J. Phys. Chem. 1987, 91, 4391-4393
439 1
Stirring Effects in Chemical Instabilities: Heterogeneity Induced Oscillations in the C102- 4- I-Reaction Michael Menzinger* and Albert Giraudi Department of Chemistry, University of Toronto, Toronto, Ontario M5S l A l , Canada (Received: January 5, 1987) When studied in a CSTR at a constant reactant concentration, the CIOz- + I- system was found to oscillate only within a narrow range of stirring rate. The waveform, amplitude, and period of the oscillations are found to depend dramatically on stirring, as is the variance (jitter) of the latter attributes. The probable role of heterogeneitiesin controlling these phenomena is discussed.
The effect of stirring on chemical instabilitiesI4 demonstrates, on one hand, that the reactor content is not homogeneous and, on the other, that the nonlinear dynamics depends sensitively on its degree of segregation. While the origin of heterogeneities in a CSTR is now understood2-they derive primarily from the incomplete mixing of separate (Le. nonpremixed NPM) reactant feedstreams-much remains to be learned, however, about their dynamical consequences. In the past, stirring effects have been studied primarily in the C 1 0+ ~ I- reaction, where the role of heterogeneities in reducing the domain of bistability is well documented.Ig2 In their original work on the C10,- I- reaction, Dateo et aL5 reported bistability and limit cycle oscillations, believed to occur in a homogeneous medium, and summarized their findings by a cross-shaped bifurcation diagram in concentration space at constant residence time and stirring rate. Luo and Epstein4 recently reexamined the oscillatory and nonoscillatory regions of the same reaction at several compositions, flow rates, and stirring rates and established that, in addition to reactant composition, stirring as well as pumping could act as bifurcation parameters. Except for an extremely narrow window in composition and flow rate, premixing (PM) effectively suppressed the oscillations. Instead of the two steady states recognized in the past, Luo and Epstein identified three steady labelled LI, MI, and HI, two of which (LI and MI) are claimed to belong to the thermodynamic branch. In this article we report the remarkable effect of stirring rate S on the shape, period, and amplitude of oscillations at a given composition and residence time. We find that oscillations occur under NPM conditions only within a narrow stirring window S,,, < S < S max, outside of which the system settles into two different steady states. This demonstrates that the oscillations, rather than being due to a limit cycle supported by the homogeneous system, are in some way induced by heterogeneities. In addition, we analyze the surprisingly large irregularities of the oscillation period and amplitude in order to learn more about the interaction of external heterogeneous noise with the dynamical system. The experiment was conducted in a standard2 CSTR (volume = 36 cm3, thermostated at T = 24.3 0.2 "C), stirred by a 16-mm glass propeller whose rate could be accurately adjusted over the range of S = 50-2000 rpm. The emf of a standard213 Pt macroelectrode (0.025-cm-diameter Pt wire, 10 mm long) was monitored relative to a Hg/HgS04 reference. Three reactant feedstreams (1) KI, the buffer, (2) N a 2 S 0 4+ H2SO4, and (3) NaC102 + N a O H were pumped at equal rates by a peristaltic pump. Feeds (1+2) were routinely combined prior to entry into
+
+
(1) Roux, J. C.; DeKepper, P.; Boissonade, . Phys. Lett. 1983, 97A, 168. Roux, J. C.; Saadaoui, H.; DeKepper, P.; Boissonade, J. Springer Proc. Phys. 1984, 1 , 70. (2) Menzinger, M.; Boukalouch, M.; DeKepper, P.; Boissonade, J.; Roux, J. C.; Saadaoui, H. J . Phys. Chem. 1986, 90, 313. (3) Menzinger, M.; Jankowski, P. J . Phys. Chem. 1986, 90, 1217. (4) Lou, Y.; Epstein, I . R. J . Chem. Phys. 1986, 85, 5733. (5) Dateo, C.; Orban, M.; DeKepper, P.; Epstein, I. R. J . Am. Chem. SOC. 1982, 104, 591 1 .
0022-3654/87/2091-4391$01.50/0
the reactor. The resulting flows (1+2) and (3), however, could be made to enter the reactor separately (nonpremixed, NPM mode) through two narrow ports at the bottom of the reactor, separated by 1.0 cm, or they could be combined in a Y-shaped mixing capillary 6 cm from the CSTR (PM mode). The transit time from mixer to reactor was 0.5-2.5 s, depending on the flow rate. To enlarge the oscillatory region and to minimize the occurrence of iodine pre~ipitation,~ we chose the Na2S0, + H2S04 buffer to produce pH 1.1 in the reactor. Initial concentrations in the reactor are quoted in Figure 1. The remarkable influence of the stirring rate S on the oscillations is illustrated in Figure 1. In this recording S was decreased in discrete steps. It is found that (1) oscillations occur only within a narrow range, 300 < S, < S < S, < 415 rpm; (2) the period, amplitude, and waveform all depend dramatically on S; and (3) the oscillations are surprisingly jittery: the high amplitude oscillation near S,, has an extremely irregular long period ( ( T ) = 260 s, standard deviation u7 = ( ( ( 7 ) - T , ) ~ ) = ~ /100 ~ s) and constant amplitude, while the rapid, low-amplitude oscillations are characterized by a nearly constant period but near Smin considerable irregularities of the amplitude. (4) In the nonoscillatory regime S > S,,, beyond the upper stirring limit, the potential depends very sensitively on stirring. It decreases from t = 375 mV to c = 180 mV as S is varied from 1400 to 500 rpm, in qualitative argeement with previous observation^.'^^,^ Supplementary experiments show that (5) premixing suppresses the oscillations at all stirring rates (see, however, ref 4), (6) both bifurcations at S,,, and S,,, occur without hysteresis, and (7) the intermediate stationary state MI4 obtained at S > S,, ( t = 180 mV) is monostable and excitable, either by injecting I- or by monemtarily suppressing the Cloy flow. The transient induced by a supercritical perturbation is similar in shape to that of the high-amplitude oscillations near S, (Figure 1). In this connection it should be pointed out that, in the course of the high-amplitude oscillation (Figure 1) near S,,, the system resides mostly on an intermediate potential (the intermediate or MI state identified4 as a second thermodynamic branch), from which it undergoes at irregular intervals large-amplitude excursions to the low-potential (flow branch, HI4) and high-potential (thermodynamic branch, Li4) states. For the same point in concentration space as the one presented in Figure 1, the stirring effect was studied quantitatively at several residence times T . Figure 2 displays the S dependence of (a) the average amplitude ( A ( S ) ) ,(b) its relative variance u A ( S ) / ( A ( S ) ) , (c) the average period ( ~ ( sand ) ) ,(d) its variance u,(S). The arrows in the (7(S))plot indicate points just outside the oscillatory regime and they serve to bracket the values of Smin and S,,,. Typically, ( 7 ) and ur increase rapidly with S while ( A ) and uA level out as S approaches S,,, from below. The bifurcation at S,,, is characterized by a vanishing6 of ( A )and the sharp rise of the relative dispersion of the amplitude uA/ ( A ) ,accompanied by low and nearly constant values of (+)) and uT. The rise in ( 7 ) and u7 near S,,, is particularly pronounced at the low residence time T = 285 s. The bifurcation near S,,, is clearly of 0 1987 American Chemical Society
4392
The Journal of Physical Chemistry, Vol. 91 No. 16, 1987
Menzinger and Giraudi
~
1”r1
30 415
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’r
s.333
I
E
I
0
1 IO
I
I 20
1
I 30
I 40
I
I
I
50
TIME (min) Figure 1. The effect of stirring on oscillations. [KI] = 1.72 X lo-’ M, [NaCLOJ = 5.7
M, p H 1.1; ko = T’= 3.1
X
S /rpm
( 6 ) The amplitude ( A @ ) )does not follow the parabolic law that characterizes a Hopf bifurcation since the latter is restricted to the immediate neighborhood of the bifurcation point. ( 7 ) (a) Puhl, A., private communication. (b) Puhl, A.; Nicolis, G . submitted to J . Chem. Phys. (8) Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields; Springer: New York, 1983. (9) Schuster, H. G. Deterministic Chaos; Physik Verlag: Weinheim, 1984. (10) Olsen, L. F.; Degn, H. Nature (London) 1977, 267, 177.
lo-’ s-’
S/r pm
Figure 2. Stirring rate dependence of (a) mean amplitude ( A ) ,(b) Its relative variance A / ( A ) ,(c) mean period for 3 residence times T [in seconds]. Concentrations as in Figure 1.
the supercritical Hopf type.6 On the other hand, the sharp rise of ( ~ (near 9 )S,, especially at T = 285 s, suggests the proximity in parameter space of an infinite period (saddle-loop or homoclinic orbit) b i f u r c a t i ~ n . ~ ~ ’ ~ ~ The irregularities of period and amplitude raise the question wether they reflect deterministic chaos or the result of external stochastic noise. To probe this point we present in Figure 3 the first return maps9 of periodt0 T ~ =+f ( T~N ) and amplitude ANfl = g(A,), prepared from two time series near S,,, and S,,,, respectively. In the case of deterministic chaos this procedure is expected to yield an iterative map of well-defined shape>10 while in the case of stochastically dominated deviations from regularity one expects a random distribution of points. All four panels of Figure 3 show that the latter is the case. Hence we are led t o conclude that the observed irregularities are stochastic in nature. Turbulent mixing is known to give rise to a broad distribution of irregularly shaped vortices of the entering material embedded
X
(T), and
(d) variance u7 of period,
in an essentially homogeneous bulk.” The distribution function of these heterogeneities and the degree of segregation depend sensitively on stirring. The existence of an upper stirring limit S,,, in the present experiments demonstrates that a minimum degree of segregation is required for oscillations to occur and that the oscillations depend in some way on the heterogeneity. The lower limit, on the other hand, shows that the segregation must not be too extreme for oscillations to occur. The principal problem in the present context is to understand the role which heterogeneity plays in generating the oscillations and the associated bifurcation phenomena. Given that the Pt electrode responds primarily to -In [I-], the first 10 min of the recording in Figure 1 show that decreased stirring strongly enhances the [I-] concentration in the bulk. How this is possible can be seen by considering the reaction mechanism proposed by Epstein and Kustin12which involves two competing pathways for C102- consumption: Under well-mixed conditions (i.e., at very high S and/or PM conditions) the reactants are in intimate contact from the start and they are converted via the net reaction C102- 41- 4H’ C1- + 212 + 2 H 2 0 (1)
+
+
-
which is known to be autocatalytic in I, and inhibited by 1’. When (1 1) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewocd Cliffs, NJ, 1962. (12) Epstein, I. R.; Kustin, K. J . Phys. Chem. 1985, 89, 2275.
Stirring Effects in the C l o y
+ I- Reaction
The Journal of Physical Chemistry, Vol. 91, No. 16, 1987 4393
S.400
S.310
E
a 0.91 \ . . .
+ 2
. . . .
3.8 0.8
AN /
0.9 A
I
i I2
~
Figure 3. First return maps of period N + 1 = f ( ~ and ~ of ) amplitude = g(AN) for oscillations near S,,, (top) and S,,, (bottom).
A,,,
the system is on a thermodynamic branch, where the I2 concentration is high, reaction 1 is virtually instantaneous and hence the system remains on that branch, characterized by low [I-] and [C102-] and high [I2]. If, on the other hand, the reactant feedstreams are not premixed and the stirring is slow, reaction 1 is limited by the mixing process, and the competing C10,consuming pathway 5c102-
+ 21, + 2 H 2 0
-
5C1-
+ 4103- + 4H’
(2)
takes over. In this mode reactant iodide is consumed less rapidly by subsequent reactions than it is by (1) and consequently the average [I-] concentration rises and the electrochemical potential drops. This crossover between (1) and (2) is evident in Figure 1 from the decrease of the electrode potential with S at S > S,,,. It is reasonable to assume that this trend continues into the region below S,,,. One mechanism by which oscillations may arise in a system that remains in a steady state under homogeneous (high S) conditions is through the action of the S-dependent bulk concentration [I-] discussed a moment ago. This [I-] can act as a homogeneously distributed bifurcation parameter whose tuning via S gives rise to the birth and the death of a limit cycle in an otherwise homogeneous medium. In this picture the irregularities in amplitude and period (Figure 1, 2b, and 2d) can be thought of as being due to random perturbations of the limit cycle by discrete packets of reactants before they have been mixed into the bulk. The jitter in 7 near S,,, is plausible in view of the noise susceptibility of trajectories in the vicinity of a homoclinic orbit,9 generally referred to as “sensitive dependence on initial conditions”. Several (homogeneous) two-dimensional dynamical models are known to exhibit the bifurcation structure (supercritical Hopf and a nearby a saddle-loop or infinite period bifurcation) observed in the present experiments. Examples are the cubic autocatalysis A 2X 3X, followed by catalyst decay X C, as analyzed by Gray and Scott,16 and a more specific mode analyzed by
+
-
-
~
Othmer.13 The former illustrates how exotic bifurcation behavior, including the above parameter dependence of the oscillation attributes, may arise from a simple and fundamental kinetic paradigm. Puhl and Nicolis” have recently been able to predict stirring effects very similar to the ones reported here. They based their analysis on the normal form (to which dynamical equations reduce in the vicinity of a bifurcation point, through the adiabatic elimination of fast modes), derived from an extended version of the cubic autocatalytic model. That normal form was subsequently subjected to a previously developed analysis which accounts for the effect of incomplete mixing. The results confirm the above view, that stirring may tune the system through a bifurcation sequence that preexists already in the homogeneous case. An alternate scenario for the heterogeneity induced oscillations is based on localized excitations from a deterministically monostable but excitable state.I4 It differs from the previous model through the random nature of the oscillations. That this might be an appropriate picture, at least in the neighborhood of the upper stirring limit, is suggested by the confirmed excitability of the steady state beyond S,,,, the similarity of the transients above S,, with the oscillations at S < S,,,, and the stochastic nature of the iterative maps (Figure 3). According to this picture, supercritical I- heterogeneities, generated at random times by incomplete mixing, cause local excitations which nucleate and grow into transients14 on a large scale, followed by a return to the same ~ steady state from which excitation occurred. This would account for the extremely irregular intervals between transients. The conspicuous noise on the intermediate (MI) potential (see Figure 1) reflects the relaxation of subcritical perturbations. Indeed, closer examination of this noise reveals a great number of sawtooth-shaped transients composed of a sudden rise and a slower, , the irregular periods reflect the first monotonic decay. Near S passage time of supercritical perturbations. As S decreases, the average size of the heterogeneities increases, and the excitation threshold as well as the amplitude of the response-transient decrease due to the above-mentioned increase of [I-] in the bulk. The rate of arrival of the supercritical perturbations rises rapidly with decreasing S, and the point is soon reached where the system suffers an excitation as soon as it has recovered from the refractory portion of the transient.l4 This would account for the simultaneous plays a fundamental decrease of (7)and u7 with S. E~citability’~ role in wave propagation in chemical and neuronal systems. A quantitative study of this type of noise-induced limit cycle was performed by Treutlein and S ~ h u l t e n for ’ ~ the Bonhoeffer-Van de Pol model of nerve impulses. The similarity of their oscillations and those observed in this work in striking. Our experimental evidence does, however, not permit us to distinguish between the two dynamical alternatives described above. This work confirms the previous suggestion3that heterogeneities may dramatically affect the dynamics of chemical oscillations and demonstrates that it can even introduce bifurcation structure that is absent from the homogeneous case.
Acknowledgment. We express our thanks to Andreas Puhl and Prof. G. Nicolis for helpful correspondence and for sending the manuscript of ref 17 and to Prof I. Epstein for making ref 4 available prior to publication. This work is supported by the NSERC of Canada. (1 3) Othmer, H. G. In Oscillations and Travelling Waves in Chemical Systems, Field, R. W., Burger, M., Ed.; Wiley: New York, 1985. (14) Field, R. W.; Noyes, R. Faraday Symp. Chem. SOC.1974, 9, 21. ( 1 5 ) Treutlein, H.; Schulten, Ber. Bunsen-Ges. Phys. Chem. 1985,89, 710. (16) Gray, P.; Scott, S. K. Chem. Eng. Sci. 1984, 39, 1087. (17) Puhl, A,; Nicolis, G. J . Chem. Phys., submitted for publication.
