Download Calculation of Noise Resistance from Simultaneous Electrochemical

J. E l e c t r o c h e m .
Soc.,
Vol. 141, No. 11, November 1994 9 The Electrochemical Society, Inc.
During electrolysis, confocal images were taken at time intervals
of 5, 10, 30, 60, 180, 300, and 600 s. The four time sequential
images of 0, 10, 60, and 120 s shown in Fig. 3 are single confocal
slices, recorded at an intermediate depth across the rough specimen surface. The contrast mechanism is reflected light intensity, so
that high spikes in these images correspond to positions where the
focus of the confocal microscope coincides with the surface of the
specimen. The width of the peaks in these images is a measure of
the granularity of the specimen surface, and there is clearly no
large change in that parameter in these four images. The average
height of the peaks in these four images is a measure of the overall
reflectivity of the surface as the process proceeds, and the reduction of average peak height corresponds to the change from a bright
copper surface at 0 s, to a dull brown at 120 s. The scanned area
in these images is slightly larger than the area of the electrode, and
the flat area at the edges corresponds to the Teflon ring around the
electrode. Signals emanating from it are excluded from the confocal
images.
Figure 4 shows a depth profile of the surface after electrolysis,
using the same procedure and contrast mechanism as in Fig. 2.
The surfacejs significantly smoother than that shown in Fig. 2, with
a roughness range of 0 to 45 #m.
Conclusions
The experimental results demonstrate the power of CSBLM in
the monitoring of electrode surfaces during electrolysis and in
providing quantitative 3-D information about surface roughness.
Low laser light intensity and the absence of probes virtually eliminates externa~ influence.with the surface process(es). The tech-
L155
nique appears to be very promising in microstructure-controlled
surface preparation processes.
Acknowledgments
This research was supported by the Natural Sciences and Engineering Research Council of Canada. Software for postprocessing
and the display of images was provided by Waterloo Scientific
Incorporated, Canada.
Manuscript submitted May 16, 1994; revised manuscript received Aug. 22, 1994.
The University of Waterloo assisted in meeting the publication
costs of this article.
REFERENCES
1. A. G. Martin, R. C. Bhardwaj, and J. O'M. Bockris, J. AppL
Electrochem., 23, 531 (1993).
2. S. L. You, M. Arendt, A. J. Bard, B. Evans, C. Tsai, J. Sarathy,
and J. C. Campbell, This Journal, 141,402 (1994).
3. Images of Materials, D. B. Williams, A. R. Pelton, and R. Gronsky, Editors, Oxford University Press, Inc., New York (1991).
4. A. E. Dixon, S. Damaskinos, M. R. Atkinson, and L. H. Cao, in
Scanning Microscopy Instrumentation, Pro& SPIE, 1556,
G. S. Kino, Editor, p. 144 (1992).
5. A. E. Dixon, S. Damaskinos, and M. R. Atkinson, Nature, 351,
551 (1991).
6. The Electrolytic and Chemical Polishing of Metals, W. J. McG.
Tegart, Pergamon Press, Inc., Elmsford, New York (1959).
Calculation of Noise Resistance from Simultaneous
Electrochemical Voltage and Current Noise Data
Gordon R Bierwagen*
Department of Polymers and Coatings, North Dakota State University, Fargo, North Dakota 58105
ABSTRACT
A derivation of the noise resistance appropriate to locally stationary electrochemical noise measurements is presented. The equation
for noise resistance, R~ = ~v/~, where the ~v and cr~values are the standard deviations of the voltage and current noise signals, is shown
to be derivable from the low frequency limit of spectral density functions for the potential and current noise. The results are especially
pertinent to the study of corrosion in organic coatings/metal substrate systems.
There has been considerable use of a derived noise resistance,
Rn, for the experimental study of corrosion by electrochemical
noise (ECN) methods. ~ In many of the analyses of ECN data in
corroding systems, especially in organic coating/metal substrate
systems, a statistically derived "noise resistance," R~, is used in
interpreting the results from the time records of simultaneously
measured voltage (measured as the potential difference developed
between the two electrodes) noise, Vn(t), and current noise,/,(t). 2-8
Values for R~ are calculated from these time records as Rn = ~v/cr~,
where the ~v aod ~ values are the standard deviations of the
voltage (potential difference) and current noise signals, respectively, from their means <V> and </> as measured over a time
interval | Rn values are often identified with the resistance to
charge transfer or the passage of current of a coating on the surface of a corroding metal. 3-6 This result seems intuitively correct in
that it is an expression that follows Ohm's law, and all of the data
thus measured using this relationship seem to track with other
measures of coating resistance, such as dc resistance and the zero
frequency limit of the coating impedance, Z(0) = R. We have been
using this value also in our studies of coated-metal systems, 9'1~but
we have not been totally satisfied with the intuitive, Ohm's law
arguments for the use of Rn presented in Ref. 1, or for that matter,
in any of the references we have examined.
To this end, the literature on noise and other random processes
was reviewed with respect to developing a properly derived expression for Rn. 11-14We present a derivation which is based on a
* Electrochemical Society Active Member.
first-principles analysis of noise phenomena and is an improvement
on the present derivations and use of Rn in the present ECN
lite ratu re. 1-10
If we have a noise signal x described by a continuous probability
density function f(x) that is stationary, its moments <xm>, defined
by
<xm> = f==xmf(x)dx
[1]
have the characteristic of being time independent. If we let <x> be
the mean of xand <x2> the mean of x z, the variable (x - <x'>) has
a mean of zero, but its root mean square, called the (standard
deviation) squared or variance of x, equal to
cr~ = <x2> - <x> 2
[2]
is nonzero. If we let x(t) be the value of the noise at time t, and x(t +
-r) as its value at a time (t + -r), the autocorrelation function Ax(~r) of
x is given as ~2
Ax('~) = f ~ x(t)x(t + v)dT = <x(t)x(t + "r)>
[3]
The spectral intensity function (or power spectral density function)
Sx(e) for such a situation is known to be related to Ax(T) by the
Weiner-Khintchine theorem ~2'15
Sx(o~) = ~1 ~=~ Ax(T)e_i,oTdr = 2 f l ~Ax(~) cos (e-~)d~
[4]
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J. Electrochem. Soc., Vol. 141, No. 11, November 1994 9 The Electrochemical Society, Inc.
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The requirements for using this derivation are consistent with the
manner in which the data are acquired by Skerry and co-workers, 2-4
and to the way our group takes its data. 9'1~ Exactly the same expression for R~ for this circuit can be derived in a similar manner
from Eq. 27 through 31 of chapter 13 of Pathria. 12
If we do not use the dc offset data, we have a modified noise-resistance parameter, R,~, given by
R
~
m
~r~ + < V>2
I.
v(t)
/
Fig, 1. LR circuit for Nyquist noise estimation.
with the latter true since A~(T) is an even function. Equation 4 can
be used as the basis for the derivation given below, and could
actually be the starting point of this work, but we included the
previous discussion and definitions of terms for the sake of clarity
to those working in this area.
This also implies from Eq. 4 (see Ref. 15, Eq. 10.40)
[5]
where the modified noise resistance R,~ (defined by Eq. 10A), is
now a function of the squares of the average potential difference
and current as well as their variance. It, however, makes clear the
case of cathodic protection where an imposed potential is in place
to inhibit corrosion reactions. In the case of large mean potential
and/or mean current, we see their effects on the resistance as
measured by noise analysis. If part of the noise circuit under study
allows high current to flow under low potential, i.e. the equivalent of
the measurement of a coated metal system with a pinhole in one
part of the coating, the equivalent noise resistance will be "low" if
the dc-set data are not considered, but high if the dc-offset data are
considered. Thus the Rn value from electrochemical noise measurements as calculated by Eq. 10 will be most heavily influenced by
the intact coating, while the R~ as calculated by Eq. 10A will be
strongly affected by the imperfections and holes in the coating. The
absolute value of the difference, ARN, between these noise resistance values
z~a, = IRn - R*I
One can also write for a specific time interval @ that ~
'fi'
<x> e = ~
x(t)dt
[6]
For a stationary noise source, we have
S~(0) = lim 2e<x2>e = 2@(0-~ + <x>~)
@-->oo
[7]
where 0-e is used as was 0- in Eq. 5.
Thermal noise in electrical circuits, originally modeled by a rhythmic lamellar (RL) circuit, was the situation first considered by these
methods, initially by Nyquist. ~ For these historical reasons, and
because of the fact that this circuit is the most studied in the noise
literature, we consider this circuit first, even though it might not be
exactly representative for corrosion. Based on this circuit model, we
can write the following expression ~1'~2for the inductance L in the
equivalent circuit shown in Fig. 1 as
Sx(o~ ) -- ~
2
Sv(0)
R~ = Si(0)
as calculated by the two equations may thus give an estimate of the
degree of imperfections in a coating. This will be examined in future
experiments in our laboratory.
The thermal-noise circuit of Fig. 1 is not directly appropriate
to corrosion studies, but Peebles ~7 extended this analysis to the
circuit shown in Fig. 2, the Randles equivalent circuit often used
in the modeling of corrosion processes, la In his analysis, only thermal noise is considered. However, one can extend Peebles'
arguments to the specific situation of Fig. 2 and calculate Rn value
from the zero frequency spectral power densities for this equivalent
circuit as
Rn = ~ =
[Sv(0)]
[S,(O) J
,,,-,olimL',
+
R
ct~
2
~ ] = (Rs + R~tg)2 [12]
[8]
where S~(co) and Sv(0) are the spectral density functions for the
current noise and voltage noise, respectively, and R~ is the noise
resistance of this circuit, the specific quantity we are seeking to
derive and the parameter we have defined above. This circuit is
also considered by Pathria ~2 in a similar noise analysis and is the
basis for the work of Nyquist for the thermal noise due to a resistor.
This implies we can write for the low-frequency limit
Sv(0) -->
[11]
s~(0)
R~+ ~ 2 L 2
S,(0) - ~
[10A]
R~2 = o7 + <t>2
2 0 ( ~ + <V> 2) _ 2 + <V>2
20(m2 ~ < ~
- 0-~ + </>2 [9]
since the Os cancel. If we look at the dc-offset measurements of V
and I (Le., subtracting the dc current / [=average current], and dc
potential difference V [=average potential difference] values from
the measured ! and V values) in electrochemical noise measurements of both potential and current, we have < V> = </> = 0 while
leaving the 0-v and o:~ values unaffected in Eq. 10 [see also Bendat, ~6 pp. 94-5). We will then have the result
20{r~
R~2 = ~
Conclusions and Future Work
A derivation of the noise resistance, Rn, appropriate for use in
analyzing electrochemical noise data is presented, and the appropriate assumptions for its calculation are described. The derivation
is based on concepts developed in the studies of "noise" in electrical circuits and extended to the case of the electrochemical noise.
It is shown that Rn can be calculated from the ratio of the standard
deviation of the potential noise divided by the standard deviation of
the current noise for locally stationary noise data. We intend to
extend our analytic considerations along these lines, as recent
Vlt)
[10]
or
R n ~
v
0- I
which is the desired result, the equation quoted by Eden et al., 1 and
used by most workers in ECN studies of coatings, but never derived
properly. It is the low-frequency limit result for a stationary noise
source. This latter requirement is locally satisfied for a system that
is stationary for the data acquisition interval used in measuring the
voltage and current noise time series used in computing 0-v and 0-~.
II-
i
Fig. 2. Equivalent circuit for corrosion noise in coated system. Rs = solution resistance, R~tg = coating resistance, C =
double-layer capacitance.
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J. Electrochem. Soc., Vol. 141, No. 11, November 1994 9 The Electrochemical Society, Inc.
studies indicate the value of such methods, 19as well as the usefulness of extended use of power-spectral-density analysis to ECN
data.2~ Equations 10 and 10A constitute a set of equations to
extract meaningful low-frequency results from noise data, especially noise-resistance data appropriate for the study of the corrosion protective properties of coatings. Equation 10 seems appropriate for extracting information on the properties of hole-free coatings
and Eq. 10A seems appropriate for the study of actively corroding,
failing, coatings with open imperfections in their structure. Equation 12 may be a term for estimating the degree of imperfections in
a coating contributing to corrosion.
L157
5. C. T. Chen and B. S. Skerry, Corrosion, 47, 598 (1991).
6. F. Mansfeld and H. Xiao, This Journal, 140, 2205 (1993).
7. F. Mansfeld, H. Xiao, and L-T. Han, Abstract 80, pp. 138-139,
The Electrochemical Society Extended Abstracts, Vol. 93-2,
New Orleans, LA, Oct. 10-15, 1993.
8. E. M. Farrell, W. Cox, R. H. Stott, D. A. Eden, J. L. Dawson, and
G. C. Wood, High Temp. Tech., 3, 15 (1985).
9. G. P. Bierwagen, D. J. Mills, and D. E. Tallman, in Proceedings
of 12th International Corrosion Congress, Vol. 6, pp. 42084218, Paper 576, Houston, TX (1993).
10. D. J. Mills, G. P. Bierwagen, D. E. Tallman, and B. S, Skerry, in
Proceedings of 12th International Corrosion Congress,
Acknowledgments
This work was partially supported by the Office of Naval Research under Grant No. N00014-93-10013, A. J. Sedriks, Program
Officer. The helpful criticisms of Dennis Tallman, Brian Skerry, Douglas Mills, and the reviewers are also acknowledged.
Manuscript submitted March 18, 1994; revised manuscript received Aug. 5, 1994.
North Dakota State University assisted in meeting the publication
costs of this article.
REFERENCES
1. D.A. Eden, K. Hladky, D. G. John, and J. L. Dawson, Paper 274
presented at CORROSION~86, NACE, Houston, TX, March
1986.
2. D. A. Eden, M. Hoffman, and B. S. Skerry, ACS Symposium
Series, Vol. 322, Polymeric Materials for Corrosion Control,
R. A. Dicke and F. L. Floyd, Editors, American Chemical
Society, Washington, DC (1986).
3. B. S. Skerry and D. A. Eden, Prog. Organic. Coatings, 15, 269
(1987).
4. B. S. Skerry and D. A. Eden, ibid., 19, 379 (1991).
Vol. 1, pp. 182-193, Paper 486, Houston, TX (1993).
11. A. Van Der Zeil, Noise-Sources, Characterization, Measurement, Prentice Hall, Englewood Cliffs, NJ (1970).
12. R. K. Pathria, Statistical Mechanics, Pergamon Press, Elmsford, New York (1972).
13. S. L. Marple, Digital Spectral Analysis, Chap. 4, pp. 111-129,
Prentice Hall, Englewood Cliffs, NJ (1987).
14. A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal
Processing, Prentice Hall, Englewood Cliffs, NJ (1989).
15. M. K. Ochi, Applied Probability and Stochastic Processes in
Engineering and Physical Sciences, Wiley, New York (1990).
16. J. S. Bendat, Principles and Applications of Random Noise
Theory, Wiley, New York (1958).
17. P. Z. Peebles, Probability, Random Variables, and Random
Signal Principles, McGraw-Hill, New York (1980).
18. D. D. Macdonald and M. C. H. McKubrie, in Impedance Spectroscopy, J. R. Macdonald, Editor, Chap. 4.3, p. 301, WileyInterscience, New York (1987).
19. M. Moon and B. S. Skerry, J. Coatings Tech., Accepted for
publication.
20. C. Gabrielli, F. Huet, and M. Keddam, J. Chem. Phys., 99, 7232
(1993).
21. C. Gabrielli, F. Huet, and M. Keddam, ibid., 99, 7240 (1993).
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