J. Phys. Chem. 1982, 86, 3907-3910
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Complex Periodic and Aperiodic Oscillation in the Chlorite-Thiosulfate Reaction' Mlklds OrbLn2 and Irvlng R. Epsteln' Depettment of Chmlsby, Brandels University, Weltham, Massachusetts 02254 (Recelved: May 7, 1982; In Flnal Form: August 6, 1982)
In a stirred tank reactor at 25 "C,the reaction between chlorite and thiosulfate ions exhibits a rich variety of oscillatory phenomena. On decreasing the flow rate at an appropriate fixed input composition, one observes periodic large amplitude oscillations (L), then a range of flow rates in which each period contains one large and one small amplitude peak (L + S), followed by L + 2S, L + 35,...,L + 16s and fiially pure small amplitude oscillations (S or L + 4).Between each pair of periodic oscillatory ranges (L + nS,L + (n + 1)s) there exists a narrow range of flow rates in which an apparently random mixture of L + nS and L + (n + 1)s oscillations is found. The entire sequence may also be observed by varying the temperature at fixed composition and flow rate. The implications of these results for the possible existence of chemical chaos in this system are discussed.
Introduction While turbulence has long been a subject of considerable study in the physics of fluids, interest in the analogous chemical phenomenon of sustained, deterministic, aperiodic time dependence of species concentrations is much more recent, dating from a suggestion by Ruelle3 in 1973 that oscillatory chemical reactions might, under appropriate circumstances, exhibit "chemical chaos". Since that conjecture, several investigators, most notably R o s ~ l e r , ~ have constructed sets of "kinetic-type" differential equations which display chaotic behavior, though these equations do not describe the kinetics of any known chemical system. Tyson5 has examined a simplified model of the Belousov-Zhabotinskii (BZ) reaction6 and argued on qualitative grounds that it should be capable of generating chaos. Perhaps the most direct theoretical support for the existence of chaos in a real chemical system is Turner's numerical calculation' on a four-variable Oregonator model8 of the BZ reaction. He obtains, as a function of residence time of the reactants in a flow reactor (CSTR), a series of alternating periodic and chaotic regimes. Such a sequence of periodic and aperiodic states in the BZ reaction in a CSTR has been found in experiments by Hudson et Vidal et al.,l0and Turner et al.' One sees in general different types of complex periodic behavior, e.g., n large and m small amplitude peaks (nL + mS) per period. As the flow rate of species into the reactor is varied, the system shifts from one type of periodic behavior to another, with an intervening range of flow rates in which an apparently random mixture of the two types of behavior is observed. Maselkoll has also reported aperiodic behavior, though of a more complex nature, in a BZ system in a CSTR. (1)Part 13 in the series Systematic Design of Chemical Oscillators. Part 12: Orbin, M.; Epstein, I. R. J. Am. Chem. SOC., in press. (2)Permanent address: Institute of Inorganic and Analytical Chemistry, L. EBtv6s University, H-1443Budapest, Hungary. (3)Ruelle, D. Trans. N.Y. Acad. Sci. 1971,35,66-71. (4)RBssler, 0.E. Z. Naturforsch. A 1976,3I,259-64, 1664-70. ( 5 ) Tyson, J. J. J. Math. Biol. 1978,5 , 351-6. (6)Belousov, B. P. Ref. Radiats. Med. 1959,1958,145-7;Zhabotinsky, A. M. Biofizika, 1964,9,306-11. (7)Turner, J. S.;Roux, J.-C.;McCormick, W. D.; Swinney, H. L. Phys. Lett. A 1981,85,9-12. (8)Field, R. J. J. Chem. Phys. 1975,63,2289-96. (9)(a) Schmitz, R. A.; Graziani, K. R.; Hudson, J. L. J. Chem. Phys. 1977,67,304&4. (b) Hudson, J. L.; Hart, M.; Marinko, D. J. Chem. Phys. 1979,71,1601-6. (c) Hudson, J. L.; Mankin, J. C. Zbid. 1981,74,6171-7. (10)(a) Vidal, C.; Roux, J. C.; Rossi, A.; Bachelart, S. C. R. Acad. Sci. Paris, Ser. C 1979,289,73-6. (b)Vidal, C.; Roux, J. C.; Bachelart, S.; Rossi, A. Ann. N.Y. Acad. Sci. 1980,357,377-96. (11)Maselko, J. Chem. Phys. 1980,51,473-80. 0022-3654/82/ 2086-3907$01.25/0
In spite of these results, some controversy exists as to whether the calculated and observed chaos is inherent in the dynamics of the BZ reaction or whether it is artifactual, due, for example, to uncontrolled numerical and experimental fluctuations. In particular, Ganapathisubramanian and Noyes12 have recently carried out calculations on a seven-variable Oregonator model13which suggest that, if chaos is possible in this system, its range of existence would be far too small to be observed experimentally in the absence of external perturbations. An earlier calculation by Showalter et al.13 gave qualitative agreement with the complex periodic behavior found experimentallyg8 but produced chaos only when the error parameter in the numerical integration was made so large as to produce spurious results. Neither of these calculations, however, was performed over a range of values for the rate constants and input concentrations. Thus, the question of whether genuine chaos is inherent in the dynamics of the BZ reaction, and by implication in oscillatory chemical systems in general, is still an open one. In order to probe more thoroughly the nature of chemical chaos, it is highly desirable to have additional systems which display the phenomenon. We report here an experimental study of complex oscillations in a totally distinct system, the chlorite-thiosulfate r e a ~ t i 0 n . l ~This system exhibits a rich variety of oscillatory modes, similar in nature to those observed in the BZ system. We have examined this behavior over a somewhat wider range of constraint parameters than has been investigated in studies of the BZ reaction, and our results shed some light on the factors which may affect the observation of chemical chaos. Experimental Section The apparatus, which has been described elsewhere,15 consists of a Pyrex continuous stirred tank reactor of 20.55 mL volume, with no air space between the solution surface and the reactor cap. The reactor is fed by a Sage 975 peristaltic pump with three polyethylene tubes for input and a single tube for the outflow. The sodium chlorite (freshly prepared for each experiment from a stock solution stabilized by addition of M NaOH) and sodium thiosulfate solutions occupy two of the input feeds, while the third contains an appropriate acetate or phthalate (12)Ganapathisubramanian, N.; Noyes, R. M. J. Chem. Phys. 1982, 76,1770-4. (13)Showalter, K.; Noyes, R. M.; Bar-Eli, K. J. Chem. Phys. 1978,69, 2514-24. (14)Orbin, M.; De Kepper, P.; Epstein, I. R. J.Phys. Chem. 1982,86, 431-3. (15)De Kepper, P.; Epstein, I. R. J. Am. Chem. Sot. 1981,103,6121-7.
0 1982 American Chemical Society
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The Journal of Physical Chemistry. Vol. 86, No. 20, 1982
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Flgure 1. Complex perlodic osclilatlons with [CiO,-], = 5 X lo4 M, [S2O;-], = 3 X lo4 M, pH 4, T = 25.0 OC: (a) L, T = 5.9 min; (b) L S, T = 9.5 min; (c) L 2S, T = 10.8 min; (d) L 3S,T = 13.5 min; (e) L 4S, T = 15.8 min; (f) L 6S, T = 20.6 min; (a) L 12S, T = 26.3 min; (h) S, T = 47.3 min.
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buffer. The reactor is stirred at 1000 rpm by a magnetic stirrer. A constant temperature water bath maintains the temperature to f O . l "C. The reaction is monitored by a platinum electrode with a mercury-mercurous sulfate electrode as reference. In a typical experiment, the input concentrations and temperature were held fixed and the flow rate was gradually varied from zero to the pump's maximum value allowing sufficient time for the system to reach a stable stationary or oscillatory state at each flow rate. The flow rate was then decreased back to zero in the same manner in order to search for hysteresk phenomena. We estimate the reproducibility of the flow rate to be f0.5%. Results In a recent paper14 reporting the discovery of the chloritethiosulfate oscillator, we noted the existence, for certain input concentrations, of complex periodic oscillations, with each period containing both large and small amplitude peaks. The more thorough study of this behavior that we are now reporting reveals a startling richness in the oscillatory modes of this system. At an appropriate set of fixed input concentrations and temperature, increasing the residence time r (reciprocal of the flow rate ko)brings the system successively from a steady state to a region of simple, broad, large amplitude oscillations (L), to a region in which each period of oscillation contains one large amplitude peak and one narrow, small amplitude peak (L + S),to regions characterized by (L + 251, (L+ 3S), ..., until these regions of complex oscillationsgive way to simple, small amplitude oscillations. A typical sequence of these periodic oscillation regimes is shown in Figure 1. The number of S peaks accompanying the single L peak in each cycle varies with the input con-
Flgure 2. Aperiodic osciilatlons with input concentrations and temperatures as in Figure 1: (a) (L, L S) T = 6.8 min; (b) (L S, L 2S), T = 10.5 min; (c) (L 8S, L 4- 9S), T = 23.6 min; (d) [S202-],, = 4 X lo4 M and other constraints unchanged: (L 3S, L 4S, L 5S), T = 11.42 min.
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centrations. The range of ko in which a given mode of oscillation occurs is quite reproducible, and no hysteresis has been observed. We have described above the periodic oscillatory behavior of this system. However, aperiodic behavior is also observed. In a narrow, but reproducible range of flow rates between each neighboring pair (L + nS,L + (n + 1)s)of complex periodic modes, we find a range of k, in which apparently random mixtures of L + nS and L (n + 1)s groupings appears. Some examples, under the same conditions as in Figure 1, are shown in Figure 2. This alternation of periodic and aperiodic regions as a function of ko is analogous to the behavior observed in experiments on the BZ system*" and predicted by Turner's calculation.' Both the periodic and aperiodic oscillations are stable for at least 12 h, so long as the temperature remains constant. All modes observed are also stable to even relatively large perturbations, such as those produced by temporarily removing one of the input tubes from its reservoir tank. Studies of chemical chaos in the BZ r e a c t i ~ n ' . ~have J~ generally been performed a t a single fixed input composition with the flow rate as the only variable constraint. Hudson et al.9breport that lowering the temperature from 25 to 20 OC a t fixed composition displaces the observed behavior to lower flow rates. We have investigated the effects of varying both input composition and temperature as well as flow rate on complex oscillations in the chlorite-thiosulfate system. These results are of significance in attempting to understand both the general dynamics of the system and the specific question of whether chaos is inherent in those dynamics. In Figure 3, we show a phase diagram obtained by varying the input thiosulfate concentration as well as the flow rate. We observe that the complex oscillations occur
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The Journal of phvsical Chemlstty, Vol. 86, No. 20, 1982 3900
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TABLE I: Modes of Oscillation Observed as a Function of Temperaturea
simple osc
z 5
simple oscillation 0
simple oscillation 0
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rate x io3 (s-')
Flgure 3. A section of the phase diagram of the chlorite-thiosulfate system in the [S,032-]o-k, plane with [CQ-1, = 5 X lo4 M, pH 4, T = 25.0 OC. "SIMPLE"and "simple" denote pure L and pure S osclatbns,respecthre(y. Verticei segments (1) show flow rate at which periodic L nS pattern appears with the ratio above the segments; m, chaotic reglon with mixed pattern of L n S , L (n 1)s. (L (n 2)s); v,b w potenUai steady state (SS I); A, Mgh potential steady state (SS 11); 0 , blstablllty (SS I/SS 11).
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[ClO;],
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Flgure 4. A section of the phase diagram in the [S,O~-]o-[CiO,-], plane. The two solkl lineg enckse the osclliatocy region. The dashed line separates regions of simple and complex oaciiiatbn. Fixed constraints: k , = 2 X lo4 s-', pH 4, T = 25.0 OC.
T,"C
mode of oscillationb
22.0 22.8 24 0 24.4 25.0 25.2 25.4 25.6 25.8 26.1 26.6
L L+S L+S L+S,L+2S L i.2s L + 2s L i2 s , L + 3s L + 3s L + 35, L + 4 s L + 4s, L + 5 s S
a [cia;], = 5 x 10-4M , [S,O,*-I, = 4 x 10-4M, PH 4, k o = 2.05 X s-', A single letter (L or S) signifies simple periodic oscillation; L + nS, L + (n + 1)s signifies aperiodic oscillation in which the two different types of cycles are mixed in a random fashion.
tion in this system at pH 4 and a flow rate of ko = 1.91 X lo9 s-l. Such limits of oscillation have not yet been established in the BZ system, though Turner et aL7suggest that the progression of modes observed in that system proceeds in different directions as a function of k, a t the opposite boundaries of the complex oscillation region. At pH 2 or 3, no complex oscillations were observed a t any composition tried. At pH 5, the region of complex oscillation is shifted to higher chlorite and thiosulfate concentrations. For example, at 25 "C and pH 5, complex oscillation is observed at [C1Opl0= 0.002 M, [SZO,2-], = 0.0016 M, and ko = 2 X s-l. The parameters for which the various modes of behavior appear are reproducible and all transitions occur without any observable hysteresis. In several of our early experiments an apparent lack of reproducibility of the transition point between two modes was traced to a small drift in the temperature of the system. When the temperature was controlled more precisely, the irreproducibility vanished. The above observation together with Hudson et al.'s reportgb of a significant temperature dependence of the complex oscillation in the BZ system led us to carry out a study of the effect of varying the temperature in the present system. In Table I we summarize the results of changing the temperature while the flow rate, pH, and input concentrations are held fixed. The data, which represent the most thorough study to date of the temperature dependence of complex oscillation, are in good qualitative agreement with the sparser observations of Hudson et al., though the chlorite-thiosulfate system appears considerably more temperature sensitive than the BZ reaction. The fact that changing the temperature by only 4 "C causes the system to go through the entire sequence of states accessible by varying the flow rate from zero to infinity is of particular interest. In both the temperature and the flow rate variation experiments the ranges of aperiodic behavior were considerably narrower than those of the complex periodic oscillations.
only in a relatively narrow range of [S2032-],, though outaide this range either of two stationary states, bistability, or either of two simple oscillatory states may be found. This phase diagram is among the richest yet Discussion studied in dynamic systems, rivalled only by the chlorAside from a rather limited set of results reported by ite-iodide-iodate-arsenite systemz6and some of the gasOlsen and Degnl8 on the horseradish peroxidase system, phase CSTR reactions reported by Gray,17though neither the present work represents the only study of aperiodic of those systems has been shown to exhibit aperiodic beoscillations in a homogeneous non-BZ system. In terms havior. systematic investigation of the effects of varying the Figure 4 is a phase diagram in the [C102-]~-[S~O~-l~of constraints on the oscillatory behavior, it is probably fair plane showing the regions of simple and complex oscillato say that the chlorite-thiosulfate system now ranks as the most thoroughly studied "chaotic" reaction to date. (16) O r b e , M.;Dateo, C. E.; De Kepper, P.; Epstain, I. R. J . Am. Chem. SOC., in press. (17)Gray, P.Ber. Bunsenges. Phys. Chem. 1980,84, 309-15.
(18) Olsen, L. F.;Degn, H.Nature (London) 1977,267, 177-8.
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The Journal of Physical Chemistry, Vol. 86, No. 20, 1982
Experimentally, the C102--S20~-reaction seems to provide an almost ideal system for studying aperiodic phenomena. The system does not appear to be sensitive to any trace impurity in the way that the BZ reaction depends upon the concentration of the bromide which is always present in the reactant b r ~ m a t e .The ~ large and small amplitude oscillations in the present system differ sufficiently in waveform and amplitude that no ambiguity arises as to their identification. The period of oscillation is significantly longer than in the BZ reaction, but not so long as to require inconveniently long experimental runs. The slower variation makes the reaction easier to follow and reduces the effects of any problems due to the response time of the electrode. Also, a rich variety of complex modes may be observed ranging from L + S up to L + 16s. We now ask whether the results presented afford a clear resolution of the question whether chaos is inherent in the dynamics of certain oscillatory chemical reactions or whether such aperiodic behavior is in some sense an experimental artifact. Unfortunately, a significant measure of ambiguity remains. The sequence of complex periodic and aperiodic states found strongly resembles that obtained in e x p e r i m e n t ~ ~and ~ ~ calculations7 J~ on the BZ system, which would seem to support the interpretation of chaos as a general phenomenon in some if not all oscillatory reaction dynamics. On the other hand, a number of our results can also be interpreted to support the notion that the aperiodic behavior observed is generated by uncontrolled (and perhaps uncontrollable) fluctuations in the values of the experimental constraints. The widths of the chaotic regions when either flow rate or temperature is varied are always much narrower than the widths of the periodic regions. When ko is varied, chaotic regions typically extend over a range of 0 . 5 4 % of the total flow rate, while as a function of temperature the broadest chaotic regions are only 0.1-0.2 "C wide. These widths are slightly larger than, but close to, the limits of our experimental accuracy. While in a given experiment, no hysteresis is observed and the chaotic regions are reproducible, we found in some early trials that when older, more flexible tubing was used in the peristaltic pump, the widths of the chaotic regions were greater than in experiments with newer, more rigid tubing. In order to test the effects of small fluctuations in flow rate of the sort that might be more likely with the older tubing, the following experiment was performed. A stable L 2s pattern was established. The system was then perturbed by removing one of the three input tubes from its reservoir for about 3 s in each 2 min. The reservoir used was alternated so that the average composition remained constant. The total fluctuation thus introduced was about 0.5% of the total reactor volume. This experiment was continued for a period of 2 h, during which time several L + 3s cycles were observed at apparently random intervals. Each L + nS type of periodic oscillation represents a different stable "phase" in an appropriate region of the constraint space. It is then clear that as a constraint such as ko is varied the system must pass through a critical point at which the regions of L + nS and L + ( n + 1)sstability meet. Exactly at this point (and in some very small region around it) fluctuations in the system will drive it randomly
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between the two phases, even if the experimenter has the best possible control over the constraints. Such behavior, however, is not the deterministic chaos which has been suggested to occur in the BZ reaction and more abstract models. In principle, it should be straightforward to perform an experiment to distinguish between the random perturbation between two neighboring modes described above and true deterministic chaos. One need only repeat the same experiment with the same initial conditions and observe whether or not the sequence of L + nS and L + (n + 1)s cycles in the different trials is the same, as it should be for true chaos. However, one of the characteristic features of deterministic chaos is its extreme sensitivity to very small changes in the initial conditions. Thus, even the most carefully controlled repetition of experiments would be unlikely to lead to reproducible sequences of oscillations. One of the strongest arguments for true chaos in the BZ reaction is the existence of a complex periodic-aperiodic sequence as a function of flow rate in the calculations of Turner et aL7 However, Showalter et al.13 have pointed out the possibility that numerical errors can artificially induce chaos in a simulation of that sort, and no study of the effects of varying the error parameter are reported in A model which produced similar Turner's ~alculation.~ aperiodic phenomena (independent of the error parameter) in the present system would be persuasive evidence in favor of the genuineness and generality of chemical chaos. Unfortunately, however, not even a tentative mechanistic proposal has been put forth to account for oscillations in the chlorite-thiosulfate reaction. While the phenomena reported here correspond to the complex oscillations seen in the BZ system, certain modes of behavior found in the latter reaction have not yet been observed in the chloritethiosulfate system. These include periodic behavior in which each cycle includes more than one large amplitude oscillation (nL + mS) and sequences of modes in which the number of small amplitude oscillations per cycle either increasesgJOor decreases7 with ko depending upon the values of [C1O~l0 and [Sz032-l,. We have not attempted to analyze our data in terms of phase portraits, power spectra, or Poincar6 sections. While such methods can provide useful insights into chaotic systems and may prove enlightening when the concentration vs. time plots are difficult to interpret, they cannot (or at least should not) produce dynamic structure which is not already present in those latter plots. As noted above, the nature of the oscillations in this system renders the concentration-time plots particularly transparent. While the present study has not yielded a definitive answer as to the true nature of chemical chaos, it does provide another experimental example and a broad range of data on that system. The remaining ambiguity only gives further emphasis to the need for mechanistic studies on this and related systems. Acknowledgment. We thank Professor Harry Swinney for enlightening discussions on the nature of chaos, Professor Kenneth Kustin for a critical reading of the manuscript, and Mohamed Alamgir for assistance with several of the experiments. This work was supported by Grant CHE 7905911 from the National Science Foundation.
