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Ind. Eng. Chem. Res. 1998, 37, 2172-2179
Chaotic Oscillations on Arrays of Iron Electrodes Zhihao Fei† and John L. Hudson* Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesville, Virginia 22903-2442
Experiments have been carried out with arrays of iron electrodes in sulfuric acid. The arrays can be used to study surface patterns since the currents of the individual electrodes in the array, as well as the total current, are measured. Most of the experiments were done with an impinging jet system at a constant potential, under which conditions high-frequency periodic and chaotic oscillations occur. It was seen that the complexity of the oscillations increased as the system size (the number of electrodes in the arrays) increased; oscillations changed from period 1 to period 2 to a simple chaos and then to a more complex chaos. This increase in complexity is similar to that found in earlier work in which the size of single electrodes was increased. In the case of low-dimensional chaos on the arrays, the overall dimension (that of the total current) was the same as the local dimensions (those of the individual electrodes); the overall dimension appears to be higher than the local dimensions with more complex chaos. The experiments are done under conditions under which the coupling between electrodes whose spacing was that of the greatest distance in the largest array is not strong enough to synchronize periodic oscillations. Introduction Spatiotemporal patterns are known to arise autonomously in a variety of chemically reacting systems including liquid-phase reactions, gas-solid heterogeneous catalytic reactions, and as shall be discussed in this paper, electrochemical reactions. The patterns are produced by the interaction of nonlinear chemical reaction and coupling among sites. The coupling can be local such as that produced by diffusion, or it can be longer range, even global; in global coupling a change in a condition or rate of reaction at one location is felt equally at all other locations. An example of global coupling is a fluid-solid reacting system where reaction at any location on the solid produces a concentration change in the fluid of finite capacity and the change in concentration then influences the rate of reaction at all sites on the surface. Electrochemical reactions are a type of heterogeneous reacting system; both local and global coupling can exist. There is also a coupling, that of migration through the electric field in the electrolyte, which is long-range but not global; the coupling is intermediate between local diffusive coupling and global coupling. It is long-range because changes are felt everywhere on the electrode surface, but it is not global because the effect decreases with distance. Our interest here is in the patterns that occur in the oscillatory regime, particularly under chaotic conditions. Consideration is given to the question of how coupling among sites affects the dynamic behavior and how increasing the size of a system, which influences the ease of coupling relative to the rate of reaction, can lead to increased complexity. The experiments are carried out using arrays of electrodes of varying numbers of elements. The arrays consist of a number of small disks which are made from the ends of wires embedded in an * Author to whom correspondence should be addressed. E-mail:
[email protected]. Fax: 804 982 2658. Telephone: 804 924 6275. † Present address: Xytel, 1001 Cambridge Dr., Elk Grove Village, IL 60007.
insulator. The current in each of the electrodes is measured independently, and waves and moving patterns can be characterized from the time series of current from many sites. An array of electrodes behaves approximately like a larger, single electrode of the same total area; the analogy is not perfect, of course, since there is a current distribution on each single electrode. This similarity exists because much of the coupling in the electrochemical system is through the electrolyte and long-range effects are important. Long-range interactions have received attention recently in electrochemical (Fla¨tgen and Krischer, 1995; Mazouz et al., 1997a,b) and gas-solid heterogeneous reaction systems (Veser et al., 1993; Middya et al., 1994; Falcke et al., 1995; Mertens et al., 1994). Oscillations in current and/or potential occur commonly during electrochemical reactions (Hudson and Tsotsis, 1994), including the electrodissolution of metals (Lee et al., 1985, Haim et al., 1992, Otterstedt et al., 1996a,b) and electrocatalytic reactions (Albahadily and Schell, 1991; Koper and Sluyters, 1993). The anodic electrodissolution of iron in acidic solution is one of the most highly studied systems. Under potentiostatic conditions autonomous current oscillations can occur (Franck, 1951; Russell and Newman, 1987). Depending on system parameters, notably applied potential and mass-transfer coefficient between the electrolyte and electrode, two general types of oscillations have been seen. In one region of parameter space, near the Flade potential above which the electrode is passive, slow (period of seconds to minutes) relaxation oscillations can occur (Franck and Fitzhugh, 1960). The dynamics and the spatiotemporal patterns associated with the relatively slow active-passive periodic relaxation oscillations have been studied by many investigators. Surface waves occurring during oscillatory conditions have been studied on both disk (Pigeaud and Kirkpatrick, 1969; Hudson et al., 1994) and ring (Sayer and Hudson, 1995) electrodes; bifurcations such as spatiotemporal period doublings have been seen. The surface patterns during
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Figure 1. Configurations of electrode arrays: (a) single electrode; (b) 2 × 2 array; (c) 4 × 4 array; (d) 61-electrode array. All diameters ) 0.5 mm.
active-passive oscillations have also been studied using arrays of electrodes (Fei et al., 1996). In a second region of parameter space, high-frequency (10-500 Hz), often chaotic oscillations have been observed (Wang and Hudson, 1991), and it is this type of oscillation that is investigated in this paper. The oscillations occur under conditions in which the masstransfer coefficient between electrolyte and electrode is elevated relative to a stagnant solution, and the experiments are done with an impinging jet system. No passivation of the electrode occurs. The oscillations in the high-frequency region can be either periodic or chaotic. Wang and Hudson (1991) have shown that there is an increase in complexity of the behavior as the electrode size is increased. They used a series of electrodes (employing a rotating disk electrode system) and showed that, as the area of the electrode was increased, the behavior changed from periodic to chaotic and a further increase produced an increase in the dimensionality, or complexity, of the chaotic attractors. Here we carry out similar experiments, but rather than increasing the size of a single electrode, we increase the number of identical electrodes. Sets of electrodes of diameter 0.5 mm are used, and the number of electrodes is increased in steps from 1 to 61. We compare the total current of all the electrodes to that obtained on the series of single electrodes. We also investigate the behavior of the current from the individual electrodes. Thus information on spatial variations of the current can be obtained. Experimental Section The configurations of the arrays, consisting of 1-61 electrodes, are shown in Figure 1. The diameter of the jet is 5 mm, and the distance from the jet to the
electrode surface is 6 mm. Each electrode is made from pure iron wire (Aldrich Chemical Co., Inc. 99.99+%) of diameter 0.5 mm. The distance between the wires is less than 0.05 mm. The electrodes are embedded in epoxy, and reaction takes place only on the ends. The electrode array faces downward. The electrode surface is polished with 400-grit silicon carbide sandpaper before each experiment. The electrodes and epoxy are thus sanded to a flat plane. The surface is then washed with deionized water and dried with compressed air. The electrochemical reaction is controlled by a potentiostat (EG&G Princeton Applied Research, model 273). Electrodes in the array link to working electrode jacks of the potentiostat through a ZRA box (zero-resistance ammeters). The ZRA circuitry consists of an operational amplifier and associated feedback circuitry such that currents in the range 10-9-10-2 A may be measured for each electrode in an array without altering the polarization potential of the electrodes. All electrodes of the array, up to 61, are held at the same potential. A pentium PC installed with a 32-channel data acquisition board (Keithley DAS-1800HC2) was used for data sampling. One channel was used for measurement of the total current (through all individual electrodes) and a second for the potential of the working electrode (the array); both of these signals come directly from the potentiostat. A third channel is used to record the total current (again, through all the elements) using a ZRA; this total current measurement is simply a check on the total current signal obtained from the potentiostat. Twenty-nine channels are thus available for measurement of the currents of the individual electrodes. Thus for the smaller arrays, the currents of all electrodes were measured. The 61-electrode array is the exception since not all currents could be sampled. Note that current is flowing through all 61 electrodes but that, due to the limitations of the 32-channel acquisition board, only some of the individual currents were sampled. (A subsequent increase in the number of channels allows the sampling of up to 80 channels.) Since a selection had to be made, the current was measured on the three major diagonals consisting of 9 elements each; this required 25 channels since the center point is common to all three diagonals. The sampling frequency of each channel is 2500 Hz. Experiments are carried out in 1 M H2SO4 solution. The volume is 300 mL. The reference electrode is a standard Hg/Hg2SO4/K2SO4 electrode in a capillary, the end of which is below the array (next to the jet) and about 6 mm from the surface. The counter electrode is a cylindrical platinum foil in the plane of and encircling the array. Results Effect of Array Size on Dynamics. All of the experiments were done in the region of parameter space in which high-frequency periodic or chaotic oscillations occur. The experiments are done under potentiostatic (constant potential) conditions, and the total currents and current of the individual elements are measured. In this section we consider the total current on the arrays, that is, the sum of all the individual currents. The results of a series of experiments in which the main parameter being changed is the number of electrodes in the array are shown in Figures 2-4. The numbers of electrodes, all of the same individual size, are 1, 4, 16, and 61. The jet speed is the same in all
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Figure 3. Total current of 4 × 4 array, E ) -360 mV, jet speed ) 18 cm/s: (a) time series; (b) attractor.
Figure 2. Time series, total current: (a) single electrode, diameter ) 0.5 mm, E ) -440 mV, jet speed ) 18 cm/s; (b) 2 × 2 array, E ) -430 mV, jet speed ) 18 cm/s.
cases, 18 cm/s. (The potential is somewhat different for these four experiments only because the parameter range in which high-frequency oscillations occurs changes with the reaction area. The potential range in which the oscillations occurs increases as the reaction area increases. Nevertheless, the relative potential, that is, the position within the oscillatory range, is the same in the four cases, viz., the potential is held somewhat above the midpoint of the oscillatory range.) Thus the dominant parameter being changed is the number of electrodes. For the single electrode and the 2 × 2 array, only periodic oscillations are found; see Figure 2, where period 1 and period 2 oscillations are seen for the single electrode and the 4 × 4 array, respectively. As the electrode number is increased, chaotic oscillations develop. An example of these chaotic oscillations is shown for the 4 × 4 array in Figure 3, where the time series and an attractor may be seen. Although we have not varied the parameter (the number of electrodes) finely enough to determine the bifurcation structure, it is likely that a period doubling sequence to chaos occurs as the number of electrodes is increased. The time series and attractor for the hexagonal array of 61 electrodes are shown in Figure 4. As can be seen from both the time series and the attractor, the complexity of the signal increased further as the number of electrodes was increased from 16 to 61. These results are consistent with the results obtained with a single electrode with a rotating disk in which it was seen that
Figure 4. Total current of 61-electrode array, E ) -50 mV, jet speed ) 18 cm/s: (a) time series; (b) attractor.
the complexity of the signals increased as the area of the single electrode was increased. The information dimensions (Kostelich and Swinney, 1987; Badii and Politi, 1985) of the chaotic attractors have been calculated and are shown in Figure 5 as a function of the number of electrodes in the array. Also
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Figure 5. Information dimension versus electrode surface area or electrode number in the array. Dimensions of single electrodes from Wang and Hudson (1991).
shown on the figure are the dimensions of the single electrode obtained by increasing the area of the electrode. For comparison, the same abscissa, the total area of the reaction surface, whether that of a single electrode or the array, is used. Although the work in the present paper with the arrays has only two chaotic attractors, and thus dimensions, it is noted that the attractor dimension appears to be approximately the same for a single electrode and an array of electrodes of the same total area. Current of Individual Electrodes. In the above we saw that the complexity of the signals increased as the number of electrodes increased. The electrodes are all the same size and are held at the same potential. Nevertheless, it is expected that there are slight differences among the electrode surfaces; thus we are investigating the case of nominally identical, but slightly differing, coupled electrodes. In order to gain additional information on the nature of the dynamics, we have measured the current signals on individual electrodes in the arrays. We consider first the 4 × 4 array, which was the largest of the arrays studied for which periodic oscillations were observed. (The 4 × 4 array can produce both periodic and chaotic oscillations, depending on the applied potential.) A time series of total current in the periodic regime is shown in Figure 6a. Although there is a maximum of four electrodes in any direction, some limited information can be obtained on spatial patterns from the measurements of the individual electrodes. The electrodes in the array are numbered from 0 to 15 with the first row 0-3, the second 4-7, etc. The autocorrelations for seven of the individual electrodes (1-7) and the cross correlations of six of them with number 1 are shown in Figure 6b,c. Thus the data shown in Figure 6 are for the top two rows. It is seen that the individual electrodes all oscillate at the same frequency but that there are some phase lags among some of them. Some of the electrodes are approximately synchronized. The corner electrodes, of which only electrode 3 is shown, are not synchronized to any of the others in the array. However, the sets of three electrodes {1, 4, 5} and {2, 6, 7} are almost synchronized with each other. These sets are in the upper left corner and upper right corner as shown in the diagram of Figure 1, omitting the corner
Figure 6. Periodic oscillations of 4 × 4 array, E ) -375 mV, jet speed ) 18 cm/s: (a) time series of total current; (b) autocorrelation of each electrode; (c) cross correlation of electrode 1 versus each electrode.
electrodes. Coupling is both by migration through the electric field and by convection; it is thus reasonable that these sets should be synchronized since the elements are adjacent to each other and are most strongly coupled by convection. Flow from the jet passes, for example, from 5 over 1 and 4 (similarly from 6 to 2 and 7. Although there is also flow in the direction from 5 to 0 and 6 to 3, that is, diagonally outward, the slightly greater separation distance apparently decreases the interaction sufficiently that these electrodes are not synchronized. We turn now to the chaotic behavior on both the 4 × 4 and the 61-electrode arrays. Time signals from the individual electrodes in the chaotic regime have also been measured. In the chaotic region the time series from each of the individual electrodes is also chaotic. These time series are qualitatively the same as those for the total current which were shown in Figures 3a and 4a for the 4 × 4 and the 61-electrode arrays, respectively. No pair of signals from the individual electrodes is exactly synchronized. The coupling in the chaotic region is not strong enough to overcome differences among the electrodes, noise, and the inherent separation of trajectories characteristic of chaotic behavior.
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Figure 7. Chaotic oscillations of 4 × 4 and 61-electrode arrays, jet speed ) 18 cm/s: (a) autocorrelation of each electrode, 4 × 4 array, E ) -360 mV; (b) cross correlation of electrode 1 versus each electrode, 4 × 4 array, E ) -360 mV; (c) autocorrelation of each electrode, 61-electrode array, E ) -50 mV; (d) cross correlation of electrode 1 versus each electrode, 61-electrode array, E ) -50 mV.
Autocorrelations and cross correlations for the chaotic signals from the 4 × 4 and hexagonal 61-electrode arrays are shown in Figure 7. The autocorrelations for the individual electrodes for the 4 × 4 array are very similar although the amplitudes vary slightly. Those for the 61-electrode array show greater variation. The autocorrelation for the 4 × 4 array, Figure 7a, is typical for a low-dimension chaotic system; the correlation dies off with increasing lag k; that for the 61-electrode array, Figure 7c, dies off even more quickly since the dimension (complexity) of the chaos is higher. Note that in the cross correlations, Figure 7b,d, no two curves are
exactly the same; this is most pronounced in Figure 7d. The signals from the individual electrodes are not synchronized, and no signal is strongly correlated to any other. Important characteristics of chaotic behavior are the dimensionality of the signals, related to the number of equations required to describe it; of interest are the dimensionality of the entire system (in our case the total current) and that locally (here the individual electrodes) and how the local and overall dimensions are related. To this end we constructed attractors and calculated one type of dimension from them. Attractors constructed from the time series for four of the individual electrodes for the 4 × 4 array and the 61-electrode hexagonal array are shown in Figures 8 and 9, respectively. The general shapes of the attractors for each of the individual electrodes are similar, that is, the attractors from the signals from the individual electrodes for a given condition are similar to each other; but they are certainly not identical, as can be seen by comparing Figure 8a-d or 9a-d. Furthermore, the attractors for the individual currents from the 4 × 4 array have the same general features as that obtained from the total current. For the 61-electrode array, the individual attractors appear to have a somewhat less complicated structure than that of the total current; this is most noticeable in Figure 9a. The information dimensions for the individual attractors were calculated and compared to the dimension obtained from the total current. The information dimension is one measure of the complexity of signals; although there are other measures of dimensions, including correlation dimension and even embedding dimension (Mikhailov and Loskutov, 1996) which can be calculated, the information dimension has proven to be a useful quantity. For the 4 × 4 array, the dimension of the total current is about 2.5. This is a mean value taken from several experiments with the array; almost all values fell in the range 2.4-2.6. The individual currents also gave a mean value of 2.5; here the variation was slightly greater, but almost all fell in the range 2.35-2.65. Thus to the accuracy of the experiments all individual currents have the same value of information dimension as the total current, and the dimension of the system can be obtained from a signal at any location. This is the expected result for a system in which the spatial correlation among sites is high (for related discussion, see Mikhailov and Loskutov (1996)). We shall return to the degree of interaction among the electrodes below. The information dimensions were also calculated from the individual signals obtained with the hexagonal 61electrode array. Again data from many experiments were used and means taken. The mean dimension for the total current was 3.4; the variation among runs was from 3.2 to 3.6. There was a greater variation, more scatter, in the data from the individual electrodes; the mean value was 2.9 with values being obtained from 2.6 to 3.8. Although the scatter is somewhat greater, it is seen that the mean value of the dimension calculated from individual electrodes is about 0.5 below the mean value for the total current. Therefore, it is seen that the information dimension obtained with data from an individual site is below that of the entire system. This may indicate that the interactions in this system die out over the length scales of the electrodes. We return to this point below in the discussion.
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Figure 8. Attractors from individual electrodes of 4 × 4 array, E ) -360 mV, jet speed ) 18 cm/s (the attractor from the total current was shown in Figure 3): (a) electrode 3; (b) electrode 4; (c) electrode 5; (d) electrode 7.
Interactions among Electrodes. We have seen above that the geometry and size of the electrode arrays influence the nature of the system dynamics. The oscillations on individual electrodes are affected by those of the others, and coupling occurs both through the electric field and by convection and diffusion. In order to aid in the understanding of the interactions among the electrodes in the arrays, we have carried out some simple experiments with small numbers of electrodes and have varied the coupling strength by changing the distance between, and the relative positions of, the electrodes. Here we summarized the results which are most relevant to an interpretation of the abovedescribed experiments; details can be found in Fei (1997). A series of experiments was carried out with two electrodes using the same impinging jet apparatus as was used with the configurations shown in Figure 1. All conditions except the configuration of the working electrode were the same as that used for the experiments described above; the jet speed and the positions of the reference and counter electrodes were the same. The two electrodes had individual diameters of 0.5 mm, that is, were the same size as those used in the experiments described above. In the first set of experiments the individual electrodes were placed on a diameter of the electrode support and were equidistant from the centerline of the jet flow. The distance between the two electrodes was varied systematically; the distances between electrodes were as follows: less than 0.05 mm, the same as used in the array studies; 1 mm; 2 mm; and 3 mm. The oscillations under these conditions are always periodic since arrays of this total area will not yield
chaotic behavior, at least not under the conditions of our experiments. For the first two configurations, that is, up to and including a separation of 1 mm, the oscillations on the two electrodes have the same frequency and phase; the total current is then, of course, a simple periodic oscillation. As the separation is increased to 2 mm and above, the oscillations of the two individual electrodes are no longer phase-locked; they have a slightly different frequency and, of course, cannot be in phase. The total current under these conditions is quasiperiodic. We carried out some similar experiments with two (and four) electrodes with a rotating disk apparatus and where again the distance between (among) the electrodes was varied; similar results were obtained with the rotating disk as with the impinging jet except that the separation distance at which phase locking no longer occurred was greater with the rotating disk. The likely reason, of course, is that the azimuthal flow of the rotating disk increases coupling among the electrodes. (Note that neither the rotating disk nor the impinging jet is a uniformly accessible surface (Newman, 1973)swith constant mass-transfer coefficientsunder the conditions of these experiments where the entire surface is not active.) We see that mass transfer via convection is important. In order to investigate the effect of convection further, we carried out some experiments with the impinging jet off center so that convection was asymmetric. The electrode spacing was 1 mm. The jet center was placed 3 mm off center so that flow occurred from one electrode (1) to the other electrode (2). The signal from electrode 1 led that of electrode 2. Reversing the direction of flow causes electrode 2 to lead electrode 1. Putting the flow
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Figure 9. Attractors from individual electrodes of 61-electrode array, E ) -50 mV, jet speed ) 18 cm/s (the attractor from the total current was shown in Figure 4): (a) electrode 2; (b) electrode 3; (c) electrode 5; (d) electrode 7.
transverse to the line on which the two electrodes are placed so that the flow does not cause convection from one to the other causes a decoupling of the time series; the resulting total time series is quasiperiodic. Discussion There have been many theoretical and simulation studies carried out with arrays of coupled chaotic oscillators, both with arrays of coupled maps (Kaneko, 1989, 1994) and with coupled ordinary differential equations (Zanette and Mikhailov, 1997; Kocarev and Parlitz, 1996). In almost all of these studies the individual elements, the maps or the differential equations, are identical, that is, each element is governed by the same local equation with the same values of the parameters. The coupling among elements in the arrays is usually taken to be local (diffusive), in which individual elements are directly coupled only to their immediate neighbors (Kocarev and Parlitz, 1996; Kaneko, 1989), or global, in which some mean is taken over the system and this mean fed back to each element of the array (Kaneko, 1994; Zanette and Mikhailov, 1997). In the simulation studies a random distribution of initial conditions is typically taken. The chaotic nature of the individual elements tends to keep the elements unsynchronized since any slight differences are magnified by the sensitive dependence of the state on the initial conditions. Coupling, whether local or global, tends to bring the states together, and thus, whether the elements are synchronized or not depends on the relative strengths of the effects. It has been shown that synchronization, either of all elements or of subsets of
elements, can occur. Furthermore, for locally coupled large-size systems, the complexity of the chaos, as measured by the number of positive Lyapunov exponents or the dimension of the chaos, can scale with the system size (Chate´, 1995); interpretations of the increase in complexity are based on the ideas of more spatial modes coming into play or of the system size becoming larger than the spatial correlation length. In chemically reacting systems both local and global coupling can be significant. Diffusion is an important coupling mechanism in many fluid-phase and surface reactions, and a global coupling, through for example mixing in a reservoir, can be important in others. In the present study, as would be the case in any chemically reacting system with coupled sites, we are dealing with nominally identical coupled elements. Although the electrodes are designed to be the same size with the same properties, certain variations among them are unavoidable. Thus the sizes likely vary slightly, the reactivity and grain structure of the wire ends are different, and there are likely some minor variations among the distances among the elements. Thus we are dealing with an array of coupled, slightly different oscillators. The two main mechanisms of coupling are through the electric field and by convection. The former is a nonlocal coupling because changes at any location on the array are felt almost instantaneously at all other locations; the electric field in the electrolyte is governed approximately by the LaPlace equation. Although this coupling is not local, it is also not global since the effect does die out with distance. Thus, the coupling through
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the electric field is strongest between neighboring electrodes but does occur among all the electrodes on the array, the effect decreasing with increasing distance. Convection also plays a role in this system. Since we are using an impinging jet, flow is approximately toward the electrode and radially outward from the center. Elements near the center of the array influence those on the outside to a greater extent than conversely. (As noted above, this is not a uniformly accessible surface.) The arrays are small enough that there is coupling among all the elements. In the periodic region, some of the elements are even synchronized by the coupling. In the chaotic region, the nature of the chaos, that is, the sensitive dependence on initial conditions and thus disturbances, as well as the noise and variations in the system, is strong enough that no two elements are synchronized. The degree of chaos, or the dimension of the overall chaotic attractor, increases as the system size increases. The experiments with arrays of individual electrodes can be viewed as a model system for studying variations along the surface of single, larger electrodes. We note, for example, that the increase in dimension of the arrays as the number of elements was increased was the same as that found earlier with a series of single electrodes of increasing size. Information about spatial variations can thus be gathered with the arrays which is not easily available with other configurations since the current, and thus rate of reaction, can be obtained at each position on the surface independently. Acknowledgment This work was supported in part by grants from the National Science Foundation and from the Chevron Oil Company. Literature Cited Albahadily, F. N.; Schell, M. Observation of Several Different Temporal Patterns in the Oxidation of Formic Acid at a Rotating Platinum-Disk Electrode in an Acidic Medium. J. Electroanal. Chem. 1991, 308, 151. Badii, R.; Politi A. Statistical Description of Chaotic Attractors: the Dimension Functions. J. Stat. Phys. 1985, 40, 725. Chate´, H. On the Analysis of Spatiotemporally Chaotic Data. Physica D 1995, 86, 238. Falcke, M. Cluster Formation, Standing Waves, and Stripe Patterns in Oscillatory Active Media with Local and Global Coupling. Phys. Rev. 1995, 52, 763. Falcke, M.; Engel, H. Pattern Formation during the CO oxidation on Pt(110) Surfaces under Global Coupling. J. Chem. Phys. 1994, 101, 6255. Fei, Z. Spatiotemporal Dynamics of Iron Sulfuric Acid Electrochemical Reaction System. Ph.D Dissertation, University of Virginia, 1997. Fei, Z.; Kelly, R. G.; Hudson, J. L. Spatiotemporal Patterns on Electrode Arrays. J. Phys. Chem. 1996, 100, 18986. Fla¨tgen, G.; Krischer, K. Accelerating Fronts on an Electrochemical System due to Global Coupling. Phys. Rev. E 1995, 51, 3997. Franck, U. F. U ¨ ber die Aktivierungsausbreitung auf passiven Eisenelektroden. Z. Elektrochem. 1951, 55, 154. Franck, U. F.; Fitzhugh, R. Periodische Elektrodenprozesse und ihre Beschreibung durch ein mathematisches Model. Z. Elektrochem. 1960, 65, 156.
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Received for review September 18, 1997 Revised manuscript received December 8, 1997 Accepted December 11, 1997 IE970655H
