Fractal characteristic analysis of electrochemical noise with wavelet transform

Corrosion Science 48 (2006) 1337–1367 www.elsevier.com/locate/corsci

Fractal characteristic analysis of electrochemical noise with wavelet transform X.F. Liu

a,b,*

, H.G. Wang a, H.C. Gu

b

a

b

Xi’an High-technology Institute, Xi’an 710025, PR China State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, PR China Received 23 June 2004; accepted 1 June 2005 Available online 15 August 2005

Abstract In the study, the fractal parameters were introduced to describe the irregular characteristic of electrochemical noise, which were calculated with wavelet transform to overcome some defects of other methods. Compared with other fractal parameters, wavelet standard deviation (STD) exponent (Hws) had smaller deviation and could describe the fractal characteristics of electrochemical noises in a wide range, therefore Hws was chosen to evaluate general and local irregularity of potential signals generated from 7075-T76 aluminum alloy in 3.5% NaCl solution without and with different inhibitors respectively. The results showed that the smaller general Hws value, the more frequently potential fluctuated on the electrode surface, and the less effect of inhibitor in the solution. The difference between general Hws and 1 reflected the condition of surface passivity. Local Hws could effectively extract the local feature of electrochemical noise and quantitatively describe the change of its fractal characteristic with time, which was a promising index to analyze electrochemical noise in practice.  2005 Elsevier Ltd. All rights reserved. Keywords: Electrochemical noise; Wavelet transform; Fractal; Standard deviation of wavelet coefficient; General and local feature

* Corresponding author. Address: State Key Laboratory for Mechanical Behavior of Materials, XiÕan Jiaotong University, XiÕan 710049, PR China. Fax: +86 029 323 7910. E-mail address: [email protected] (X.F. Liu).

0010-938X/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.corsci.2005.06.001

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1. Introduction Mandelbrot introduced the concept of fractals in terms of statistical self-similarity, the idea that the shape of an object did not define its size [1]. The original example was the length of a rocky coastline, a map of a rocky coastline gave no indication of its scale. Many natural phenomena have since been shown to exhibit statistical self-similarity. Examples include earthquakes, fragments, river networks, and mineral deposits [2]. Fractals are mathematical sets with a high degree of geometrical complexity, which can model many kinds of time series. So Mandelbrot and Van Ness extended the concept of statistical self-similarity to time series, and the classic example is a fractal Brownian motion (fBm) [3]. The fractal dimension is an important characteristic of fractals because it contains information about their geometric structure. It has become an effective tool to study complex sets. There are many definitions for the fractal dimensions of a fractal set [4,5], but their calculation is not so easy [6]. For simplification, some parameters were proposed to describe fractal characteristic, for example, Hurster index (Hu) and Hausdorff exponent (Ha), whose relationships with fractal dimension were also discussed. The estimation of fractal parameter as a constant has been extensively studied [7,8]. In certain modeling applications, treating the self-similarity parameter as a constant seemed justifiable. However for many phenomena with self-similar behavior, the natures of the self-similarity change as these phenomena evolve. To model such these data sufficiently, the fractal parameter must be allowed to vary as a function of time. For such processes, the local fractal parameter function (H(t)) delivers important, even decisive, information regarding their behavior. Therefore it is desirable to develop a procedure for H(t) estimation. Recently electrochemical noise has been researched widely for the detection and evaluation of localized and general corrosion behavior [9,10]. For electrochemical noise technique, one of its principal advantages is that it can be used without disturbing of the system under investigation. In addition, it is more sensitive to localized corrosion processes than some traditional techniques, which produce little information. In the literatures, two main methods for the mathematical treatment of electrochemical noise have been implemented: the statistical method in time domain and the spectral method in frequency domain [11–16]. Although these techniques were useful for analyzing stationary phenomena and some information was obtained, they were limited in analyzing non-stationary signals. Some electrochemical noises originating from corrosion processes were generally considered to be non-stationary in character, for example large amplitude localized transients. Wavelet analysis is a relatively new mathematical tool used to supplement conventional Fourier analysis, and does not have some of the inherent inadequacies related to Fourier approach [17,18]. Previous wavelet analysis of electrochemical noise has included monitoring corrosion process with wavelet transform to distinguish corrosion mechanisms such as pitting and crevice corrosion [19,20]. Another study used wavelet transform methods to analyze current noise data from different electrochemical systems using the energy distribution at different scales to measure the activity of

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the corrosion events at different scale values [21]. The noise resistance was calculated as a ratio of the standard derivation of the reconstructed potential noise to the reconstructed current noise [22]. The effect of some inhibitors on aluminum alloy was analyzed with the standard deviation of wavelet coefficients [23]. For an electrochemical noise signal originating from corrosion process, wavelet transform could decompose it into a series of components at different scales and locations. Each component was defined by a set of wavelet coefficients, which was the similarity between the noise signal and a wavelet function, and then contained information about the time scale characteristic of the associated corrosion event. Therefore, wavelet transform was suitable to analyze the self-similarity of a time series [24], that is, fractal characteristic, although some fractal characteristics were analyzed with chaotic theory [25], whose computation was complex and difficult to understand. For a time series of electrochemical noise originating from corrosion process in practical engineering, it is usually non-stationary and its feature changes frequently as environmental media and surface condition of electrode material vary with time, therefore its fractal characteristic should also change. Although general fractal parameter could give the useful information about electrochemical noise, local fractal parameter describing the characteristic at a certain interval is more interesting, from which the local information about electrochemical noise is valuable in practice, and helpful to evaluate and understand corrosion process. In this study, it demonstrated that wavelet transform could be used to evaluate general and localized fractal characteristic of electrochemical noises, and a series of simulating signals with known features were presented to check the effectiveness of the evaluating procedure, furthermore with which general and localized fractal characteristic were analyzed about potential noises resulting from 7075-T76 aluminum alloy in 3.5% NaCl solution without and with different inhibitors, which were related to corrosion behaviors of the material and the effect of inhibitors, so as to illustrate that the fractal parameters representing the characteristic of electrochemical noise were useful in practical engineering.

2. Fractal-wavelet background 2.1. Fractal and its parameter Until today, there is no common definition of what is a fractal, but it is clear that a fractal has many differences from Euclidean shapes [26]. Mandelbort gave a mathematical definition of a fractal as a set for which the Hausdorff Brsicovich dimension strictly exceeded the topological dimension. However, he was not satisfied with this definition as it excluded sets that one would consider fractals. The fractal dimension is an important characteristic of the fractals because it contains information about their geometric structures. In practice, electrochemical noise could be divided into two kinds of time series: fractional Gaussian noise (fGn) and fractional Brownian motion (fBm). The analysis

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of their fractal characteristic, that is, self-similarity, is often related to their persistence and stationary. The persistence measures the correlations between adjacent values within a time series and can be strong, weak, or non-existent (a white noise). Values of a time series can affect other values in the time series that are not only nearby in time (short range) but also far away in time (long range). Stationary is also one of the most important aspects of a time series. A time series is said to be strictly stationary if all moments are independent of the length of the time series considered [27]. Quantifying the persistence and stationary of electrochemical noise signal could provide information about the nature of corrosion processes. In order to describe these fractal characteristics, some parameters were defined, such as fractal dimension (D), Hausdorff exponent (Ha), Hurst exponent (Hu), spectral-power exponents (b), and so on. The relationships among them and their application ranges were also discussed [27–29]. The classic example of a stationary, discontinuous time series is a Gaussian white noise. Consider a variable en, n = 1, 2, . . . , N, with a Gaussian distribution of values that are uncorrelated and random; the distribution has zero mean and a standard deviation re. A white noise is a time series constructed with yn(wn) = en. The classic example of a non-stationary time series is a BrownianPmotion, which is obtained by n summing a Gaussian white-noise sequence, y n ðBmÞ ¼ i¼1 ei . The standard deviation of a Brownian motion is given by rn ðBmÞ ¼ re n1=2

ð1Þ

A time series of electrochemical noise can be prescribed either in the time domain or in the frequency domain in terms of discrete Fourier transform, and its powerspectral density, Sm, usually has a power-law dependence on frequency S m  fmb ;

m ¼ 1; 2; . . . ; N =2

ð2Þ

where fm = m/N. The value of b is a measure of the persistence strength about the time series, which is the slope in its power-spectral density periodogram with the log–log scaling. The Brownian motion has b = 2 and a white noise has b = 0. It has been concluded [27] that the relationships among b, the persistence strength, and stationary of a time series are as following: b>1

Strong persistence

Non-stationary

1 > b > 0 Weak persistence b¼0 Uncorrelated

Stationary Stationary

b<0

Stationary

Anti-persistence

Among the wide variety of ‘‘fractal dimensions’’ in application, the definition of Hausdorff is the oldest and probably the most important. Hausdorff dimension has the advantage of being defined for any set, and is mathematically convenient, because it is based on measures, which are relatively easy to manipulate. A major disadvantage is that in many cases it is hard to calculate or estimate with computation methods. However, in order to understand the mathematics of fractals, familiarity with Hausdorff measure and dimension is essential.

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Hausdorff exponent, Ha, is one of persistence measures for a time series, which is associated the standard deviation, rn, with the y-coordinate of the time series, and the variable n with the x-coordinate, and results in [3]: rn  nHa

ð3Þ

A white noise, rn, is independent of n, thus Ha = 0. Any stationary time series, rn, by definition, must be independent of n, thus again Ha = 0. From the above equation, it was found Ha = 0.5 for a Brownian motion. 0 < Ha < 0.5 corresponds to shortrange strong persistence, and 0.5 < Ha < 1 to long-range strong persistence. Voss has used the box-counting method to obtain the general relation between the Hausdorff exponent, Ha, and the fractal dimension, D, with the result [28]: Ha ¼ 2  D

ð4Þ

Ha can also be calculated with the semi-variogram method [30]. For a Brownian motion, Ha = 0.5 and D = 1.5. For the time series, Ha is defined between 0 and 1, and D between 1 and 2. According to VossÕs theory [31], a relationship between the power b, the Hausdorff exponent Ha, and the fractal dimension D is given by b ¼ 2Ha þ 1 ¼ 5  2D

ð5Þ

Therefore, the Hausdorff exponent Ha is only applicable for the time series with 1 6 b 6 3, because Ha is defined from 0 to 1 while 1 6 D 6 2. However, the spectral-power exponent b could be a measure of the persistence strength to all time series, not just 1 6 b 6 3. An alternative approach to quantify the correlations in time series was developed by Hurst [32], who considered a river flow as a time series and determined the storage limits in an idealized reservoir. Basing on these studies he introduced empirically the concept rescaled-range (R/S) analysis, from which Hurst exponent (Hu) was obtained. It was found [27] that Hurst exponent provided a quantitative measure of the persistence strength and anti-persistence for fractional Gaussian noises (fGn, 1 6 b 6 1), the relation between the Hurst exponent Hu and the power b was expressed as b ¼ 2Hu  1

ð6Þ

Since a white noise (b = 0) is a random process that has adjacent values which are uncorrelated, it is appropriate to conclude that Hu = 0.5 implies a uncorrelated time series. It follows that 0.5 < Hu 6 1.0 implies persistence and that 0 6 Hu < 0.5 implies anti-persistence. 2.2. Fractal calculation with wavelet transform For a time series of electrochemical noise originated from corrosion process, its power b in the frequency domain is usually in the range of 0–3. So, Ha and Hu are just suitable to analyze some parts of electrochemical noises respectively, and

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their calculations are also complex. Although the power b obtained with fast Fourier transform (FFT) could be applied to all the time series of electrochemical noise, Fourier transform has some inherent disadvantages, for example, the transform results do not provide time resolution, and de-trending and windowing often have an influence on the transform results. Especially, spectral analysis requires the signal to be stationary, i.e. the statistical properties do not change with time. However, typical potential noise drifts with time and therefore it is common to remove this drift, usually by subtracting a linear regression line from the data, and some errors may be induced in the power spectrum. Likewise, windowing is another pre-analysis procedure to induce errors associated with the ends of the data sequence. Further more, Fourier transformation is suitable to analyze the periodic time series, however electrochemical noise from corrosion process always does not exhibit periodic characteristic. To overcome these disadvantages of the above fractal parameters and find a suitable parameter representing the fractal characteristic of all electrochemical noises, wavelet transform was introduced, which provided information on both the time and frequency dependence of a time series. Wavelet transform has a fractal basis and is particularly useful when applied to non-periodic multi-scaled time series, and the method could not only be applied to stationary process but also to non-stationary process [33,34]. In wavelet transform process, Mallat pyramidal algorithm [35] was used and the original time series passed through two complementary filters and emerged as two components: the approximation and the detail. The decomposition process could be iterated with successive approximations being decomposed in turn, so that one signal was broken down into many higher-resolution components, which were the approximations A2j f and details D2j f of the signal at different scales 2j, here j was the level of decomposition hierarchy, j 2 Z. Fig. 1 shows two-level signal decompoe sition (analysis) and reconstruction (synthesis) using wavelet, where GðGÞ represents e an analysis (synthesis) low pass filter and QðQÞ represents an analysis (synthesis) high pass filter, Ad2j f denotes an input signal and Ad2jþ1 f ðD2jþ1 f Þ signifies the low(high) frequency component of input signal, where 2j, 1 < j < J, denotes the scale. In the decomposition stage, a signal was filtered and convolved, and then downsampled by a factor of 2. In the reconstruction stage, two components were combined by inverse processes of the analysis stage. When wavelet transform was applied to analyze fractal characteristic of a time series, there were two kinds of useful data. One was the reconstructed detail signal, whose energy could be used to calculate Hausdorff exponent (Ha). Other was the wavelet coefficient at different scales, which were similarity between wavelet function and the time series at different scales, and should be able to represent the self-similarity of the time series. The power spectrum P(x) of a fractal time signal was represented by P ðxÞ ¼ axb ¼ ax2H a 1

ð7Þ

where a denoted a constant. The power spectrum P 2j ðxÞ of a fractal signal filtered by ^ j ðxÞ was expressed as the high pass filter w 2

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decimation by a factor of two. a

interpolation by a factor of two. b Fig. 1. Two-band signal decomposition and reconstruction by wavelet: (a) decomposition; (b) reconstruction.

    ^ j 2 P 2j ðxÞ ¼ P ðxÞw 2 x 

ð8Þ

^ j xÞ denoted the wavelet function at the scale 2j. The power ^ j ðxÞ ¼ wð2 where w 2 d spectrum P 2j ðxÞ of a discrete signal, sampled by a factor of 2j, was written as P d2j ðxÞ ¼ 2j

þ1 X

P 2j ðx þ 2j 2mpÞ

ð9Þ

m¼1

Let E2j be the energy of a high frequency signal D2j f , defined as Z j 2j þ2 p d E 2j ¼ P j ðxÞdx 2p 2j p 2

ð10Þ

By putting Eqs. (8) and (9) into (10), the following relationship was obtained: E2jþ1 ¼ 22Ha E2j

ð11Þ

Thus, the Hausdorff exponent Ha was computed by 1 E jþ1 Ha ¼ log2 2 2 E2j

ð12Þ

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For the wavelet coefficients at different scales, their variance were calculated as j

N =2 X  2 1 Vj ¼ d j;k  dj j N =2  1 n¼1

ð13Þ

where N was the number of potential noise data, dj,k was the wavelet coefficient at the scale 2j and location k2j, and dj was the mean value of dj,k at the scale 2j. It has also been shown [27] that if the original data series obeys a power law distribution (and therefore a fractal), the variance function can be expressed as a power law relation with the form: V j ð2j Þ  ð2j Þ

Hw

ð14Þ

Hw ¼ b

ð15Þ

where Hw is known as the wavelet exponent. The authors of this paper have introduced calculating the standard deviation of the wavelet coefficient at different scale so as to compare with the fluctuation level of every detail signal quantitatively [23,36], as following: ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi PN =2j  PN =2j PN =2j j 2  2 k¼1 d j;k  d j k¼1 ðd j;k  k¼1 d j;k =ðN =2 Þ STDj ¼ ¼ N =2j  1 N =2j  1 j ¼ 1; 2; . . . ; J

ð16Þ

According to the above equation, in log2 ðSTDj Þ  log2 ð2j Þ plot, the influences of inhibitors on electrochemical noise of 7075 aluminum alloy were discussed in detail. Although the authors realized the self-similarity of electrochemical noises in the paper, the fractal problem was not discussed deeply [23,36]. In fact, STDj ¼ V j ð2j Þ

1=2

1Hw

 ð2j Þ2

ð17Þ

Therefore, in log2 ðSTDj Þ  log2 ð2j Þ plot, the slope of log2 ðSTDj Þ values, Hws, estimated with least-squares regression, should be related to Hw and b as following: Hws ¼ 0.5Hw ¼ 0.5b

ð18Þ

Hws was named as wavelet STD exponent. In the following, a set of simulating signals with arbitrary values of b (b = 1, 0.5, 0, 0.5, 1, 1.5, 2, and 2.5) were generated to calculate Ha, Hw, and Hws with wavelet transform for comparison so as to choose the best parameter describing the fractal characteristic of electrochemical noise. Fractional Gaussian noise (fGn, 1 6 b 6 1) could be generated synthetically from Gaussian white noise function (b = 0) [27]. A discrete Fourier transform was taken of a Gaussian white noise, the resulting Fourier spectrum would be flat. The resulting Fourier coefficients Ym were filtered using the relation:

X.F. Liu et al. / Corrosion Science 48 (2006) 1337–1367

Y 0m ¼

mb=2 N

Ym

1345

ð19Þ

An inverse discrete Fourier transform was taken of the filtered Fourier coefficients, and the result was a fractional Gaussian noise. To remove edge effects, only the central portion was retained. Fractional Gaussian noises could be summed to give fractional Brownian motions, bfBm = 2 + bfGn. Fractional Brownian motion are associated with b values greater than 1, and the classic example of Brownian motion is b = 2, which is simply the integral of white noise. Several examples of fGn and fBm are shown in Fig. 2 with b = 1, 0.5, 0.5, 1, 1.5, 2, and 2.5. Fig. 2 shows the change in behavior of a time series with regard to the power spectral exponent. These simulating signals were decomposed with wavelet transform, using the simplest wavelet function, Haar, and then fractal dimension parameters, that is, Ha,

Fig. 2. Simulating signals with arbitrary values of b: (a) 1; (b) 0.5; (c) 0; (d) 0.5; (e) 1; (f) 1.5; (g) 2; and (h) 2.5.

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Table 1 Hausdorff exponent (Ha), wavelet variance exponent (Hw), and wavelet STD exponent (Hws) of simulating signals with arbitrary values of b in Fig. 2 Serial number

b

Ha

Hw

Hws

a b c d e f g h

1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5

0.8686 0.6951 0.4828 0.2511 0.0128 0.2252 0.4540 0.6568

0.7689 0.4234 0.0052 0.4374 0.9559 1.4369 1.8982 2.3061

0.3761 0.2034 0.0109 0.2451 0.4863 0.7268 0.9574 1.1614

Hw, and Hws, were calculated. The number of data points N and the used wavelet limit the maximum value for decomposition level, J. Using the Haar wavelet with N = 1024, it is possible to compute up to J = 9. Considering the calculation of STDj, J was chosen as 6 in the process of simulating signals in Fig. 2. After the decomposition of the noise signals, the detail signals at each level need reconstruct so as to calculate Ha, and then the energy of each detail signal was calculated, so Ha was half of the slope obtained with least-squares regression from log2 ðEnergyÞ  log2 ð2j Þ plot according to Eq. (12), shown in Table 1. Wavelet variance and STD of wavelet coefficients at each level were calculated with Eqs. (13) and (16), then Hw and Hws could be obtained with the similar method to Ha calculation from log2 ðV j Þ  log2 ð2j Þ and log2 ðSTDj Þ  log2 ð2j Þ respectively, shown in Table 1. From Table 1, it was found with least-squares regression method that the relationships among Ha, Hw, Hws, and b were as following: Ha ¼ ðb  1Þ=2;

deviation r ¼ 0.0677

Hws ¼ Ha þ 0.5; deviation r ¼ 0.0050 Hws ¼ 0.5Hw; deviation r ¼ 0.0064 Hws ¼ 0.5b; deviation r ¼ 0.0633 Hw ¼ b; deviation r ¼ 0.1266 The results showed that Ha was clearly unsuitable to describe the fractal characteristic of all simulating signals because its values were minus for the signals with b < 1, which were out of its definition, and the STD of its residuals from the regression line was bigger than HwsÕs. Hw and Hws could be used to describe all the simulating signals. However compared with Hw, Hws had smaller residuals with the relationship of b except b = 0, shown in Fig. 3(a), even if Hws was multiplied by 2 to be at the same level as Hw, its residuals from b would also be smaller than HwÕs in the range of b > 0 (Fig. 3(b)). In practice, b value of electrochemical noise is usually between 0 and 3. Therefore, Hws was chosen as the best parameter to evaluate the fractal characteristic of electrochemical noise. According to the fractal definition of b and Ha, the relationships among Hws, the persistence strength, and stationary of a time series were as following:

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Fig. 3. Absolute differences between b and corresponding Hw (+), between b/2 and corresponding Hws (*) (a), and between b and corresponding 2 Hws (*) (b) of the simulating signals in Fig. 2.

Hws > 1

Long-range strong persistence

Non-stationary

Hws ¼ 1 1 > Hws > 0.5

Brownian motion Short-range strong persistence

Non-stationary

0.5 > Hws > 0 Weak persistence Hws ¼ 0 Uncorrelated

Stationary Stationary

Hws < 0

Stationary

Anti-persistence

2.3. Estimation of local fractal parameter In practice, a time series of electrochemical noise originating from corrosion process, f(t), is often measured at discrete, equally spaced time points. Because of the

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change of environmental factors such as temperature, solution concentration, inhibitor, and so on, the characteristic of electrochemical noise would vary with time. So the fractal parameter need not only describe its general characteristic, but also evaluate the change trend of its characteristic. It is necessary to estimate its local fractal parameter. The time series could be partitioned into 2l non-overlapping subintervals of equal length, where l was an integer chosen such that 0 6 l 6 J. The 2l subintervals were of the form:  I m ¼ ðm  1Þ2J l ; m2J l ; m ¼ 1; . . . ; 2l ð20Þ b wsðtÞ was constructed, which could be regarded as For each Im, an estimation H estimating the average value of the scaling function Hws(t) over the corresponding b wsðtÞ associated with Im might be subinterval Im. The appropriate time index of H regarded as the midpoint of Im, namely (2m  1)2Jl1. At first, the full 2J observation of the time series was transformed with Haar wavelet, and a series of dj,k, the wavelet coefficients at the scale 2j and location k2j, were obtained. Then for each Im, the STD of corresponding dj,k values was calculated at each level j respectively, where the level j was no more than some positive integer J 0 (J 0 6 J)chosen as wavelet decomposition space, and the location k2j was within the subinterval Im. For each m = 1, . . . , 2l, the bivariate collections of data was defined as  ðX m ; Y m Þ ¼ flog2 ð2j Þ; log2 ðSTDðd j;k ÞÞg; k2j 2 I m ; 1 6 k 6 2J j ; 1 6 j 6 J 0 ð21Þ Then a least-squares line was fitted to (Xm, Ym), treating the Xm as the regressor meab wsðtÞ was the surements and the Ym as the response measurements. The estimate H slope in the least-squares fit. As a final and optional stage in the procedure, a curve was constructed from the b wsðtÞ by employing local polynomial smoothing. This curve collection of estimates H then served to approximate the shape of the wavelet STD exponent function Hws(t). In practical engineering, the sample size of electrochemical noise is not usually a power of two, the data can be augmented with repeating a part of original noise in order to make the sample size become a power of two, and the results are adjusted to delete these corresponding to the repeating data. If such the sample of electrochemical noise is processed with FFT, the data are often augmented with zero values, thus a large error will be induced in analysis results. So wavelet analysis is more suitable to practical application. The performance of the above algorithm for local fractal parameter, Hws(t), was tested with time series simulating five kinds of ideal electrochemical noise generated from corrosion process, which were shown in Fig. 4. The characteristics of five noise signals were as following: S1—stochastic white noise, simulating general corrosion process on the surface. S2—no fluctuation, simulating ideal passivity on the surface.

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Fig. 4. Simulating potential noise signals for calculation of local fractal parameter, Hws.

S3—there were five break/heal events with short time, whose position, amplitude, and transient time were (100, 10, 8), (300, 5, 8), (500, 15, 8), (715, 20, 8), and (900, 30, 8) respectively, simulating pitting occurrence on the inorganic passive surface. S4—there were five break/heal events with long time, whose position, amplitude, and transient time were (97, 10, 32), (300, 6, 16), (500, 15, 32), (705, 20, 32), and (900, 10, 16) respectively, simulating pitting occurrence on the organic passive surface. S5—there were two slow fluctuations, whose position, amplitude, and decay time were (100, 10, 170) and (500, 6, 185), simulating slow varying trend of potential under the passive condition. For each signal, the sample size was N = 1024 (210, J = 10) and the sample frequency was 1 Hz. In the estimation algorithm, each sample was partitioned into 32 = 25 (l = 5) subintervals, which resulted in 32 estimation of Hws(t) corresponding to the time (2m  1) * 24, m = 1, . . . , 25. Each signal was firstly transformed with Haar wavelet and the decomposition space (J 0 ) was 4. Then, the STD of dj,k corresponding to each subinterval was calculated at each level and its Hws was estimated according to Eq. (21). In constructing the estimated curves of Hws(t), the points were just connected directly, not smoothed over the discontinuity. The calculation results of the local Hws(t) for the simulating noises are shown in Fig. 5, and their general Hws values were also calculated in the same decomposition space, J 0 = 4 so as to compare with each other, shown in Table 2.

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Fig. 5. Estimation of local wavelet STD exponent (Hws) for simulating potential noises in Fig. 4.

With comparison of simulating signals and the corresponding general and local Hws values, some interesting information could be obtained. For stochastic signal, S1, its general Hws was near zero, so its adjacent data was uncorrelated. Its local Hws values were near or below zero except for two points around 528 s and 784 s, which indeed exhibited weak persistence from its shape in Fig. 4 although the signal was produced with random function in Matlab software, thus the conclusions about S1 fractal characteristic from its general and local Hws values respectively were basically the same. For the simulating passivity, S2, its general Hws indicated that it was near Brownian motion, but its local Hws did not exist because there were some zero values of the STD of dj,k at some scale levels. In fact, for S2, S3, S4, and S5, the local Hws did not exist in all simulating periods of ideal passivity. This fact more further reflected the surface condition: there was no fluctuation. For S5, because the partition interval (25 s) was more less than its slow fluctuation time (170 s and 185 s), the fluctuations were divided into some parts in the analyzing process, therefore local Hws values were just obtained at some parts, which could not describe the fluctua-

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Table 2 General wavelet STD exponent (Hws) of simulating signals in Figs. 4 and 6 Noise signal

General Hws

S1 S2 S3 S4 S5 S6

0.0022 0.8174 0.4186 0.8250 1.0860 0.1801

tion feature. However, its general Hws was enough to represent its fractal characteristic. From the general Hws values of the signals, S3 and S4, they belonged to weak persistence and short-range strong persistence respectively. Their local Hws values could not only show the persistence for each fluctuating event, but also indicate its position. The local Hws values also showed that there existed strong persistence fluctuation even in weak persistence signal, such as (300, 5, 8) in S3, and so did long-range strong persistence fluctuation in short-range strong persistence signal, such as (97, 10, 16) in S4. One fluctuation event, (500, 15, 32), was divided into two partition intervals in wavelet transform process, so there were two Hws values. In order to demonstrate the ability of local fractal parameter to reflect the feature change of a time series and the influence of interval length on estimation of local fractal parameter, simulating signal S6 was constructed with four sections of different self-similarity, which were respectively a part [1, 256] of four simulating signals with known b values, Fig. 2(c) b = 0, (e) b = 1, (f) b = 1.5, and (g) b = 2, that is, S6 [1, 256] with b = 0, S6 [257, 512] with b = 1, S6 [513, 768] with b = 1.5, S6 [769, 1024] with b = 2, shown in Fig. 6(a). In J 0 = 4 decomposition space, similar to the above calculation procedure of local Hws, its local Hws values were estimated with the interval length 32 and 64 respectively, and the results are shown in Fig. 6(b) and (c). Its general Hws was also calculated in the same decomposition space so as to compare with the results of local Hws, shown in Table 4. According to the above definition of Hws and the relationship between Hws and b, with comparison of S6, its local Hws curves and its general Hws, it was found that its general Hws could not represent the signal with varying fractal characteristic, but its local Hws values were basically able to reflect the varying trend of its characteristic although the interval length had some influence on the calculating results. Local Hws curve resulting from the interval length of 64 described the change trend of its characteristic more accurately than from the interval length of 32. For a section with steady feature, for example S6 [513, 768] and S6 [769, 1024], if the interval length was too small, local Hws values would concentrate on reflecting local feature and might influence the judgement of fractal characteristic about the section, however the statistical property of local Hws values still reflected its fractal characteristic. Therefore the interval length should be chosen reasonably so as to reflect the change of signal characteristic with time.

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Fig. 6. Simulating signal (a) with varying fractal characteristic and the estimation of its local Hws with the interval length, 32 (b) and 64 (c), respectively.

From the above discussion about the general and local Hws of the simulating signals in Figs. 4 and 6, it was shown that the local Hws could extract the varying feature of a signal with time and be more suitable to describe the signal fluctuating frequently, however the general Hws was able enough to represent the fractal characteristic of the signal with very slow fluctuation. The subinterval partition of a time series would influence the result of local Hws calculation, which should be carried out on the basis of its self-feature and usually avoid separating a local accident with distinct feature.

3. Experiment The test material 7075-T76 aluminum alloy had the following analysis of composition: Cu(1.68%), Zn(5.61%), Mg(2.16%), Mn(0.21%), Cr(0.11%), other(0.66%),

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Al(balance). The corrosion resistance of 7075-T76 aluminum alloy (Al–Zn–Mg–Cu) is determined by the oxide film of the surface and the intermetallic phases such as MgZn2, Al7Cu2Fe and Al2CuMg [36–39]. In the test electrolyte, which was 3.5% NaCl solution made of reagent grade sodium chloride and distilled water, localized corrosion such as pitting corrosion occurred easily on the surface of 7075-T76 aluminum alloy. In order to investigate the effect of inhibitors on localized corrosion of aluminum alloy, some inhibitors were added to the test electrolyte, such as phosphate, molybdate, citrate, benzimidazole/benzothiazole (MBTZ), and chromate. The total amount of inhibitors in all experimental solution was fixed at 500 ppm and the circumambient temperature was 28 ± 2C. The working electrodes were cuboid, and a hole was drilled on one part of the specimen to be connected with the wire of the measuring instrument. The exposed surface was prepared by polishing with 280-, 400-, 600-grit papers in succession and then cleaning with acetone. All the surfaces including the wire were masked with wax, then a 1 cm2 circular area of wax was removed and this exposed area of specimen was rinsed with acetone and water, and finally dried in air. A new working electrode was used in each run. A saturated calomel electrode (SCE), brought into close proximity to the working electrode by a Luggin probe, was used to measure the potential of the working electrode. After the specimen was immersed in the test solution for 24 h, its potential noise signal at open-circuit was sampled with an instrument for measuring and analyzing electrochemical noise [40]. The sampling interval was 2 s and the length of the measured record was 1024 points.

4. Results and discussion 4.1. Measurement results The measured potential noise data in the test solutions are shown in Figs. 7 and 8. When the 7075-T76 aluminum alloy was put into the 3.5% NaCl solution for 1 h, its potential fluctuated very frequently (Fig. 7(a)), but after 24 h both the amplitude and rate of potential fluctuation decreased (Fig. 7(b)). When the test solutions contained different inhibitors, electrode potential exhibited different fluctuation phenomena. In phosphate solution, potential fluctuation (Fig. 8(b)) was similar to that in the 3.5% NaCl solution for 1 h, and electrode potential fluctuated more slowly in phosphate/molybdate solution than in phosphate solution (Fig. 8(c)), and after adding citrate (Fig. 8(d)), it fluctuated further more slowly than in phosphate/molybdate solution. In phosphate/citrate solution, the potential changed very slowly (Fig. 8(e)), and then after MBTZ was added (Fig. 8(f)), the fluctuation amplitude decreased more further. In chromate and phosphate/molybdate/ citrate/MBTZ solutions, the electrode potential seldom fluctuated (Fig. 8(a) and (g)).

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Fig. 7. Potential noise signals of 7075-T76 aluminum alloy immersed in 3.5% NaCl solution after 1 h (a) and 24 h (b) respectively.

4.2. Calculation results of general fractal parameters Just as the same of the calculation procedure for the simulating signals in Fig. 2, these electrochemical noise signals were decomposed with wavelet transform, using the simplest wavelet function, Haar, and then their fractal parameters, that is, Ha, Hw, and Hws, were calculated respectively. The number of decomposition level, J, together with the sampling interval Dt, determined the scale range studied (2Dt, 2JDt). It should be noticed that the number of data points N and the used wavelets limit the maximum value of chosen J. Using the Haar wavelet with N = 1024, it was possible to compute up to J = 9. However, a higher J would imply a shorter window of data to analyze because of the boundary effects, and due to the fact that all the transients contained in the potential records last less than 300 s, it was determined that J = 6 gave a reasonable scale range to study the transients. After the decomposition of the noise signals, the detail signals at each level need reconstruct so as to calculate Ha, and then their energies was calculated respectively, so according to Eq. (12), Ha was half of the slope obtained with least-squares regression from log2 ðEnergyÞ  log2 ð2j Þ plot, shown in Fig. 9. The results are in Table 3. The wavelet variance and STD of coefficients at each level were calculated with Eqs. (13) and (16), then Hw and Hws could be obtained from log2 ðV j Þ  log2 ð2j Þ and log2 ðSTDj Þ  log2 ð2j Þ respectively, shown in Fig. 9. From Fig. 9, it was shown that the residuals of Ha evaluation, which were the vertical distances between data points and regression line, were clearly bigger than for Hws value, so Hws value should be more accurate to represent the characteristic of electrochemical noise. For all the electrochemical noises, their Hws values were calculated from Fig. 10, and shown in Table 3. These electrochemical noises also were analyzed with FFT so as to get the spectrum power, b, according to Eq. (2). Because detrending and windowing often influence the FFT results, the same pre-treat methods, linear detrending and window = 1024, were applied for all noise signals to compare with each other. The results are also shown in Table 3.

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Fig. 8. Potential noise signals of 7075-T76 aluminum alloy immersed in 3.5% NaCl solution containing chromate (a), phosphate (b), phosphate/molybdate (c), phosphate/molybdate/citrate (d), phosphate/ citrate (e), phosphate/citrate/MBTZ (f), and phosphate/molybdate/citrate/MBTZ (g) respectively.

From Table 3, it was found with least-squares regression method that the relationships among Ha, Hw, Hws, and b were as following: Ha ¼ ðb  1Þ=2;

deviation r ¼ 0.0676

Hws ¼ Ha þ 0.5;

deviation r ¼ 0.0073

Hws ¼ 0.5Hw; deviation r ¼ 0.0086 Hws ¼ 0.5b; deviation r ¼ 0.0722 Hw ¼ b;

deviation r ¼ 0.1543

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Fig. 9. The energy of detailed signals and the STD of detailed coefficients after potential noises were processed with wavelet transform for 7075-T76 aluminum alloy in 3.5% NaCl solution without inhibitor (a) and with phosphate (b).

Therefore, the results from the practical noises were consistent with theory analysis and the results from simulating signals. These results also showed that Ha was

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Table 3 Hausdorff exponent (Ha), wavelet variance exponent (Hw), wavelet STD exponent (Hws) (J = 6), and spectrum power (b) of potential noises generated from 7075-T76 aluminum alloy in 3.5% NaCl solution without and with inhibitors Serial number

Solution

Ha

Hw

Hws

b

1# 2# 3# 4# 5# 6# 7# 8# 9#

Blank (1 h) Blank (24 h) Chromate Phosphate Phosphate/molybdate Phosphate/citrate Phosphate/citrate/molybdate Phosphate/citrate/MBTZ Phosphate/citrate/molybdate/MBTZ

0.0744 0.1085 0.4834 0.0907 0.0049 0.6438 0.2467 0.5280 0.4904

0.8102 1.2118 1.9664 0.7907 0.9896 2.2853 1.4665 2.0537 1.9779

0.4154 0.6142 0.9915 0.4037 0.5031 1.1510 0.7416 1.0352 0.9973

0.8439 1.4277 2.0580 0.8334 1.1651 2.6280 1.3860 2.0830 2.0070

clearly unsuitable to describe the fractal characteristic of all electrochemical noises because of its minus values in some signals and its big dispersion, and that Hw had larger deviation with the relationship of b in comparison with Hws although it could be suitable for all the noise signals. Therefore Hws was more suitable to be as the fractal parameter of electrochemical noise in this paper, and its definition could be deduced from b, Ha, and Hu meanings, which has been explained in previous theory part. 4.3. Relationships between fractal characteristic and corrosion process According to Hws definition and the results of Table 3, the studied electrochemical noises could be divided into four kinds as following: (1) (2) (3) (4)

1# and 4#: weak persistence, stationary, 2#, 5#, and 6#: short-range strong persistence, non-stationary, 3# and 9#: Brownian motion, 7# and 8#: long-range strong persistence, non-stationary.

The characteristics of these potential noises were closely related to the summation of electrical processes occurring on the electrode surface in the test solutions. The relationship between fractal characteristic and surface condition would be analyzed in detail. Corrosion resistance of 7075-T76 aluminum alloy is not only associated with the oxide film of its surface, but also with micro-structural heterogeneity at the surface. It contains significant amounts of constituent particles such as Al7Cu2Fe, MgZn2 and Al2CuMg [36–39]. Local galvanic cells form easily on the metal surface in aqueous environments because of the difference in electrochemical activity between these heterogeneous phases and between the particles and the matrix. These cells lead to accelerated, localized corrosion (pitting) attack. It has been proved that in 3.5% NaCl solution, comparison with the matrix, MgZn2 particles act as principal anodes,

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-6

-10

2

Log (STDj )

-8

1# 5# 6# 8# 9#

-12 -14 -16

1

2

3

4

5

6

j

Log (2 )

a

2

-6

-10

2

Log (STDj )

-8

-12

2# 3# 4# 7#

-14 -16 b

1

2

3

4

5

6

j

Log (2 ) 2

Fig. 10. STD of wavelet detailed coefficients of potential noises generated from 7075-T76 aluminum alloy in 3.5% NaCl solution without and with inhibitors.

and Al7Cu2Fe as cathode to dissolve, and Al2CuMg particles also act as anodic sites, losing Mg and Al through dissolution in the early stage of corrosion and becoming more cathodic as Cu is left behind [38,39,41]. In addition to the effects of microstructural heterogeneity, the importance of passive surface films on pitting must be considered. In this study, Cl could destabilize the passive film over the matrix to promote the occurrence of pitting corrosion [42]. So, in this study, when the sample was put into 3.5% NaCl solution for 1 h, there were a lot of reacting spots on the surface because of the effect of particles and Cl, and which had little or no influence with each other, thus there was a weak correlation between adjacent values, and the moments were independent of the length of the measure time. After 24 h, because Cu was deposited on the surface from the solution to form a yellow thin film, the reacting spots decreased and potential fluctuated slowly. As a result, the correlation be-

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tween adjacent data increased and the moments were determined by the measure time, so the fractal characteristic of its potential noise was strong persistence and non-stationary. When the inhibitors were added into the test electrolyte, they influenced the surface corrosion process with different mechanism. Because of chromate powerful oxidizing properties and its adsorbability to the material surface [43], the presence of chromate in the solution, even in the presence of aggressive anions like Cl, stimulates the repair of flawed regions of the surface film and oxidizes the active sites, accommodates on the active sites and cathodic areas, and inhibits active galvanic couples, leading to the formation of a stable corrosion resistant barrier film [44]. So its potential seldom fluctuated, just stochastic fluctuating, and its amplitude was very small. Therefore, the potential noise belonged to Brownian motion. Because the main protective effect of phosphate is to react with corrosion products such as Al3+, Zn2+, and Mg2+ to deposit on the specimen surface and form the protective film [45], which may not be uniform, its inhibiting effects do not occur quickly. After 24 h immersion, there existed a lot of corrosion and deposition reaction on the surface, which had little influence on each other. Therefore, the potential noise in phosphate solution had weak persistence and stationary. After a small amount of molybdate was added in phosphate solution, the breakdown probability of the film decreased in chloride media because molybdate as an oxidizer could reinforce the oxide film of aluminum alloy surface [44]. However, because of the weaker oxidizing properties of molybdate compared to chromate and its low concentration in the test, there were still some flaws on the specimen surface, at which break/heal events and phosphate deposition occurred. So the correlation between adjacent data strengthened and the moments were related to time. Carboxylates generally provided the combined action of an easily-reduced cation and a strongly-adsorbed anion that formed a chemical bond between the carboxyl ion and the metal substrate [46]. Thus, when the citrate ion with three carboxyls was added in 3.5% NaCl solution, it could competitively adsorb on the surface of aluminum alloy instead of chloride ion. When citrate met with phosphate in the solution, they would form a synergistic film to cover the surface of aluminum alloy [36], but this film was not compact and uniform. Some ions could penetrate through it from the solution to the substrate surface or from the substrate to the solution, and further could not absorb firmly on the particles such as Al2CuMg [29]. Therefore, corrosion reaction had been occurring although its rate was very low. This process was reflected by much slower fluctuation on potential noise in phosphate/citrate solution, which could be described as long-range strong persistence in fractal characteristic. In phosphate/molybdate/citrate solution, the adsorption of citrate lowered the reactivity of the surface and facilitated oxidizing passivation of the remaining surface by molybdate, resulting in the passive film on a majority of electrode surface. But the synergistic effect of adsorption and oxidation could not form the stable film on some particles such as Al2CuMg, which easily induced pitting. So its potential noise exhibited the typical pitting characteristic, that is, break/heal events occurred on the

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surface film. In one break/heal event, its potential data were correlated and its transient time was longer than in phosphate/molybdate solution, thus its Hws was bigger than in phosphate/molybdate solution, and its potential noise had stronger persistence though in short-range. Because MBTZ containing the elements of sulfur and nitrogen could easily adsorb on copper and its alloy to form a chemisorbed layer and hinder the corrosion of copper and its alloy [47,48], MBTZ adsorbed easily on the intermetallic phases containing copper such as Al7Cu2Fe and Al2CuMg, thus it could inhibit the corrosion induced by intermetallic phases, hindered the cathodic process of corrosion on 7075 aluminum alloy and significantly drove the corrosion potential toward more negative values. As a result, in phosphate/citrate/MBTZ and phosphate/molybdate/citrate/MBTZ solution, pitting and localized corrosion were inhibited, so potentials became more stable than in phosphate/citrate and phosphate/molybdate/citrate solution respectively. In phosphate/citrate/MBTZ solution, potential still fluctuated very slowly with lower amplitude, so it exhibited long-range strong persistence. When molybdate was added into the solution, potential seldom fluctuated, similar to that in chromate solution, so its fractal characteristic belonged to Brownian motion. With comparison of Hws values of potential noises in different solutions, combined with corrosion reaction on the electrode surfaces, it was shown that Hws value could evaluate the irregularity of electrochemical noise quantitatively at the analysis scale, that is, as Hws became smaller, the corresponding potential fluctuated more frequently, and the surface was more unstable, thus the effect of the inhibitor on corrosion was weaker. The difference between Hws and 1 reflected the distance to Brownian motion for potential noise at the analysis scale, the smaller the difference, the nearer to Brownian motion, and the more stable the potential at the analysis scale. In fact, if the comparison was carried out in the Fig. 10, these views were very clear. Therefore, log2 ðSTDj Þ  log2 ð2j Þ plot could not only be used to compare the fluctuation at different scales with each other [23,36], but also calculate the fractal parameter, Hws, so as to describe the irregularity, or complexity of electrochemical noise quantitatively. The above results were coincidence with the previous studies. It has been established that the slope (a) of the potential noise power spectrum on a log vs. log scale (the parameter a being obtained from the equation PSD(V)  f a) can be used as a significant parameter to distinguish the types of corrosion [10,14]. A slope shallower than 20 dB/decade has been related to pitting corrosion process, and a slope steeper than 20 dB/decade has been associated with passivation. The parameter a was the same meaning with b in this paper, and they were different just in expression form, that is, a = 20 dB/decade was same as b = 2. So according to the previous study, there formed passive films on the sample surface in chromate, phosphate/citrate, phosphate/citrate/MBTZ, and phosphate/molybdate/citrate/MBTZ solutions, and there existed pitting corrosion in other solutions. Although the result was right, it was rougher qualitatively compared with the result from wavelet analysis, and a lot of useful information was lost and the differences among these noise signals were not discovered.

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Table 4 Comparison of wavelet STD exponent (Hws) at different scales (J = 4 and 6) of potential noises generated from 7075-T76 aluminum alloy in 3.5% NaCl solution without and with inhibitors Serial number

Solution

Hws (J = 4)

Hws (J = 6)

1# 2# 3# 4# 5# 6# 7# 8# 9#

Blank (1 h) Blank (24 h) Chromate Phosphate Phosphate/molybdate Phosphate/citrate Phosphate/citrate/molybdate Phosphate/citrate/MBTZ Phosphate/citrate/molybdate/MBTZ

0.3572 0.6666 1.0045 0.3403 0.6924 1.2317 1.0248 1.0400 1.0008

0.4154 0.6142 0.9915 0.4037 0.5031 1.1510 0.7416 1.0352 0.9973

4.4. Influence of the analysis scale on fractal evaluation The fractal parameters of all the studied potential noises were calculated at the scale 256 s (J = 6) level so as to compare with each other. In fact, self-similarity of a time series exists in a certain scale range, that is, for a time series, if the analysis scale is in this range, the scale has little or no influence on the evaluation of fractal parameter, and if out of this range, the scale would have some influence on the fractal evaluation. So Hws values of all potential noises were calculated at the scale 64 s (J = 4) level, shown in Table 4. Comparing Hws values at J = 4 level with the corresponding values at J = 6 level, it was found that except in phosphate/molybdate and phosphate/molybdate/citrate solution, there were no apparent change for Hws properties of potentials in other solution. In phosphate/molybdate and phosphate/molybdate/citrate solution, potential noises mainly consisted of a series of break/heal events of passive film, whose transient time was shorter in phosphate/molybdate solution than in phosphate/ molybdate/citrate solution. When the analysis scale became smaller from 256 s (J = 6) to 64 s (J = 4), the correlation between adjacent data in a break/heal event would become strong spontaneously, especially for potential in phosphate/molybdate/citrate solution, its fractal characteristic became long-range strong persistence from short-range strong persistence because the analysis scale (64 s) was little more or less than the transient time of a lot of events. In fact, the results were consistent with that from FFT analysis. PSD of potential noise in phosphate/molybdate/citrate solution exhibited two parts: a low-frequency plateau and a high-frequency part which fitted the following relationship: PSD(V) = Kfa, a = 2.2151, and the roll-off frequency, fc, which was the frequency to separate the two parts of the PSD, shown in Fig. 11. From the literature [49], it could be concluded from a and fc that pitting corrosion occurred with the frequency fc on the passive surface. Here, fc  1/scale and a  2Hws. Therefore, the results from FFT and wavelet analysis supported to each other, but the result from wavelet analysis was clearer than from FFT.

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Fig. 11. The power spectrum of potential noises generated from 7075-T76 aluminum alloy in 3.5% NaCl solution with phosphate/citrate/molybdate.

4.5. Analysis of local fractal characteristic The discussion in previous theory showed that the local Hws was more suitable to describe the varying feature of the signal fluctuating frequently, and the general Hws was able enough to represent the fractal characteristic of the signal with very slow fluctuation. So the potential noises, originated from 7075-T76 aluminum alloy in 3.5% NaCl solution containing phosphate, phosphate/molybdate, and phosphate/ molybdate/citrate respectively, were chosen so as to analyze their local fractal characteristics. The estimation algorithm was to partition the noise signals into 32 = 25 (l = 5) subintervals, which resulted in 32 estimates of local Hws. When the STD of dj,k for each estimate was calculated, the wavelet decomposition level j was restricted to be no more than 4 (J 0 = 4). In order to compare with each other conveniently, the results of local Hws was put together with its corresponding original noises, shown in Figs. 12–14. In phosphate solution, the general Hws of its potential noise is 0.3403 (Table 4), which indicated that the fractal characteristic of its noise was weak persistence, that is, corrosion events on the electrode surface were independent because phosphate could not form a stable and uniform passive film [50]. From Fig. 12, many of local Hws were between 0 and 0.5, so in a majority of time, the adjacent potential data were just correlated weakly, but there were a few of Hws below zero or more than 0.5, which meant that the potentials in the corresponding parts were anti-persistence

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Fig. 12. Potential noise (a) generated from 7075-T76 aluminum alloy in 3.5% NaCl solution with phosphate and the estimation of its local wavelet STD exponent (Hws) (b).

Fig. 13. Potential noise (a) generated from 7075-T76 aluminum alloy in 3.5% NaCl solution with phosphate/molybdate and the estimation of its local wavelet STD exponent (Hws) (b).

or short-range strong persistence. With comparison of the potential fluctuation and its local Hws values in phosphate solution (Fig. 12(a) and (b)), the local Hws values

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Fig. 14. Potential noise (a) generated from 7075-T76 aluminum alloy in 3.5% NaCl solution with phosphate/molybdate/citrate and the estimation of its local wavelet STD exponent (Hws) (b).

could effectively describe the change of the irregularity about potential fluctuation, for example, the potentials around 2000 s were strongly correlated, so the local Hws values of two corresponding intervals were 0.4587 and 0.8900. When both phosphate and molybdate existed in the solution, because of molybdate oxidation, the defects on the passive film decreased [44] and the potential fluctuation became slower, so the correlation between adjacent data was strengthened, and its general Hws was 0.6924. This change was also reflected with a majority of local Hws values between 0.5 and 1. However, the local Hws also pointed out that the weak persistence or anti-persistence existed about between 200 s and 300 s, and between 1800 s and the 1900 s (Fig. 13(b)), in which the corresponding potential fluctuated more frequently than other parts (Fig. 13(a)). In phosphate/molybdate/citrate solution, because of the synergistic effect of three inhibitors, the more stable film was formed on the surface of aluminum alloy [50,51], but there were a few spots, especially intermetallic particles such as Al7Cu2Fe and Al2CuMg, which was difficult to form the stable film [50]. So there were still break/heal events on the surface, however because of the influence of citrate on the capacitive behavior of the metal–environment interface, the re-passivation time became longer [52] than the analysis scale level. Thus its general Hws was 1.0248, which exhibited long-range strong persistence, and most of its local Hws was also more than 1. But at the beginning and end of noise signal (Fig. 14(a)), potential fluctuated more frequently than the middle part. This phenomenon was reflected clearly by the corresponding local Hws values (Fig. 14(b)), which were between 0 and 0.5 and below zero to exhibit weak persistence and anti-persistence respectively.

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From the application of local Hws in Figs. 12–14, the analysis showed that the conclusions from the general and local Hws were basically consistent about the fractal characteristics of these electrochemical noises. But the local Hws could quantitatively extract the local feature of electrochemical noise, and accurately display the fractal characteristics of different parts so as to analyze its corrosion process in convenience, and the general Hws just reflected the statistical property of the local Hws values. Therefore, in practice when electrochemical noise is used to measure and monitor corrosion process, the local Hws could effectively describe the change of the noise properties, especially when the data sample size is not sufficient, it can be extended with repeating measurement data so as to avoid the inherent defects of FFT. Therefore, local Hws analysis should be more suitably applied to analyze electrochemical noise in engineering. 5. Conclusion On the basis of the introduction to calculate general and local fractal parameters of simulating signals with wavelet transform, general and local fractal characteristics were analyzed about electrochemical potential noises generated from 7075-T76 aluminum alloy in 3.5% NaCl solution without and with different inhibitors respectively. It was concluded as following: • log2 ðSTDj Þ  log2 ð2j Þ plot could not only be used to compare the fluctuation of electrochemical noise at different scales with each other, but also calculate the fractal parameter, wavelet STD exponent (Hws), so as to describe the irregularity, or complexity of electrochemical noise quantitatively. • Compared with Hausdorff exponent (Ha) and wavelet variance exponent (Hw), Hws resulting from the standard deviation of wavelet detail coefficients, had smaller deviation and could describe the fractal characteristics of electrochemical noises in a wide range. Compared with the spectral power (b) from FFT analysis,Hws calculation avoids some inherent defects such as detrending and windowing. • For the potential noises in the study, the smaller Hws value, the more frequently potential fluctuated on the electrode surface, and the less effect of inhibitor in the solution. The difference between Hws and 1 reflected the condition of surface passivity. The smaller the difference, the more stable surface film at the analysis scale. • For the studied electrochemical noises, the results from Hws analysis were consistent with that from FFT analysis, but which were more distinct and quantitative than from FFT. • The results from general and local Hws were basically consistent about the fractal characteristics of these electrochemical noises. But local Hws could apparently indicate the change of the fractal characteristics with time for electrochemical noise so as to analyze its corrosion process in convenience, and the general Hws reflected the statistical property of the local Hws values. • Local Hws was more suitable to describe the varying feature of signal fluctuating frequently, however general Hws was enough to represent the fractal characteristic of the signal with very slow fluctuation.

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