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Ind. Eng. Chem. Res. 1995,34, 3246-3251
Spatiotemporal Patterns on a Ring Electrode Jon C. Sayer and John L.Hudson* Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesville, Virginia 22903-2442
We have studied the spatiotemporal dynamics of the electrodissolution of a n iron ring under potentiostatic conditions in a region of parameter space in which active-passive transitions occur. "he current (proportional to the overall dissolution rate) is measured as a function of time; patterns (due to growth and dissolution of surface films) are recorded with a microscope and a video camera. We observe the following spatiotemporal features as a parameter, the potential, is changed: (1)the current oscillates periodically and the passivation occurs through the growth of a symmetric film which propagates radially; (2) the symmetry breaks giving rise to a period-doubled oscillation in which activation-passivation occurs on alternate half sections of the ring; (3) this symmetry is broken resulting in patterns on four quadrants of the ring.
Introduction This paper, like the others in this issue, is written in recognition of the publication of the textbook Transport Phenomena by Bird, Stewart, and Lightfoot (1960). One of us first becamse acquainted with this book in the form of prepublication notes during the 1959-1960 school year. Since that time our hardback copy has served as a classroom text and as a reference; it has been, as evidenced by its condition, the most frequently used book on the shelves. The subject of the paper is the development of spatiotemporal patterns in an electrochemical system. In the systems being studied there are variations in species concentrations and in potential field both normal to and parallel to the electrode surface. Transport occurs by convection, diffusion, and migration; furthermore, there is a global coupling since variations at one location produce a change in the electric field which in turn affects conditions throughout the system. Thus this is a system in which many of the transport processes described in the text that we are recognizing play important roles. The electrochemical reaction being considered is the anodic electrodissolution of iron in acidic solution. This reaction is of interest both for practical reasons, since it is an important corrosion reaction (Kaesche, 19851, and also because of its rich temporal and spatiotemporal dynamics. Under potentiostatic conditions autonomous current oscillations can occur (Frank,1951;Bartlett and Stephenson, 1952; Podesta et al., 1979; Russell and Newman, 1987). In addition, patterns on the electrode are known to occur through the formation and dissolution of surface films. The growth of the films and the temporal variation of the current can be, of course, interrelated. The nature of the film has not been definitively determined; models postulating the presence of either a porous, thin iron sulfate film or a colloidal dispersion of iron sulfate or hydroxide have been proposed (Barcia et al., 1992). The growth of a film can passivate the surface causing a decrease in current; the dissolution of the film is associated with a rise in current. The growth and dissolution of these films, however, does not occur uniformly over the surface; rather, regions of active and of passive behavior form and the interface between the regions changes with time.
* To whom correspondence should be addressed. E-mail:
[email protected].
Many of the studies of oscillations during the electrodissolution of iron in acidic solution have been carried out under potentiostatic conditions; current is measured as a function of time. One convenient parameter in these studies is the potential of the iron electrode. Two regions of parameter space have received the most attention. 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 (Wang and Hudson, 1991). The second parameter region is near the Flade potential, which is the potential a t which the system changes from active to passive, that is, the potential a t which the current drops from large values t o approximately zero. 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 (Franck and FitzHugh, 1961). Pigeau and Kirkpatrick (1969) have investigated the surface conditions during these latter periodic relaxation oscillations. During the passivation phase of the oscillation, a zone with higher reflectance emerges at the outer rim of a circular electrode and propagates toward its center in an approximately symmetric 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. (1993) have extended these studies. They investigated the behavior as a function of the potential in the vicinity of the Flade potential; as the potential is varied the spatial symmetry of the oscillatory state breaks, giving rise t o a period-doubledoscillation. This results, after further decreases in potential, in a periodic state in which half of the electrode undergoes activation-passivation during one half of a cycle and the other half of the surface participates during the next half of the temporal cycle. Oscillations and wave propagation also occur during the electrodissolution of other metals (Lev et al., 1988; Haim et al., 1992; Otterstedt et al., 1995; Lee et al., 1985) as well as during some electrocatalytic reactions (Albahadilyand Schell, 1991;Koper and Sluyters, 1993; Flatgen and Krischer, 1995). For further examples see the recent review of Hudson and Tsotsis (1994). In this work we carry out the electrodissolution of iron on a ring electrode under potentiostatic conditions near the Flade potential. The periodic current is measured 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3247
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Figure 1. Current-voltage curve taken in cathodic direction for iron in 2 N sulfuric acid. Sweep rate 0.5 mV/s (E versus Hgmg2SO4).
as a function of time and the surface is observed with the aid of a microscope and a video camera. A spatiotemporal period doubling is seen as the potential is lowered. With further change in parameter, additional, more complicated, bifurcations take place. Experiments
Our experiments are carried out with an iron ring (from 99.9985% iron rod, Aesar Puratronic), outside diameter 12 mm and inside diameter 6 mm, which faces upward in a 2 N sulfuric acid solution. The total surface area of the electrode is 84.82 mm2. The electrode is encased in epoxy of diameter 33 mm. The counter electrode, a ring of platinum foil, surrounds the iron working electrode equidistantly. A microporous membrane, placed inside the counterelectrode, allows free travel of ions, but helps prevent convective flow from hydrogen evolution in the system. There is no stirring; the fluid is almost stagnant, but some unavoidable convection does take place as the reaction occurs. Potential is held constant relative t o a Hg/Hg2SO4 reference electrode, and the current (proportional to the overall dissolution rate) is measured at 90 Hz. Surface films are visualized with the aid of a microscope and a video camera. Results
A current-potential curve for the iron ring system is shown in Figure 1; this was made by scanning the potential at a rate of 0.5 mVls in the cathodic direction. The general shape of the curve is typical for the iron/ sulfiu-ic acid system (Kaesche, 1985). Three regions can be seen; the Tafel region around -900 mV (Hg/HgzSO4), the mass transfer limited plateau between approximately -740 and -180 mV, and the Flade potential at about -180 mV. Above the Flade potential, where the transition from active to passive behavior occurs, the electrode is covered by a thin pore-free oxide film which inhibits dissolution of the electrode. In this region oscillations occur during the scan as can be seen from the figure. The number of oscillations increases with
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Time (seconds) Figure 2. Current oscillations during anodic dissolution of iron, E = -134.7 mV (Hg/HgzSOd). (a) Sustained autonomous oscillations. (b) One activation-passivation cycle; the arrows indicate the phases corresponding to the subtracted digitized images shown in Figure 3.
decreasing scan rate. Furthermore, potential range in which the oscillatory behavior occurs depends on the sample chosen for the experiment. The sample chosen for the experiment of Figure 1 has a large range over which the oscillations occur; other samples would have a smaller region and thus fewer oscillations on a figure analogous to Figure 1. We now consider the dynamics of this system under potentiostatic conditions in the vicinity of the Flade potential. In Figure 2a, sustained oscillations obtained a t a potential of -134.7 mV (Hg/HgzSO4) are shown. The time series has the appearance of typical relaxation oscillations which are characterized by a long phase with very low current, an abrupt increase in the current (activation), and a gradual relaxation to the low current (passivation). In Figure 2b one cycle of the oscillation is shown on an expanded time scale. We recorded the visible changes which occurred during these oscillations with
3248 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
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f e d Figure 3. Differen 3 of successive snapshots the surface during the passivation of the surface at E = -134.7 mV (Hg/HgzSOd. The images correspond to the phases shown in Figure 2b.
a video camera. After digitization, consecutive images were subtracted in order to illustrate the behavior on the surface more clearly. Subtracted images corresponding to six times during the passivation are shown in Figure 3; the time corresponding t o each of these pictures is indicated on Figure 2b. The image form a wave which starts at the outer radius of the ring and progresses inward. The front is (approximately) a shrinking circle; as the front approaches the inner radius of the ring, a second front begins from the inner radius and meets the first front. We have a periodic solution p-1 with a period which we define as 217. There is full O(2)symmetry; at each instant the pattern is unchanged by rotations or reflections of the ring (Golubitsky et al., 1988). As the potential is lowered, a spatiotemporal period doubling occurs. In order to describe this period doubling, it is helpful to relate it to the observations on a disk (Hudson et al., 1993). Using a disk electrode, a spatiotemporal period doubling also occurs as the parameter (potential) is decreased. Just before the bifurcation point, periodic behavior p-1 exists with a period which we also call 217. The front in this case is a shrinking circle; during the passivation phase the radius of the circle shrinks to zero. At the bifurcation point a spatiotemporal period doubling occurs; after the bifurcation point (lower potential) the behavior is still periodic. A splitting of the center of the waves occurs. On alternating cycles the wave collapses on centers which are slightly displaced from each other. As the potential is lowered further, these centers move apart. Eventually, a potential is reached at which the centers are so far apart that the waves occur on two separate sections of the disk. There are two oscillations per period; on alternate oscillations the waves occur on alternate halves of the disk. Note that after, the bifurcation point, there is a line of symmetry (an axis); in addition, a rotation of the pattern by 180" is equivalent t o a half-period phase shift. The currents corresponding to each of the two oscillations making up a complete period are in principle the same, although in
practice they vary slightly because of heterogeneities on the electrode surface. A related spatiotemporal period doubling occurs with the ring electrode being considered here. The behavior with the ring is slightly more complicated than that of the disk. Obviously, in state 1, before the period doubling bifurcation occurs, the center of the circle whose radius is collapsing during the passivation phase is not on the electrode surface (the ring). The experiment is done by lowering the potential in discrete increments starting at E = -134.7 mV, the potential of state 1, Figure 3. The period-doubling bifurcation occurs at a potential between E = -138 and -139 mV. Before describing the bifurcation, it is convenient to describe the behavior, called state 2, which is observed at a potential of E = -143.7 mV. Thus we describe first the behavior at a potential 5 mV below the bifurcation point. The time series of state 2 is shown in Figure 4a, and two peaks corresponding t o a complete cycle are shown in Figure 4b. During the first of these peaks, a passive-active-passive cycle occurs on one half of the ring electrode. During the second peak the film growth and dissolution occurs on the other half of the ring. Subtracted images for the passivation phase are shown in Figure 5. Note that in the first half of the cycle, as seen in Figure 5a-f the passivation wave moves primarily radially inward over approximately half of the ring; there is also some growth radially outward from the inner radius and some inward growth from the ends of the half-ring. During the second half of the cycle (Figure 5g-1) corresponding to the second peak, the events occur on the other half of the ring. The transition from Figure 3 (call this state 1)to Figure 5 (state 2) appears t o be a symmetry-breaking period-doubling bifurcation in the presence of O(2) symmetry. State 2 has a period of 417, or twice that of state 1. State 2 has an axis of reflection (more or less vertical in the pictures) which is a symmetry of the states at each instant in time. There exists then an (almost) vertical line of symmetry. Furthermore, a rotation of state 2 by 180" is the same as a half-period phase shift (shift in time of
Ind. Eng. Chem. Res., Vol. 34, No. 10,1995 3249
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Figure 4. Current oscillations after spatiotemporal period doubling, E = -143.7 mV (Hg/Hg&Or). (a) Sustained oscillations. (b) T w o peaks corresponding to a complete cycle. Arrows correspond to times of images shown in Figure 5.
2ll); equivalently, reflecting across a horizontal line yields a half-period phase shift. As noted above, the transition from state 1 (E = -134.7 mV, Figures 2 and 3) to state 2 (E = -143.7 mV, Figures 3 and 4) appears to have occurred through a period-doubling bifurcation. We started at E = -134.7 mV and lowered the potential in discrete steps; state 1 persisted to E = -138 mV and the period-doubling bifurcation occurred between E = -138 and -139 mV. For values of the potential equal t o or below -139 mV (even before state 2 is reached), the spatial patterns on successive oscillations are no longer the same; however, the pattern is always the same on alternate oscillations. The period has doubled; the doubling is not seen in the current since the current is the same for each half of the full cycle. Just past the bifurcation point, the waves cover most, but not all, of the ring; there is a small inactive area. A reflection across a horizontal axis is the same as a half-period phase shift. The inactive areas on alternate oscillations continue to grow, and
With the ring electrode even more complicated spatiotemporal behavior is observed. At a lower potential (E = -148.5 mv) another type of spatiotemporal behavior occurs. Subtracted images for the passivation phase of two peaks of the cycle are shown in Figure 6. (We call this state 3.) During two peaks in the time series, the activation-passivation first occurs on two oppositely placed quadrants; this is followed by activation-passivation on the other two quadrants. As in the above examples, the active area becomes smaller during the passivation phase. Note that in Figure 6h there is still activity on the lower quadrant although the upper quadrant has been passivated; it is not known if this asymmetry is due to the dynamics or to a heterogeneity of the electrode surface. This third state also has an axis of reflection (approximately vertical in the pictures) which is a symmetry of the states at each instant in time; so state 3 has (as did state 2) a vertical reflectional symmetry. On the other hand, rotation by 180"of state 3 is not equivalent to any phase shift in time (as it was in state 2). The transition between state 2 (Figure 5 ) and state 3 (Figure 6) is complicated and certainly consists of several steps. There is no single bifurcation (such as the period-doubling bifurcation between the behavior seen in Figures 3 and 4) which can explain the transition observed. We started with a periodic solution p-1 with period 211 and full O(2) symmetry (state 1, Figure 3). A period-doubling bifurcation to a solution p-2 with period 411 and broken O(2) symmetry occurred t o state 2 with a spatial reflectional symmetry across the vertical axis and a phase shift symmetry (Figure 5). As noted by Swift and Wiesenfeld (1984) and more generally by Fiedler (1988), it is not possible for p-2 to again period double unless the phase shift symmetry is first broken. A phase shift symmetry breaking will produce another periodic solution p'-2 of period 411 which is still symmetric across the vertical axis. Then it is possible for p'-2 t o undergo a perioddoubling bifurcation to a solution p-4 of period 8II. There are two ways in which this can happen: either the spatial symmetry is preserved or it is broken. In the latter case the new periodic solution p-4 will have a half-period phase shiR; shifting by 411 is equivalent to reflecting across the vertical axis. For related results (involving coupled chemical oscillators) see, for example, Waller and Kapral (1984) or Mankin and Hudson (1986). We have not yet been able to carry out controlled experiments in the parameter range between state 2 and state 3 in order to determine the details of the transitions. We have observed a period doubling of state 2 (to period 8II) and patterns between those of states 2 and 3. These results may be consistent with the breaking of the phase shift symmetry and perioddoubling bifurcation with breaking of spatial symmetry which was mentioned above; however, because the transitions appear t o occur with very small changes in parameter, the results are neither complete nor reproducible. (The behavior seen in Figures 5 and 6 is reproducible; it is only the transitions between them which has not been ascertained.) Further work shall attempt to explain the sequence of transitions. Discussion An anodic reaction, the oxidation of iron, takes place on the working electrode which, in our case, is the ring.
3250 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
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Figure 5. Differences of successive snapshots of the surface during passivation a t E = -143.7 mV (HgkIgzSOJ. Images correspond to the phases shown in Figure 4b.
Oxide and salt films grow and recede during each passivation-activation cycle. The passivation, and likely also the activation, occurs through the movement of a wave across the electrode surface. In the simplest case shown, a symmetric wave propagates radially inward. With changes of a parameter, the potential, the symmetry is broken through a period-doublingbifurcation. Further changes in parameter produces additional symmetry breakings. The motion of these waves is influenced by diffusion and migration which carry ions to and from the reacting surface. Global coupling (through the electric field) can also play a role in the development of patterns in electrochemical systems just as it can (through mixing and heat transfer) in catalytic surface reactions (Middya et al., 1994). The working electrode (the iron ring) is held at virtual ground. The potential difference between the iron ring nd the reference electrode is held constant
by the potentiostat through changes in the potential of the counter electrode. Any changes, anywhere on the reacting surface, thus effect changes throughout the system. This global coupling likely plays an important role in the formation of the spatiotemporal patterns and in the bifurcations seen in this work. A first principles model capable of describing the phenomena observed in this experimental study would have to include transport in the fluid by diffusion and migration, the potential field, and a boundary condition at the electrode surface for the nonlinear electrodissolution reaction. The basic oscillatory state, before the f i s t bifurcation takes place, has full O(2) symmetry; the partial differential equations describing the system depend on radial and axial position as well as time. In order to describe the series of bifurcations seen here, another spatial variable, angular position, must be
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Figure 6. Differences of successive snapshots of the surface during the passivation of two peaks a t E = -148.5 mV (Hg/ Hg2S04).
added. A full mathematical description of these phenomena thus represents a considerable challenge.
Platinum-Disk Electrode in an Acidic Medium. J.Electroanal. Chem. 1991,308,151-173. Barcia, 0. E.; Mattos, 0. R.; Tribollet, B. Anodic Dissolution of Iron in Acid Sulfate Under Mass Transport Control. J. Elect m h e m . Soc. 1992,139,446-453. Bartlett, J. H.; Stephenson, L. Anodic Behavior of Iron in HzS04. J. Electrochem. Soc. 1952,99,504-512. Bud, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. Fiedler, B. Global bifurcation ofperiodic solutions with symmetry; Springer: Berlin, 1988. Flatgen, G.; Krischer, K. Accelerating Fronts in an Electrochemical System due to Global Coupling. Phys. Rev. E 1995,51,39974004. Franck, U. F. mer die Aktivierungsausbreitung auf passiven Eisenelektroden. 2.Elektrochem. 1951,55,154-160. Franck, U. F.; FitzHugh, R. Periodische Elektrodenprozesse und ihre Beschreibung durch ein Mathematisches Modell. 2.Elektrochem. 1961,65,156-168. Golubitsky, M.; Stewart, Ian; Schaeffer, David G. Singularities and Groups in Bifurcation Theory; Springer: New York, 1988; VOl. 11. Haim, D.; Lev, 0.;Pismen, L. M.; Sheintuch, M. Modeling Spatiotemporal Patterns in Anodic Nickel Dissolution. Chem. Eng. Sci. 1992,47,3907-3913. Hudson, J. L.;Tsotsis, T. T. Electrochemical Reaction Dynamics: A h v i e w . Chem. Eng. Sci. 1994,49,1493-1572. Hudson, J. L.; Tabora, J.; Krischer, K.; Kevrekidis, I. G. Spatiotemporal Period Doubling during the Electrodissolution of Iron. Phys. Lett. A 1993,179,355-363. Kaesche, H. Metallic Corrosion;National Association of Corrosion Engineers: Houston, 1985. Koper, M. T. M.; Sluyters, J. H. A Simplified Approach to the Modeling of Wave Propagation at Electrode/Electrolyte Interfaces. Electrochim. Acta 1993,38,1535-1544. Lee, H. P.; Nobe, Ken; Pearlstein, Arne J. Film Formation and Current Oscillations in the Electrodissolution of Cu in Acidic Chloride Media. J. Electrochem. SOC.1985,132, 1031-1037. Lev, 0.;Sheintuch, M.; Yarnitzky, C.; Pismen, L. M. Standing and Propagating Wave Oscillations in the Anodic Dissolution of Nickel. Nature 1988,336,458-459. Mankin, J. C.; Hudson, J. L. "he Dynamics of Coupled Nonisothermal CSTRs. Chem. Eng. Sci. 1986,41,2651-2661. Middya, U.; Luss, D.; Sheintuch, M. Spatiotemporal Motions due to Global Interaction. J. Chem. Phys. 1994,100,3568-3581. Otterstedt, R.; Jaeger, N. I.; Plath, P. Waves Accompanying Current Oscillations in the Anodic Electrodissolution of Cobalt in Phosphoric Acid Solutions. Znt. J. Bifurcations Chaos 1995, in press. Pigeau, A.; Kirkpatrick, H. B. A Correlated Potentiostatic Microscopic Study of Iron Passivation in Sulfuric Acid. Corrosion 1969,25,209-214. Podesta, J. J.; Piatti, R. C. V.; Arvia, A. J. The Potentiostatic Current Oscillations at IrodSulfuric Acid Solution Interfaces. J. Electrochem. Soc. 1979,126,1363-1367. Russell, P.; Newman, J. Anodic Dissolution of Iron in Acidic Sulfate Electrolytes. J. Electrochem. SOC.1987,134,1051-1058. SwiR, J. W.; Wiesenfield, K. Suppression of Period Doubling in Symmetric Systems. Phys. Rev. Lett. 1984,52,705-708. Waller, I.; Kapral, R. Synchronization and Chaos in Coupled Nonlinear Oscillations. Phys. Lett. 1984,105A,163-168. Wang, Y.; Hudson, J. L. Effect of Electrode Surface Area on Chaotic Attractor Dimensions. AIChE J. 1991,37,1833-1843.
Acknowledgment
Received for review December 7, 1994 Accepted June 8,1995@
This work was supported in part by the NSF and by OWARPA. We thank Marty Golubitsky and Michael Dellnitz for helpful discussions.
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Literature Cited Albahadily, F. N.; Schell, M. Observation of Several Different Temporal Patterns in the Oxidation of Formic Acid at a Rotating
Abstract published in Advance ACS Abstracts, September 1, 1995. @
