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J. Phys. Chem. 1996, 100, 18986-18991
Spatiotemporal Patterns on Electrode Arrays Z. Fei,† R. G. Kelly,‡ and J. L. Hudson*,† Department of Chemical Engineering and Department of Materials Science and Engineering, UniVersity of Virginia, CharlottesVille, Virginia 22903-2442 ReceiVed: May 3, 1996; In Final Form: October 11, 1996X
Experiments were carried out with arrays of iron electrodes in sulfuric acid solution under conditions in which slow active-passive relaxation oscillations occur. The arrays consisted of a number of small disks which were made by exposing the ends of wires embedded in an insulator. Three array geometries were used: (a) 2 × 8 which approximates a one-dimensional geometry, (b) 4 × 4 square array, and (c) 61 electrodes in a hexagonal pattern. The experiments were done potentiostatically, and the current in each electrode was measured independently; therefore, the spatiotemporal patterns which occurred were directly determined. For the oscillatory conditions, which occur at potentials above the Flade potential, a wave moves from the center of the electrode to the edge during the activation phase and another moves in the opposite direction, from edge to center, during the passivation. The velocities of these waves depend on array size and applied potential; the activation velocities are much faster than those of passivation. Both the activation and passivation wave fronts accelerate as they propagate along the array. As the potential is lowered, a spatiotemporal period doubling occurs. Long-range coupling plays an important role in the dynamics of such electrochemical reactions; the arrays of electrodes behave qualitatively similar to single electrodes of the same total surface area. The arrays can thus be used to gather information on the rate of reaction at various sites on a reacting surface.
Introduction Electrochemical reactions can exhibit rich temporal and spatial dynamics.1 For example, oscillations and spatial structure have been observed during the electrodissolution of metals2-8 as well as during some electrocatalytic reactions.9-12 One of the electrodissolution reactions, the anodic electrodissolution of iron in acidic solution, has been studied by several investigators. Under potentiostatic conditions autonomous current oscillations can occur.13-16 In addition, patterns on the electrode are known to develop through the formation and dissolution of surface films. Many of the studies of oscillations during the electrodissolution of iron in acidic solution have been carried out under potentiostatic conditions; current is then measured as a function of time. A convenient parameter in these studies is the potential of the iron electrode. Two regions of parameter space are of particular interest. The first is in a region where the mean current is almost independent of applied potential; this is known as the mass transfer limited plateau. Using a rotating disk electrode, high-frequency (0.1-1 kHz), often chaotic, oscillations have been observed.17,18 As the potential is increased, a point is reached where the the system passivates and the current drops from large values (ca. 0.3 A/cm2) to very low values (ca. 10-5 A/cm2); the potential at which this occurs is known as the Flade potential. Very close to the Flade potential the system can oscillate between active and passive conditions; this often produces slow (period of seconds to minutes) relaxation oscillations in the current.19 Pigeau and Kirkpatrick20 have investigated the surface conditions during such periodic relaxation oscillations on a circular electrode. During the passivation phase of the oscillation, a zone with higher reflectance emerges at the outer rim and propagates toward the center in an approximately symmetric †
Department of Chemical Engineering. Department of Materials Science and Engineering. X Abstract published in AdVance ACS Abstracts, November 15, 1996. ‡
S0022-3654(96)01266-X CCC: $12.00
manner; that is, the active region of the electrode surface remains approximately circular, and the diameter of this circle decreases during the passivation. Hudson et al.21 have extended these studies; the periodic current was measured as a function of time, and the surface was observed with the aid of a microscope and a video camera. A spatiotemporal period doubling was observed. With changes in the parameter (potential), the spatial symmetry of the oscillatory state broke, giving rise to a perioddoubled oscillation. This resulted, after further decreases in potential, in a periodic state in which it appeared that half of the electrode underwent activation-passivation during one half of a cycle and the other half of the surface participated during the next half of the temporal cycle. Similar studies have also been carried out on a ring electrode.22 On the ring, as on the disk, a spatiotemporal period doubling was seen as the potential was lowered. Subsequently, however, with further change in the parameter, additional, more complicated, bifurcations took place. In this paper we report the results of experiments carried out with arrays of iron electrodes in sulfuric acid solution; the arrays consist of a number of small disks, which are made by exposing the ends of wires embedded in an insulator. The current in each of the electrodes is measured independently. Therefore, when spatial patterns occur on the array, the pattern can be determined directly; information not available from visual techniques is obtained. 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. Therefore, the spatiotemporal bifurcations that occur on the single electrode occur also on the array. Furthermore, much faster measurements can be made with the array. During active-passive oscillations in the iron/sulfuric acid system, the activation is very fast whereas the passivation is slow. Both activation and passivation occur by the motion of a wave front © 1996 American Chemical Society
Spatiotemporal Patterns on Electrode Arrays
J. Phys. Chem., Vol. 100, No. 49, 1996 18987
Figure 1. Configurations of electrode arrays: (a) 2 × 8 array, (b) 4 × 4 array, and (c) 61 array. Diameter of each electrode ) 0.5 mm.
across the electrode surface. With the visual methods used in previous work20-22 and described above, only the passivation was seen. In the experiments described here, both are measurable. Experiments The configurations of the arrays consisting of 16 or 61 electrodes used in the study are shown in Figure 1. 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. In each case some of the electrodes are numbered, and these will be referred to later. The electrode array faces downward and the electrolyte is stagnant. The potential is controlled by a potentiostat (EG&G Princeton Applied Research, Model 273). Electrodes in the array are linked to working electrode jacks of the potentiostat through a multiple ZRA box (Scribner Associates, Inc.), so the currents of individual electrodes can be measured. A PC installed with a 32-channel data acquisition board (Keithley DAS-1800HC2) is used for data sampling. Both the total current and the currents of individual electrodes can be measured simultaneously. The sampling frequency 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 and counter electrode is platinum. Most of the experiments are done potentiostatically; in order to get reproducible results, the electrode is first held at a potential at which the electrode is passive (-110 mV), and the potential is swept slowly (0.5 mV/s) in the cathodic direction to the desired value and held there. Results Before turning to the measurements of the currents in the individual electrodes in the arrays, we discuss a series of experiments in which only the total current is measured. The results of two experiments done by sweeping the potential in the cathodic direction are shown in Figure 2. In Figure 2a the results obtained with an array of 16 electrodes in the 4 × 4 pattern are shown. At high potentials (E > -120 mV) the electrode is passive and no oscillations occur. As the potential is lowered, slow active-passive relaxations oscillations begin. Below the Flade potential, approximately -195 mV, the electrode is in the active state. For comparison, a similar experiment was carried out using a single, larger (diameter ) 2 mm) electrode, and the result is shown in Figure 2b. This electrode has the same total area as the array of smaller
Figure 2. Current vs applied potential (sweep rate ) 0.5 mV/s): (a) 16 (4 × 4) electrode array; diameter of each electrode ) 0.5 mm; (b) single electrode, diameter ) 2 mm.
electrodes. Note that the results of the two experiments shown in Figure 2a,b are similar, but not identical. The range in which oscillations occur is somewhat smaller in Figure 2b, and the potential at which the system becomes active is somewhat higher (-185 mV); nevertheless, the same transitions occur in the two experiments. We shall see below other strong similarities between experiments done with a single electrode and with arrays. We now turn to the oscillations obtained with the arrays. The experiments are done potentiostatically, i.e., at fixed values of the potential. We will show results as a function of the potential as this parameter is changed from high to low values, i.e., in the cathodic direction. The parameter range can be broken up into three parts. Above approximately -130 mV the electrode is passive. Between -170 and -130 mV period-one activepassive oscillations occur; in this range the frequency of the oscillations increases as the potential is decreased. Finally, at approximately -170 mV a spatiotemporal period doubling takes place. The bifurcation points were not determined exactly; the values cited are known to within 5 mV. It is also likely that
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Figure 3. Total currents on arrays: (a) 2 × 8 array, E ) -165 mV, (b) 4 × 4 array, E ) -165 mV, (c) 61 array, E ) -160 mV, periodone oscillations.
the ranges are not exactly the same for the three types of array. Nevertheless, it is noted that the existence or nonexistence of oscillations and the type of oscillation (period one or two) depends much more strongly on potential than on array configuration or total area. Period-One Oscillations. We first consider the parameter range in which period one active-passive oscillations are seen. The total current, i.e., the sum of the current through all the electrodes in a given array, is shown for each array in Figure 3. Note that the shapes of the three curves are approximately the same; there is a passive period in which the current is low, a sharp rise in current (activation), and a slower drop in current (passivation). The amplitude of the oscillations obtained with the 61-electrode array is, of course, larger than those found with either of the other two arrays, both of which have 16 electrodes, since the total area is greater. In addition, the frequency of the oscillations is different for the three cases; the period of the oscillations obtained with the largest array, Figure 3c, is the longest. Note also that it can be seen in the same figure that the passivation phase of the cycle is also longer for the 61electrode array than for either of the other two cases. Results obtained by measuring the currents of the individual electrodes in the arrays are shown in Figures 4 and 5. During both the activation and passivation phases of the periodic cycle, waves move across the electrode array; these wave patterns are shown schematically in Figure 4. For all three array types the activation begins at the center of the electrode and moves toward the outside; the individual electrodes activate sequentially from inside to outside. These activation patterns are depicted schematically in Figure 4. The passivation takes place in the opposite direction; the outside electrodes first passivate, and a wave of passivation moves inward. The passivation is followed by the longer passive state and then the cycle begins again with another activation. The waves shown in Figure 4 are all symmetric about the center of the electrode array. In the case of the 2 × 8 array, Figure 4a, the activation and passivation start and end at the center of the quasi-one-dimensional geometry. In the case of the other two arrays, the activation and passivation start and end on the center of the array. In
Figure 4. Wave directions, period-one oscillations (left column, activation; right column, passivation): (a) 2 × 8 array, E ) -165 mV; (b) 4 × 4 array, E ) -165 mV; (c) 61 array, E ) -160 mV.
Figure 5. Currents of individual electrodes on the 61 array (E ) -160 mV, period-one oscillations): (a) activation and passivation, (b) activation (expanded scale).
Figure 4c, for example, they start and end on the electrode designated as no. 4.
Spatiotemporal Patterns on Electrode Arrays
Figure 6. Total current on arrays: (a) 2 × 8 array, E ) -190 mV; (b) 4 × 4 array, E ) -190 mV, (c) 61 array, E ) -160 mV, periodtwo oscillations.
The individual currents of seven of the electrodes of the 61electrode array are shown in Figure 5. Figure 5a shows an entire activation-passivation process. The activation is too fast to distinguish the curves during the activation process on this time scale. However, it can be seen that the slower passivation process occurs sequentially on the electrodes of the array and that this passivation occurs from the outer electrodes to the center, ending on the center electrode no. 4. An expanded time scale depiction of the activation is shown in Figure 5b. Note that the activation occurs in approximately 0.003 s. The activation begins on the center electrode, no. 4, and propagates outward). Period-Two Oscillations. We now consider the behavior at lower potentials, past the point where a spatiotemporal period doubling takes place. Time series of the total current for the three arrays are shown in Figure 6. In each case, a period now consists of two cycles of activation-passivation. Thus in Figure 6c, for example, the period is approximately 1.2 s. The spatiotemporal period doubling takes place in the following manner: above the bifurcation point, approximately -170 mV, the oscillations are period one. The waves are symmetric about the center of the electrode array as described above. For example, on the 61-electrode array, the waves are centered on electrode no. 4. As the potential is lowered, the center splits, and on alternating activation-passivation cycles the center of the waves is on electrode no. 3, then no. 5, then back to no. 3. As the potential is lowered further, the centers move farther apart. Examples are shown in Figure 7 for the 2 × 8 array and in Figure 8 for the 61-electrode array, respectively. Time series for one complete period on the 61-electrode array are shown in Figure 9; this corresponds to Figure 8a,b. In Figure 9a the first activation-passivation corresponding to Figure 8a can be seen. Since the activation is fast, it is shown in an expanded time scale in Figure 9b. The activation wave emanating from electrode no. 1 on the left side of the array is seen; this is followed by the passivation collapsing on that same electrode. The second part of the period is shown in Figure 9c,d. This time the activity is centered on electrode no. 7 on the right side of the array. This is then repeated, i.e., the behavior of Figure 9a,b is again seen.
J. Phys. Chem., Vol. 100, No. 49, 1996 18989
Figure 7. Wave directions, 2 × 8 array; period-two oscillations (left column, activation; right column, passivation): (a) E ) -190 mV, cycle one; (b) E ) -190 mV, cycle two.
Figure 8. Wave directions, 61 array; period-two oscillations (left column, activation; right column, passivation): (a) E ) -185 mV, cycle two; (b) E ) -185 mV, cycle two.
Wave Speeds. The wave speeds can be determined determined from the signals of current on the individual electrodes such as those given in Figures 5 and 9. We calculated the speeds by noting the time on each electrode at which the current reaches a value approximately midway between the passive and active values. For example, in Figure 5b for the activation the time was noted at which the current of each electrode reached a value of 2.5 mA. The speed was then found from the ratio of the distance between electrode centers (6 mm) and the time differences. Consider first the 61-electrode array. The time series are given in Figure 5 for the period-one oscillation obtained at a potential of -160 mV. The wave speed for the passivation increases from an initial vaue of approximately 6 mm/s to a final value of 60 mm/s; i.e., the wave accelerated from 6 to 60 mm/s. The speeds during the activation phase are much faster, and these increase from approximately 550 to 650 mm/s. The period-doubled oscillations were obtained at a lower potential,
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Fei et al. × 8 configuration the passivation waves accelerated from 10 to 40 mm/s as the waves progressed along the array during the period-one oscillations; the activation wave accelerated from approximately 70 to 240 mm/s. The activation waves are slower than those obtained with the 61-electrode array whereas the passivation are approximately the same. Furthermore, the 4 × 4 electrode array, which of course has the same total area as the 2 × 8 array, yielded wave speeds about a factor of 3 and 10 lower than the 2 × 8 array for activation and passivation, respectively; the speeds for the 4 × 4 array are, of course, the least well-known since only one pair of electrodes can be used for the calculation of the speed of each wave. Discussion
Figure 9. Currents of individual electrodes on the 61 array (E ) -185 mV, period-two oscillations): (a) cycle one, activation and passivation, (b) cycle one, activation (expanded scale): (c) cycle two, activation and passivation; (d) cycle two, activation (expanded scale).
i.e., E ) -185 mV. Under these conditions the speeds of the activation waves were approximately the same as those obtained for the period-one oscillations; however, the velocities of the passivation waves increased by a factor of four as the potential was lowered. Such a result would be expected due to the increased difficulty of forming a passive film at lower potentials. The wave speeds depended on both total electrode area and on geometry. Two 16-electrode arrays were used. With the 2
Activation-passivation relaxation oscillations occur on an iron electrode in sulfuric acid solution. The cycle, beginning with an active surface, likely consists of the following steps: metal dissolution occurs and iron sulfate forms on the surface, a passive oxide layer grows under the salt layer passivating the surface, and the salt film dissolves leaving the oxide layer.23 The activation, acid dissolution of oxide, follows. During the oscillations the surface is not uniform, but rather waves propagate across the surface in both the activation and passivation stages. The size of the electrode plays a role in the time required for the activation and passivation. For larger electrodes the times are longer and, in addition, so is the period of an entire oscillatory cycle. This can be seen by comparing, for example, the periods obtained in this work with those in an experiment carried out with larger surface areas where longer periods were obtained.21 We have shown how arrays of electrodes can be used to gather information on the spatiotemporal patterns. Since the current through the electrodes can be individually measured, the rate of reaction as a function of both position and time is obtained. The array of electrodes behaves similarly to a single electrode of the same shape and total area. The two do not behave exactly the same, of course, because the current distribution on a single electrode will not be the same as that on a series of electrodes separated by an insulator, even if all the electrodes are at the same potentialsas they are in our experiments; for a discussion of primary and secondary current distributions on electrodes see, for example, ref 24. Nevertheless, the behavior of the processes described here is strongly influenced by long-range interactions through the electrolyte. (Global interactions have received considerable attention recently in reaction-diffusion systems.25) Since the long-range interactions are important, the insulating regions between the electrodes do not alter the qualitative behavior of the surface patterns and the oscillations. The similarities between the behavior of a single, larger electrode and an array of smaller electrodes with the same total area can be seen both in sweep experiments and in those done potentiostatically. In the former, one sees the same transitions from passive to oscillatory to constant active behavior. In the latter, one sees the same transitions and also the period-doubling bifurcation. Furthermore, the wave patterns on the surface of the single electrode and on the array are the same. We have previously carried out experiments on single disk electrodes and have observed a spatiotemporal period-doubling bifurcation similar to that seen here.21 Wave patterns similar to those shown in this paper were seen. Thus, the patterns obtained with the electrode arrays are representative of patterns on a single electrode. With the array of electrodes, much faster patterns can be measured because the current is measured at each location and the current signals can be measured at a high
Spatiotemporal Patterns on Electrode Arrays frequency; with the single electrodes only video measurements have been made, and these are limited by the frequency of the video, 30 Hz. Furthermore, the video method is obviously limited to systems in which there is a visible pattern, and this limitation does not extend to the electrode arrays. Far from the period-doubling bifurcation point there was visible activity only on half of the electrode surface. In these experiments with the arrays, it is seen that there is activity most of the time over the entire array, even when the center of the waves is far to one side. This can be seen, for example, in Figures 7 and 8. The wave speed was obtained for both the activation and passivation phases of the cycle. The activation is at least an order of magnitude faster than the passivation. Both wave speeds increase with increasing electrode surface area. Furthermore, both the passivation and the activation waves acelerate as they propagate across the surface. Accelerating potential fronts on iron wires have been previously measured using banks of reference electrodes;26 accelerating fronts have been observed during the activation phase in both bistable and oscillatory conditions in the system cobalt/phosphoric acid;7 and an accelerating front has been observed during the reduction of peroxodisulfate on silver in the bistable parameter region.11 In this present work, we see that such accelerations occur during both the activation and passivation phases of oscillations. We have seen how waves can propagate across an electrode surface, even when that surface is not continuous. Spatiotemporal patterns occur similar to those which arise on the single, larger electrode. The arrays can be used to gather direct information on the rate of reaction at various sites on the reacting surface. Acknowledgment. This work was supported in part from grants from the National Science Foundation, from the Chevron Oil Co. (D. Townley), and the Mobil Oil Co. We acknowledge with great appreciation our long, interesting, and stimulating association with John Ross and the many contributions he has made to the field of nonlinear dynamics in chemistry.
J. Phys. Chem., Vol. 100, No. 49, 1996 18991 References and Notes (1) Hudson, J. L; Tsotsis, T. T. Chem. Eng. Sci. 1994, 49, 1493. (2) Lev, O.; Sheintuch, M.; Yarnitzky, C.; Pismen, L. M. Nature 1988, 336, 458. (3) Haim, D.; Lev, O.; Pismen, M.; Sheintuch, M. Chem. Eng. Sci. 1992, 47, 3907. (4) Haim, D.; Lev, O.; Pismen, M.; Sheintuch, M. J. Phys. Chem. 1992, 96, 2676. (5) Otterstedt, R. D.; Jaeger, N. I.; Plath, P. J. Int. J. Bifurcation Chaos 1994, 4, 1265. (6) Lee, H. P.; Nobe, K.; Pearlstein, A. J. Electrochem. Soc. 1985, 132, 1031. (7) Otterstedt, R. D.; Plath, P. J.; Jaeger, N. I.; Sayer, J. C.; Hudson, J. L. Chem. Eng. Sci. 1996, 51, 1747. (8) Otterstedt, R. D.; Plath, P. J.; Jaeger, N. I.; Hudson, J. L. Faraday Trans. 1996, 92, 2933. (9) Albahadily, F. N.; Schell, M. J. Electroanal. Chem. 1991, 308, 151. (10) Koper, M. T. M.; Sluyters, J. H. Electrochim. Acta 1993, 38, 1535. (11) Fla¨tgen, G.; Krischer, K. Phys. ReV. E 1995, 51, 3997. (12) Fla¨tgen, G.; Krischer, K. J. Chem. Phys. 1995, 103, 5428. (13) Franck, U. F. Z. Elektroch. 1951, 55, 154. (14) Russell, P.; Newman, J. J. Electrochem. Soc. 1987, 134, 1051. (15) Bartlett, J. H.; Stephenson, L. J. Electrochem. Soc. 1952, 99, 504. (16) Podesta, J. J.; Piatti, R. C. V.; Arvia, A. J. J. Electrochem. Soc. 1979, 126, 1363. (17) Diem, C. B.; Hudson, J. L. AIChE J. 1987, 33, 218. (18) Wang, Y.; Hudson, J. L. AIChE J. 1991, 37, 1833. (19) Franck, U. F.; FitzHugh, R. Z. Elektrochem. 1960, 65, 156. (20) Pigeau, A.; Kirkpatrick, H. B. Corrosion 1969, 25, 209. (21) Hudson, J. L.; Tabora, J.; Krischer, K.; Kevrekidis, I. G. Phys. Lett. A 1994, 179, 355.22. (22) Sayer, J. C.; Hudson, J. L. Ind. Eng. Chem. Res. 1995, 34, 3246. (23) Alkire, R.; Ernsberger, D.; Beck, T. R. J. Electrochem. Soc. 1978, 125, 1382. (24) Newman, J. Electrochemical Systems; Prentice-Hall, Inc.; Englewood Cliffs, NJ, 1973; Chapter 18. (25) Middya, U.; Luss, D.; Sheintuch, M. J. Chem. Phys. 1994, 100, 3568. (26) Sullivan, B. G. 1990 Thesis, University of Virginia, Charlottesville, VA.
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