fractional noise electrochemical

8/19/2019 fractional noise electrochemical

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Journal of Vibration and Control

DOI: 10.1177/10775463070874382008; 14; 1443Journal of Vibration and Control

Yangquan Chen, Rongtao Sun, Anhong Zhou and Nikita Zaveri

Fractional Order Signal Processing of Electrochemical Noises

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Fractional Order Signal Processing of

Electrochemical Noises

YANGQUAN CHENRONGTAO SUNCenter for Self-Organizing and Intelligent Systems, Department of Electrical and Computer Engineering, Utah State University, Logan, UT 84322-4120 USA ([email protected])

ANHONG ZHOU

NIKITA ZAVERI Department of Biological and Irrigation Engineering, Utah State University, 4105 Old Main Hill, Logan, UT 84322-4105 USA ([email protected])

(Received 14 December 20051 accepted 4 October 2006)

Abstract: The corrosion processes of stainless steel under different solutions were examined using electro-

chemical noise (ECN). Using rescaled range analysis, we demonstrated that ECN signals produced by corro-

sion processes have non-stationary and self-similar properties. The comparison and analysis of ECN signals

in both the time and frequency domains showed that conventional methods failed to sufficiently distinguish

between the ECN signals obtained under different solutions. Therefore, we introduced the use of fractional

Fourier transforms, a powerful tool for the time-frequency analysis of self-similar signals, to process ECN

signals that can better describe the corrosion behaviours of the electrode in different solutions.

Keywords: Electrochemical noise, stainless steel, self-similar signals, rescaled range analysis, fractional Fourier

transform, spectral noise impedance

1. INTRODUCTION

One of the most important properties of any material, which is used as a bioimplant is safety.

Metals and alloys are widely used as biomedical materials in medical and dental devices, and

the biocompatibility of a metallic alloy is closely associated with the interaction of the alloy

with the surrounding environment. Metal release from the implant into the surrounding tissue

may occur as a consequence of various mechanisms, which may have either a mechanicalnature (for example, due to wear phenomena) or an electrochemical nature (such as corrosion

processes). The implantation of a metal object into the body inevitably leads to some degree

of local tissue response and, depending on the material utilized, can also induce a reaction

in cells distant from the site of the surgery. These reactions may be merely moderate or

transient, but in more severe cases, serious tissue damage with permanent morphological

and structural changes can occur.

Journal of Vibration and Control, 14(9–10): 1443–1456, 2008 DOI: 10.1177/1077546307087438112008 SAGE Publications Los Angeles, London, New Delhi, SingaporeFigures 1–5, 7 appear in color online: http://jvc.sagepub.com


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1444 Y. CHEN ET AL.

Corrosion is defined as a chemical or an electrochemical reaction between a material,

usually a metal, and its environment, that produces a deterioration of the metal and its prop-

erties.

Stainless steel was identified early on as a suitable material for orthopaedic implants.

It remains one of the most frequently used biomaterials for implants to the present day, be-

cause of its well-suited mechanical properties and excellent clinical track record. In addition,

stainless steel has very low corrosion resistance and has low production costs.

Electrochemical noise (ECN) is a technique whereby biocorrosion information is ex-

tracted from fluctuations in either potential or current, observed on a corroding electrode.

In this study, the corrosion behaviors of the stainless steel electrode in three artificial saliva

solutions were studied using the zero resistance ammeter (ZRA). In the ZRA measurement

configuration, two electrodes, the working electrode (WE) and the counter electrode (CE),

which are identical in construction (i.e., materials and size) are immersed in the solution of

interest. The fluctuation of the potential of the WE and CE versus the reference electrode(RE) is measured, as well as the coupling current between the WE and CE. The ZRA is sim-

ply a current to voltage converter, giving a voltage output proportional to the current flowing

between its two input terminals while imposing a ’zero’ voltage drop to the external circuit.

The ZRA is an application for the measurement of the galvanic coupling current of dissimilar

metals. Here, the coupling current is measured between two stainless steel electrodes.

Figure 1 shows an example of electrochemical noise responses obtained from a stainless

steel electrode which was exposed to the artificial saliva solution for 5 minutes. This ECN

data typically consists of three sets of measurements: The corrosion potential of the work-

ing electrode (WE), the corrosion potential of the counter electrode (CE), and the coupling

current between the WE and CE. In this study, we use the potential of the WE for signalprocessing.

A stochastic model of this type of ECN signal, obtained from the electrolysis current

during bubble evolution, was reported by Gabrielli et al. (1985). The experimental power

spectral density (PSD) was in agreement with the theoretical model. Therefore, the PSD of

the fluctuations generated by a stainless steel electrode at the corrosion potential can be used

for measurement of the corrosion rate.

Existing empirical ECN analysis methods, using statistical or Fourier spectral methods

(Gabrielli et al., 1985), were developed under the assumption that the stochastic model has

a Gaussian distribution. However, these signals generally have significant impulsion in theirwaves in the time domain, and their autocorrelation may have thick tails, which are typical

self-similar properties. This article presents rescaled range (R/S) analysis to show the self-

similar property of ECN signals of stainless steel. Hurst (1951, 1965) developed rescaled

range analysis as a statistical method to analyze long records of natural phenomena.

Recently, the fractional Fourier transform has been found to be a powerful tool for time-

frequency analysis of self-similar signals. It has been successfully used in a range of appli-

cations, such as optical systems and optical signal processing (Ozaktas et al., 1994), swept-

frequency filters (Almeida, 1994), time-variant filtering and multiplexing (Ozaktas et al.,

1994), pattern recognition (Mendlovic et al., 1995), and the study of time-frequency distrib-

utions (Fonollosa and Nikias, 1994). The fractional Fourier transform algorithm used in thisarticle was obtained from the work of Ozaktas et al. (1996). For signals with time-bandwidth

product N, this algorithm computes the fractional transform in O ( N log N ) time.




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FRACTIONAL ORDER SIGNAL PROCESSING OF ELECTROCHEMICAL NOISES 1445

Figure 1. An example of ECN measurement. The top plot is the potential noise of the WE electrode, the

middle plot is the potential noise of the CE electrode, and the bottom plot is the corresponding coupling

current between WE and CE. This example shows Tomasi’s artificial saliva solution, with stainless steel

electrodes.

This article is based on previous work (Sun et al., 20061 Sun, 2007), and is organized as

follows: Section 2 describes the experimental approach, and Section 3 exploits R/S analy-

sis to show the self-similar properties of the signals. In Section 4, experiments in both the

time and frequency domains are presented. Corrosion rates in three different solutions are

successfully described using the fractional Fourier transform. Conclusions are given in Sec-

tion 5.

2. EXPERIMENTAL APPROACH

2.1. Experimental Setup

Stainless Steel was used for both the working and counter electrodes. Polishing pads were

used to clean the stainless steel surface before the start of each experiment. The electrodes

were then thoroughly rinsed off with distilled water before being made ready for use. An

Ag/AgCl reference electrode (CH Instruments, TX) was used. All measurements were per-

formed at room temperature.

A VMP2/Z (PAR, TN) electrochemical testing station was used for ECN measurement.The zero resistance ammeter technique was a built-in function of the multichannel poten-

tiostat used. Parameters of the ECN measurement experiments are given in Table 1. ECN

measurements were conducted for 30 minutes for each of the three solutions used.




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1446 Y. CHEN ET AL.

Table 1. Experimental conditions used.

Software used EC-Lab for windows v9.01

CE vs. WE compliance range –10 V to +10 V

Electrode connection StandardElectrode surface ares 0.001 cm2

tR1 (h:m:s) 0 : 00 : 10.0000

dtR1 (s) 0.5000

ti (h:m:s) 0: 30 : 0.0000

I range Auto

Bandwidth 5 MHz

Table 2. Chemical composition of Jenkin’s artificial saliva solution (solution A).

Constituents grams/250mL

NaCl 0.2125

KCl 0.3000

CaCl2.2H2O 0.0375

MgCl2.6H2O 0.0125

K2HPO4 0.0875

KSCN 0.0250

NaF 0.0025

H2O2 0.0750

Sorbic Acid 0.0125

Table 3. Chemical composition of Tomasi’s artificial saliva solution (solution B).


NaCl 011685

KCl 012400

CaCl2.2H2O 0102925

MgCl2.6H2O 01010125

K2HPO4 0102275

Table 4. Chemical composition of the NaCl artificial saliva solution (solution C).


NaCl 215

2.2. Test Solutions

Three different types of simulated saliva solutions were used for the ECN measurement. Theconstituents of each solution are listed in Tables 2 to 4.




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Table 5. The log(R/S), fractional dimension, and Hurst parameter values for the three

different solutions, obtained by R/S analysis.

Solution A Solution B Solution C

log(R/S) 214114 213936 214502D 110919 111532 111660

H 019081 018468 018340

3. RESCALED RANGE ANALYSIS

Self-similar random processes were proposed by Mandelbrot et al. (1968) for modelling

of long-term behavior. A rescaled range ( R / S ) statistics method was subsequently proposed

(Feder, 1988) for evaluation of the Hurst exponent ( H ) in order to identify the occurance of

self-similar properties. It was shown that 015 2 H 2 1 indicates the presence of self-similarproperties, and when 0 2 H 2 015, there is antipersistence.

Two factors are utilized in this analysis. First, the range R, which is the difference

between the minimum and maximum ’accumulated’ values or cumulative sum of X 3t 4 5 6 for

the natural phenomenon at discrete integer-valued time t over a time span 5 . Second, the

standard deviation S , estimated from the observed values X i3t 6. Hurst found that the ratio

R7S is described for a large number of natural phenomena by the empirical relationship

R7S 2 3c 3 56 H (1)

where 5 is the time span, and H is the Hurst exponent. Hurst set the coefficient c equal to

0.5. R and S are defined as

R356 2 max14t 45

X 3t 4 5 65 min14t 45

X 3t 4 5 6 (2)

and

S 21

1

5

5

2t 51 383t 65 68 754

2

5 12

(3)

where

68 75 21

5

52t 51

83t 6 (4)

and

X 3t 4 5 6 2t

2u51883u65 68 7 5 9 1 (5)

The relationship between the Hurst exponent and the fractal dimension is simply D 225 H . Table 5 gives the Hurst parameters of the ECN data for the three solutions used here.




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1448 Y. CHEN ET AL.

It can be seen that all three are between 0.5 and 1, from which we can tell that the ECN

signals produced by the corrosion process in stainless steel have self-similar properties.

4. TIME-FREQUENCY ANALYSIS OF ECN SIGNALS

4.1. Time Domain

We determine the following factors for the data produced using each solution: Mean, vari-

ance, skewness, kurtosis, and noise resistance. The mean of the potential measurements is

equal to

E 2

1

N

N

2k 51

E [k ] (6)

where E [k ] is the potential value.

Variance is a measurement of the average AC power in the signal, sometimes referred to

as noise power, and determined as

S 2 1 N

N 2k 51

3 E n [k ]62 1 (7)

Skewness is a non-dimensional measurement of the symmetry of a distribution. A zerovalue means that the distribution is symmetrical about the mean. A positive value indicates

that there is a tail in the positive direction and a negative value implies the presence of tail in

the negative direction. A time record consisting of unidirectional transients will typically be

heavily skewed, and this may be useful for detection of transients associated with metastable

pitting. The skewness is defined as

skewness 2 1n

N 2k 51

1 E n [k ] 5 E

6 E n [k ]2

531 (8)

Kurtosis is a measurement of the extent to which the data are peaked or flat, relative to a

normal distribution. Data sets with higher kurtosis tend to have a more distinct peak near the

mean, decline rapidly, and have heavy tails. Data sets with lower kurtosis are flatter near the

mean, rather than having a sharp peak. Positive kurtosis indicates a “peaked” distribution,

and negative kurtosis indicates a “flat” distribution.

kurtosis 2 1n

N 2k 51

1 E n [k ] 5 E

6 E n [k ]2

541 (9)

Noise resistance can according to Cottis and Turgoose (1999) be used to yield a corrosion

rate measurement with the LPR, EN, and EIS techniques. These resistances are related to

the corrosion rate by the Stern-Geary linear approximation to the Butler-Volmer equation,




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Table 6. Four statistical components: Mean, Variance, Skewness, Kurtosis and the noise

resistance of the three different solutions.

Solution A Solution B Solution C

Mean 012388 5013192 5013808Variance 113896 1054 212726 1055 411023 1056Skewness 5114296 118382 014142Kurtosis 319443 417590 215538

Noise resistance 2215420 2014687 2613401

R p 2 Rn 29 E

9iapplied 2 a c

21303 3icorr 6 7 a c8 2 B

3icorr 6 (10)

where R p is a polarization resistance obtained from the LPR and EIS techniques, Rn is a

reaction resistance obtained from the EN technique, 9E is the incremental change in poten-

tial measured due to the incremental change in applied current density (9iapplied ), B is the

Stern-Geary constant, a and c are the anodic and cathodic Tafel constants, respectively,

and icorr is the corrosion current density, from which a corrosion rate may be calculated using

Faraday’s law. The Stern-Geary constant (determined from the Tafel constants) is the only

variable that is normally not measured, but is commonly assumed to have a value between

0.020 and 0.030 V/decade.

The electrochemical noise resistance can be obtained as

Rn 2V

I (11)

where V and I are the standard deviations of potential and current, respectively, for a

given time record.

The mean value of the potential noise in Table 6 indicates that the potential noise was

greater in solution B than in solution A, and greatest in solution C. The variance values in-

dicate that the noise power showed the opposite pattern (highest for solution A to lowest for

solution C), which means the corrosion rate decreases from solution A to solution C. The

skewness shows that solution A has a negative tail, while solutions B and C have positivetails. The kurtosis values show that all the potential noises are “peaked” distributions, mean-

ing they have large fluctuations. The noise resistance, derived by the conventional method,

shows that solution B has the smallest noise resistance, which indicates the highest corro-

sion rate. The conclusions from the noise resistance and variance values are contradictory1

therefore, we cannot tell the corrosion rate using these conventional methods.

4.2. Frequency domain

The Fractional Fourier transform has complexity similar to the fast Fourier transform algo-

rithm. In self-similar random process applications, it is possible to improve performance by

the use of the fractional Fourier transform instead of the ordinary Fourier transform. Since

the fractional transform can be computed in about the same time as the ordinary transform,




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1450 Y. CHEN ET AL.

these performance improvements come without additional cost. In some cases, filtering in a

fractional Fourier domain, rather than the ordinary Fourier domain, allows one to decrease

the mean square error in estimating a distorted and noisy signal.

The fractional Fourier transform is used to determine the power spectrum density of the

potential noise. The at h-order fractional Fourier transform 8F a f 93 x 6 of the function f 3 x 6can be defined for 0 2 a 2 2 as

F a 9

f 3 x 6 8F a f 9 3 x 6

5 Ba3 x 4 x

6 f 3 x 6d x (12)

Ba3 x 4 x 6 A exp

i3 x 2 cot 5 2 x x csc x 2 cot (13)

A exp35i sgn3sin674 i726

sin12

(14)

where

2 a2

(15)

and i is the imaginary unit.

The kernel approaches B0 3 x 4 x 6 3 x 5 x 6 and B2 3 x 4 x 6 3 x x 6. This

definition is easily extended beyond the interval [524 2] by making use of the facts that F 4 jis the identity operator for any integer j, and that the fractional Fourier transform operator is

additive in index, that is, F

a1

F a2

2 F a1a2

. The Hermite-Gaussian functions

F a 9 n 3 x 6

2 e 5ian2 n 3 x 6 (16) n 3 x 6 2

214

2nn! H n

2 x

exp

75 x 28 (17)where H n3 x 6 is the n

t h-order Hermite polynomial, are a complete set of Eigenfunctions of

the fractional Fourier transform. The spectral expansion of the linear transform kernel is

Ba7 x 4 x

8 2 2n20

e5ian

2 n 3 x 6 n7 x 81 (18)

Second and higher dimensional transforms have separable kernels, so that most results easily

generalize to higher dimensions (Lohmann, 19931 Ozaktas and Mendlovic, 1993 a,b).

Figure 2 shows that there is an increase in corrosion potential of approximately 0.1 V

between solutions, with solution A having the highest corrosion potential. Figure 3 provides

a qualitative comparison between the PSD of potential noise calculations for conventional

fast Fourier transforms and fractional Fourier transforms.

The fast Fourier transform (FFT) of the potential noise shown in Figure 2 is depicted inFigure 3, while Figure 4 shows the fractional Fourier transform (FrFT). There is an increase

in the magnitude of potential noise from solution A to solution C, which is reflected in the

FrFT plot. It is hard to recognize the increase between solutions A and B in the FFT plot, and




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Figure 2. Potential noise of stainless steel WE tested in three different solutions for 30 minutes.

the FFT spectrum has unexpected fluctuations. Figure 4, however, we can clearly see that

the magnitude of the FFT of potential noise in solution C is larger than that of solution B,

while the magnitude of the FFT of potential noise in solution B is larger than that of solution

A. Thus, the changes in the ECN signals (due to an increase in the rate of corrosion) resulted

in a decrease in the magnitude of the FFT.

Figure 5 shows the potential noise power spectrum as determined using different values

of a in the fractional Fourier transform. We can see that the passband becomes shorter as

a increases from 0.1 to 0.8. The plot also shows that the potential noise is dominated by

low frequency components1 this is because it is usually the low frequency information that

is useful for noise impedance computations. A rough estimate of noise impedance can be

obtained from the fractional Fourier transform diagrams by comparing the magnitude.

4.3. Spectral Noise Impedance

Figure 6 shows the equivalent circuit of a simultaneous ECN measurement, and is described

by Eden et al. (1986). Z 13 f 6 and Z 23 f 6 are the electrochemical equivalent impedances of the

two electrodes, 13t 6 and 23t 6 are the Thevenin equivalent EN sources associated with each

electrode, and 3t 6 and i 3t 6 are the measured potential noise and current noise, respectively.

Bertocci et al. (1997) defined the spectral noise impedance as

Rsn 3 f 6 2

S 3 f 6

S i 3 f 6 (19)




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1452 Y. CHEN ET AL.

Figure 3. FFT power spectrum of potential noise for stainless steel WE in the three different solutions.

Figure 4. FrFT power spectrum of potential noise for stainless steel WE in the three different solutions,

using a = 0.5.




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Figure 5. Power spectrum of potential noise of a stainless steel WE in three different solutions, using

fractional Fourier transforms. Fractional order a varies from 0.1 to 0.8.

where S 3 f 6 is the power spectral density of the potential noise and S i 3 f 6 is the power

spectral density of the current noise.

For identical impedances Z 13 f 6 2 Z 23 f 6 2 Z 3 f 6, the spectral noise impedance can beexpressed as

Rsn 3 f 6 2 Z 3 f 6 1 (20)

Bertocci et al. (1997), derived the relationship between noise resistance Rn and spectral noiseimpedance Rsn as

Rn 2 f max

f minS i 3 f 6 R

2sn 3 f 6 d f f max

f minS i 3 f 6 d f

12

(21)

where f is the frequency in Hertz, f min is the lower limit of the frequency (equal to two

times the inverse measurement time), and f max is the higher limit frequency (equal to half the

sampling frequency). If f min is sufficiently low, Rn can be expressed as

Rn 2 Rsn 3 f 06 1 (22)




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1454 Y. CHEN ET AL.

Figure 6. Equivalent circuit of a simultaneous ECN measurement.

Figure 7. Spectral noise impedances of solutions A, B and C, as derived using the fractional Fouriertransform.




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Here, we use the fractional Fourier transform for the spectral noise impedance. Figure 7

shows that the spectral noise impedance of solution C is the largest of the three impedances,

and the spectral noise impedance of solution A is the smallest. According to equation (10),

the corrosion rate is inversely proportional to the noise impedance. Therefore, both Figure 7

and the mathematics confirm that the corrosion rate of solution A is higher than that of

solution B, and the corrosion rate of solution B is higher than that of solution C.

5. CONCLUSION

Of the three different simulated saliva solutions, solution A causes the highest corrosion

rate, with solution B producing the second-highest rate and solution C causing the slowest

corrosion.

Analysis of ECN signals in both the time and frequency domains have been presented. R / S analysis shows that the ECN signals have self-similar properties, which cause conven-

tional analysis methods to perform poorly. It has been shown in this article that fractional

Fourier transforms can successfully be used for analysis of electrochemical noise data, in ad-

dition to being a useful technique for analysis of self-similar signals in general. Impedance

data obtained from the fractional Fourier transform can yield the corrosion rate.

The R / S analysis and fractional Fourier transform are two examples of fractional order

signal processing that can be applied to electrochemical data.

Acknowledgement. This work was supported by USU College of Engineering “Skunk works” Seed Grant program. We

wish to thank Nephi Zufelt for preparing the Ti electrodes.

REFERENCES

Almeida, L.B., 1994, “The fractional Fourier transform and time-frequency representations,” IEEE Transactions onSignal Processing 42, 3084–3091.

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