Stochastical Analysis of a Saline Electrochemical System by Michelson Interferometry

O. Sarmientoa, D. Mayorgab, J. Uruchurtub, J. Marbanb, E. Sarmientoc

a Instituto Tecnológico de Zacatepec-ITZ, Depto. Metalmecánica, Calzada Tecnológico

No. 27, Zacatepec Morelos, México. b Centro de Investigaciones en Ingeniería y Ciencias Aplicadas-CIICAp. Av. Universidad

1001, Chamilpa, Cuernavaca Morelos, México. c Universidad Tecnológica Emiliano Zapata-UTEZ, Área Mecánica Industrial, Av.

Universidad Tecnológica No.1, Emiliano Zapata Morelos, México.

In this work, the state-space diagrams, Lyapunov exponents and correlation dimension are obtained to characterize the corrosion process of an aluminum sample immersed in a solution of distilled water to 0.5, 1.5 and 3% NaCl. The stochastic analysis of the system is made from data sets obtained from Michelson interferometric patterns recorded during the optical monitoring of the corrosion process, which are digitally processed and converted into time series associated with the response of the optical pattern obtained. Moreover, the technique of spectral analysis (FFT) and calculation of the Hurst exponent (H) are used to establish a relationship between frequency components and morphology of corrosion in an entirely qualitative and where only distinguishes corrosion process.

Introduction Aluminum usually develops pits at a corrosion potential of the same type than the one which would exist when the metal is exposed to an environment containing chlorides; this particularity is advantageous due to the fact that if anyone whishes to follow the natural electrochemical potential of aluminum when pitting is presented, it will not be necessary then to displace it to the pitting potential, as its open-circuit potential is the same pitting one in the presence of chlorides. On the other hand pitting phenomenon is related with an apparently random pattern with eventual transitory jumps associated to the passive layer breaking, which explains why this phenomenon, and ordinary one not only for aluminum, had been considered as a stochastic one (1, 13). The extensive development of techniques dedicated to investigate electrochemical reactions has its fundaments in the study of electrochemical processes fluctuations known as electrochemical noise (EN) (2). During the last years, studies performed on current and potential fluctuations have been used to characterize corrosion processes, giving as a result a new approach to oscillations presented in this type of phenomenon (3-9). On the other hand, several analytic tools have been developed to detect and characterize the hidden features on the apparently random data of electrochemical systems, and reveal in this way the properties which might represent their fluctuant behavior. Among these tools, chaos theory is included, which have been developed to characterize non-linear and non-periodic systems; this theory has been also applied during the last years to study and characterize the behavior of several biological, chemical and physical systems, including

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the electrochemical processes (10-12, 15). As an example, on (13) it is proposed that corrosion potential behavior of aluminum at long term and due to chlorides presence is ruled by at least one strange attractor in a region of phase space (chaos theory concepts); these terms can be defined as the values where a system approaches (attractor) into a mathematical space containing the values of trajectories of variables which represent the phenomenon (phase space). This analysis allows making a distinction of a stochastic behavior (random) from another deterministic. Although concepts and models needed to characterize a chaotic behavior are usually complex, the association of low dimension chaotic dynamics to the presence of strange attractors has allowed its quantification (14). There are several methods useful to provide the estimation of strange attractors properties which can be characterize, for example, by its fractal presence into a spatial representation (Correlation Dimension), or by the divergence degree of repeated patterns using Lyapunov Exponents. On (13, 14), studies of electrochemical measurements which include analysis of noise fluctuations in a potential registered during aluminum free corrosion into a saline solution are described; the analysis of electrochemical noise spontaneously generated during the experiments were performed by several methods as those mentioned above, and such analysis revealed the form and stochastic and fractal nature of their signals. Previously, important non-intrusive alternatives to evaluate metallic corrosion in situ have been reported being Michelson interferometry one of those (16-18). The method is a relevant optical technique due to its capabilities for metrology, as it is able to measure and detect surface alterations in the order of λ/2, being λ the used laser source wavelength (19); the monitoring and evaluation of corrosion processes from interferometric patterns consecutively registered is combined with other non-linear analysis techniques, and a representation of the “hidden” information also contained in such processes is feasible (20, 21). Other techniques applied on electrochemistry which can be combined with optical ones are Scale Rank Analysis (R/S) and Power Spectral Density Analysis (PSD). The first was originally proposed by Hurst (22) and applied by Mandelbrot and Wallis to determined the time series fractal feature (23); a description and applications of the (R/S) technique can be found in (24-29). The second method is the most common way to analyze registered on time data transformed to frequency domain by using the Fourier Fast Transform (FFT) (30, 31). As explained above and with eventual application of dynamic description techniques, the purpose of this study was to analyze the surface profiles obtained from Michelson interferograms registered during the corrosion process of an aluminum sample immersed into saline solution with different NaCl concentrations; by using different Chaos Theory analysis tools to compare and correlate changes related to the corrosion process dynamic and the processes which take place at its surface.

Experimental Procedure The samples used in our experiments were fabricated with 20x20x1.5 mm3

dimensions from commercial aluminum (99.5% pure), capped into MC-40 plastic resin and manually polished bright in one of its faces. As was previously reporte in (16-18), the utilized experimental setup consisted of a Michelson interferometer (Figure 1), where a He-Ne laser beam (75 mW, 632.8 nm) is amplified approximately to a ~10mm diameter

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with a biconvex lens (f=50 mm). A 50/50 beam splitter divides the laser beam into two components; one is directed and reflected back from a mirror to the beam splitter (reference beam), and the other illuminates the sample (object beam), previously fixed into an electrochemical cell (10x10x8 cm3), and situated in the other interferometer arm. The optical signal at the interferometer output is also amplified with a 40X microscope objective and displayed on a translucent observation screen; interferograms are then registered with a CCD camera located behind the screen and finally recorded in a PC for further analysis. Into the electrochemical cell located in the interferometer object arm, besides enclosing the sample under investigation (working electrode), it also contained a reference electrode and an auxiliary or counter-electrode (Calomel and graphite, respectively) immersed in a NaCl solution.

Figure 1. Optical-electrochemical experimental setup for an aluminum sample (16,18).

At time series acquisition, 8000 data samples are registered at a 200 ms/date sampling

rate. A HP Benchlink data logger and a Newport optical power detector coupled to a PC are used. The beam power reflected from the sample (object beam) is used as a time series parameter, because it is difficult during the system study to control more than one variable. The stochastic time series analysis is performed by using the Chaos Data Analyzer software. In order to obtain the corrosion process characterization, an ACM Gill 8AC potentiostat is used to control the electrochemical polarization tests. The combination of potentiodynamic polarization curves and the progressive and consecutive interferograms recording using a CCD KP-M2R camera, allows identifying of electrochemical parameters associated. The experimental curve was obtained applying a fast screening potential in the -1200 mV to 1500 mV range, at a 270 mV/min rate, and the utilized electrolyte was a solution of distilled water and NaCl (250 ml) at different mass concentrations (0.5, 1.5 and 3%). Each electrochemical and optical test had the same time interval, as they were simultaneously performed; every interferogram was registered on real-time (video rate), so each recorded image correspond to an specific point at the electrochemical polarization curve. An application software was used for image processing; each interferfogram was processed to obtain intensity maxima profiles (1x230 pixels vectors) corresponding to ∼78.53 mm2 transverse sections illuminated by the object beam. Assuming the metal surface exposed to corrosion changes on time, those changes might be observed in the registered intensity profiles; these are used to characterize the metal surface by using the Hurst exponent and FFT techniques.

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Results The time series associated to the laser beam power reflected from the aluminum

sample surface exposed to a NaCl solution are presented at Figure 2; three different time series were obtained for different NaCl concentrations: a) 0.5%, b) 1.5% and c) 3.0% respectively. Considering the obtained data, the state space diagrams were calculated (Figure 2d-f); these consist of graphics of the variable of interest as a function of the delay of the same variable derived on time (1 delay). This analysis allows distinguishing a stochastic behavior from a deterministic one; however for a deterministic system the trajectories would adopt a particular form which topology determines the level of behavior organization of the system.

Figure 2. Time series and state space diagrams for an aluminum simple for different NaCl concentrations: a) and d) 0.5%; b) and e) 1.5%; c) and f) 3.0%.

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As it can be seen at Figure 2d-f, graphics present a particular form as there exist

separations and approaches confined only to a region of the phase space (or state space), which allows to consider them as strange attractors. The characteristic attractor for the chaotic behavior of a system might reflects in its geometry such a situation, so it should exhibit two opposed tendencies: neighbor trajectories might converge to the attractor and contrary to this, to reflect the state of sensitivity to initial conditions, trajectories should diverge separating more and more; all these features can be determine by calculation of the correlation dimension and Lyapunov exponent, respectively. The calculation of the attractor dimensionality consists, basically, to built a function C(ε), containing a probability that two arbitrary points of the orbit are closer than ε, which is arbitrary chose. The correlation dimension (Eq. 1) is defined as the curve slope C(ε) vs. ε, being C(ε) the correlation integral defined by Eq. 2, where H is the Heaviside function; a non-integer value from the correlation dimension indicates that the data correspond, probably, to a fractal (32):

�� = ����→0 ��� �(�)��� � [1]

�(�) = lim�→∞ 1�2 � ��� − ��� − ������,�=1�≠�

[2]

The Lyapunov exponents are the most important parameters used to characterize the

attractor properties for a dynamic system. The maximum Lyapunov exponent (λ) is estimated by approximately following the evolution of chosen pairs form data points, being this a measure of the rate index where trajectories pass close one to each other in the phase space where they diverge (33, 34). For chaotic systems, at least one Lyapunov exponent might be positive (λ>0), i.e. the attractor exhibit one chaotic behavior in at least one direction along the phase spaces. For periodic systems, all the Lyapunov exponents are negative (λ<0). The Lyapunov exponent is zero (λ=0), close to a bifurcation or a kind of existing equilibrium state (15, 33, 35, 36). Eq. 3 can be used to determine the Lyapunov exponents.

λ = lim�→∞ 1�� �����=1 ����+1�� � [3]

Although the sensitivity to initial conditions which present the three obtained time

series (Figure. 2) at similar conditions seems to obey a deterministic systems, their long-term behavior can be also unpredictable. At Figure. 3, the correlation dimension and Lyapunov exponents estimated at different analysis dimensions are shown.

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Figure 3. a) Correlation dimension, b) Lyapunov exponents. On Figure 3a, two zones are clearly distinguished for the three systems: at the first one, it is observed how the correlation dimension tends to saturate for the first 5 analysis dimensions at a ∼1.04 saturation value; the second zone occurs for higher analysis dimensions as the saturation value increases to ∼6.03. Figure 3b represents Lyapunov exponents for the three analyzed systems. A more positive value for the exponent λ∼0.989 is observed at an analysis dimension equal to 2; subsequently the tendency points for the three systems to more ordered states as the analysis dimension increases, which may indicates an intrinsic electrochemical noise level regulated by chloride ions (37). The critical point on the correlation dimension and Lyapunov exponent determination lies on the analysis dimension choice; for chaotic systems the correlation dimension tends to saturate to a certain value as the analysis dimension increases (embedding dimension), and for random systems there is not such saturation. On the other hand, is enough for the highest exponent to be positive to consider the system an unpredictable one (chaotic). At Figure 4 the polarization cycle curve for the aluminum sample immersed into distilled water at 3% NaCl is presented; this result contributes to characterize the corrosion process dynamics for different polarization conditions, and to correlate the parameters obtained from Hurst analysis and FFT by means of the optical Michelson interferograms.

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Figure 4. Polarization curve of aluminum in H2O at 3% NaCl. The solid curve shows the corrosion potential (electrochemical method), and the dashed curve one represents the current density obtained as a function of the orthogonal displacement (optical method). On Figure 5, Michelson interferograms obtained for different potentials are shown, as also as their Fourier spectra corresponding to their digitally transformed intensity profiles (see inner windows). The FFT analysis shows that any signal may be expressed as a sum of sinusoidal functions of several frequencies and amplitudes and where their components, although dispersed in a time domain, may have a restricted situation or a characteristic similarity in the frequency domain. The method consists in splitting the data series for a variable of interest in an infinite sum of harmonic components (sines and cosines). As a result the contributions associated to each of these components in a Fourier spectrum can be obtained; the FFT is defined in terms of frequency by the next expression:

�(�) = � �(�) ∙ �−�2���∞−∞ �� [4]

The Fourier spectra obtained from the digital transformation of the interferograms to intensity profiles, represent the corrosion process as an electrochemical potential function at different polarizations curve zones as for example: active, passive, instability and pitting conditions.

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Figure 5. Interferograms, Fourier spectra and aluminum profile intensities at 3% NaCl. a) -910, b)-107, c) 435 and d) 1458 mV. The transition from the active zone to the pitting corrosion condition along the passive region modifies the optical signal of the interferogram due to the system dynamics for different potentials. The changes of a quasi-periodic signal, apparent noise and the number of transitions are clearly observed at Figure 5. The effect of the intensity profiles

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taken from the interference patterns on the behavior of Fourier spectra, can be explained by factors that modify the polarization and the main process taken place at the metal, where non-conductive substance layers can be formed on its surface, and increases the current density on non-cover sections, and on the other hand causes a high polarization on covered sections. Besides, the analysis establishes a direct relation between the frequency components, as a consequence of the events taken place at the metallic surface, essentially on the contact surface (metal-liquid border) or in the layers or adjacent corrosion products.

On Figure 6, the Hurst exponents (H), Lyapunov exponents (λ) and correlation dimension (CD) are presented calculated from the digital transformation to intensity profiles of the Michelson interferograms. The Hurst exponent detects persistence changes on the data series, where H=0.5 values would indicate that the analyze time series is a random one, and so every event included on the series is independent from the other. On the other hand, if 0.5<H<1 it would indicates that the series would tend to be persistent, i.e. that every event included on the series would be mutually related, in other words the appearing of an event would delayed or inhibits the beginning of a following event on the series. The definition of the Hurst exponent is closely linked to the Fractal Dimension concept (Df) by: �� = 2 − � [5]

The H exponent has a great versatility, as it has been used as a parameter for

corrosion studies on the study of corrosion, when pitting morphology generated on localized corrosion processes is evaluated. For the calculation of Hurst exponent, one of the most commonly used methods is the Rescaled Range Method (R/S), defined by Ec. 6,

� �� =1��(�� ��⁄ )

��=1 [6]

where R represents the difference between the maximum and minimum value of the variable, S is the standard deviations of the time series; a detailed description of this method can be found at (38). It has been shown that the existing relation between R/S y H is:

� �� ∝ �� ; � =log�� �� �

log(�) [7]

where τ is the measured time period and H is the Hurst exponent.

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Figure 6. Hurst and Lyapunov exponents, and Correlation Dimension for aluminum at 3% NaCl. The Lyapunov exponents and the correlation dimension were calculated based on the mutual information function, which determines the optimal value of delay time for the state space reconstruction (39); the minimum optimal analysis dimension (40) were determined using the false nearest neighbors method (FNN). Application software was used, where the R/S algorithm method to obtain the Hurst exponent values was implemented. The exponent values on Figure 6 correspond to different potentials obtained for different times of the potentiodynamic curve of the corrosion process (Figure 4). It can be appreciated that Lyapunov values hold into a λ= 0.45-0.515 (-910 to 500 mV) range, corresponding to the different zones, active (-910 mV), pre-passive and passive (-700 a -660 mV) of the polarization curve; it can be interpreted as some kind of reorganization onto the metallic surface. The highest positive exponent λ=0.528 is given around 750 mV (∼7 min) of the anodic oxidation region to subsequently decay to potentials above 1000 mV. A similar behavior is observed on the correlation dimension graphic for the same potentials, except that highest correlation dimension DC=4.537 is obtained for a 1250 mV potential value. However, the correlation dimension values are kept over the 4 value, which suggests a requirement of at least 5 differential equations to describe the process dynamics. An interesting correlation is observed mainly for the Hurst values and the interferograms registered at different potentials of the process. The H exponents change from persistent to less persistent as the potential increases among 0.94 and 0.78 values during the first 6 minutes (-910 a 500 mV), revealing in this way the observed changes in the interferometric patterns. The behavior of H exponent may be associated to the existing instability of changes among active, pre-passive and passive zones, which may represent a short-memory phenomenon, and also be reflected in the spectral analysis (see Figure 5). At higher positive potentials (transpassive zone), Hurst values tend to an average value of 0.85±0.02, which may be associated to a long-term memory effect phenomenon, and so the system tends to more ordered states.

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Conclusions The simultaneous experimental application of optical and electrochemical methods, together with applications of non-linear dynamics description techniques, allows a visual tracking useful to characterize the corrosion process dynamics from the digital transformation to intensity profiles of Michelson interferograms. The presented results show the advantage of utilizing chaos theory analysis tools in order to compare and correlate changes related to the corrosion processes dynamics. According to our results, the analyzed corrosion process dynamics is a chaotic one.

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