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J. Phys. Chem. 1991, 95, 2130-2138
FEATURE ARTICLE Chemical Oscillatlons, Chaos, and Fluctuations in Flow Reactors F. W. Schneider' and A. F. Miinster Institute of Physical Chemistry, University of Wiirzburg, Marcusstrasse 911 1 , 0-8700Wiirzburg, FRG (Received: August 9, 1990; In Final Form: October 30, 1990)
We present experiments using the Belousov-Zhabotinsky chemical oscillator in a compact CSTR (continuous flow stirred tank reactor) which (i) verify earlier observations of deterministic chaos by Swinney and co-workers at low flow rates using purified malonic acid, (ii) show Farey-ordered oscillations in the mixed mode region at high flow rates whose maximal Lyapounov exponents are positive due to statistical noise rather than deterministic chaos, and (iii) demonstrate the excitabilityof a steady state close to a Hopf bifurcation. The latter amplifies superimposed Gaussian distributed fluctuations on the flow rate. In experiments classified as "fluctuation chemistry", we investigate the effects of external fluctuations by perturbing the flow rate and of internal micromixing noise by stirring at various rates. Good agreement between experiment and calculations based on the extended Oregonator (using the Field-Forsterling rate constants) is achieved except in the chaotic region. We use a number of measures to characterize the dynamics: Fourier spectra, attractor construction by the singular value decomposition method, Poincars sections, one-dimensional maps, maximum Lyapounov exponents, and correlation dimensions. Comparisons with other work are given.
I. Introduction Interest in chemical aperiodicities has been increasing ever since the first reports of chemical chaos appeared in the literature.' Periodic and aperiodic chemical reactions may be observed in nonequilibrium nonlinear systems far from chemical equilibrium. In order to maintain the reaction far from equilibrium, a continuous flow stirred tank reactor (CSTR) is used. The in-flowing reactant solutions are mixed with the bulk liquid in the reactor by vigorous stirring. Mixing may be divided roughly into two processes, namely macromixing which describes the hydrodynamic convection of one fluid through the other* on a macroscopic length scale and micromixing. In the micromixing process, small local concentration variations are produced by stretching and folding3 of incompletely mixed liquid elements which subsequently disappear by molecular diffusion. These concentration variations are called fluctuations. They show measurable amplitudes and various lifetimes. Importantly, a nonlinear mechanism is very sensitive toward these fluctuations when their lifetimes are comparable with or larger than the lifetimes of the individual nonlinear steps. There is another type of noise which may be classified as external noise. The latter is introduced by statistical variations in the external bifurcation parameters such as the flow rate or the temperature, which may fluctuate about their average values. In order to study the behavior of the system toward external noise we superimpose random perturbations on the feed rate of our piston pump. This procedure allows quantitative studies in fluctuation chemistry. With regard to aperiodic motions a fundamental problem and question arises: Is the observed aperiodicity due to deterministic chaos or is it a consequence of the above types of interactive noise or do all phenomena contribute? Deterministic chaos4vsmay arise ( I ) Schmitz, R. A,; Graziani, K. R.; Hudson, J. L. J . Chem. fhys. 1977, 67, 3040. (2) Villermaux, J. In Spatial Inhomogeneities and Transient Behaviour; Gray, P:, Nicolis, G., Baras, F., Borckmans, P., Scott, K., Eds.; Manchester University Press: 1990 p 119. (3) Ottino, J . M.; Leony, C. W.; Rising, H.; Swanson, P. D. Narure 1988, 333, 419. (4) Schuster, H. G. Deferministic Chaos, 2nd ed.; VCH Verlag: Weinheim, 1988.
0022-365419112095-2130%02.50/0
from various sources, for example, from the intrinsic chemistry or from coupling between two independent nonlinear oscillators@ even if the oscillators themselves are periodic. Characteristically, the smallest change in initial conditions will produce a different chaotic time series. This represents the well-known sensitivity toward initial conditions. Such systems are very sensitive toward fluctuations in general. As a specific experimental example we discuss the occurrence of chaos and amplified noise in the Belousov-Zhabotinsky reaction which is probably the most thoroughly studied nonlinear chemical oscillator. We shall show that the reported chaotic time series at high flow rates may be interpreted as amplified noise instead in the BZ reaction. At low flow rates, however, deterministic chaos has been shown to exist by Swinney and co-~orkers"'-'~as verified in this work. The observed BZ chaos is intrinsic to the chemical mechanism. A dramatic amplification of flow rate noise is demonstrated when the BZ reaction is in an excitable steady state close to a Hopf bifurcation at high flow rates. There are a number of measures that characterize deterministic c h a ~ s . ~For . ' ~ chaos the experimental Fourier spectra are broad and the autocorrelation functions show a decline in their amplitudes with time. The phase portrait of chaos is the so-called (5) Berg6, P.; Pomeau, Y.; Vidal, C. Order within Chaos; Wiley: New York, 1986. (6) Schreiber, I.; Marek, M. fhysica D 1982, SD,258; Phys. Left. 1982, 91, 263. (7) Crowley, M. F.; Field, R. J. J . Phys. Chem. 1986, 88, 762. (8) Ameodo, A. Personal communication. (9) Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcafions of Vecror Fields; Applied Mathematical Sciences Vol. 42; Springer: Berlin, 1986. (10) Swinney, H. L.; Roux, J. C. In Non Equilibrium Dynamics in Chemical Sysrems; Vidal, C., Pacault, A., Eds.; Springer: Berlin, 1984; pp 1 24- 140. ( 1 I ) Turner, J. S.;Roux, J. C.; McCormick, W. D.; Swinney, H. L. fhys. Letr. 1981, 85A, 9. (12) Simoyi, R. H.; Wolf, A,; Swinney, H. L. Phys. Reu. Left. 1982,49, 245. (13) Coffman, K. G.; McCormick, W. D.; Noszticzius, Z.; Simoyi, R. H.; Swinney, H. L. J . Chem. fhys. 1987, 86, 119. (14) Roux, J. C.; Turner, J. S.; McCormick, W. D.; Swinney, H. L. In Nonlinear Problems: Presenr and Future; Bishop, A. R., Campbell, D. K., Nicolaenko, B., Eds.; North Holland: Amsterdam, 1982; p 409.
0 1991 American Chemical Society
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Feature Article “strange” attractor whereas periodic and quasiperiodic motion corresponds to a limit cycle and a torus, respectively. In order to construct the attractor one may use the method of the singular value decomposition (SVDI5) which eliminates the effect of instrumental noise. Several techniques are available to study the properties of the strange attractor. The PoincarE section is obtained from the intersections X , of the trajectories with a ( n 1)-dimensional surface producing a continuous curve of densely located points for chaotic motion. One may construct a onedimensional map by plotting successive trajectory values (X,, and Xel)as they pass through the PoincarE section. If the resulting one-dimensional map is a continuous curve with a t least one extremum, deterministic chaos is indicated since X,,+, is uniquely determined for every value of X,, according to X,,,,=AX,,). A unique indication of chaos is the observation of the so-called routes to chaos such as period doubling, intermittency, or:the destruction (wrinkling) of a torus in the case of quasiperiodic motion. One of the quantitatiue measures for chaos is provided by the largest Lyapounov exponent (A,, ). A positive value of A,, indicates that the trajectories comprising an attractorI6J7 diverge over a long time average which is characteristic for chaotic motion. This divergence describes the rapid loss of the system’s memory of its previous history as time evolves. For periodic motion A,, is equal to zero, indicating that the limit cycle conserves its information in time. Characteristically, large fluctuations in the reactor cause the A,, values to become positive even for periodic motion.’*-21 Therefore, for a positive value of A,, a distinction between noisy periodic motion and chaos is difficult. Finally, the correlation dimension D,,, of a chaotic attractor assumes a “fractal” (noninteger) value above 2.0, indicating the self-similarity of sets of chaotic t r a j e c t ~ r i e s . ~ ~ * ~ ~ Some nonlinear oscillators display remarkably complex periodic oscillations. The sequence of peaks in these complex oscillation patterns may be described by the Farey arithmetic when a bifurcation parameter is varied as shown in models (refs 24 and 25, also evident in ref 26) and in experiments such as the Mn-catalyzed BZ reaction,27in an electrochemical oscillator,28and in the present work. Farey ordered patterns may be generated by frequencylocking; i.e., the system contains two commensurate frequencies wIand w2. A Farey pattern may be characterized by the so-called firing number p / q , where p and q are integers. The Farey sum of two firing numbers pl/ql and p 2 / q 2is defined as the ratio of the sums of the two numerators and denominators: PI P2 PI + P2 - e-=41 92 41 + 92 The Farey sums represent rational approximations of irrational numbers. In principle they may show an arbitrarily resolved fine structure. However, experimental fluctuations prevent the separate (15) King, G.;Jones, R.; Broomhead, D. S.In Proceedings on Chaos 1987, Nucl. Phys. E. (Proc. Suppl.) 1987. (16) Benettin, G.;Galgani, L. Invited paper at the international School on Intrinsic Stochasticity in Plasmas, Cargese, France, 1979. (17) Benettin, G.;Galgani, L.; Giorgilli, A.; Strelcyn, J. M. Meccanica 1980, 15, 9. (18) Schneider, F. W.; Kruel, Th. M.; Freund, A. In Proceedings in Nonlinear Science: Sfruciure, Coherence and Chaos in Dynamical Sysiems (1986 Workshop); Christiansen, P. L., Parmentier, R. D., Eds.; Manchester University Press: 1989; p 601-606. (19) Freund, A.; Kruel. Th. M.; Schneider. F. W. In From Chemical to Biological Organisaiion; Markus, M., MOller. S. C., Nicolis, G.,Eds.; Springer: Berlin, 1988; pp 49-60. (20) Herzel, E.; Ebeling, W. Z . Narurforsch. 1987, 4 2 2 , 136. (21) Harding, R. H.; Ross, J. J . Chem. Phys. 1988,89,4743. (22) Grassberger, P.; Procaccia, I. Phys. Rev. Leu. 1983, 50, 346. (23) Grassberger, P.; Procaccia, I. Physica 1983, 90, 189. (24) Bar-Eli, K.; Noyes, R. M. J . Chem. Phys. 1988,88, 3646. (25) Argoul, F.; Arneodo, A.; Richetti, P.; Roux, J. C. J . Chem. Phys. 1987, 86, 3325. (b) Richetti, P.; Roux, J. C.; Argoul, F.; Arneodo, A. J . Chem. Phys. 1987, 86, 3339. (26) Kai, T.; Tomita, K. Prog. Theor. Phys. 1979, 61, 57. (27) Maselko, J.; Swinney, H. L. J . Chem. Phys. 1986, 85, 6430. (28) Albahadily, F. N.; Ringland, J.; Schell, M. J . Chem. Phys. 1989,90, 813.
observation of closely located Farey patterns. If the flow rate is adjusted in a range between two experimentally resolvable Farey oscillations, fluctuations may produce an aperiodic time series which is a consequence of statistical switching between the neighboring attractors. This is why an explanation of the periodic-aperiodic sequences in terms of deterministic chaos bears the risk of overinterpretation in this case. It is important to note that, in contrast to the universal sequences, Farey sequences do not represent a route to chaos, since they refer to periodic trajectories only. It is possible, of course, that a given theoretical model might show both chaos as well as Farey sequences although in different parameter region^.^^.'^ Among several suggested models2Sb-73we favor the extended Oregonator model3I for the BZ reaction since it reproduces the Farey ordered oscillations at high flow rates. On the other hand, the Oregonator model is in need of modification, since it is too simple regarding the organic part of the reacti~n.’~In order to compare nonlinear dynamic models with experiment in a realistic way it is desirable to superimpose fluctuations on the model parameters such as on the flow rate and/or on the other variables. In the numerical integrations the fluctuations may be Gaussian, Poisson, or uniformly distributed. In this work we use Gaussian distributed fluctuations imposed on the flow rate. In order to study the effects of superimposed fluctuations in an experiment one may impose random noise of large average amplitude on the feed rate of a piston pump which pumps the reactant solutions into the reactor. This may be precisely done via a stepping motor under the control of a laboratory computer. These perturbation experiments are part of a novel field in nonlinear dynamics which may be called fluctuation chemistry.33 Deterministic chaos has been reported in the chlorite-thiosulfate system,34in a single enzyme system,35 in g l y c o l y ~ i s ,in~ ~het*~~ erogeneous and in an electrochemical r e a ~ t i o n . ~ ’ The BZ reaction has been intensively tested for c ~ ~ o s . ~ J * ~ ~ * ~ ~ Our contributions to the field of aperiodicity concern mainly the BZ reaction which will be discussed in the following. 11. Continuous Flow Stirred Tank Reactor (CSTR)
I . The CSTR as a Noise Generator. The continuous flow stirred tank reactor (CSTR) is the type of reactor most commonly used in the study of periodic and aperiodic phenomena in chemical reactions. A fill reactor which is more economical has also been e m p l ~ y e d . ~In~ ~the~ ’fill reactor the system is far from equilibrium only in the initial stages of the reaction, since the reactor has no outflow. (29) Barkley, D. E. J . Chem. Phys. 1988,89, 5547. (30) Lindberg, D.; Turner, J. S.; Barkley, D. J . Chem. Phys. 1990, 92, 3238. (31) Showalter, K.; Noyes, R. M.; Bar-Eli, K. J . Chem. Phys. 1978,69, 2514. (32) Gybrgyi, L.; Rempe, S.L.; Field, R. J. Submitted for publication in J . Phys. Chem. (33) Nicolis, G. See ref 2, p 507. (34) Orban, M.; Epstein, 1. R. J. Phys. Chem. 1982,86, 3907. (35) Olson, L. F.; Degn, H. Nature 1977, 267, 177. (36) Boiteux, A.; Goldbeter, A.; Hess, B. Proc. Nail. Acad. Sci. U.S.A. 1975, 72, 3829. (37) Markus, M.; Kuschmitz, D.; Hess, B. FEBS Lerr. 1984, 172, 235. (38) Imbiehl, R.; Cox, M. P.; Ertl, G.J. Chem. Phys. 1986, 84, 3519. (39) Jaeger, W. I.; Moller, K.; Plath, P. 2.Naiurforsch. 1981, A36, 1012. (40) Wicke, E.; Onken, H. U. Personal communication. (41) Bassett, M. R.; Hudson, J. L. J . Phys. Chem. 1989, 93, 2731. (42) Hudson, J. L.; Hart, M.; Marinko, D. J. J . Chem. Phys. 1979, 71, 1601.
(43) Roux, J. C.; Rossi, A. In Non Equilibrium Dynamics in Chemical Systems; Vidal, C . , Pacault, A, Eds.;Springer: Berlin, 1984; p 141. (44) Wegmann, K.; Rbsler, 0. E. 2.Nafurforsch. 1978, 330, 1179. (45) Roux, J. C. Physica 1983, 7 0 , 57. (46) Roux, J. C.; Rossi, A.; Bachelart, S.; Vidal, C. Phys. k i t . A 1980, 77, 391. (47) Rbssler, 0. E.; Wegmann, K. Narure 1978, 271, 89. (48) Vidal, C.; Roux, J. C.; Rossi, A.; Bachelart, S. Seances Acad. Sci. Ser. C 1979, 289, 73. (49) Vidal, C.; Roux, J. C.; Bachelart, S.;Rossi, A. Ann. N . Y. Acad. Sci. 1980, 357, 377. (50) Vidal, C.; Bachelart, S.; Rossi, A. J . Phys. (Les Ulis, Fr.) 1982, 113, 7.
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themi s t o r
T
1,3 cn
1
+icn+
IC
Figure 1. Model of the compact (1.42 mL) CSTR, which is a squareshaped magnetically stirred (1 cm) spectrophotometric cell. The light beam passes unhindered between stirrer and tubes. Syringes, feed tubes, and the reactor are thermostated.
A CSTR is a device well-known to chemical e n g i n e e r ~ . ~ I It -~~ was rediscovered in the mid-1970s for use in chemical oscillationsM and independently in nonlinear enzymatic nucleic acid synthesis.55 Since a CSTR is an experimental mixing device, the individual geometry of a given reactor is very important regarding mixing efficiency. Unfortunately, a standardized reactor does not yet exist for direct comparisons of experiments done in different laboratories. In any reactor regardless of design large spatial concentration fluctuations occur in the neighborhood of the entry of the reactant solutions. In other words, a CSTR is a “noise generator”. Specifically, the mixing of the in-flowing reactant solutions occurs via the macromixing and a subsequent micromixing process where the transition between the two processes is somewhat diffuse. In accord with other work we consider the stretch and fold process of small liquid elements by stirring as being part of the micromixing process together with molecular diffusion. Micromixing may affect the rates of the nonlinear steps if the micromixing time is comparable with or longer than the lifetimes of these steps. 2. Compact and Noncompact Reactors. By experience, optimal mixing may be achieved if the reactor has a cylindrical shape with built-in baffles which ensure turbulent mixing.2 High energy dissipation is brought about by vigorous stirring. We call such an efficient mixing device a compact reactor. Usually the thin inflow tubes of reactant solutions are directed onto the blades of the stirrer. For spectrophotometricdetection, where cylindrical reactors lead to experimental problems since they act like optical lenses, one may use a shortened square-shaped spectrophotometric cell (1 cm path length) as a compact reactor with a rounded-out bottom and a plastic stopper containing a small dome to allow the passage of any accumulated air bubbles (Figure 1). In this case magnetic stirring may become necessary. The advantage of magnetic stirring is the high energy dissipation during the stirring process. Stirring speed is usually between 1000 and 1500 rpm. Concentration fluctuations in compact reactors may have a broad distribution of lifetimes from the millisecond to second range. ( 5 1) Levenspiel, D. Chemical Reaction Engineering Wiley: New York, 1972. (52) Ark, R.Introduction to the Analysis of Chemical Reactors; Prentice Hall: Englewood Cliffs, NJ, 1965. (53) Lapidus, L., Amundson, N. R. Eds. Chemical Reactor Theory. A Reuiew; Prentice Hall: Englewood Cliffs, NJ, 1977. (54) Marek, M.; Stuchl, J. Biophys. Chem. 1975, 3, 241. ( 5 5 ) Schneider, F. W. Biopolymers 1976, IS, 9. ‘
Schneider and Munster In a large CSTR it is conceivable that the macromixing process may generate concentration gradients due to long-lived dead spaces (e.g., in the neck of a reactor or in the inlet port^^^,^') in which independent chemical oscillators may form. Mass convection and diffusion through the contact areas between subvolumes and the main volume of the reactor could occur in principle and thereby produce deterministic chaos in the chemical reaction. We define such a reactor as a noncompact CSTR. In reality any compact reactor contains a degree of “noncompactedness”due to insufficient macromixing in the immediate vicinity of the inlet tubes. So far we are not aware of any published chemical chaos that may have been produced in a noncompact reactor. 3. Micromixing As Measured by CARS. Micromixing of liquids has been described3 as a stretch and fold process of liquid elements. This procedure is followed by molecular diffusion between bulk and liquid elements whose size has decreased below A number of liquid elements may be the Kolm~gorov-limit.~~ locally formed per unit time at various points in the reactor by vigorous stirring of the inflowing liquids. The production of liquid elements is consistent with our recent experiments on the nonreactive mixing of two liquids (toluene and p-xylene) in a CSTR in which the local distributions and deviations from the average composition were determined by CARS (coherent anti-Stokes Raman ~ c a t t e r i n g )spectroscopy ~~ as follows. Two crossed pulsed laser beams (10 Hz) create a microvolume (-8 pL) of illumination at their crossing point in which a CARS signal of toluene (1004 cm-I) is generatedm within 9 ns recorded every 0.1 s. This illumination volume may be arbitrarily moved inside the reactor with an effective observation window of 10 pm diameter and a length resolution of -0.8 mm. First, in the vicinity of the inflow tubes at an intermediate flow rate (k = 0.85 mi&) macromixing occurs as evidenced by the large values of the measured standard deviations from the average concentration. Due to the rapid rotatory motion of the asymmetric stirrer, the macromixed liquid is preferentially transported into one corner of our square-shaped reactor. Here the standard deviation is equally large, since the composition of the liquid changes dramatically in the CARS interaction volume from pulse to pulse. At a larger distance from the inlet tube the micromixing process has produced a larger number of small liquid elements by stretching and folding. Therefore, the CARS signal has been generated from a composition which is more uniform; producing a lower standard deviation from the average composition. At some distance from the inlet ports depending on the reactor geometry, flow rate, and stirring speed, mixing proceeds to completion as seen from the diminishing values of the standard deviation. Although the liquid elements are thought to be formed by a stretch and fold process, it is unlikely in a stirred reactor that they are self-similar in the sense of an “ordered” fractal. Their distribution is more likely to be that of a percolation cluster6’which may represent “fractal-like” spatial structures. Thus it is evident that the liquid mixture in a compact CSTR is far from being homogeneous in local concentrations during the macromixing and micromixing processes even if low flow rates and high stirring rates are employed. 111. The BZ Reaction in the CSTR 1 . Experimental Method. The BZ reaction is the metal ion
catalyzed oxidation of malonic acid by bromate ion in a strongly acidic aqueous medium.62 The BZ oscillator is known to be a nonsinusoidal relaxation oscillator which oscillates (“relaxes”) between two regions of its manifold (potential surface). All participating species oscillate with different amplitudes and phases. Periods of oscillation are typically between 1 and 3 min depending Gyiirgyi, L.;Field, R. J. J. Phys. Chem. 1989, 93, 2865. GyBrgyi, L.; Field, R. J. J. Chem. Phys. 1989, 91, 6131. Kolmogorov, A. N. J . Fluid Mech. 1962, 13, 82. Kraus, H.P.; Schneider, F. W. To be published. Schneider, F. W. In Nonlinear Raman Spectroscopy and its Chemical Applications; Kiefer, W., Long, D. A., Eds.; D. Reidel: Dordrect, 1982; p (56) (57) (58) (59) (60)
445.
(61) Kopelman, R. Science 1988, 241, ,1620. (62) Field, R. J. J. Chem. Educ. 1972, 49, 308.
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Feature Article on flow rate, temperature, and concentrations. Our BZ measurements were carried out in a 1.4-mL CSTR (Figure l ) within a narrow concentration range as used in several laboratories. The reactor is fed by three identical IO-mL gas-tight Hamilton syringes with Teflon-coated barrels. The syringes contain the following reactant solutions for the experiments a t high flow rates (corresponding to region I1 in Figure 2) at 25 OC: 0.9 M malonic acid and 0.003 M Ce(N03)3;0.263 M KBr03; 1.12 M H2S04. For low flow rates (region I) a t 28 “C, the following syringe concentration were used: 0.75 M malonic acid and 2.5 X lo-’ M Ce3(S04),; 0.3 M KBrO,; 0.6 M H2S04 (to obtain the reactor concentrations divide by 3). The different temperatures are those of other literature studies in order to facilitate experimental comparisons. Our steady-state experiments (region 111) were performed at the same concentrations and temperatures as those of region 11. On the basis of the work by Noszticzius et al.,63 the malonic acid was purified by several recrystallizations from acetone/H2S04mixtures. The syringes are simultaneously moved by a self-designed and very precise syringe pump which is driven by a stepping motor. Each step of the motor moves the syringe piston by 100 nm which corresponds to 0.06 pL flow volume per pulse. The period between two motor steps can be electronically controlled with high precision. Typical stepping frequencies for our measurements of the BZ reaction were in the range of 5-230 Hz. Therefore, the liquid flow into the reactor is practically constant. The CSTR is placed in a spectrophotometer which records the integrated light intensity. All components are thermostated. In order to control the temperature, a small thermistor is mounted inside the reactor. Mixing of reactant solutions is caused by an unsymmetric magnetic stirrer which is driven by a small motor located below the reactor bottom. The Ce4+ concentration is monitored at an observation wavelength of 350 nm. The spectrophotometric data are transmitted to a x / t recorder and to a laboratory computer in digitized form at a sampling frequency of 2 Hz. Calculations and simulations were done on a Siemens 7860L computer. It is also possible to measure the Br- concentration with a Br--selective electrode or the electrochemical potential with a platinum electrode. We prefer the rapid response and noninvasiveness of the spectrophotometric method (Hitachi U3210) whose repeatability (0.001 OD units) is comparable with that of the electrodes. A comparison between the spectrophotometric method and the platinum electrode did not show any difference in the typical reaction times as used in this study. Therefore, the weak illumination used in the spectrophotometric method has no effect on the observed dynamics. 2. Numerical Simulations. Model calculations for the BZ reaction were carried out with the modified Oreg~nator.~’ This model turns out to be a good qualitative model at high flow rates only, since it does not reproduce the experimentally found chaos at low flow rates when the “new” rate constants are used.30 A
X
C
+Y
k
X
k
+Yd 2P ki
+ W =kiX + Z’
+P k3 = 3
kl = 2.0, k2 = 200 X
k7 = 8
lo6, k4 = 2
X
X
lo4, k8 = 8.9 X lo3
The total organic part of the BZ reaction is represented by the last step with the two constants k l , and g = 0.45. All rate constants are those given by Field and Fiirsterling@ except for k2 = 200 which was chosen to reproduce the experimentally observed Farey sequences. On the other hand, quantitative agreement with the experimental upper critical flow rate (upper Hopf bifurcation) is poor (factor of 10 discrepancy). In order to simulate CSTR conditions, flow terms are added to the system of differential equations for the in- and outflow of reactants and the outflow of intermediates and reaction products. The “residence time” in the CSTR is equal to the reciprocal flow rate constant kp The concentrations in the calculations were identical with those of region I1 (section 111.1). Our numerical integrations were done by an implicit Taylor integrator (written by F. Buchholtz) which is adequate for stiff differential equation systems. Because of the stiffness of the extended Oregonator model one must choose small integration time steps (lo4 s) and small error limits (lo-*). 3. Attractors and Invariant Measures of Chaos. Attractors. A dissipative dynamical system is defined by a complete set of linearly independent variables X,(t) at any time t . Each variable corresponds to a coordinate in phase space. Therefore, a single point X ( t ) = (Xl(t),X2(r),...,Xi(r)l in phase space uniquely defines the system at t . As the system evolves with time the set of points in phase space generate a curve, the so-called trajectory. After a finite time, the trajectories will find themselves on a stable curve in phase space, the attractor. In principle, the attractor may be obtained by plotting all X ( t ) of the system in an n-dimensional space where n is the “embeddng dimension” for the attractor. In general n is always larger than the total number of independent variables. In an experiment usually one single variable is measured. Therefore, the attractor must be reconstructed from the time series of this single variable. This is possible, since each variable reflects the total information of the system.65 A number of methods have been developed to reconstruct the attractor, such as the delay-time method,65the derivative method,& and the SVD method.Is We favor the SVD method since it works well for stiff systems and eliminates the effects of white noise. Stiff systems contain individual steps whose rates vary over many orders of magnitude. Lyapounov Exponents. The spectrum of Lyapounov exponents describes the exponential convergence to or divergence from an attractor in all its dimensions. Thus, the number of Lyapounov exponents of a given system is equal to its dimension. As an example, consider a three-variable system which gives rise to three different Lyapounov exponents. A stable node or focus is characterized by three negative exponents (-,-,-); a limit cycle corresponds to Lyapounov exponents with signs (0,-,-), whereas a torus, Le., a quasiperiodic attractor, gives rise to two zero and one negative Lyapounov exponents (O,O,-). In contrast to these periodic attractors strange attractors are characterized by an exponential divergence of trajectories from the attractor in at least one dimension of the flow. Therefore, their Lyapounov exponents are of the signs (+,O,-). As seen from the above example the largest Lyapounov exponent A,, allows a distinction between periodic or quasiperiodic motion on the one hand (Amx = 0) and chaotic motion on the other (A,, > 0). From our experimental data we approximated A,, in the following manner: First the attractor is constructed according to the S.VD method. We then applied the Wolf-Swift-Swinney-Vastano (WSSV)67 method to determine Amax as
kg
2X
kq
A
+P kll
Z’-gY
k9 = 3 X IO3, k l o = 1
+C
X
IO-* where L(t,) is the length of a vector which connects two points
kll = 0.2 (64) Field, R. J.; Wrsterling. H. D. J . Phys. Chem. 1986, 90, 5400. (65) Takens, F. Lecture Notes in Muthemofics; Springer: Berlin, 1981;
where A = BrOY, Z’ = Ce4+,C = Ce3+,P = HOBr, W = BrOz’, X = HBr02, and Y = Br-.
Vol. . -. . R98. - - -.
(63) Noszticziug, Z.; McCormick, W. D.; Swinney, H. L. J . Phys. Chem. 1987. 91, 5129.
(66) Packard, N. H.; Crutchfield. J. M.; Farmer, J. D.; Shaw, R. S.Phys. Rev. Leu. 1980, 45, 712. (67) Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. L. Physic0 1985, lbD, 285.
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on two different trajectories at time ti. A starting vector between points on closely lying trajectories is chosen whose length is minimal. This vector is evolved until a chosen cutoff distance is reached. This procedure is repeated with a new starting vector of minimal length whose direction is close to the former one. The n e value for A, is calculated from the long time average obtained from a large number of vectors. This method works well for uniform attractors where the density of points is homogeneous 8 along the attractor. Experimental attractors, however, are often nonuniform as seen from the slow convergence of A,, to its final value with time. The WSSV method is not applicable if an attractor is strongly nonuniform. It is seen from various models that interactive noise may cause ,A, to become positive even for -I I 1 periodic and quasiperiodic motion.18-21*68*69 Therefore, a positive -2 -1 1 value for A, is not necessarily a proof for deterministic chaos in an experiment. Figure 2. Experimental bifurcation diagram for the BZ reaction. ConThe Correlation Dimension. A powerful tool to characterize centrations and temperature are those given in section Ill. 1. The maxima the geometric properties of an attractor is provided by its diand minima of the optical density at 3 5 0 nm (Ce4+) are plotted vs the mension. Periodic motion corresponds to a limit cycle, i.e., a closed logarithm of the flow rate. The oscillationsstart with a Hopf bifurcation curve in phase space whose dimension is equal to one. A quaat kf= 1.5 X 1O-* m i d . In region I, states of period doubling and chaos siperiodic attractor gives rise to a dimension of 2.0, since it fills are observed. Farey ordered nonchaotic mixed-mode oscillations are observed in region 11. PI and PI* are simple period 1 oscillations where the surface of a torus. Strange attractors are made up by sets the frequency of PI* is roughly twice the frequency of P,. Region 111 of self-similar trajectories. Therefore, their dimension must be marks a focus which is excitable in the neighborhood of the upper Hopf a fractal, noninteger value. As a consequence of the Kaplan-Yorke bifurcation. The latter occurs at k f = 0.38 min-I. conje~ture’~ this value must be greater than 2.0. There are a number of different definitions of the dimension: cillations at higher flow rates. The latter oscillatory region contains The Hausdorff dimension D, the information dimension u and a rich substructure of bifurcations representing mixed mode osthe correlation dimension DCorr where D > u > D,,,. From our cillations whose sequence is determined by the Farey arithmetic. experimental data we calculated the correlation dimension D,,, Small-amplitude oscillations of period 1 ( P I * )are observed in a according to the algorithm of Grassberger and P r o c a ~ c i a ~ ~ i ~narrow ~ range between the Farey ordered oscillations and the upper Hopf bifurcation (HB). The latter occurs experimentally at kf = 0.256 min ( T =~ 3.91 min). Region I11 represents an excitable steady state (focus) beyond the upper Hopf bifurcation. A Hopf bifurcation marks the transition between a limit cycle and a stable where C(r) is the correlation integral: focus4v9where the focus represents a steady state reached by a damped oscillation. We demonstrate the excitability of the steady state (region 111) by experimentally superimposing Gaussian i#j distributed noise on the flow rate. At very low flow rates below region I ( 0. N is the number of data points, and Nrefis been investigated for its sensitivity toward fluctuations. the number of randomly chosen reference points. Thus C(r)counts 5. Experimental Observations in the BZ Reaction. Deterthe number of points that lie within the sphere around Xi. The ministic Chaos at LAW Flow Rates (Region I). The region of embedding dimension is increased until D,,, reaches a constant small-amplitude oscillations at low flow rates (residence times value. D,,, is determined from the slope of a plot of log C(r) vs above 30 min) has previously been investigated by Swinney and log r. In the absence of noise, the log-log plot always shows a who observed chaotic oscillations as well as period single scaling region. Noisy periodic and quasiperiodic motion doubling up to period 8 in an Erlenmeyer-shaped 3 1.O-mL reactor. gives rise to a break point in the log-log plot where the upper We have verified the existence of chaos in a 1.4-mL compact region yields D,,,, for the attractor. Since chaos is sensitive to square-shaped reactor. changes in initial conditions, the effects of noise are present over It is important to note that commercially available malonic acid the whole range of the attractor. Therefore, chaos does not show must be purified for the experiments on chaos. Without sufficient a break point in the log-log pot. In the case of large noise, purification, chaos could not be observed as pointed out by however, the break point is shifted toward higher values of r for Noszticzius et al.;63 Le., exclusively periodic time sequences (PI periodic motion until it finally disappears at large noise levels. = period 1) are then obtained between T , of~ 23 and 63 min. A distinction between chaos and noisy motion is no longer possible For decreasing flow rates (increasing residence times) from 3.1 in the presence of large noise. Noise may cause D,, to rise above X 2.0 even for periodic and quasiperiodic m ~ t i ~ n . ~ ~ * ~ ~ * ~ ~ m i d (32 min) we observed PI which is followed by P2 and Psvia period doubling (Figure 2). Further lowering of the 4. Bifurcation Diagram. In order to gain an overview of the flow rate produced the chaotic series C and P3 which is followed BZ reaction we show the experimental bifurcation diagram in by occasional patterns of P6 and an additional chaotic sequence which the maximum and minimum Ce4+ concentrations of an C’. Eventually, PI’ (with smaller amplitudes than PI) is obtained oscillation are plotted versus k p The concentrations used to without any further resolved states between C’ and PI’. A Hopf determine the bifurcation diagram (Figure 2) are those of region m i d where the PI’ osbifurcation is observed at 1.5 X I in section 111.1. There are three flow rate regions: region I shows cillations give way to a focus. low-amplitude oscillations and narrow regions of deterministic The correlation dimensions for the observed noisy P2 and P3 chaos at low flow rates and region I1 shows high-amplitude ososcillations were determined to be between 1.4 and 1.6. For the aperiodic attractors C and C’ the values for D,, were in the range (68) Kruel, Th. M.;Freund, A.; Schneider, F. W. J. Chem. Phys. 1990, of 1.8-2.1. The latter may be interpreted as an average between 91, 416. periodic (&, = 1) and chaotic (D” > 2.0) motion originating (69) Harding, R. H.;Sevcikova, H.;Ross, J. J . Chem. Phys. I=, 89, from the fluctuations generated in any CSTR by micromixing 4737. which may cause the system to statistically switch between periodic (70) Kaplan, J.; Yorke, J. A. In Lecture Notes in Mathematics; Peitgen, H. 0.. Walter, H. O., Eds.; Springer: Berlin, 1978; Vol. 730, p 228. and chaotic states. Chaos is evidenced by the albove observation
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The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 2135
Feature Article
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Figure 4. Amplification of fluctuations. At 1.05 ki Gaussian distributed
0
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noise was imposed on the experimental flow rate. The measured standard deviation from the average OD (350 nm) of Ce4+ (steady state) is plotted vs the standard deviation of the imposed perturbation. Beyond a critical perturbation amplitude ( N13%) the response becomes larger than the perturbation. In this case fluctuations are amplified. The line indicates equal perturbation and response amplitudes. It is seen from the intercept of the curve that the size of the "natural" fluctuations is about 2.5%.
(71) Li, T.; Yorke. J. A. Ann. Math. Monthly 1975, 82, 985. (72) Field, R. J.; KBrb, E.; Noyes, R. M. J . Am. Chem. SOC.1972, 94,
for "noisy" limit cycles (Figure 3). On the other hand, the A,, values increased dramatically for time series measured at kf values between two neighboring Farey sequences. We consider these particular aperiodic time sequences to be statistical mixtures of the neighboring patterns as evidenced from their Fourier spectra, Poincare sections of their attractors, and one-dimensional maps (not shown). Calculations using the extended Oregonator with 3% (Gaussian) noise superimposed on the flow rate show a similar behavior with respect to the above measures. Thus we conclude that aperiodic motion observed at high flow rates is statistical in nature whereas aperiodic patterns at low flow rates arise from deterministic chaos. 6. Toward a Fluctuation Chemistry.33 The effects of fluctuations are profound on nonlinear dynamical systems particularly in the neighborhood of bifurcation points75where noise-induced transitions may occur. The response of nonlinear reactions toward noise will also be nonlinear and fluctuations may be strongly amplified. There are several ways to superimpose fluctuations on an experimental system. Internal (micromixing) noise due to stirring is always present in a CSTR. Its amplitude and frequency can be controlled by the stirring rate. External fluctuations may be superimposed by random variations of parameters like the temperature and the flow rate. For experimental reasons we apply rectangular pulses of constant time intervals but varying amplitudes to the average flow rate. The amplitudes of the applied pulses are allowed to change according to a Gaussian distribution with a given variance. Flow Rate Fluctuations-Periodic Oscillations (Region II). In order to show the effects of excessive external noise on an experimental limit cycle, we experimentally superimposed a noise amplitude of f7% (pulse duration 5 s) on the average flow rate of 0.155 min-l for the Farey pattern 2l as an example. The response was a strongly aperiodic time series. Its Fourier spectrum is relatively broad and it shows a number of individual frequencies. For this experiment A,, was calculated to be positive (0,0171) similarly to chaotic motion. Interestingly, the first zero (to)in the autocorrelation function is identical in the unperturbed and in the perturbed experiment (to = 12.5 s; not shown). A computer simulation using the modified Oregonator with superimposed fluctuations (f7%) yields a similar Fourier spectrum and value for A,,. Therefore, this aperiodicity arises solely from the statistical switching between contiguous periodic Farey attractors and not from any deterministic chaos. Excitability of the Steady State (Region III). Fluctuation experiments were performed at an experimental steady state that was located slightly above the upper experimental Hopf bifurcation, Le., approximately 5% above the critical flow rate. The latter occurs at kP = 0.256 min-' for "natural" noise in the reactor. Steady states are termed excitable, when a small experimental perturbation causes a sudden and large change in the response
(73) Gyargyi, L.; Field, R. J. J . Phys. Chem. 1988, 92, 7079; J . Chem. Phys. 1990, 93, 2159. (74) Farey, J. Philos. Mag. J. (London) 1816, 47, 385.
(75) Horsthemke, W.; Lefever, R. Noise Induced Transitions: Theory and Applications in Physics, Chemistry and Biology; Springer: Berlin, 1984.
Figure 3. A,
versus flow rate in the BZ reaction. Dark squares are Farey patterns starting with 2'. Light squares are statistical mixtures of adjacent Farey patterns. The patterns 2223and Z3 show relatively high A, values which may be substantially reduced if H2S04is premixed in all three syringes.
of the route to chaos and by the onedimensional map (not shown). Furthermore, the presence of P3 is commonly taken as an indication of chaos,71since P3 is produced by a tangent bifurcation which marks the transition from a periodic window to chaos as the bifurcation parameter is changed. Some reduced version^^^^.'^ of the FKN mechanism72for the BZ reaction show period doubling, chaos, and the universal sequence. However, these models have been criticized as being chemically ~nreasonable.7~A chemically realistic model will have to include elementary steps involving a number of organic intermediate~.~~ Experimental Farey Patterns at High Flow Rates (Region II). Our investigations at high flow rates showed that the sequence of the multipeaked oscillation patterns is determined by the Farey arithmetic" as already observed by Maselko and Swinney for the Mn-catalyzed BZ reaction.27 In the following we illustrate the situation for the present Ce-catalyzed BZ reaction within the kf range from 0.154 to 0.205 min-l. The multimode oscillations are made up of large- and small-amplitude oscillations. Ls represents a Farey pattern where L and S are the numbers of large- and small-amplitude peaks in a given time pattern, respectively. The firing number F is defined as F = S / ( L S) = p / q
+
In the above flow rate range we observed the following patterns: 2', 2122,22, 2223,23 with the corresponding Farey ordered firing numbers 1/3,3/7,2/4,5/9, and 315, respectively. These periodic patterns are separated by regimes of aperiodic oscillations. For all the observed periodic patterns it is seen that the Fourier spectra are characterized by at least two frequencies which are commensurate. In a very narrow range of flow rates adjacent to the upper Hopf bifurcation we observed pure P I * oscillations whose amplitudes are similar to the small amplitudes in the Farey patterns. Their Fourier spectra show a single frequency with low-intensity overtones. The Farey patterns are found between the high-amplitude PIoscillations at intermediate flow rates and the low-amplitude PI* oscillations at higher flow rates. The frequency of the latter PI*oscillations is roughly twice that o f the former. For the Farey patterns the calculated experimental Lyapounov exponents show relatively low positive values which are typical
8649. .. . .
Schneider and Monster
2136 The Journal of Physical Chemistry, Vol. 95, No. 6, I991 of the system which returns to its previous state. When the standard deviation of the flow rate noise was increased to 12%, aperiodic oscillations of high amplitudes together with bursts occurred. Optimal fluctuation effects were observed for pulse lengths corresponding to the resonance frequency of the focus (about 5 s pulse length). As the average pulse period decreased, fewer bursts were observed and the average deviation from the steady-state concentration of Ce4+ decreased. In Figure 4 the standard deviation from the steady state is plotted vs the standard deviation of the superimposed Gaussian fluctuations for some experiments at kf ic: 1.05kS. It is seen that the response amplitude becomes larger than the perturbation if the perturbation amplitude exceeds a threshold value of about 12.5%. This phenomenon is an example of the amplification of fluctuations in the vicinity of a Hopf bifurcation. Simulations with the extended Oregonator yield a similar curve (not shown). Internal Fluctuations Due to Stirring (Micromixing) (Region 10. Internal concentration fluctuations are dependent on the stirring rate. In general, high-speed stirring should produce smaller sized and shorter lived fluctuations than low-speed stirring in compact CSTR. Strong stirring effects are expected in a CSTR for nonlinear reactions whose fast steps are comparable in rate with the average fluctuationfrequency in the reactor. We expect the average noise frequency to be proportional to the stirring rate to a first approximation. Smaller liquid elements should have shorter lifetimes than larger elements. It seems likely on physical grounds that the average amplitude of the fluctuations should decrease with stirring rate. Puhl and Nicolis investigated the effect of incomplete mixing by normal form analysis of a strongly nonlinear simplified chemical They found that noise-dependent bifurcations will not be introduced if the mixing time is much shorter than the lifetimes of the chemical reaction steps. If the chemical reactions occur on a similar time scale as the mixing process, however, new bifurcations may be introduced by mixing in principle. Since micromixing occurs on a time scale comparable to the rates of the fast steps, strong mixing effects are expected to occur in the BZ reaction. We earlier investigated the BZ reaction as a function of stirring rate (175-2000 rpm)18-'9at several flow rates and determined the experimental Fourier spectra, attractors, A, values, and the correlation dimen~i0ns.I~ The results confirmed the expectation that a decrease in stirring rate produces a greater aperiodicity according to all of the above measures. On the other hand, high stirring rates lead to a decrease in the positive values of A, (Figure 5a) and in a decline of Dmrrbelow 2.0 (Figure 5b) a t kf = 0.145 mi&, indicating the presence of noisy periodic motion. 7. Comparison with Other Work. Low Flow Rate in the BZ Reaction: Deterministic Chaos. The CSTR experiments of Swinney et aI.'&l4 are the most often cited examples of deterministic chaos in a chemical reaction. Using identical conditions, our results were in general agreement although some differences in the sequence of states were observed93 (see also section 5 ) . Coupled Oscillators Producing Chaos. Gyorgyi and Field56-57 suggested that deterministic chaos may arise from the diffusive coupling of two independent oscillators in different volumes in a noncompact reactor such as in the bulk reaction mixture and in the tips of the CSTR inlet ports. We have tested different diameters for the inlet tubes in the presence and absence of a dome in the reactor lid. We observed neither qualitative nor quantitative differences in the appearance of the chaotic patterns in the BZ reaction at low flow rates. Therefore, internal diffusive coupling does not seem to be an important mode of interaction in our reactor. However, diffusive coupling does produce chaos in two separate mass coupled reactor^.^^^^ High Flow Rate in the BZ Reaction: Periodic-Aperiodic Sequences. (a) Deterministic chaos was reported in early work by Hudson et aL4*at residence times of -6 min. These experiments were done in a 26.4-mL CSTR at 25 OC. The reactor was (76) Puhl, A.: Nicolis, G.J . Chem. Phys. 1987, 87, 1070. (77) Nakajima, K.;Sawada, Y.J . Chem. Phys. 1980, 72, 2231.
=e 0'0151 * A
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Figure 5. Stirring rate dependence for kf = 0.145 min-' ( 2 5 "C) near the onset of the Farey patterns (11): (a, top) A, versus stirring rate (rpm). A,, declines to about 0.002. The two points at 2000 rpm correspond to inefficient mixing in our CSTR. (b, bottom) Dmn vs stirring rate (rpm). Most values are below 2.0, indicating noisy periodic motion and not chaos.
fed by a four-channel peristaltic pump. Four Plexiglas baffles ensured turbulent mixing of the inflowing solutions while the reaction mixture was stirred by a glass stirrer a t 2000 rpm. In earlier experiments, chaos was reported by Schmitz et al. in the ferroin-catalyzed BZ reaction 1. The Ce-catalyzed reaction showed various periodic and aperiodic oscillation patterns. At rrCE = 6.35 min an oscillation pattern was observed (Figures 3-5 in ref 42) in the potential of a platinum electrode which may be classified as 1'. At rrCE= 5.68 min the authors obtained a l 2 pattern. At a residence time of 6.01 min, aperiodic oscillations were obtained between those two patterns which appeared to be a statistical mixture of the 1' and 1*mixed-mode oscillations. The authors interpreted this sequence of oscillation patterns as a periodic-chaotic sequence and used the Fourier spectrum and the return map to support this assertion. On the other hand, it is certain that in the aperiodic time series at r,, = 6.01 min the variations in the flow rate were sufficiently large to move the system back and forth between the basins of the two neighboring periodic Farey attractors. This process may produce a positive Lyapunov exponent which, in this case, is indicative of amplified noise. (b) We draw similar conclusions from the detailed experimental work of Argoul et al?5 who report the existence of periodic-chaotic sequences as a function of flow rate in the mixed-mode region of the BZ reaction at 43 "C. Despite their argumentation for deterministic chaos we suggest an interpretation of their results in terms of statistical noise since they state that the fluctuations in the flow rate were of the same order of magnitude as the parameter range they expl~red?~ In the large part they base their argumentation for the periodic-chaotic sequences on a new reduced version of the FKN mechanism. However, Gy6rgyi and Field73 have criticized their model for chemical reasons. Moreover, at least some of Argoul et al.'s calculated microscopic chaos in their periodic-chaotic sequences was reported by Gy6rgyi and Field to be a numerical artifact. Using the periodic Oregonator with superimposed fluctuations (f3%) we have simulated periodicaperiodic time series which show great similarity with Argoul et al.'s experimental results. Thus the 'unavoidable" statistical fluctuations are most likely the basis for the aperiodic sequences found in their experiments. (c) Maselko and Swinney2' observed exclusively periodic Farey sequences without any chaos in the Mn-catalyzed BZ reaction which they interpreted as frequency locking on a torus. Our present work shows a similar behavior for the Ce-catalyzed BZ
Feature Article reaction a t high flow rates which, however, seems to be more sensitive toward experimental fluctuations than the Mn-catalyzed reaction. It is likely that the underlying chemical mechanisms are similar for the two systems considering the phenomenological similarity they exhibit. This chemical analogy is consistent with our conclusion that, at high flow rates, chemical chaos has not been shown to exist in the BZ reaction, although this particular plausibility argument is not a proof. Quasiperiodic Route to Chaos. The transition from quasiperiodicity to chaos occurs via the wrinkling of a torus as studied extensively in circle mapsO4 Hydrodynamic experiments have clearly shown this r o ~ t e , whereas ~ ~ , ~ the ~ situation in the BZ r e a c t i ~ is n not ~ ~ uniquely ~ ~ ~ resolvable. Inspection of the power spectrum does not provide a clear distinction between experimental quasiperiodicity and chaos."I Stirring Effects. Stirring effects are very important in any nonlinear chemical reaction through the macro- and micromixing process. Stirring effects were reported earlier by RuofPa3 in the BZ reaction in a batch reactor and by Roux et al.84in the bistable ClO;/I- system in a CSTR where noise-induced transitions were observed. An interpretation of mixing effects in the latter system was given by Boissonade and D e K e p ~ e r . In ~ ~the same system Luo and Epstei# observed a shift from the oscillatory region to a noisy steady state, when the stirring rate was increased. Menzinger and JankowskiE7observed important premixing effects in the feedstreams. In a recent paper Dutt and MenzingeP demonstrate the different dynamical roles played by stirring and premixing of reactants. They emphasize the importance of heterogeneous effects on chemical instabilities. In the BZ reaction spatially distributed concentration fluctuations were found by Menzinger and J a n k o w ~ k in i ~a~stirred batch reactor where effects of the electrode surface may become important. studied the effect of stirring rate in a particular Argoul et BZ experiment (flow rate 0.45 mL/min) in which they observed a transition from an aperiodic to a periodic state upon increasing the stirring rate from 600 to IO00 rpm, respectively. We conclude that coupling of the effective fluctuations with individual steps in the reaction was optimal at -600 rpm in this case. As a consequence, an aperiodic state is observed at 600 rpm. An increase in stirring rate produces smaller liquid elements of necessarily shorter lifetimes. Thus the coupling between fluctuations and the chemical oscillator may be less effective at 1000 rpm. Therefore, the system stabilizes in the basin of one of the neighboring Farey attractors which represents the observed periodic state. On the other hand, at a higher flow rate of 0.55 mL/min, Argoul et al. observed a transition from a periodic to an aperiodic state (Figure 16a in ref 25) when the stirring rate was raised from 600 to 1000 rpm, respectively. They stated that this "experiment may cast a doubt on the deterministic interpretation of the observed nonperiodic states if we associate stirring rate with noise". We did not observe a transition from a periodic to an aperiodic state in identical experiments in our compact reactor. However, a possible interpretation may be offered in terms of the above-mentioned coupling effectiveness between the average fluctuation frequency and the rates of the individual reaction steps. In other words, coupling seems to be more effective at 1000 rpm than at 600 rpm at the higher flow rate. More work needs to be done to further resolve this point. (78) Dubois, M.; Bergs, P.; Croquette, V. J . Phys. (Paris) Lett. 1982.43, L295. (79) Swinney, H. L.; Gollub, J. P. J . Fluid Mech. 1979. 94, 103. (80) ROUX,J. C.; Rossi, A. In Non Equilibrium Dynamics in Chemical Systems; Vidal, C., Pacault, A., Eds.; Springer: Berlin, 1984; pp 141-145. (81) Dumont, R. S.; Brunner, P. J . Chem. fhys. 1988,88, 1481. (82) Ruoff, P. Chem. fhys. Lett. 1982, 90, 76. (83) Ruoff, P.; Noyes, R. M. J . fhys. Chem. 1986, 90, 4700. (84) Roux, J. C.; DeKepper. P.; Boissonade, J. Phys. Lett. A 1983, 97A, 168. (85) Boissonade, J.;.DeKepper, P. J . Chem. Phys. 1987, 87, 210. (86) LUO,Y.; Epstein, 1. R. J . Chem. Phys. 1986, 85, 5733. (87) Menzinger, M.; Jankowski. P. J. J . Phys. Chem. 1986, 90, 1217. (88) Dutt, A. K.; Menzinger, M. J . Phys. Chem. 1990, 91,4867. (89) Menzinger. M.; Jankowski, P. J. J . Phys. Chem. 1986, 90, 6865.
The Journal of Physical Chemistry, Vol. 95, No. 6,1991 2131
IV. Discussion and Conclusions At long residence times (>30 min) deterministic chaos may be observed in the BZ reaction if purified (recrystallized) malonic acid is used in general agreement with experiments done by Swinney and co-workers. The chaos observed may be considered as being intrinsic to the chemical mechanism. At short residence times (4-7 min) one observes the well known periodic-aperiodic sequence in the so-called mixed mode region at a constant stimng rate of 1500 rpm. The ordering of the observed periodic oscillation patterns follows the Farey arithmetic as also predicted by the extended Oregonator model. When the flow rate is adjusted between two resolvable Farey oscillations, any sufficiently large fluctuations push the bifurcation parameter back and forth between the basins of attraction of the two oscillators. The resulting time series have an aperiodic appearance and may give rise to positive Lyapounov exponents and fractal correlation dimensions above 2.0, although the aperiodicity is statistical in nature. Furthermore, the one-dimensional maps constructed from the PoincarB sections of their attractors represent clouds of random points. The statistical origin of the aperiodic time series is s u p ported by simulations with a periodic model, namely the extended Oregonator with Gaussian distributed noise which is superimposed on the flow rate. The simulations yield periodic-aperiodic sequences with similar A, values as those observed in experiments whereas the noise-free Oregonator model exclusively produces periodic Farey attractors with ,A, = 0 and DWrr= 1. We distinguish between two kinds of noise in a chemical mechanism: External noise superimposed on external parameters like the flow rate and internal micromixing or stirring noise. In the calculations the former acts on all linear flow terms whereas the latter acts on all chemical species including the nonlinear terms in the rate equations. In the steady state as well as in the oscillatory region close to a Hopf bifurcation, dramatic fluctuation effects are also found by superimposition of Gaussian distributed noise on the flow rate. The steady state is shown to be excitable by the appearance of aperiodic oscillations with intermittent bursts which originate from statistical crossings of the upper Hopf bifurcation. In general, fluctuations are amplified only in the vicinity of a bifurcation since noise may induce transitions between the focus and the oscillatory region. At large distances from a bifurcation fluctuations are damped. Noise amplification in a focus close to a Hopf bifurcation is most efficient if the "pulse period" of the superimposed flow noise is close to the natural resonance frequency of the focus. The experiments on excitability are successfully modeled by the periodic Oregonator with superimposed flow fluctuations. In the oscillatory region of the BZ reaction one may use a similar perturbation frequency of the flow fluctuations as that in the steady state, in agreement with simulations. Stirring and "natural" CSTR noise are interrelated: A distribution of noise frequencies is likely to occur, where the noise frequency is approximtely proportional to the stirring rate. Fluctuations caused by micromixing act upon all chemical species including intermediates. Since the nonlinear steps of the mechanism are particularly sensitive toward fluctuations close to a bifurcation, stirring effects will be most important if the lifetimes of the local concentration variations are comparable to the rates of the nonlinear steps. At stirring rates below 600 rpm this condition is fulfilled in our compact reactor at kf = 0.125 m i d and optimal coupling between the fluctuations and the mechanism is achieved. As a consequence, aperiodic oscillations are observed. An increase in stirring rate above 600 rpm decreases the lifetimes of the liquid filaments and leads to less effective coupling; Le., the time series observed become more periodic. The scale of the stirring noise covers a wide time range between some milliseconds up to seconds. Flow rate fluctuations which are shorter than one second have little effect on the BZ reaction. The "optimal" flow fluctuation frequency is comparable to the resonance frequency of the focus; Le., the time scale of experimental flow rate fluctuations is in a range between 5 and 10 s. In contrast to stirring fluctuations, flow rate fluctuations act
2138
J. Phys. Chem. 1991, 95, 2138-2143
only indirectly on the nonlinear steps. The amplitudes of the flow rate noise must be chosen in the percent range in experiments and simulations whereas numerical simulations of stirring noise lead to profound effects when the noise amplitudes are in the ppm range. For large noise (flow or micromixing) a distinction between amplified fluctuations and deterministic chaos is no longer possible as also shown in Brusselator calculations,’**68the Duffing equation,m and the Selkov model.m The effects of deliberately imposed fluctuations and the stirring effects may be well understood on the basis of the extended Oregonator model with superimposed fluctuations on the flow rate or any other bifurcation parameter. Furthermore, the Oregonatorso may formally explain the observed1”l4 deterministic chaos at low flow rates if the rate constants are chosen accordingly. At high flow rates the observed periodic-aperiodic sequences are explained by the extended Oregonator as Farey ordered periodic states including noise. Thus we do not find it necessary to explain these findings by intrinsic deterministic chaos in the BZ reaction at high flow rates in the mixed-mode region. In current work we found that the methylene blue-sulfideoxygen oscillator shows a subcritical Hopf bifurcation at high flow rates which is particularly sensitive toward external fluctuations.w In addition, in the oscillatory chemiluminescence of luminol the critical flow rate at the transition from oscillations to a steady state was found to dramatically vary with stirring ratea9’ Since stirring effects are of great importance in nonlinear chemical kinetics, it seems desirable to use a “standard reactor” (90) Resch, P.; Milnster, A. F.; Schneider, F. W. Submitted for publication in J . Phys. Chem. (91) Amrehn, J.; Schneider, F. W. To be published.
for comparison of experiments done in different laboratories. A compact reactor with cylindrical shape and baffles has been suggested as a standard. Another standard reactor design might be a simple I-cm spectrophotometric cell with 1.5 cm height (achieved by an elongated Teflon stopper) which is stirred with high energy dissipation by an asymmetric magnetic stirring bar at 1500 rpm. Such a relatively simple reactor could be built for spectrophotometric observation in any laboratory. A possible role of periodicity in nature is to act as a pacemaker. However, periodicity may be pathological in neural networks.92 Periodicities are obscured by large fluctuations. On the other hand, deterministic chaos in nature may act as a facile absorber of large external and internal fluctuations. Large fluctuations increase the dimensionality of chaotic motion. Conventional kinetics has given valuable information about the dynamics of elementary steps. The complexities of nonlinear dynamics have now captured the imagination of many experimentalists and theorists. Further progress in this new discipline of physical chemistry will depend on the interplay between well-defined and precise experiments and computer simulations of chemically reasonable theoretical models. Acknowledgment. We gratefully acknowledge helpful discussions with A. Arneodo, H. P. Kraus, F. Buchholtz, A. Freund, Th. M. Kruel, and H. L. Swinney. We thank I. R. Epstein, R. J. Field, J. Ross, and J. Villermaux for preprints of their work. We also thank the Volkswagenstiftung and the Fonds der Chemischen Industrie for partial financial support. (92) West, B. J.; Goldberger, A. L. Am. Sei. 1987, 75, 357. (93) Schneider, F. W.; Miinster, A. M. To be published.
ARTICLES Quenchhg of Excited Cd(’PJ) Atoms by Alkane Hydrocarbons Shunzo Yamamoto* and Hiroyuki Hokamura Department of Chemistry, Faculty of Science, Okayama University. 3- 1-1, Tsushima-naka, Okayama 700, Japan (Received: April 24, 1990; In Final Form: July 27, 1990)
The intensity of 326.1-nm Cd vapor resonance radiation was measured at various temperatures between 280 and 485 ‘C and at various pressures of some alkanes. The quenching rate constants were determined on the basis of Stern-Volmer plots, using values of the imprisonment lifetime of Cd(’P,) estimated from the quenching rate of Cd(’P,) by hydrogen. The temperature dependence of the quenching rate constant for ethane, ethane-& propane, butane, and isobutane is expressed by the three-parameter equation k , = A + B exp(-E/RT). At 485 ‘C the second term of the above equation dominates, and the k , value shows a dependence on C-H bond strength. This dependence can be explained by a mechanism that allows hydrogen atom abstraction by an excited cadmium atom exclusively from the weakest C-H bond. Introduction A number of investigations of the mercury,’-3 cadmium,“ and zinc’ photosensitized decompositions of alkanes have been made ( I ) Cvetanovic, R. J. Prog. React. Kine?. 1964, 2, 39. (2) Calvert, J. G.; Pitts, J. N., Jr. Photochemistry; Wiley: New York, 1966; p 434. (3) Campbell, J . M.;Strausz, 0. P.; Gunning, H.E. J . Am. Chem. Soc. 1973, 95, 740. (4) McAlduff, E. J.; Yuan, Y. H.J . Photochem. 1976, 5 , 797. (5) Konar, R. S.; Darwent, B. deB. J . Chem. Soc., Faraday Trans. I 1978, 74, 1545. ( 6 ) Konar, R. S.; Darwent, B. deB. Ind. J . Chem. 1979, 18A, 95.
by means of product analysis. The primary processes were suggested to be C-H bond rupture, with the resultant formation of alkyl radicals and metal hydride in the cases of cadmium and zinc and formation of alkyl radicals and hydrogen atoms in the case of mercury. For m e r c ~ r yand ~ , ~zinc,I0 remarkable dependences (7) Yamamoto, S.; Kozasa, M.; Sueishi, Y . ;Nishimura, N. Bull. Chem. Soc. Jpn. 1988, 61, 3439. (8) Holroyd, R. A.; Klein, G . W.J . Am. Chem. Soc. 1963, 67, 2273. (9) Gunning, H. E.; Campbell, J.; Sandhu, H.;Strausz, 0. P. J . Am. Chem. SOC.1973, 95, 746. (10) Yamamoto, S.; Nishimura, N. Bull. Chem. Soc. Jpn. 1982,55, 1395.
0022-3654/91/2095-2138%02.50/0 0 1991 American Chemical Society
