J . Phys. Chem. 1992, 96, 8667-8671
8667
Experlments on Interacting Electrochemical Oscillators Y. Wangt and J. L. Hudson* Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22903-2442 (Received: March 12, 1992; In Final Form: May 1 1 , 1992)
Experimental studies were carried out with coupled electrochemical oscillators. The first set of experiments involved the electrodissolution of iron in sulfuric acid. One, two, or three electrodes were embedded in the end of a rotating disk. We show examples of coupling of periodic oscillators leading to chaos and of coupled chaotic oscillators leading to higher chaos.
Introduction The coupling of chemical reactions which are occurring at two or more different locations can lead to more complicated behavior than that which can take place at the single reaction site. The coupled sites might be stirred reactors which are exchanging mass or heat, or they might be individual catalyst particles in a packed bed or wen multiple locations on a single catalyst particle. If the individual sites are at steady state, coupling can produce patterns in concentration or temperature. Tsotsis 1983, for example, has shown how patterns can arise via interaction of lumped reactors. If the individual sites are oscillating, the coupling can lead to wen richer behavior. In this case the coupling of two dissimilar oscillating reactors can lead to phase locking or resonance at frequency ratios of 1 to 1, 1 to 2, etc., or to quasiperiodicity, in which case the oscillations in the two reactors have incommensurate frequencies. The coupling can also lead to transitions to chaos. Such phenomena have been studied using mathematical models in some detail by several investigators such as Boissonade et al. (1990), Taylor and Kevrekidis (1991), Boukalouch et al. (1987), Crowley and Field (1986), Crowley and Epstein (1989), and Ravi Kumar et al. (1983). Bar-Eli (1984a,b, 1985, 1990) has carried out extensive mathematical studies of two (and more) continuous stirred reactors (CSTRs) coupled through the transfer of material. He has shown, among other things, that under some conditions the coupling of two oscillators can lead to an extinction of the apcillations; this is often referred to as "phasedeath". There have also been numerical studies of coupled identical reaction cells in which symmetry breaking, and subsequently other phenomena such as chaos, can occur thrsugh the exchange of a passive species such as mass in the simple irreversible exothermic reaction (Schreiber and Marek, 1982; Mankin and Hudson, 1986). Experimental studies of coupled chemical oscillators, however, are much more difficult to carry out. Coupling of two CSTRs is complicated by the necessity of maintaining uniform conditions in each of the individual cells. Nakajima and Sawada (1980, 1981) and Stuchl and Marek (1982) performed early studies of coupled cells with the Below-Zhabotinskii (B-Z) reaction, and this latter work was later fxpanded to include forcing effects (Dolnik et al., 1987, Dolnik and Marek, 1988). Bar-Eli and Rcuveni (1985) demonstrated experimentally that coupled oscillators can exhibit inhomogeneous stable stationary states. Some of the difficulties associated with coupling of chemical cells were alleviated by Crowley and Field (1986) who coupled two B-Z reactors electrically. Two recent studies with well-designed chemical cells with mass interchange have led to important advancements. Boukalouch et al. (1987) have studied the chlorite-iodide reaction in two CSTRs coupled through a variable aperature which allowed careful control of total mass exchange; they demonstrated in an unsymmetric regime that oscillations can be produced in parameter regions where both (uncoupled) reactors would be at steady state. Crowley and Epstein (1989) used needle valve ports for careful control between two reactors; they observed
T
Y
(C)
(d)
Figure 1. Schematic of electrode configuration.
with the B-Z reaction, among other things, in and out of phase entrainment, phase death, and multistability. Weiner et al. (1991) studied the " a 1 oscillator (B-Z reaction without malonic acid) in coupled cells including a time delay. The coupling of chemical oscillations in other systems and geometries is also of considerable importance, particularly since such studies may help explain the mechanisms of the interactions among sites. For example, in gassolid (catalytic) reacting systems coupling among reacting sites can occur via heat conduction or surface diffusion or through the gas phase. Brown et al. (1985) have investigated steady-state thermal communication between two particles of supported catalyst, during hydrogen oxidation. Onken and Wolf (1988) have studied coupling of chemical oscillators via heat conduction: the exwriments were done with a Si02disk, two regions of which cont'ained active platinum/Si02 Resent address: Jackson Laboratory, E.1. du Pont de Ncmours & Co., Inc., Deepwater, NJ 08023. sites. Onken and Wicke (1989) observed mutual interaction of 0022-3654/92/2096-8667$03.00/00 1992 American Chemical Society
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8668 The Journal of Physical Chemistry, Vol. 96, No. 21, 1992
Time (I)
0.5
0.0
Time (s)
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I
I
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Figure 2. Time series: R = 2 mm; E = -0.37 V (Hg/Hg,SO.,); 400 rpm.
reaction rate oscillations on catalyst pellets during the oxidation of CO, and Jaeger et al. (1986) studied thermal coupling among three areas of an amorphous alumina-supportedcatalyst during the heterogeneous catalytic oxidation of ethanol. Schiith et al. (1990) have developed a model of thermal coupling among 10 randomly placed oscillators on a 10 X 10 matrix; Gerhardt and Schuster (1989) developed a cellular automaton model of coupling on a reacting surface. Coupling through the gas phase has also been considered by Tsai et al. (1988) between two platinum wires and by Ehsasi et al. (1990) using two Pd(ll0) single crystals. In a series of two papers we describe some experimental results with a different type of chemical oscillator, viz., coupled electrochemical reaction sites. (The second paper is Bell et al. (1992).) Each of the two systems considered in the two papers has its own characteristics. The first experiments, which are described in this paper, were done with one, two, and three iron electrodes embedded in the end of a rotating disk. These experiments follow some earlier dynamic experiments we carried out with single iron rotating disk electrodes (Wang and Hudson, 1991). In that earlier work, series of runs were made with electrodes of varying size. It was found that the complexity of the chaotic oscillation increased with increasing electrode diameter. One possible explanation for the increase in complexity is through the interaction of reacting sites on the surface of the electrode surface; the number of such interactions would increase with increasing electrode area. We show in the two papers that some of the phenomena discussed above in the introduction can also occur in coupled electrochemical systems. Electrochemicalreactions have some features which make then convenient for studies of coupled chemical oscillators. A signal, either current or potential depending on the conditions, can be easily measured. Sometimes, as in our second set of experiments, each electrode can yield an independent signal (current) and the potential of each electrode can be controlled
Figure 3. Attractors corresponding to Figure 2: 7 = 0.4 ms. The plane used to make the sections is defined by the equation Z (t - 8 7 ) = 45 mA and Z (t - 8 7 ) = 64 mA for the two and three electrode configurations, respectively. Two Poincare sections are shown, one for each direction crossing the plane.
individually. To some extent, the coupling strength can be controlled by placement of the electrodes. The idea of coupled electrochemical oscillators is not new. Franck and Meunier (1953) carried out such experiments and showed the effect of variable coupling by adjusting the distance between two autonomously oscillating wire electrodes.
Experiments The experiments described in this paper were done with iron rod electrodes embedded in a 33-mm-diameter Teflon cylinder, with the iron exposed at the end. A schematic of the electrode configuration can be seen in Figure 1. Either one, two, or three electrodes, of diameter 2 mm, are used, in all cases the electrodes in a given experiment are at the same radial position and are placed either symmetrically (Figure 1a-c) or unsymmetrically (Figure Id) about the center point. The working electrodes are all held at the same potential, and only the total current is measured. The counter electrode was 0.1-"-thick platinum foil of 25 X 50 mm cross section, and the reference electrode was Hg/Hg2S04,each placed in side chambers of a cell which contained 300 mL of 1.O M H2S04held at 25 OC. ReSUlts
First the results shown in Figure 2 are considered. The 2"diameter iron rods are placed close together at radial positions R = 2 mm from the center of the Teflon disk. The electrodes are symmetrically placed. The single electrode yields a periodic oscillation, as can be seen in the time series (Figure 2) or the attractor (Figure 3) made by the method of time delays (Packard
The Journal of Physical Chemistry, Vol. 96, No. 21, 1992 8669
Interacting Electrochemical Oscillator Experiments
d=2.2
8
d=2.6
0 d=3.1
d=3.5 46-60
Figure 4. Attractors and Poincar6 sections: R = 2 mm; E = -0.44 V (Hg/Hg2S04); 400 rpm; 7 = 0.4 ms. The plane used to make the sections is defined by the equation I ( t - 167) = 18 mA for the single electrode, by I ( t - 207) = 45 mA for the two electrodes, and by I ( t 167) = 61 mA for the three electrodes.
et al., 1980; Takens, 1981). These oscillations are occurring on the mass-transfer limited plateau, that is, in the region where the currents are at their highest values. The results for the two and three electrode configurationscan also be seen in Figures 2 and 3. The current shown is the total current to the two or three electrodes, respectively. The attractors in Figure 3, particularly that for the three electrode case, appear to be those of low order chaos. The PoincarC sections of the attractors are also shown in Figure 3. (Two Poincar6 sections are shown, one for each direction of traversal of the plane.) The sections appear to be almost onedimension curves which is also consistent with low order chaos. Information dimensions for these attractors were calculated using the nearest-neighbor method (Badii and Politi, 1985; Kostelich and Swinney, 1987; Kostelich, 1990); comments on the use of this method for the calculation of dimensions for attractors such as those seen here can be found in Wang and Hudson (1991). The dimension for the attractor of Figure 3c (three electrodes) is 2.5. Coupled chaotic oscillators, obtained using the same geometries as those used for the previous result, are shown in Figure 4. (Only the potential differs from the conditions of the previous two figures.) The attractors for the one, two, and three electrode systems, and also cross sections, are shown. The calculated dimensions for the one, two, and three electrode systems are 2.2,2.6, and 3.1, respectively. Thus, under the conditions of these experiments, the single electrode exhibits low-dimensional chaotic behavior, as evidenced by the value of 2.2 and also by the shape of the attractor and PoincarC sections. A dimension of slightly greater than two is obtained in now classical sets of three ordinary differential equations (ODES) yielding chaos, such as the Rossler attractor (Rijssler, 1976a,b). The two electrodes yield an attractor dimension of 2.6 and the three electrodes a dimension of 3.1. In
Figure 5. Attractors: R = 10 mm; E = -0.253 V (Hg/Hg2S04); 200 rpm; 7 = 0.4 ms.
both cases, therefore, we see a somewhat more complex chaotic behavior. Since the currents in the latter two cases are the sums of the currents for the two or three electrodes, respectively, it is seen that some coupling of the electrodes is occurring. If no coupling were present, the dimensions would be 2 X 2.2 = 4.4 and 3 X 2.2 = 6.6, respectively. We show one final set of results with this system in Figure 5. The electrodes are now placed at a greater radial position, viz., at a radius of 10 mm. The attractor obtained for a single electrode is shown in Figure 5a(top). Note that it resembles that seen above in Figure 3a(top); however, in order to obtain roughly the same result as in Figure 3a(top), the rotational rate had to be decreased since the electrode is placed at a greater radial position. (The mass-transfer coefficient increases with increasing rotational rate and with increasing radial position. Note that although we are using a rotating disk, we are not operating under conditions of a uniformly accessible surface since the electrodes are not positioned in the center of the rotating disk.) The results obtained with two electrodes placed 180° apart are seen in Figure 5b(middle). The attractor (and the time series from which it was made) is very close to that of the single electrode case. The two electrodes are locked on a single periodic cycle. There is certainly some coupling; no coupling would most likely yield a quasiperiodic attractor since the two electrodes, although very similar, cannot be exactly identical. (Note again that the current in these experiments is the sum of the current of the two electrodes.) In a comparison of Figures 3b(middle), and Sb(middle), it is seen that the coupling is different in the two cases. In the configuration of Figure 3, the streamlines leaving one electrode pass over the other whereas this is not the case for the conditions of Figure 5. Bringing the two electrodes closer together, but still at a radial position of 10 mm, as in Figure Sc(bottom), produces chaotic behavior. In the configuration of Figure Sc(b0ttom) the coupling is nonsymmetrical since the fluid flow leaving one electrode passes
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8670 The Journal of Physical Chemistry, Vol. 96, No. 21, 1992
over the other, but not conversely. The only definite conclusion that can be drawn from Figure 5 is that the coupling is different in the two cases of Figure Sb(middle), c(bottom); this implies that mass-transfer coupling does play some important role. There is not enough information in the results to say which of the two cases is more strongly coupled. One might think that configuration 5c would result in stronger coupling since the electrodes are closer together. However, since the coupling is asymmetric and in fact only stronger in one direction, the coupling in Figure 5c(bottom) may be weaker. This latter possibility could very well explain the results shown since very strong coupling can lead to phase locking and a simple oscillation as shown in Figure Sb(midd1e).
Concluding Remarks We have shown in this paper that coupling does occur among electrodes embedded in the end of a rotating disk and that this coupling can lead to chaos (for coupled periodic oscillators) and to chaos of higher dimension (for coupled chaotic oscillators). A brief discussion of the coupling phenomena, and the possible role of transport of reaction products or intermediates among the electrodes and of ohmic drop, is given at the end of the following paper. We do note here, however, the results shown in Figure 5b,c (middle, bottom). The only difference between these two experiments is the proximity of the two electrodes which affects both possible contributions to the coupling. It is seen that transitions to chaos only occurred when the streamlines are such that species from one electrode can flow directly to the other electrode. Clearly transport of mass between and among the electrodes plays an important role in the coupling of the electrochemical oscillators. It is seen that an increase in complexity can cccur as the number of reacting sites or electrodes increases from one to two to three. The coupling among sites most likely contributes to the increase in dynamic complexity which was seen as the size of a single electrode surface was increased in a previous study (Wang and Hudson, 1991). A limitation of the present study, however, is that only the total current to the multiple electrodes was measured. (This limitation could and should be removed by a redesign of the apparatus.) Thus, we do not know how each individual site is affected by the others but rather know only the composite behavior. A knowledge of the currents a t each electrode would help in understanding not only the coupled system being studied but perhaps also the effect of the earlier single disk size experiments. In the subsequent study, part two of this sequence, two electrodes in a different configuration are considered, the current of each of two electrodes is measured separately and the potential of each electrode controlled individually. Acknowledgment. This work was supported in part by the
National Science Foundation, the North Atlantic Treaty Organization, and the Center for Innovative Technology (Commonwealth of Virginia). We thank N. I. Jaeger for discussions. Registry No. Fe, 7439-89-6; HzS04, 7664-93-9.
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Brown, J. R.; D’Netto, G. A,; Schmitz, R. A. Spatial Effects and Oscillations in Heterogeneous Catalytic Reactions. In Temporal Order, Rensing, L., Jaeger, N. I., Eds.,Springer Verlag: Berlin, 1985. Crowley, Michael F.; Eptein, Irving R. Experimental and Theoretical Studies of a Coupled Chemical Oscillator: Phase Death, Multistability, and In- and Out-of-Phase Entrainment. J. Phys. Chem. 1989, 93, 2496-2502.
Crowley, Michael F.; Field, R. J. Electrically Coupled Belousov-Zhabotinskii Oscillators. 1. Experiments and Simulations. J. Phys. Chem. 1986, 90, 1907-1915.
Doldk, M.; Marek, M. Extinction of Oscillatorsin Forced and Coupled Reaction Cells. J. Phys. Chem. 1988, 92, 2452-2555. Doldk, M.; Padu s6kov6, E.; Marek, M. Periodic and Aperiodic Regimes in Coupled Reaction Cells with Pulse Forcing. J. Phys. Chem. 1987, 91, 4407-4410.
Ehsasi, M.; Frank, 0.; Block, J. H.Coupled Chemical Oscillators in Catalytic Oxidation of CO on Pd (1 10) Surfaces. Chem. Phys. Lett. 1990, 165, 115-1 19.
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Kostelich, E. J. Software for CalculatingAttractor Dimension Using the Nearest Neighbor Algorithm. Unpublished (Feb 1990). Kostelich, E. J.; Swinney, H. L. In Chaos and Related Nonlinear Phenomena; Procaccia, I., Shapiro, M., Eds.; Plenum: New York, 1987. Mankin, J. C.; Hudson, J. L. The Dynamics of Coupled Nonisothermal Continuous Stirred Tank Reactors. Chem. Eng. Sci. 1986, 41, 265 1-2661.
Nakajima, K.; Sawada, Y. Experimental studies on the weak coupling of oscillatory chemical reaction systems. J . Chem. Phys. 1980, 72, 223 1-2234.
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Bar-Eli, K. On the Stability of Coupled Chemical Oscillators. Physica 1985, 14d, 242-252.
Onken. H. U.; Wolf, E. E. Coupled Chemical Oscillatorson a Pt/SiO, Catalyst Disk. Chem. Eng. Sei. 1988, 43, 2251-2256. Packard, N. H.; Crutchfield, J. P.,Farmer, J. D.; Shaw, R. S . Phys. Rev. Lett. 1980, 45, 712-716.
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Taylor, M. A.; Kevrikidis, I. G. Some common dynamic features of coupled reacting systems. Physica D 1991, 51, 274-292.
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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
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