1907
J . Phys. Chem. 1986,90, 1907-1915
Electrically Coupled Belousov-Zhabotinskii Oscillators. 1. Experiments and Simulations Michael F. Crowley and Richard J. Field* Department of Chemistry, University of Montana, Missoula, Montana 5981 2 (Received: May 29, 1985; In Final Form: November 26, 1985)
Experiments have been carried out with electrically coupled Belousov-Zhabotinskii oscillators. The driving force for the coupling is the instantaneous, relative redox-potential difference between the oscillators resulting from different values of [Ce(IV)]/ [Ce(III)], the oscillating variable of highest concentration, in each oscillator. The actual coupling results from a current flow between large-area working electrodes in each oscillator that drives [Ce(IV)]/ [Ce(III)] toward the same value in both oscillators. The major redox processes occurring as a result of the current flow involve Ce(IV) in one oscillator and Ce(II1) in the other. Electrical coupling has the advantages that it does not involve mass transfer between the oscillators, that it affects substantiallyonly one dynamic variable, [Ce(IV)]/[Ce(III)], that it is easily understood by using the Oregonator model of the Belousov-Zhabotinskii chemistry, and that the dynamic phenomena observed mainly depend upon the gross features of the chemical dynamics rather than on the details of the chemistry. The phenomena observed in electrical-coupling experiments include the following: drifting, 1:1 in-phase and out-of-phase entrainment, entrainment of the two oscillators at other integer frequency ratios, e.g. 2:l or 3:2, quasiperiodicity,chaos, suppression of oscillations in one or both oscillators, and a form of bursting. The behavior of electrically couplkd Belousov-Zhabotinskii oscillators is accurately simulated by the Oregonator model, and the root causes of the observed phenomena, especially the chaos, are apparent within the dynamic structure of the Oregonator.
The coupling of chemical oscillators is of considerable and general theoretical and practical interest.'g2 The coupling of Belousov-Zhabotinskii (BZ) oscillator^^-^ in particular has been the subject of a number of experimental and computational investigations as it is presently the best understood chemical oscillator. Experiments have been carried out by Marek and Stuch15 using coupled CSTR reactors and by Fujii and Sawada6 as well as by Nakajima and Sawada' using coupled batch reactors. CSTR experiments are preferred as one may obtain with them much better control over experimental variables such as reactant concentrations. Some phenomena observed in these experiments include entrainment of the slower oscillator by the faster, synchronization, phase-difference locking, and rhythm splitting in which one oscillator suppresses an occasional cycle of the other. There have been theoretical predictions8 of chaosg in coupled oscillators, and indications of chaos were found7 experimentally, although not pursued. Stuchl and Marek'O studied systems containing up to seven coupled BZ oscillators in the CSTR mode. In all of these experiments the coupling was by mass (reaction mixture) transfer through windows of various configurations connecting two reactors. Bar-Eli"-13 used a version of the FKN3p4 mechanism of the BZ reaction to explore computationally the behavior of mass-transfer-coupled BZ oscillators. H e found evidence of several of the phenomena observed experimentally. There are practical problems associated with the coupling of chemical oscillators by reaction-mixture transfer. It is difficult to understand quantitatively and to model the diffusive and convective forces involved in the coupling. Bar-Eli's calculations'* (1) Rehmus, P.; Ross, J. In Oscillations and Traveling Waves in Chemical Systems, Field, R. J., Burger, M., Eds.; Wiley-Interscience: New York, 1985; p 287. (2) Othmer, H., Ed. Lecture Notes in Biomathematics, Springer-Verlag: New York, in press. (3) Field, R. J.; Koros, E.; Noyes, R. M. J . Am. Chem. SOC.1972, 94, 8649. (4) Field, R. J. In Oscillations and Traueling Waves in Chemical Sysfems, Field, R. J., Burger, M., Eds.; Wiley-Interscience: New York, 1985; p 55. (5) Marek, M.; Stuchl, I . Biophys. Chem. 1975, 3, 241. (6) Fujii, H.; Sawada, Y . J . Chem. Phys. 1978, 69, 3830. (7) Nakajima, K.; Sawada, Y. J . Chem. Phys. 1980, 72, 2231. (8) Tomita, K.; Kai, T. Prog. Theor. Phys. 1979, 61, 54. (9) Haken, H., Ed. Chaos and Order in Nature; Springer-Verlag: New York, 1981. (10) Stuchl, I.; Marek, M. J . Chem. Phys. 1982, 7 7 , 2956. (11) Bar-Eli, K. J . Phys. Chem. 1984, 88, 3616. (12) Bar-Eli, K. J . Phys. Chem. 1984, 88, 6174. (13) Bar-Eli, K. J . Phys. Chem. 1985,89, 2852.
0022-3654/86/2090-1907$01.50/0
show that violation of the CSTR rapid, uniform-mixing as~ u m p t i o n , as ' ~ must occur practically when coupling is by mass transfer, considerably affects coupled behavior. Another problem is that all reactant, product, catalyst, and intermediate species are exchanged when coupling is by mass transfer, making numerical modeling of the coupling difficult and qualitative mathematical analysis even more difficult. Finally, the direct physical connection between reactors in mass-transfer coupling makes it essentially impossible to carry out and interpret experiments in which the coupled oscillators have significantly different chemical composition. Thus the oscillators cannot be very different from each other in period, waveform, etc. Marek and StuchlScontrolled the periods of their coupled oscillators by maintaining them at different temperatures, and Nakajima and Sawada7 did the same by varying the stirring rate in each reactor. In both of these cases the chemical composition of the two oscillators was assumed to be the same. In order to avoid problems resulting from mass-transfer coupling, we have explored the electrical coupling of BZ oscillators. Such coupling also can be easily understood in terms of the Ore g ~ n a t o rmodel '~ of the BZ reaction. The classic BZ reaction is the metal-ion- (cerium ion is used here) catalyzed oxidation of easily brominated organic materials (e.g. malonic acid or the acetylacetone used here) by bromate ion in a strongly acid medium. The metal-ion catalyst oscillates between its oxidized and reduced forms (Ce(1V) and Ce(II1) here) during this reaction. An oscillatory redox potential can be measured which is related mainly to [Ce(IV)]/[Ce(III)], although in some cases there may be contributions from the HOBr/Br2/Br- couples16 or couples involving intermediate species.I7 The basic concept of an electrical-coupling experiment has been discussed by Crowley and Field,18 who carried out calculations indicating that it could be done and that some interesting phenomena, especially chaos, should be expected to appear. The basic principle is that large-area platinum electrodes are placed in each of the reactors to be coupled, and the circuit is completed by a wire between the electrodes and by an ion bridge between the ~~
(14) Denbigh, K. G.; Turner, J. C. R. Chemical Reactor Theory; Cambridge University Press: Cambridge, 1984; 3rd ed. (15) Field, R. J.; Noyes, R. M. J . Chem. Phys. 1974, 60, 1877. (16) Field, R. J.; Boyd, P. M. J . Phys. Chem. 1985, 89, 3707. (17) Orbln, M.; Koros, E. J . Phys. Chem. 1978, 82, 1672. (18) Crowley, M. F.; Field, R. J. In Nonlinear Phenomena in Chemical Dynamics, Vidal, C., Pacault, A., Eds.; Springer-Verlag: N e w York, 1981; p 147.
0 1986 American Chemical Society
1908 The Journal of Physical Chemistry, Vol. 90, No. 9, 1986
Crowley and Field
n
reactors. The effect of this coupling is that a current flows between the reactors which is proportional to the instantaneous difference in redox potential in them. Assuming that the oscillatory potential is mainly related to [Ce(IV)]/[Ce(III)] and that the redox reactions occurring at the electrodes mainly involve Ce(II1) and Ce(IV), the effect of the current flow is to drive [Ce(IV)]/[Ce(III)] toward the same value in both reactors. Thus the oscillators are coupled chemically as well as electrically. The chemical effect of electrical coupling is assumed to be entirely on the metal-ion catalyst, whose concentration is several orders of magnitude higher than that of any other oscillatory species. For example, reaction 1 may occur in the first reactor while reaction 2 occurs in the second reactor. This chemistry e-
+ Ce(IV)
Ce(II1)
-
-
Ce(II1)
(1)
+ e-
(2)
Ce(IV)
is easily modeled by adding a single term to the d[Ce(IV)]/dt = -d[Ce(III)]/dt mass-action kinetic equation for each reactor. The form of this term is given by the Nernst equation, which we believe to be valid in our experiments with large-area electrodes, relatively low currents, and rapid stirring. In reactor 1 we have d[Ce(WlI - -d[Ce(WI, dt dt
1-91
Ci e m 4
a l =
where the subscripts refer to reactors 1 and 2, R is the resistance of the coupling circuit, and I is the instantaneous coupling current. A proportionality constant converts from coulomb/s to M/s for a reactor of specific volume. Equation 3 assumes that Ohm’s law is valid, that the electrode reactions of Ce(II1) and Ce(IV) are faster than all other electrode reactions so that essentially all of the coupling current is carried by reactions of Ce(II1) and Ce(IV), and that the current density is sufficiently small and stirring sufficiently good that the electrodes do not become polarized. There is nothing in our experimental results to suggest that these assumptions are not true. There are a number of factors which complicate electricalcoupling experiments. The natural coupling potential is usually less than 100 mV. We have to amplify this potential in our experiments. The resistance of the coupling circuit is finite and mainly located in ion-transport processes in solution. For coupling to occur at the low currents attainable, the concentration of the metal-ion catalyst must be low and the volumes of the CSTRs small so that the small amount of electrochemistry occurring can have an effect. The reactors must also be small (and close together) to keep the distance between the working electrodes, and thus the resistance of the coupling circuit, as small as possible. Stirring must be rapid in small reactors equipped with large working electrodes as well as a number of smaller sensing electrodes and inlets for the CSTR feedstreams. Gas bubbles must be avoided as even small ones significantly affect the residence time in a small reactor, and they also tend to stick to the electrodes, causing several problems, e.g. increased coupling resistance and erratic monitoring-potential readings. We have not completely solved the ion-transport resistance problem. Instead sensing electrodes were placed in each reactor and a circuit constructed which amplifies the potential difference between the reactors, inverts it, and applys it to the working electrodes. This procedure mimics a coupling circuit of less than the real resistance. It is possible with this modification to carry out electrical-coupling experiments. Furthermore, the same apparatus can be used to carry out experiments in which a single oscillator is perturbed by an applied potential of any desired waveform. We are undertaking such experiments. Here we report only coupling experiments in which two oscillators perturb each other.
Experimental Section Design of the CSTR reactors used in electrical-coupling experiments is dominated by the need to keep the reactor volumes
--
Figure 1. Side view (a, top) and top view (b, bottom) of electrically coupled Belousov-Zhabotinskii oscillators: (1) glass stirrer shaft, (2) Teflon top layer and upper bearing for stirrer shaft, (3) Plexiglas overflow reservoir/aspirator trough, (4) aspirator outlet, (5) Teflon lower stirrer bearing and overflow vent to aspirator trough, ( 6 ) Plexiglas reactor cavity, (7) platinum-gauze working electrode, (8) glass stirrer body, (9) connecting port between the reactors, (10) 25-pm Millipore filter barrier, (1 1 ) center cell for the flowing sulfuric acid solution, (12) inlet for center cell sulfuric acid wash solution, (13) platinum wire lead to working electrode, (14) Tygon cell floor and septum, (1 5) aluminum base, (1 6) CSTR feed tubes for reactants, (17) Ag/AgBr monitoring electrode, (18) shiny platinum electrode used as input to the coupling circuit, (19) shiny platinum electrode for monitoring oscillations, (20) outlet for center cell wash solution, (21) reference cell for reactor monitoring electrodes, (22) gravity feed inlet for center cell wash solution.
small and the resistance of the coupling circuit low. Top- and side-view schematics of our reactors are shown in parts a and b of Figure 1. The numbers in the following discussion refer to this figure. The reactors themselves (6) are each cut into a solid block of Plexiglas and have volumes of about 2.5 mL with the glass stirrers (8) in place. To keep the junction potential (resistance) between the reactors small, ion transport is through a large-area window connecting the reactors (9) when they are clamped together. Convective transport between the reactors is eliminated by two pieces of millipore (25 bm) filter paper, one located in the wall of each reactor. These two pieces of filter paper are separated by a 2-mm gap (the center cell, 11) through which flows (1 2) a H,SO, solution of the same concentration as in the reaction mixtures, usually -3 M. This minimizes diffusive coupling between the reactors but allows current to be carried by H30+. There is undoubtably some diffusive loss of reactants out of the CSTRs into the center cell where they are washed away,
Electrically Coupled Belousov-Zhabotinskii Oscillators
The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 1909 l a r g e W o r k i n g Electrodes
Reference Electrode
AL EIternsl Resistance lo
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Adiusl CovPling Strength
Output Power Source
Input Power Source
Figure 3. Schematic diagram of coupling circuit for the electrically
coupled BZ oscillator experiment. Figure 2. Schematic diagram of CSTR pumping and potentiometric
monitoring systems for the electrically coupled BZ oscillator experiment. but we assume that this loss is small. The reference electrode for potentiometric measurements in the reactors is located in the reservoir which supplies HzS04solution to the center cell. The tops of the reactors are conical to facilitate the escape of gas bubbles through the CSTR overflow vent (3), which also serves as the entrance for the glass stirrer drive (1). The overflow from both reactors and the center cell is collected (3) and removed by an aspirator (4). The stirrers are rotated in opposite directions by a single motor and a gear arrangement. Stirring is maintained rapid enough that faster stirring does not affect the results obtained. The entire wall surface (except for the window) in each reactor is covered by a platini~ed’~ platinum gauze (52 mesh with an area of about 5 cm2) that serves as the working electrode. The floor of both reactors is made of a single sheet of Tygon (14) clamped to the bottom of the reactors by an aluminum plate (15). All sensing electrodes are inserted (1 7-19) through the reactor floors. The CSTR inlets are syringe needles also inserted through the reactor floors (1 6 ) , and pumping is by a Buchler four-channel (two to each reactor) peristaltic pump. We assume the pumping rate to be the same for both reactors. A schematic diagram of the CSTR pumping and potentiometric monitoring systems is shown in Figure 2. Each reactor is equipped with two shiny platinum electrodes and a single Ag/AgBr electrode. One Pt electrode monitors the redox potential. The Ag/AgBr electrode monitors [Br-1. Potentials are read off of these electrodes at 1-s intervals, put through a bank of A-D converters (one for each electrode), and collected by a LSI 11/23 laboratory computer for storage on a Corvus 20-megabyte Winchester disk. Output is to a Houston Instruments DP-11 digital plotter, and data can be transferred to the University of Montana DEC-2060 mainframe computer for analysis. The second Pt electrode in each reactor is used to control the coupling circuit. The potential difference between them is amplified, inverted, and applied to the working electrodes. A schematic diagram of the circuit used to do this is shown in Figure 3. Use of the amplified potential is a virtual reduction of the working resistance of the coupling circuit. The desired coupling strength is obtained by placing a variable resistance in series with the working electrodes as is shown in Figure 3. As expected, the degree of coupling increases as the amplifier gain is increased or the variable external resistance (which is normally larger than the internal resistance of the coupling circuit) is decreased. To avoid the development of potentials among the various electrodes, separate and isolated power supplies are used for the A-D converters and for the input and output sides of the amplifier. The latter are further isolated by an opto-coupler which converts an input potential to light whose intensity is proportional to the input potential. A photocell then converts the light to an output potential that has no direct electrical connection to the input
potential. Thus information is passed by the opto-coupler from the input side to the output side of the coupling amplifier with no direct electrical connection. We are confident that the potential difference sensed by the electrodes accurately reflects the difference in bulk redox potential between the reactors and that this potential is amplified without distortion before being applied to the working electrodes. The usual organic substrates used in the BZ reaction, e.g. malonic acid, cannot be used here because of the production of carbon dioxide. It was found after a long search that a reaction mixture composed of cerium ion, bromate ion, sulfuric acid, and acetylacetone is well behaved in CSTR experiments. Reproducible and stable oscillations can be obtained over a range of reactant concentrations without the production of carbon dioxide bubbles. The chemistry of the BZ reaction with acetylacetone (AA) has been reviewed by Field.3 The materials used were liquid acetylacetone (2,4-pentanedione, 99+%, Aldrich Gold Label), Ce2(S04)3.8H20(99.99%, Aldrich), KBrO,, and 95% H2S04 (both Baker Analyzed Grade). All were used without further purification except for KBr03, which was slurried several timesZowith methyl alcohol to remove KBr and then dried at 140 “ C to remove residual methyl alcohol. Experiments with Ce(IV)(NH,), (so,),and other cerium(1V) salts proved to be erratic, presumably due to problems related to the coordination state of Ce(IV).3,6 Solutions were made with distilled water. Their concentrations were known to three significant figures. There are two feedstreams to each reactor, and a reservoir for each feedstream. Residence times were usually of the order of 5-10 min. The “A” solution contains AA and the cerium salt dissolved in stock H2S04solution, and the “B” solution contains KBr03 dissolved in the stock H2SO4. The concentration of stock H2S04 was 2.73 M. The ranges of reservoir concentrations used were as follows: [AA], 0.03 to 0.07 M; [Ce(IV)] = 0.00016 M; and [KBrOJ, 0.072 to 0.120 M. These concentrations are diluted by a factor of two in the CSTRs. The [AA] has to be kept below 0.07 M. Otherwise a white substance, presumably a brominated derivative of AA, precipitates on the walls of the reactors and the electrodes. This material leads to erratic potential readings and can be removed only by hand cleaning of the surfaces. Figure 4 shows the Pt-electrode response in two uncoupled reactors driven by the same pump but with different reservoir concentrations. The pumping rate is varied until a setting (2-10) is found for which oscillation occurs in both reactors. Initially the pump was set at 10 to fill the reactors, and then pumping was stopped, allowing both reactors to operate in the batch mode. The upper reactor passed through a transient oscillatory regime, much as seen by Koros et aLZ’ The pumping rate was then sequentially changed at the times shown in Figure 4. The behaviors of the two oscillators were quite different. Finally, at a pump setting of 3, sustained oscillationsof quite different forms in each oscillator ~~
(19) Meites, L.; Thomas, H. C. Advanced Analytical Chemistry; McGraw-Hill: New York, 1958; p 76.
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(20) Field, R. J.; Raghavan, N. V.; Brummer, J. G. J . Phys. Chem. 1982, 86, 2443. (21) Koros, E.; Orbln, M.; Nagy, 2s. J . Phys. Chem. 1973, 77, 3122.
Electrically Cnupled Belousov-%habolinskii Oscillators
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The Journal ojPhysica1 Chemisrry. Val. 90. A'o. Y. l ! M
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Figure 6.
Coupling of nearly identical oscillators w i t h f , = 1.02. (a) Identification of the three traces is as in Figure 5 . I:l in-phase entrainment (synchronization) occurred when the coupling resistance was decreased from R = m to R = 5 kR at about 3200 s. The synchronized nature of the entrainment is indicated by the flat relative potential trace. (b) Power spectra of bath oscillators before and after coupling. Note the indentical frequencies and overtones of the two oscillators when they are synchronized. The feedstream concentrations were the same as in Figure 5 , and the residence times were 10.0 and 10.6 mi". coupled oscillator. Synchronization continued in Figure 6 as the coupling resistance was decreased until a t I kR (not shown) oscillation stopped in both reactors with one in a high [Ce(lV)] state and the other in a low [Ce(IV)] state. Figure 7 shows the result of coupling somewhat less similar oscillators. The value of fois I.17:l. As the coupling resistance is decreased through the values I MR, I O kC2, 6 kR, and 4 kR, the periods of the oscillators change from 42.5 and 36.4 s to 44 and 39 s. Entrainment or synchronization is not observed for these coupling strengths, but the power spectra of the slower oscillator are increasingly perturbed by the faster oscillator. The power spectra of the faster oscillator are essentially unaffected. W e refer lo as drifting this situation in which very weakly coupled oscillators perturb each other without loss of their fundamental periodicity or their becoming entrained. At 2 kR I:Iout-of-phase entrainment occurs at a period o f 4 4 s, a value intermediate to those of the uncoupled oscillators. The power spectra of entrained or synchronized oscillators are clean, identical, and show only one fundamental frequency and its harmonics. At I LO (not shown) large-amplitude oscillations cease in one reactor while continuing in the other. The stabilized oscillator does show small oscillations driven by current flow from the other reactor. but the chemical limit cycle is suppressed. Behaviors become more elaborate when less similar oscillators are coupled. Besides having a large difference info, dissimilar oscillators may also differ in amplitude and mean redox potential. Differences in mean potential may be a s large as the amplitude of oscillation. This causes current to flow more often in one direction than the other and occasionaly to flow always in one direction with only differences in magnitude. If we define the oscillator with the higher mean potential as the upper oscillator, coupling affects more strongly the peaks of the upper oscillator and the troughs of the lower oscillator. Not surprisingly, coupling usually affects more strongly the oscillator with the smaller amplitude. The faster oscillator has in general a lower mean potential
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Figure 7. The behavior of two nearly identical oscillatwh i/" : I I 1, s coupling is increased through the sequence R = m, 10 b!l ( I k i l 4 i s ! , and finally 2 kn,at which point I:I entrainment OCCUI'S. Iiuth #tic i < t l . , < potential lraces and power spectra are shown for each c~mciIlawtal milt coupling resistance. The power spectra (especially thew .,I ihc S I ~ ~ U F I . lower amplitude oscillator) become increasingly noisy :I, tl:c ~ w p l m g strength is increased and the two oscillators perturb each ollicr tnioie. The power spectra of both oscillators become very clean i l h e i i I : I ~ n trainment occurs at R = 2 kR. The feedstream concrnti.ilims ncrr a, follows: solution A, [ A A ] = 0.04 M, [Cc,(SO,),] = ( I l i ~ i l l X Z P I , m.1 [HI] =,2.13 M: solution B. [KBrO,] = 0.0958 M and [ I I ' I = L I t M The residence times were 15 and I6 mi". This differcmc r ~ i ~ ,flow lt~ small changes in tubing Characteristics in the two pumpiatg chailni-la
and a larger amplitude. The faster oscillator is less perturbed by coupling than the slower oscillator as the effect of the coupling current on [Ce(lV)]/[Ce(Ill)] compared to the effect of chcniical reaction is less than in a slower oscillator. Under suitable conditions dissimilar oscillators also drift, entrain l : l , and synchronize. Beyond this we have also found 2:I, 3:1, and 3:2 entrainment, quasi-periodic' oscillation, chaos? a kind
Crowley and Field
1912 The Journal of Physical Chemisrry, Vol. 90, No. 9, 1986
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Figure 9. A continuation of the experiment in Figure 8 for coupling resistances of 6 and 5 kR. At 6 kR the power spectrum of the slower oscillator is chaotic. The major peaks and overtones have disappeared, and the base line is several orders of magnitude higher than the power spectra of nanchaatic oscillators. The source of the chaos is apparent in the redox potential trace of the slower oscillator as random periods spent at the peaks of its cycle. At 5 kR 3:l entrainment occurs and the power spectra are again clean with peaks rising about 4 orders of magnitude above the base line.
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Coupling of dissimilar oscillators with = 2.52 through the series of coupling resistances: H = -, 15 kR, I O kR, and 7 kR. The uncoupled redox potential mcillaiionr have magnitudes of 120 and 85 mV, and the difference in m a n redox potential was 60 mV. No entrainment (even at 5:2 with fo = 2 52) was observed for these coupling resistances. The power spectra do became noisier as the two oscillators increasingly perturb each other at lower coupling resistances. Feedstream concentrations were as follows: reactor I (slower oscillator), solution A, [AA] = 0.030 M. [Ce,(SO,),J = 0.00084 M. and [H+] = 2.73 M: solution B. [KBrO,] = 0.072 M and [H'] = 2.73 M: reactor 2 (faster oscillator), solution A. [AA] = 0.070 M, [Ce,(SO,),] = O.oOn84 M, and [H+] = 2.73 M; solution B. [KBrO,] = 0.084 M and [H+] = 2.73 M. Residence times were 10 and 1 I mi". Figure 8.
of bursting,",25 and annihilation of oscillation in one or both reactors. Figure 8 shows the coupling of two oscillators with fo= 2.521 (approximately 5 2 ) . Three behaviors are seen as the coupling resistance is reduced in steps from 1 MR to 5 kR. Power spectra are shown in Figure 8 for coupling resistances of 1 MR (R = m ) as well as 15, IO, and 7 kR. Potential traces are not shown for I5 kCl as the effects of coupling are too subtle to be seen by inspection. However, the power spectrum of the slower oscillator at I5 kR does show contributions from the fundamental frequency of the faster oscillator. By I O kR, and especially by 7 kR, contributions at frequencies both related and unrelated to the fundamental frequency of either oscillator begin to appear in both oscillators. The area of these contributions grows at the expense of the fundamental frequencies. This effect is especially noticeable in the faster oscillator and implies quasi-periodic ( I ) behavior in (24) Jam, R. D.: Vanecek, D. 1.: Field. R. J. 1. Chem. Phys. 1980. 73, 3132. (25) Rinzel, 1.: Troy, W. C. J. Chem. Phys. 1982, 76. 1775.
which the oscillations are wmplex but can still be represented by a power spectrum with a finite number of contributing frequendes. The value off, the ratio of the fundamental frequencies of the two oscillators, in the range of 10 to 7 kR first decreases below 2.5 and then increases above 2.5, but 5:2 entrainment is not observed. At 6 kR the very chaotic power spectra shown in Figure 9 appear. The fundamental peaks and harmonics disappear, and the base line noise moves up several orders of magnitude. The harmonics in a power spearum are necessary to take into accaunt deviation in the shape of a periodic function from the shape of a sine wave. Their absence also indicates a lack of periodicity. The empirical source of the loss of periodicity is apparent in the potential traces shown in Figure 9. Occasional cycles of the slower oscillator are held in the high [Ce(IV)]/[Ce(IIl)] state by current flow from the faster oscillator. Not only is whether or not a particular cycle held up apparently random, but the length of time for which it is held up also is. These two effects together cause an essentially complete loss of periodicity, which can be reproduced by the Oregonator (vide infra) and understood2e32 in terms of the dynamic structure of the Oregonator. When the coupling resistance in (9) is reduced lo 5 kR, randomness disappears as the oscillators entrain 3 : l . The power spectra regain their sharpness and harmonics reappear. This (26) Crowley, M. F. Ph.D. Thesis, University of Montana, 1985. (27) Crowley, M. F.; Field. R. J.. manuscript in preparation. (28) Baker. P. K.: Field, R. J. 1. Phys. Chem. 1985, 89. 118. (29) Smale, S. Bull. Am. Math. Soe. 1967, 73. 747. See also Lcvi, M. Mem. Am. Morh. Soc. 1981,244. (30) Crowley, M. F.: Field, R. J. In Leerum Noresin Biomorhentarics. H. York, in press. Othmer, Ed.: Springer-Verlag: NEW ( 3 1 ) Mankin. Jr; Ldmbda. P.; Iludbon. J. L C'hmmtrol Reorrron b n g i n e e r i n g l l o l o n . We,. J.. Georgakb. C.. Eds.. American Chemical Se c t m Washinetan. DC. 19x2. ACS Svmn Sn No. 196. Hudson. J L : Han. M..Marcnka,D. 3. Chem. Phys. 1919,'71, 1601. (32) ROUX.J.-C.: Simoyi. R H : Swinncy,H. Physic0 D 1983,8D, 257.
The Journal of Physical Chemistry. Vol YO, No. 9, 1 Y86
Electrically Coupled Belousov-Zhabotinskii Oscillators
1'
1
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/
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8 ONE OSCILLATOR STOPPED V BOTH OSCILLRTORS STOPPED @ BOTH 0 RND V RRE STRBLE Figure 10. A schematic diagram of coupling trends observed experimentally in a coupling strength ( p ) vs. fo plane. The ordinate is the reciprocal of the variable external resistance shown in Figure 3 in kQ. T h e gain of the amplifier was held constant.
fundamental trend of drifting, quasi-periodicity, chaos, and finally entrained behavior or annihilation of oscillation as the coupling is increased is a dominant feature of our experiments. It is reproduced (vide infra) by coupled Oregonator models. At even higher coupling strengths oscillation is sometimes annihilated in one or both reactors. A bursting phenomena24is sometimes associated with annihilation in which periods of oscillation in both reactors are interspersed with periods of quiescence. Bursts of oscillation do not necessarily begin at the same time in both reactors. We believe that this phenomenon is at least partially related to the inhibition of oscillation by accumulation of a product formed during oscillation. Such a model has worked well to explain burstiiig in other CS'TR e x p e r i m e ~ i t sand , ~ ~ the production of such an inhibiting species during oscillation in the A A system has been inferred by Heilweil and E p ~ t e i n . ~ ' An approximate diagram of observed coupling trends a s h and the coupling strength are varied is shown in Figure 10. Bands of drifting, synchronization, I : 1, 2: 1, and 3: 1 entrainment, chaos, annihilation of oscillations, and bursting are observed. In general, oscillators tend to entrain at values offhigher than theirf,. Figure 10 is rough but shows the trends and possibilities in electrically coupled oscillator experiments. Considerably more work is required to flesh out its detail.
Simulation of Observed Coupling Behavior The results of coupled BZ oscillator experiments are useful mainly to the extent that they can be understood in terms of the basic chemistry and especially the dynamic structure of the BZ reaction. We find that the Oregonator modeli5not only is able to simulate our experimental results but is also susceptible to methods of qualitative mathematical analysis that show the fundamental dynamic structure leading to the observed phenomena. Mathematical results will appear e l s e ~ h e r e . We ~ ~ will *~ present here only the results of numerical simulation work. (33) Heilweil, E. J.; Epstein, I. R. J . Phys. Chem. 1979, 83, 1359.
1913
There has been some criticism of the Oregonator model recently. Noszticzius et al.34objected to its basic form and to the idea of Br- control of the oscillations, which along with b i ~ t a b i l i t yand ~~ the completion of a negative feedback loop3his its mc st basic feature. These criticisms were based upon their inability to rationalize as Br- controlled the behavior of several modified and perturbed BZ oscillators. However, all of these systems have now been rationalized with no change in the basic structure of the Oregonator. The general problem has been treated by no ye^.^' Particular experiments have been interpreted by Field,2 Field and , ~ ~ Koros et al.39 The idea of Boyd,6 Ruoff and S c h w i t t e r ~and Br- control is now accepted by Noszticzius himself.40 As much as 90% of the Br- produced indirectly by Ce(IV) comes from the reactions of organic intermediates with HOBr and not from bromine-containing organic specie^.^' This does not, however, affect the basic dynamic structure of the Oregonator. The net result of this controversy has been to emphasize the validity and robustness of the basic Oregonator dynamic framework and the importance of HOBr and the hydrolysis of Br2 within that dynamic structure. Questions concerning appropriate parameter values for the Oregonator have been raised by T y ~ o n . ~These ~ have been discussed by B a ~ E l i : ~Schwitters," and Field and Boyd.16 While there is still considerable uncertainty about proper identification of the Oregonator parameters with particular elementary chemical reactions involved in the BZ mechanism, especially concerning the effects of r e v e r ~ i b i l i t y , a~ parametrization ~%~~-~~ introduced by Tyson4' gives good agreement with experiment and is used here. It is our experience that the qualitative behavior of the Oregonator is remarkably insensitive to the parameter values used if certain ratio^'^,^^ of individual rate constant values are maintained. We believe that little doubt can remain concerning the usefulness of the Oregonator as an excellent qualitative and, properly parametrized, a surprisingly good quantitative model of the BZ reaction. The basic form of the Oregonator is given by eq Ml-M5. In
A+Y-X+P
x +Y A +X
+P 2X + Z
-+
---*
P
X+X-A+P
z 'fy
(MI)
(M3)
(M4)
(M5)
eq Ml-M5, X HBrO,, Y Br-, Z 2Ce(IV), P HOBr, and A E Br03-. The parameterfdescribes the net stoichiometry of Br- regeneration from HOBr and Ce(1V). As with any simple model of a complex system, some care and thought must be used when the Oregonator is applied a particular set of experittiental data. Most importantly, it must be ascertained that the approximations involved with the form of the model used __ (34) Nosztic7ius, Z.: Farkas, H.;Schelly, 2. A. .I. C h e m Phys. 1984, 80, 6062. (35) DeKepper, P.; Boissanade, J. In Oscillations and Traveling Waues in Chemical Sysrems, Field, R. J., Burger, M., Eds.; Wiley-Interscience: New York, 1985; Chapter 7. (36) Tyson, J. J. J . Chem. Phys. 1975, 62, 1010. (37) Noyes, R. M. J . Chem. Phj.9. 1984, 80, 6071. (38) Ruoff, P.; Schwitters, B. J . Phys. Chem. 1984, 88, 6424. (39) Koros, E.; Varga, M.; Gydrgyi, L. J . Phys. Chem. 1984, 88, 4116. Varga, M.; GyBrgyi, L.; Koros, E. .I. Phys. G e m . 1985, 89, 1019. (40) Noszticzius, Z.; Gisplr, V.; Forsterling, H.-D. J . Am. Chem. SOC. 1985, 107, 2314. (41) Varga, M.; Gydrgyi, L.: Koros, E. J . Am. Chem. SOC.1985, 207,4780. (42) Tyson, J. J. In Oscillations and Traveling Waces in Chemical Systems, Field, R. J., Burger, M., Eds.; Wiley-Interscience: New York, 1985; Chapter 4. (43) Bar-Eli, K.; Ronkin, J. J . Phys. Chem. 1984, 88, 2844. ~ ~ ~ ~ (44) Schwitters, B., University of Oslo, private communication. (45) Field, R. J. J . Chem. Phys. 1975, 63, 2289. (46) Showalter, K.; Noyes, R. M.; Bar-Eli, K. J . Chem. Phys. 1978, 69, 2514. (47) Tyson, J. J. J . Phys. Chem. 1982, 86, 3006. -_____I_____.__
E914
The Journal
of Physical Chemistry, Vol. 90, No. 9, 1986
IABLE I: Scaling and Parameters Used in Eq 5 _______parameter definition
Crowley and Field
amroximate value [”)
........ . . ..
,
. . . .......... ...............................
are appropriate for the particular experiments under consideration. r or example, conservation of metal-ion catalyst (Ce(1V) Ce( 1 1 1 ) ) is often not taken into account in the Oregonator. lt is a:wm~.d instead that only a relatively small fraction of the Ce(II1) i,; ever luidized to Ce(1V). Thus [Ce(III)] is assumed to be cimtant. It should be apparent, though, that this approach cannot work liere The primary effect of electrical coupling is on [Ce(lV!J/[Ce(III)], and we find that the Oregonator is unable to :,:produce some important features of our experiments unless c wrrvatioii of metal-ion catalyst is enforced. In general, ’ ~!:in1.i~:~tiw agreement between the Oregonator and experiment iii!pi i b . c - 4 . l with this c o ~ i s t r a i n t . ~ ~ . ~ ~ G c I Y C liere a scaling of the Oregonator equations due to ,hich requires that c‘ = [Ce(III)] [Ce(IV)] = total c-.lncentration. WP assume that the effect of electrical
+
+
d
‘ (iciul!liiig) dZ’(coup1ing) . .=._I_.__._____..._dt
.I.
- P In
{
(C - Z ) Z ’ Z ( C - Z?
}
(4)
v i*i:lblesrelated to reactor 1 are (X,Y,Z,C) and those 2 are (X’,Y’,Z’,C?. The quantity p is a coupling i c r :lrxl i s equal to RT/.RVY, where 3 ’2 is the resistance $ 4 i!-c!A < ~ ~ i i pcircuit, ~ i ~ gVis the volume of the reactors (assumed 1 0 t)p v i u , j i ) , : ~ n d9 is the Faraday. The coupling parameter is 7 m i ) at “iand increases as Yf decreases. When coupling is i ~ ! 4 i i ~ l r Z~ :~\ I C the scaled differential equations describing the !,.i i x ) it*:lrtqr
~3
*IO
. %
I’,? = u ( l - 2z
p,l’ = bgz
u) - J(.Y
-J(X
-
y)
+ y)
( ‘Ti,’/ 7 J i ’ =
2.u’(l - 2 2 3 - b’z’- p(To’/C’) In { ( l - z)z’/z(l - z ” ) ] The scaling factors used in eq 5 are given in Table I. The pai;triieter values used are those of Tyson.” The scaled variables aii: defined by x = X / X o ,y = Y / Y o ,z = Z/Z,, and T = TITo. 1 he parameter A E Br0,- enters into the time scaling, and the factor ( T d / To) is used so that the scaled time is the same in both reactors when [BrO,-] is not. The parameters C, b, g, and A a r e related to the concentrations of the principal reactants in a particular reactor. Oscillations of quite varied period and amplitude can be obtained experimentally by varying [AA], which appears only in b. Thus b and b’ were most often varied in experiments and simulations and often had different values in the t a o reactors. The other parameters were kept at their experiiwntal values and were usually the same in both reactors. The values of b and h’were adjusted by relatively small amounts to obtain agreement between the experimental and simulated periods (4 the uncoupled oscillators. Agreement between calculated and experimental values of [Cc(IV)]/[Ce(III)Jis good. In both experiments and simulations, t h e effect of increasing [AA] is to increase the frequency and to l!,,:r-ise the mean potential of the oscillations. Also in agreement i! 11 ruiwicrient. sirtiulatioiis show the effect of coupling mainly 8
Figure 11. Results o f numerical simulations of electrically coupled BZ oscillators using eq 5 . For the upper oscillator: ( 6 k , + 4 k l o ) = 0.03. A = 0.03. C = 0.00005, g = 0.28, and [H+] = 1 .OO. For the lower oscillator ( 6 k , + 4 k l o ) = 0.10, A = 0.30, c‘ = 0.0001, g = 0.30, arid [H+] = 1.00. Other parameter values are as in Table I . The calculated behaviors at various values of p are (a) p = 0, uricoupled; (b) li = 2 X IO-’, drifting: (c) p = 2.4 X 10 quasiperiodicity: (d) p = 2 . 6 X IO-’, chaos; (e) p = 2.7 X I O 7 : 5 entminnient: (f) p = 2.8 X 10 ‘,chaos; (g) p = 3.0 X l W R . 3:2 entrainment: (11) iJ = 4 . 5 x 10 4:3 e i i t r ~ ~ i n n i e ~ ~ t .
’.
’,
’,
The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 1915
Electrically Coupled Belousov-Zhabotinskii Oscillators Lo 0
-00 e3
v
v
v
I
/
v
,
20
Figure 12. State diagram in p (coupling parameter) and ratio of uncoupled frequencies (jb) calculated from eq 5. Parameter values for both oscillators were as follows: A = 0.01. C = 0.001, g = 0.35, and [ H'] = 1. One oscillator, the faster, was kept a t (6kg 4kIo) = 0.05 and b = 0.10. The period of the other oscillator was changed by varying (6k9 4klo) between 0.017 and 0.05. This approximates changing [AA] experimentally. The values of the other parameters were as in Figure 11. The qualitative agreement of this diagram with the experimental diagram in Figure 10 is quite good. The symbols used for the various phenomena are the same as in Figure 10.
+
+
at the peaks of the slower oscillator, which are lengthened by Ce(1V) reduction resulting from current flow from the other oscillator. In both experiment and simulation, the troughs of the faster oscillator may also be affected at high coupling strengths. This high degree of agreement between experiments and simulations does not occur unless the conservation of metal-ion catalyst is enforced. With this constraint essentially all of our experiments can be understood at least qualitatively. It is useful to consider the qualitative features of the coupling. The faster oscillator has a lower mean potential than the slower oscillator. Thus the maximum difference in potential between them occurs when the slower oscillator is at a peak and the faster oscillator at a trough. The coupling current is at a maximum here and at a minimum at the inverse point. At the maximum current flow Ce(1V) is rapidly removed from the slower oscillator, and this makes it difficult for it to get past its peak. Much Ce(1V) that normally would react with organic material to produce Brreacts instead at the working electrode yielding no Br-. Process B3*4in that reactor becomes balanced against the electrode reaction. It is not inhibited by Br- produced indirectly from Ce(1V). This balance continues until the faster oscillator (which is less affected by the current flow) gets through its trough. Of course, if the coupling is strong enough, oscillations may be annihilated with one reactor in a high Ce(1V) state and the other in a high Ce(II1) state. Annihilation rather than 1:l entrainment or synchronization is often observed in our experiments at high coupling strengths. It is not reproduced in the simulations unless metal-ion
conservation is enforced. Plots of log ((C- Z)Z')/(Z(C'- Z')) and log ( Z ' / Z ) vs. time in Oregonator models with and without metal-ion conservation demonstrate the source of this e f f e ~ t . ~ ~ , ~ ' , ~ ~ The calculated coupling currents are higher when depletion of Ce(II1) is considered, especially if more than 10% of the Ce(II1) is oxidized to Ce(1V). The model reproduces well the variety and relationships among the coupling phenomena observed experimentally. Figure 1 1 shows a sequence of simulations with increasing p that shows the same trend as normally seen experimentally. The value offo was 1.38. As p is increased from zero, the following phenomena appear in sequence: uncoupled, drifting, quasiperiodicity, chaos, 7:5 entrainment, chaos, 3:2 entrainment, 4:3 entrainment, and finally (not shown) suppression of oscillation in one or both reactors. This is the same trend as seen in our experiments. Figure 12 shows a diagram of simulated coupling trends calculated a s h and p are varied and equivalent to the experimental diagram shown in Figure 10. It is difficult to compare Figures 10 and 12 in detail because the quantitative effect of the coupling amplifier on the virtual coupling resistance is not known. We point out, though, that in Figures 10 and 12 fundamentally the same phenomena appear in similar orders asfo and the coupling strength are varied. In both our experiments and calculations, the coupling strength increases by about a factor of ten between when the effects of coupling are first apparent and when the coupling leads to the suppression of oscillations in one or both reactors. Chaos in this chemical system can be qualitatively understood26,27in terms of a phase-plane analysis2*of two electrically coupled Oregonator~.'~It can also be shown that this model probably contains a Smale's h o r s e s h ~ efor ~ ~parameter .~~ values closely related to experiment. This is very strong evidence that the chaos observed here results from a fundamental property of the governing dynamic equations rather than experimental noise. Despite the excellent work that has been done with chaos in the BZ r e a ~ t i o n , ~we ' , ~believe ~ that this is the first case in which the phenomenon can be reproduced and understood in terms of the properties of a model closely related to the experimental system. Thus it is very likely that the present aperiodicity is not a result of environmental perturbation or poorly understood details of the BZ chemistry. These poorly understood chemical details are mainly associated with the bromide ion regeneration reactions which are intimately linked with the organic chemistry of process C of the FKN m e ~ h a n i s m .It~ is very difficult to control process C experimentally or to model its details.
Conclusion We have demonstrated the feasibility of electrical-coupling experiments in the BZ reaction. The results of such experiments are interesting and instructive. The Oregonator model supplies a good framework within which our experiments can be-interpreted in such a way that the dynamics of the coupling and its basic features can be understood in terms of the chemistry of the BZ reaction.
Acknowledgment. This work was partially supported by the National Science Foundation under Grant CHE80-23755. We thank the University of Montana Computer Center for computing facilities and advice as well as the Deutsche Forschungsgemeinshaft for a visiting professorship to R.J.F. at Wurzburg University. Registry No. Ce, 7440-45- 1 ; Pt, 7440-06-4; acetylacetone, 123-54-6.
