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Taylor, M. A.; Kevrikidis, I. G. Some common dynamic features of coupled reacting systems. Physica D 1991, 51, 274-292.
RBssler, 0. E. Z . Naturforsch. 1976, 31A, 1664-1670. Rbsler, 0. E., Z . Naturforsch. 1976, 31A, 259-264.
Tsai, P.K.; Brian Maple, M.; Herz, R. K. Coupled Catalytic Oscillators: CO Oxidation Over Polycrystalline Pt. J. Catal. 1988, 113,453-465.
Schreiber, I.; Marek, M. Strange Attractors in Coupled Reaction-Diffusion Cells. Physica 1982, SD,258-272.
Tsotsis, T. T. Spatially patterned states in systems of interacting lumped reactors. Chem. Eng. Sci. 1983, 38, 701-717.
Schiith, F.; Song, X.;Schmidt, L. D.; Wicke, E. J . Chem. Phys. 1990, 92,145-756.
Wang, Y.; Hudson, J. L. AIChE J . 1991, 37, 1833-1843.
Stuchl, Ivan; Marek, Milos. Dissipative structures in coupled cells: Experiments. J. Chem. Phys. 1982, 77, 2956-2963.
Weiner, J.; Holz, R.; Schneider, F. W.; Bar-Eli, K. Mutually Coupled Oscillators with Time Delay. J. Phys. Chem., 1991, submitted for publication.
Takens, F. Lect. Notes Math. 1981, 898, 366-381.
Coupled Oscillating Cobalt Electrodes J. C. Bell,+ N. I. Jaeger,: and J. L. Hudson*,+ Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22903-2442, and Fachbereich 2, BiologielChemie. Universitat Bremen, 0-2800 Bremen 33, Germany (Received: May 11, 1992)
Experimental studies have been carried out with two different coupled electrochemicaloscillators. The first study was described in the preceding article. In this present paper we consider cobalt electrodes in hydrochloric acid/chromic acid electrolytes. Two electrodes, one of which was embedded in a rotating disk and the other of which was in a nonrotating disk parallel to the first, were used. Several types of phenomena due to coupling were observed. Each electrode can drive the other. Phase-locked oscillations were observed when neither, either, or both of the electrodes were held at a potential where oscillations occur independently. Furthermore, both chaos and extinction of oscillations via coupling were observed.
Introduction In a series of two papers we present some experimentalstudies on the coupling of electrochemical oscillators. In the first publication (Wang and Hudson, 1991), experiments done with one, two, and three iron electrodes embedded in the end of a rotating disk were described. We show in that paper that the coupling of electrochemical oscillators can lead to chaos and that the coupling of chaotic oscillators can yield higher chaos. In the second set of experiments, described in this paper, the reaction sites are cobalt electrodes (in hydrochloric acid/chromic acid electrolytes) which are on a rotating disk and a nonrotating disk, respectively. The potential of each of the electrodes can be controlled individually, and the current of each is measured separately. We thus have greater control over the two oscillators. We show that phenomena such as phase-locking, generation of chaos through coupling, and the extinction of oscillations via coupling can occur in coupled electrochemical systems. Experiments The experiments were carried out in a mixture of 0.6 M C r 0 3 and 1.O N HCl. The electrodes were made from 99.998% pure cobalt rods, 0.5 cm in diameter, set in Teflon sleeves of outer diameter 2.0 cm. The counter electrode was a 25 X 50 mm sheet of platinum foil; an SCE served as a reference electrode. Potentials were controlled with a PAR 273 potentiostat/galvanostat, and current was recorded both with an analog recorder and with a digital recorder at a sampling rate of 100 Hz. A schematic of the two electrodes used in the coupled experiments is shown in Figure 1. Two disks, the upper of which is rotatable and the lower of which is stationary, are placed concentrically with parallel working surfaces. For the coupled experiments each electrode is as described above; i.e., it is a cobalt rod embedded in a Teflon sheath. For some experiments carried out to investigate the effect of the altered flow field, either the top or bottom contained a cobalt electrode, whereas the opposite surface was entirely Teflon. The 'University of Virginia. Universitit Bremen.
rotation rate was 75 rpm. For the experiments with two electrodes and those with one electrode facing a blank disk, the gap between the two disks was 4.0 mm. Only a single gap size was used in these experiments. The optimum 4-mm size was selected; at this condition both electrodes oscillate autonomously, the degree of interaction is substantial, and yet axisymmetric flow appeared by visualization to be attainable. (Information on flow near an enclosed rotating disk can be found, for example, in Schlichting (1960).) In the coupled experiments each of the two electrodes was controlled by a PAR potentiostat and had separate counter and working electrodes. In this manner both working (cobalt) electrodes were held at virtual ground. The current in each of the electrodes was measured independently. Additional details can be found in Bell (1991). ReSults
Single Free Disk. For reference we present here a few results obtained with a single, free (standard) rotating disk electrode. Only experiments done potentiostatically will be reported. Current as a function of (constant) potential is shown in Figure 2. This current is the net current produced by the anodic dissolution of cobalt and the cathodic reduction of chromic acid. There is a slight drift of current over the course of an experiment. The results reported for the steady states are mean values over the first 30 min for which there is a drift in the anodic direction of approximately 1.5-2.0 mA. The minimum and maximum values in the oscillating region were taken from the largest amplitude oscillation observed at each potential. For the potential range -15 to +80 mV (SCE) oscillations occur. The type of oscillation depends on the potential. A short nonoscillating transient exists before the oscillations begin, and the duration of this transient increases with increasing potential. (This may be caused by the approach to a Hopf bifurcation as the potential is increased, however, the detailed structure of neither the upper nor the lower transition into oscillations has been analyzed in sufficient detail to determine the nature of the transitions.) Furthermore, the type of oscillation also can change slowly over the course of an experiment as the surface of the cobalt
0022-365419212096467 1$03.00/0 0 1992 American Chemical Society
8672 The Journal of Physical Chemistry, Vol. 96, No. 21, 1992 w
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electrode changes. Nevertheless, any type of given behavior normally exists for sufficiently long times (order of 15 min or more) so that quantitative characterization can be carried out. Further information on the behavior of the single rotating electrode can be found in Hudson et al. (1988) and Bell (1991). A discussion of the chemistry of the reaction can also be found
in the former. In addition, a mechanism for the oscillations is proposed on the basis of pitting corrosion of a film formed during the cathodic reduction of chromic acid. Single Electrode Opposite a Blank Disk It is clear from Figure 1 that the flow fields, and thus the mass-transfer coefficients, of both the upper and lower disks differ from those near a single disk rotating alone in solution. The flow is upward and radmlly outward on the rotating disk but radially inward and axially away from the lower stationary disk. (For a discussion of mass transfer to a rotating disk opposite a fixed boundary, see Lehmkuhl and Hudson (1971).) Therefore, before investigating the coupled electrodes, we studied the behavior of both upper and lower electrodes opposite blank disks; the latter were simply solid Teflon disks on which, of course, no electrochemical reaction could take place. Current as a function of potential for the upper and lower disks are given in Figure 3. The top diagram, for the rotating electrode opposite a blank disk, might be compared to Figure 2 (from a free rotating disk). Because of the altered flow field, the oscillatory region has been shifted slightly toward the cathodic region and has been made slightly smaller. On the other hand, the nonrotating bottom disk has an oscillatory region which covers a smaller range of potentials; nevertheless, oscillations do occur over a definite range of potentials. We have not studied the two diagrams of Figure 3 in great detail. Our purpose was to find oscillatory behavior which could be used in the coupled experiments. As above, a nonoscillatory transient precedes the oscillations on both the top and bottom
Coupled Oscillating Cobalt Electrodes
The Journal of Physical Chemistry, Vol. 96, No. 21, 1992 8673
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electrodes, and the duration of this transient increases with increasing potential. The oscillatory behaviors reported here are those first observed after the transient was completed. Oscillations are shown for three different potentials for the top disk in Figure 4a-c and for two potentials for the bottom disk in Figure 5a.b. The oscillations on the top disk were qualitatively very similar to those obtained with a single rotating disk; the dynamics of the latter case were described in Hudson et al. (1988). The oscillations on the bottom, stationary disk were always period 1, Note that the oscillations on the bottom disk have a simpler form than those on the top. Couplea Electrodes. We now present results for cases in which both the top and bottom electrodes are held at a fixed potential and the current is measured on each. All the results here were obtained at a fued rotating disk rate of 75 rpm and a gap between disks of 4 mm. The two parameters which were varied were the potentials of the rotating and the stationary disks. The parameter space studied was over the potential range for each disk of -60 to +60 mV (SCE). The oscillatory behaviors reported here are those first observed after the nonoscillatory transient. The parameter space (-60 to +60 mV) by (-60 to +60 mV) was investigated by carrying out experiments at intervals of 10 mV for the center of the space and at intervals of 20 mV elsewhere. A variety of types of steady, periodic, and chaotic states was observed. However, we do not have enough information to represent the detailed bifurcation structure in the two parameters. We therefore l i t the presentation to showing several of the types of behavior which were observed. For reference we note that the ranges of oscillatory behavior for the top and bottom electrodes acting alone are -20 to +40 mV and 0 to +20 mV (SCE), respectively. In Figure 6, time series from the top and bottom electrodes for potentials of -50 and +10 mV, respectively, are shown. The oscillations on the top disk are much weaker than those of the bottom. The conditions are such that, acting alone, the top would not oscillate, whereas the bottom would. The result shows that events on the bottom disk influence those on the top, a condition obviously necessary for coupling. The top disk oscillations can also produce oscillations on the bottom. An example is shown in Figure 7 for which the potentials are -30 and -50 mV, top and bottom, respectively. These potentials are, for both disks, below those for which oscillations w u r on the separate disks; thus, in this case the interaction is more
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complicated. Nevertheless, behavior similar to that shown in Figure 7 occurs for potential (mV) pairs of the top and bottom electrodes of (-10, -50), (-10, -30), (-10, -20), (-10, 0), and (-30, +40). It is clear that each electrode can, under some conditions, drive the other. Another type of behavior is shown in Figure 8 for potentials of -50 and +40 mV, top and bottom, respectively. The potential of the top disk is below its oscillatory range whereas that of the bottom is above its range. When coupled, both currents oscillate substantially. The oscillations are locked with a single peak per
8674 The Journal of Physical Chemistry, Vol. 96, No. 21, 1992 a)
Bell et al. a)
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Figure 10. Oscillations: (a) top electrode, E = -30 mV, below single disk oscillatory range; (b) bottom electrode, E = -10 mV, below single disk oscillatory range.
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cycle on the top and two per cycle on the bottom. Figure 9 shows yet another type of behavior for top and bottom potentials of -30 and -30 mV, respectively. Here again both potentials are outside the range where single disk oscillations occur, but here both are below their range. (In Figure 8 one was above and one below.) A second example where both potentials are below the single disk range is shown in Figure 10. (The potentials are -30 and -10 mV, top and bottom, respectively, or for each only 10 mV below the minimum of its oscillatory range.) Similar
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time series can be obtained when one of the potentials (here the top potential) is inside its oscillatory range, as can be seen in Figure 11. We now turn to a discussion of the behavior observed when both disks are held at potentials where they would independently oscillate. Simple oscillations do, of course, also m u r in this range. Two more complicated types of behavior, viz., chaos and extinction of oscillations, have also been seen. An example of chaotic behavior is shown in Figures 12-14 for top and bottom disk potentials of +10 and +20 mV. The time
The Journal of Physical Chemistry, Vol. 96, No. 21, 1992 8675
Coupled Oscillating Cobalt Electrodes
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Figure 14. Poincard sections from Figure 13 attractors (T = s): (a) top electrode, E = +10 mV, I ( t ) = -5 mA, increasing Z ( t ) ; (b) bottom electrode, E = +20 mV, I ( f ) = -1 mA, increasing I ( t ) .
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series for both electrodes are shown in Figure 12. The signal of the bottom electrode appears more irregular than that of the top. This can also be seen in the attractors, Figure 13. Poincare sections made from Figure 13 are shown in Figure 14. The behavior is likely low-order chaos. A final example is shown in Figure 15. The potentials of the top and bottom electrodes are +10 and 0 mV, respectively, where each electrode oscillates independently. The oscillations for each of the independently operating electrodes are shown as limit cycles
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in state space, When the electrodes are coupled, the oscillations are extinguished; that is, both electrodes exhibit a steady current. The steady-state behavior of each of the coupled electrodes is shown as X on the figure. Concluding Remarks The emphasis of this study is on the dynamics of coupled chemical oscillators. We have demonstrated that coupling can
8616 The Journal of Physical Chemistry, Vol. 96,No. 21, 1992
occur between electrochemical oscillators and that many of the phenomena common to coupled systems can be produced in the two systems which were investigated. We have found transitions to chaos and to higher order chaos, phase-locking, extinction of oscillations, and the production of oscillations through coupling of two steady currents. Electrochemical oscillators have produced a rich variety of dynamic phenomena in uncoupled systems. It is likely that coupled electrochemical oscillators can be used for detailed tweparameter bifurcation studies. For direct comparisons of experimental results to analyses, one of the two parameters should be the coupling strength. In the case of the two disk electrodes in the second study described in this paper, the gap width between the electrodes might have served as this parameter. Unfortunately, however, the range of gap widths over which each electrode oscillated independently was very small, so all the results presented in this paper were obtained with a single gap width of 4 mm. The disk which rotated, of course, could produce oscillations for any arbitrary gap width; however, the nonrotating disk, since its flow was produced by the rotating one, oscillated only for certain gap widths. Changing the gap width thus changed not only the coupling strength, it also changed significantly the basic dynamics of the individual oscillators. The rotation of both of the disks would alleviate this deficiency since both would then produce oscillations for arbitrary gap widths. The mechanism of the coupling is certainly complicated and was not investigated in detail in this study. Two contributors to the coupling are the ohmic drop in the electrolyte and the transport of products or intermediates among the electrodes. The electrolyte concentration was 1 .Oand 1.6 M in the two sets of experiments, respectively. Although the ohmic drop may be small, it may be large enough to produce significant coupling effects. Transport of species may also play an important role. The changed electrolyte concentration can alter the potentials at which oscillations occur. For example, in the cobalt system, a decrease of the concentration of hydrochloric acid shifts the oscillating regime in the anodic direction and a decrease of the concentration of chromic acid causes a slight shift of the oscillating regime in the
Additions and Corrections cathodic direction. This shift follows rules which point to pitting corrosion (Hudson et al., 1988). In addition, diffusion and migration of charged species contribute to changes of concentrations near the electrodes. It is likely, in fact, that some combination of ohmic drop and transport leads to the coupling. Steady and oscillating currents give rise to potential gradients in the electrolyte between the two electrodes which will be balanced by the migration of ions; this is the concept of coupling through local current which was forwarded by Franck and Meunier (1953)to explain traveling excitations on passive iron waves and the coupling of oscillating Co wires.
Acknowledgment. This work was supported in part by the North Atlantic Treaty Organization, the National Science Foundation, the Center for Innovative Technology (Commonwealth of Virginia), and the Alexander von Humboldt Stiftung. Registry No. Co, 7440-48-4; HCI, 7647-01-0; CrO,, 1333-82-0; chromic acid, 7738-94-5.
References Bell, J. C. The Dynamics at Cobalt Electrodes in Chromic Acid El%trolytes. Ph.D. Dissertation, University of Virginia, 1991. Franck, U. F.; Meunier, L. Gekoppelte periodische Elektrodenvorgange. Z . Naturforsch. 1953, 86, 396-406. Hudson, J. L.; Bell, Joseph C.; Jaeger, Nils I. Potentiostatic Current Oscillations of Cobalt Electrodes in Hydrochloric Acid/Chromic Acid Electrolytes. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 1383-1 387. Lehmkuhl, G. D.; Hudson, J. L. Flow and Mass Transfer mear an Enclosed Rotating Disk: Experiment. Chem. Eng. Sci. 1971, 26, 1592-1600. Schlichting, H. Boundary Layer Theory; McGraw-Hill: New York, 1960. Wang, Y.; Hudson, J. L. Experiments on Interacting Electrochemical Oscillators. J . Phys. Chem., preceding paper in this issue.
ADDITIONS AND CORRECTIONS 1992, Volume 96 I. Rosenberg,* J. R. Brock,* and A. HeUer*: Collection Optics of Ti02 Photocatalyst on Hollow Glass Microbeads Floating on Oil Slicks. Page 3423. In the Introduction, first paragraph, the third sentence should read correctly as follows: ...the hole oxidizes absorbed water to OH radicals and protons; ....
