26]. Later, this time domain relationship has been extended into the frequency domain to produce noise impedance spectra [710].
Major effort has been made in order to understand the theoretical background of the time domain noise resistance [1114, 3] and the frequency domain noise impedance spectra [9] since a better theoretical understanding of the techniques is very important for recognising their advantages, limitations and prerequisite conditions. However, disagreement still exists regarding the validity of some of these mathematical derivations[1517].
This paper presents a mathematical deviation to interpret the time domain noise resistance as a statistical noise resistance [3, 1819]. Errors in a previous analysis on the physical meaning of noise resistance made by Chen et al. [13] is also analysed in the appendix of this paper.
The basic idea of the statistical linear polarisation [3, 18] is that the difference in random potential noise between two 'identical' electrodes (Figure 1) works as an internal polarisation voltage signal, producing current noise. That is, the current noise is driven by the difference of random potential noise. The polarisation is linear because the two electrodes are identical, and are thus supposed to have very similar (identical) corrosion potentials, and so polarisation is in the vicinity of the corrosion potential and in the linear polarisation zone. The noise resistance, which is deduced from statistical analysis of potential and current noise signals, is reasonably equivalent to linear polarisation resistance which is measured using an externally applied potential signal. Of course, the noise experimentally measured between working electrodes and reference electrode differs from this internal polarisation signal, but statistically they are well related. This is shown in the following mathematical derivation of noise resistance.
Figure 1. A schematic diagram showing the dual electrode system and noise measurement arrangement.
Electrodes a and b (Figure 1), are assumed to be identical and exist in the same environment. Voltage and current are recorded using high input impedance voltage and zero resistance currentmeters. It can be assumed that all electrochemical properties of the two identical electrodes are the same. For example, the electrodes have the same surface area, corrosion potential (Vcorr), corrosion current density (icorr), DC potential shift (VDC), Tafel slopes (ba,bc) and polarisation resistance (Rp) etc.. However, the electrochemical potentials of the two electrodes, at a certain time t, are not necessary identical due to random signals of electrochemical noise [20].
For electrode 'a', potential at time 't' can be written,
Va(t) = VDC(t) + Va,noise(t) 
(1) 
where VDC(t) is the DC component of the potential noise and Va,noise(t) is the potential noise of electrode 'a' at time 't'.
For electrode 'b', similarly
Vb(t) = VDC(t) + Vb,noise(t) 
(2) 
By substituting (2) into (1), the potential difference between electrodes 'a' and 'b' can be deduced:
Va(t)  Vb(t)= Va,noise(t)  Vb,noise(t) 
(3) 
Obviously, the difference between Va(t) and Vb(t) is due to the difference in random potential noise, which could produce current noise between electrodes 'a' and 'b', Inoise(t).
If the combined potential of the coupled electrode system at time t is Vsystem(t), which is determined using a high input impedance voltage meter simultaneously with current noise recording, and Va(t) < Vsystem(t) < Vb(t), i.e. at time t electrode a is anodically polarised and electrode b is cathodically polarised, the overpotential for electrode 'a', D Va(t), is
D Va(t) = Vsystem(t)  Va(t) 
(4) 
Similarly, the absolute value of the overpotential for electrode 'b', D Vb(t), is
� D Vb(t)� = Vb(t)  Vsystem(t) 
(5) 
The amplitude of electrochemical noise is very small, so D Va(t) and D Vb(t) are much smaller than 10 mV. If the corrosion reactions are totally activationcontrolled and corrosion is uniform, then polarisation is linear and the SternGeary equation can be used to calculate the corrosion current for each of the two electrodes. From electrode a, the corrosion current can be calculated by an equation,
icorr(t)= 
(6) 
where inoise(t) is the noise current at time t.
Similarly for electrode b, the corrosion current can be calculated by the equation,
icorr(t)= 
(7) 
Rewriting and combining equations (7) and (6),
D Va(t) + � D Vb(t)� = 2 � 
(8) 
The SternGeary equation can also be written as
icorr(t) = 
(9) 
Substituting (9) into (8) gives
[D Va(t) + � D Vb(t)]�] /2 = inoise(t) � Rp 
(10) 
Substituting (3), (4) and (5) into (10) gives,
[Vb,noise(t)  Va,noise(t)]/2 = inoise(t) � Rp 
(11) 
The standard deviations of both sides of the equation (11) during the testing period should be equal:
s {[Vb,noise  Va,noise]/2} = s {inoise � Rp} 
(12) 
At any time t, the potential measured by the voltage meter in Figure 1 is the combined potential of the coupled electrode system, Vsystem(t). After removing DC trend in the system potential, the noise component, Vnoise, system(t) can be deduced :
Vnoise, system(t) =Vsystem(t)  VDC, system(t) 
(13) 
Because the system is under linear polarisation and the anodic and cathodic polarisation have the same slope (1/Rp), the combined system noise potential should be the mean of the two individual electrodes. Vnoise, system(t) can be written as,
Vnoise, system(t) = [Vb,noise(t) + Va,noise(t)]/2 
(14) 
The standard deviations of both sides of the equation 14 during the testing period should be equal:
s Vnoise, system = s {(Vb,noise + Va,noise)/2} 
(15) 
Vb,noise(t) and Va,noise(t) are independent identical random signals. Statistically, it is expected that,
s {(Vb,noise  Va,noise)/2} = s {(Vb,noise + Va,noise)/2} 
(16) 
Substituting equations 12 and 14 into 16 gives,
s Vnoise, system = s {inoise � Rp} 
(17) 
The standard deviation of system noise voltage noise is
s Vnoise, system = 
(18) 
where n is the number of data points recorded during the testing period. m is the mean of potential in the time period, which should be zero in value because the potential noise is a random signal.
Similarly,
s {inoise � Rp} = = Rp � s inoise 
(19) 
Substituting equations 19 into 17 gives,
s Vnoise, system / s inoise = Rp 
(20) 

i.e. s V / s i = Rp 
(21) 
Equation 21 indicates that the ratio of the standard deviation of the potential noise and the standard deviation of current noise indeed equals to the linear polarisation resistance. In this theoretical analysis, several prerequisite conditions were suggested and used: (i). All electrochemical properties of the two identical electrodes are the same. (ii). Potential noise consists of random signals, and is small in magnitude, although its origin is not clearly known. (iii). Corrosion reactions are totally activationcontrolled and corrosion is uniform. (iv). DC components and shifts in potential and current records are removed. (v). Noise components in potential and current records are independent identical random signals.
This analysis explained the prerequisite conditions that the noise resistance technique is valid. For example it explained why the removal of DC components in potential and current noise records is important in the calculation of noise resistance. Equation (16) explained why the noise resistance is called statistical linear polarisation resistance.
This analysis suggests that the noise resistance is actually a special form of linear polarisation resistance. Obviously it has similar limitations to polarisation resistance techniques.
A theoretical analysis on the electrochemical noise resistance is carried out based on basic electrochemical corrosion theory. It is proved that the noise resistance is equivalent to the polarisation resistance and is in fact a special form of linear polarisation resistance, namely statistical linear polarisation resistance. The difference in random potential noise between two 'identical' electrodes works as an internal polarisation signal which linearly polarises both electrodes, producing current noise.
A theoretical analysis on the electrochemical noise resistance has been made by Chen and Bogaerts in a paper named 'The physical meaning of noise resistance' [13]. Figure 2 shows the configuration of the working electrodes used in that theoretical analysis. Working electrodes 1 and 2 are identical working electrodes. Vt is the instantaneous corrosion potential of the two working electrode system at time t (i.e. the voltage noise) which is measured by a high resistance voltage meter (via a reference electrode). It is the instantaneous galvanic current which is measured by a zero resistance ammeter.
Figure 2. The assembly of the working electrodes in the analysis of noise resistance [13]
Chen and Bogaerts analysed working electrode 1 according to the ButlerVolmer equation and charge conservation law under three assumptions [13]. At any time t, currents entering and leaving the working electrode 1 were related by an equation (equation 5 in the reference 13):
(i) 
However, a contradiction arises when similar analysis applies to working electrode 2.
The working electrode 2 is identical to the working electrode 1. It is reasonable to assume that at time t electrode 2 behaves the same as electrode 1, i.e. electrode 2 has the same electrochemical equilibrium potentials (Ve,a, Ve,c), the same exchange current densities (io,a, io,c), the same Tafel parameters (ba, bc) and the same anodic and cathodic areas (Aa, Ac) etc.. At time t and under potential Vt, current entering and leaving the working electrode 2 should also be related by a similar equation:

(ii) 
Equations (i) and (ii) are basically identical. It and I't are equal in value and different in direction (as shown in Figure 2). In this case, if electrode 1 is anodically polarised (It > 0), then electrode 2 is also anodically polarised (I't > 0) and so It + I't > 0. Obviously the dual electrode system will not be balanced in electric charge because there is no auxiliary electrode to absorb (or provide) electrons from (or to) the system. This system is against the charge conservation law.
The analysis in reference 13 can not address this problem. The reason causing this problem is that reference 13 did not consider the two identical electrodes as a whole system. In that analysis, electrode 2 was actually treated as an auxiliary electrode to absorb (or provide) electrons. So the analysis and conclusion in the reference 13 is irrelevant to the electrochemical noise of a dual electrode system. The analysis and conclusion in reference 13 applies only to a single electrode which is under small polarisation.
1. D. A. Eden, K. Hladky, D. G. John and J. L. Dawson, Corrosion 86, Paper 274, NACE (1986)
2. F. Mansfeld and H. Xiao, J. Electrochem. Soc., vol.140, p2205 (1993)
3. Y. J. Tan, B. Kinsella and S. Bailey, Corrosion 96, paper 352, NACE (1996).
4. Y. J. Tan, S. Bailey and B. Kinsella, Corros. Sci. vol.38, 1681 (1996)
5. U. Bertocci et al, J. Electrochem. Soc., vol.144, p37 (1997)
6. Y. J. Tan, S. Bailey and B. Kinsella, British Corrosion Journal, vol.32, no.3 (1997)
7. J. L. Dawson, D. A. Eden and R. N. Carr, United States of America Patent No. 5425867
8. H. Xiao and F. Mansfeld, J. Electrochem. Soc., vol.141, p2332 (1994)
9. U. Bertocci et al, J. Electrochem. Soc., vol.144, p31 (1997)
10. F. Mansfeld and C. C. Lee, Corros. Sci., vol.39, p1141 (1997)
11. D. A. Eden and A. N. Rothwell, Corrosion 92, Paper 292 NACE (1992)
12. G. P. Bierwagen, J. Electrochem. Soc., vol.141, L155 (1994)
13. J. F. Chen and W. F. Bogaerts, Corros. Sci., vol.37, p1839 (1995)
14. R. A. Cottis, S. Turgoose and J. Mendoza, Electrochemical noise measurement for corrosion applications, ASTM STP 1277, J. R. Kearns et al. (ed.), p93 (1996)
15. F. Huet, J. Electrochem. Soc., 142, p2861 (1995)
16. F. Mansfeld and H. Xiao, J. Electrochem. Soc., vol.141, p1403 (1994)
17. R. A. Cottis, G. Bagley and M. M. AlAnsari, Proc. Electrochemical Noise Analysis Session, Research in Progress Symposium, Corrosion 98, NACE (1998)
18. Y. J. Tan, Ph.D thesis, Curtin University of Technology, Australia, pp6978 (1996)
19. Y. Tan, S. Bailey and B. Kinsella, Corros. Sci., vol.40, p513 (1998)
20. W. P. Iverson, J. Electrochem. Soc., vol.115, 617 (1968)