6405
J . Phys. Chem. 1993,97, 6405-6411
Excitations Induced by Fluctuations: An Explanation of Stirring Effects and Chaos in Closed Anaerobic Classical Belousov-Zhabotinsky Systems Peter Ruoff Department of Chemistry, Rogaland University Center, Box 2557, Ullandhaug, 4004 Stavanger, Norway Received: November 13, 1992; In Final Form: March 2, 1993
In this paper, the effect of stirring and the appearance of chaotic oscillations in closed oxygen-free.BelousovZhabotinsky systems are explained in terms of fluctuation-mediated excitations.
TABLE I: Composition of Studied BZ Systems
Introduction Theclassical Belousov-Zhabotinsky (BZ) reaction is the metal ion catalyzed oxidation and bromination of an organic substrate by bromate ion in acidic solutions.14 The BZ reaction has attracted researchers for more than two decades, and even in closed systems a variety of unusual complex behaviors can be observed that are normally expected to be found in biological ~ystems.~ Such behaviors include, for example, concentration oscillations of reaction intermediates: traveling waves of oxidation' or reduction8pulses, or excitability?JO Chemical chaos'' was first observed in BZ systems run in continuously stirred tank reactors12 (CSTRs), and the phenomenon is now widely studied by a variety of groups with many different chemical systems.*3 In most of the CSTR experiments, chaotic behavior has been recognized and interpreted as deterministicchaos; Le., the chaotic response is not attributed to the influence of random noise but to the deterministic (but highly irregular) dynamics generated by the coupling of component processes in the reacting chemical system. In this paper, we will use the word "chaos" to describe either deterministic chaos or a system's ability to mediate/amplify stochastic fluctuations of an environment in the form of an irregular response. Due to work by Field, K6riis, and Noyes (FKN),14the chemical mechanism of the oscillatory BZ reaction is now well understood. An important characteristic of the mechanism is that a key role is attributed to bromide ion, an intermediate of the reaction. Bromide ion has a control function; i.e., the kinetic state of the system depends on its concentration. Above a certain critical bromide ion concentration, [BrIcrit,the system is in a reduced state ("state/process A"),15 and bromide ion is consumed by its reaction with bromate, HBr02, and HOBr, forming Br2, which reacts with the organic substrate RH. 5Br-
+ BrO; + 6H'
Br,
+ RH
-
-
BrR
3Br2
+ 3H20
+ H+ + Br-
During process A, bromide ion is consumed, and when it drops below [BI-]&~, an autocatalytic production of HBr02 starts. At this point, the BZ system switches rapidly to an oxidized state ("statelprocess B"),15 where the reduced form of the metal ion catalyst Mn+is oxidized to M("+l)+according to the following stoichiometric component process: BrOT
+ 4M"' + 5H+
-
4M("+')+
+ HOBr + 2 H 2 0
(B)
Process B starts a set of complicatedand still not fully understood radical reactions forming (again) bromide ion, which quenches process Band throws the system back to a reduced state A. Thus, oscillations in a classical BZ reaction are understood in terms of a switching mechanism between states A and B. 0022-365419312097-6405$04.00/0
applied stock solution v01,mL 4 1.25 X 1t2M Ce(1V) in 2.27 M H2S04 1.75 M malonic acid in water 4 10 0.25 M KBrO3 in water 2.27 M
10
final concn,M 2.1 x 10-3 0.29
0.10 0.95
A closely connected phenomenon to the switching between states A and B is the excitability property of BZ systems. Excitability was first theoretically predicted by Field and N0yesg and later found experimentallyin closed stirred systems by Rwff.lo The excitability of BZ systems is assumed to be a key function in the understanding of chemical wave phenomena observed in unstirred solutions.~6J7 Many18 of the experimentallystudied excitable systems are in a steady state where the bromide ion concentration is just above [Br]crit. Although this steady stateis locally stable, a perturbation that drives the bromide ionconcentrationbelow [Br]critwill result in a single oxidation spike or, at stronger perturbations, in a train of oscillations?J0J5 As a result, BZ systems where the bromide ion concentrationis near [ B r ] d tmay become extremelysensitive to small local variations in the bromide ion concentration. In this paper, we show that such local variations may be the source of stirring effects observed in batch systems that are run in an inert atmosphere of argon or nitrogen. We also show that chaotic behavior found under batch conditionscan be explained similarly;Le., experimentallyobserved chaoticspikesare explained as an "all-or-none" l9 filtering of "concentration noise" by the excitability property of the BZ reaction.
Material and Methods
Experiments. All experiments were performed isothermally at 25 OC (kO.1 "C) in a batch reactor in an inert atmosphere of argon. Details of the reactor can be found in ref 20. Oscillations were recorded using a bright platinum electrode against a Ag/ AgCl double junction reference electrode (the outer electrolyte was 1 M sulfuric acid, which was also the reaction medium). Stirring of the reaction solution was provided by means of either a magnetic stirrer with a Teflon-coated stirring bar or a stainless steel propeller driven from above. In both cases, the same reactor was used, and no significant differencesin dynamic behaviors between these two ways of stirring were observed. All chemicals were of analytical quality and used without further purification. Stock solutions of reagents were prepared in water or in appropriate sulfuric acid solutions (Table I). The oscillator was started by mixing reactants in the following order: sulfuric acid, malonic acid, and KBrO3. At this stage, stirring was started and wet argon was bubbled through the mixture for about 5 min. The oscillator was started by adding the Ce(1V) solution. After the addition of Ce(IV), Ar bubbling was stopped, but argonwasstillallowedto flow above thesurfaceofthereacting (b
1993 American Chemical Society
6406 The Journal of Physical Chemistry. Vol. 97, No. 24, 1993
I
Ruoff
Atter the call’s lifetime T the content of all cells is mixed and the new cell concantrations are determined lrom : Cli (t+7) =
(l*~)~C,,,,,,,
(t+r)
I 350 rpm I
750 rpm
5 Min
,
I
350rpm
I
> Time
Figure 2. Stirring effect in a classical Cc(1V)-catalyzed BZ reaction under an argon atmosphere (after Li and Li).2s
Figure 1. Cell model of stirring applied to the Oregonator. c&) is the in cell “f’.In addition, cells 1, 2, concentration of chemical species “in 3, ..., G~are excited.
solution to assure that the Ar pressure above the solution was higher than the pressure of the outer atmosphere. The flow rate of Ar gas was approximately 1 L/min. Escape of Ar from the otherwise closed reactor occurred through a paper stopper. Computatiolrs. Doubleprecision computationswere performed on Sun SPARC stations, on Hewlett-Packard 9000/700 computers, and on a Cray Y-MP computer by integrating the rate equations of the Oregonator model with the FORTRAN subroutine LSODE.21 Double-precision floating point representation on the Sun and H P workstations is 64 bit and on the Cray 128 bit, leading to approximately 15 and 30 digits of precision, respectively. A comparison between Cray, Sun SPARC, and HP-9000 calculations showed slight differences in the concentration-time plots, especiallyfor chaotic oscillations near the bifurcation point between the oscillatory and the nonoscillatory excitable steady state. These differences can probablybeattributed to thedifferent numerical precisions used in the three computers. They do not affect our conclusions. Examples are given in the supplementary material and will be referred to later in the text. Model of the BZ Reaction. We use the original Oregonator22 model with the rate constant values of Field and F6rsterlingz3 A+Y+X+P
A +X
+
2X
2X-.A+P Z’fY
+Z
(1)
(3) (4) (5)
where kl = 1.3 M-l s-I, k2 = 2 X lo6 M-l s-l, k3 = 34 M-l s-l, k4 = 3 X lo3M-l s-l, k5 = 0.02 s-l, andfis a time constant with values 0.8-1.6. The concentration variable assignment is the following: A = BrO3-, X = HBr02, Y = B r , Z = 2Ce(IV), and P = HOBr. The kinetic active components are X, Y, and Z with [A] = 0.1. Model of Stirring. Figure 1 considers the following stirring model (“cell model”). The reacting solution is assumed to consist of N short-lived fluid packages (cells), which are allowed to exist for a certain lifetime T . Each of the fluid packages contains the reacting species of the BZ reaction. Fluid packages mix only at theendoftheir lifetimes. Aftermixing, theaverageconcentration
values of all chemical species are determined, and new concentration values of the reacting components are assigned to each package. These concentrationsare allowed to fluctuate randomly around the average value. In addition, a certain number of packages (k) are allowed to undergo excitation;i.e., the bromide ion concentration is below the critical value with high bromous acid (HBr02) concentrations. These excited cells have the following concentrations: [XI,, = 5.3 X 1 V M, [YlCx= 3.2 X 10-8 M, and [Z]., = 1.7 X M. To include fluctuations in the calculations, a uniform random number generator 8 based on three linear congruentialgenerators has been ~ s e d . 2The ~ period length of 9 is practically infinity, returning at each call a random number in the interval (0,1). Concentration fluctuations around an average concentration Caverage are generated by the following relationship where K is a scaling factor ( K E (0, 1)) and “*” indicates the random multiplication of +1 or -1. When Crandom has been determined for all cells, a new ”cycle” of cells starts. Stirring Effects
ExperimentalFindings. Figure 2 shows a typical stirring effect in a classical anaerobic malonic acid BZ reaction? the frequency of the oscillationsincreases as the rate of stirring decreases. Very often it has been observed that the decrease in period is also accompanied by a decrease in amplitude where oscillations stay at more positive electrode potentials (Figure 2). Virtually the same or similar stirring effects have been reported for other oscillatory systems by Ruoff,lo Ruoff and Schwitters,26 Sevcik and Adamcikova,*’ Menzinger and co-workers,28 and Noszticzius et al.29b Oscillations can also be revived by decreasing the stirring rate when the system is in an excitable steady state. This has been observed in closed methylmalonic acid26and in oxalic BZ systems. Figure 3 shows stirring-inducedexcitations and stirring-induced quenchingof oscillationsin a classical Ce(1V) malonic acid system. At high stirring rates (here 700 rpm), the system settles in a reduced nonoscillatory(excitable) steady state. When the stirring rate is reduced to 300 rpm, oscillations reappear immediately (stirring-induced excitation). Model Calculations. One of the apparent effects of stirring is that high stirring rates will diminish the random concentration fluctuations around the average concentration of a chemical species “f,while slow stirring rates will increase these fluctuations. Thisappears to be in agreement with the experimentalobservations by Menzinger and co-workers28and R~off,~O*~l where electro-
The Journal of Physical Chemistry, Vol. 97, No. 24, 1993 6407
Excitations Induced by Fluctuations
-1
‘
700 rpm
Pt-Polential (+)
+b
- 300 ipm
1 Min U
300rpm
-
700 rpm A 3 0 0 rpm 1700 rpm 4 5
-+
300 rpm
7Wrpm
-
F i i 3. Closed BZ system under an argon atmosphere showing both chaosandstirringeffects. (A) Chaoticoscillationsareobservedatconstant stirring speed (two other identical systems are shown in Figure 6): 1, the changeof stirring speed from 700to 300rpm causes immediate excitation and induction of oscillations; 2, increase of stirring speed to 700 rpm leads to a nonoscillatory excitable steady state. (B) 3 and 4, a single excitation spike can be created by first decreasing and then increasing the stirring speed;5, oscillationsreappear by decreasing the stirring speed to 300 rpm. -1
,
I
I
-1 0
600
1200
1800
2400
3000
Time, s
-4 . ! , , , , , , , , , ! 0 600
1200
1800 Time, s
2400
3000
Figure 4. Calculations using the cell model: f = 1.O,N = 600,K = 0, T = 1.0 s. During time intervals lo00 and 2000 s, &x is set to 50 to simulate the effect of decreased stirring speed; otherwise, &x = 0.
chemical potentials of slowly stirred BZ reactions appear more “noisy” than those of corresponding systems with high stirring rates. Thus, an increase of random fluctuations leads to short-lived local domains inside the reacting solution with bromide ion concentrations below and above the average bromide ion concentration. There will also be local Yexcited”domains in the reacting solution where the bromide ion concentration is below [BrIcfit. In these excited domains, HBrO2 is formed autocatalytically and bromide ion is consumed accordingly, due to the reaction between HBrO2 and B r . These local excitations can be considered as “nuclei” of the oxidized state (eq B) appearing in the reacting solution when most of the reaction solution is still in the reduced state. It is the total amount of excited nuclei compared to the amount of available bromide ion from other parts of the reacting solution (together with their mixing behavior) that determines whether local excitations will surviveand “ignite” the whole solution or whether local excited areas get “quenched” by bromide ion. When the average bromide ion concentration is high, excited packages normally get quenched when they mix and react with other packages that contain higher concentrations of bromide ion. At high bromide ion concentrations, the number of oxidizing nuclei is low but is expected to increase as the bromide ion concentration approaches [Br]&,. At lower average bromide ion concentration, the number of oxidized nuclei may be sufficient to “ignite” the reacting solution and drive the system into the oxidized state B. Figure 4 shows a simulation using the cell model (Figure 1). The computed behavior is very similar to the experimental observation^^^^' when it is assumed that the number of excited cells increases at lower stirring rates.
Figure5. Simulatedcease of oscillations by increased stirring rates using the cell model: f = 1.0, N = 600,K = 0,T = 1.Os. During time intervals lo00 and 2000 s, n, = 30 (reflecting high stirring speed). At other times, rbx = 55 (reflecting low stirring speed). According to the results in Figure 3, simulationsshould predict a nonoscillatory response at high stirring rates. This behavior is shown in Figure 5 : at sufficiently low hX(corresponding to low stirring noise at high stirring rates) the system goes to a nonoscillatoryreduced, excitable steady state with [ B r ] slightly above [Br],it. As in Figure 4, the induction of oscillations at low stirring rates (corresponding to higher stirrer noise levels) occurs because the number of excited cells (fluid packages) is now large enough to excite the whole reaction solution. At lower kx, the excitedcells are not sufficientto “ignite”the whole solution, and the systemremains in a (nonoscillatory)reduced steady state. Chaos in Batch Systems
Experhnb. The first indicationof irregular/chaotic behavior in closed BZ systems and its possible relation to excitability was reported by Field.32 In these experiments, irregular oscillations have been observed at the end of the oscillatory region when the BZ reaction enters a nonoscillatory excitable (reduced) steady state. Ruoff’ found that an increase of the initial Ce(1V) concentration accelerates the appearance of the chaotic30 and nonoscillatory excitable region. Figure 3A shows such irregular oscillations at 700 rpm. Figure 6 shows a system with the same initial composition as in Figure 3 but with two different stirring rates (300 and 1200 rpm). It is during the approach to the nonoscillatory excitable steady state that irregular oscillations are observed. As shown by earlier experiment^,^' BZ systems with low initial Ce(1V) concentrations oscillate regularly but become erratic when the initial Ce(1V) concentration is increased. A certain amount of initial Ce(1V) is necessary to “push” the system sufficiently near the nonoscillatory excitable steady state/ bifurcation border in order to observe chaos. Figure 7 shows the return maps of the chaotic spikes shown in Figure 6. These return maps are very similar to return maps of period lengths in the chaotic chlorite-iodide reaction.288Our results clearly show that the observed chaos is of stochastic rather than deterministic nature. Model Calculations. Figure 8 shows simulations of chaotic oscillations using the cell model. As found in the experiment,
Ruoff
640% The Journal of Physical Chemistry, Vol. 97, No. 24, 1993
Figwe 6. Experimental trace of chaotic oscillations in a closed BZ system? (A) 1200 rpm and (B) 300 rpm stirring speed. Note that oscillations are more "noisy" at lower stirring speeds.
0
2
4
6 8 1 0 1 2 Xn, Mln
oscillations get more pronounced for high f and high K values. It may be noted that, in an ordinary, homogeneous system without the introduction of mixing, the bifurcation border occurs at f = 2.401 where it separates the oscillatory state (f = 2.401) from the nonoscillatory excitable steady state (f = 2.402). Even at lowfaod K values, the Oregonator can generate chaotic spikes. This is shown in Figure 10, where the system approaches an excitable steady state by a slight increase of theffactor. Also here (as indicated experimentally when using higher initial Ce (IV) concentrations)$l an increasedf factor "pushes" the system toward the nonoscillatory excitable steady state. As the system approaches this nonoscillatory excitable steady state, the more irregular/chaotic the oscillations/excitations become.
Discussion
Return map of generated spikes shown in Figure 6A,B. X, is the time between spike n + 1 and spike n, while X,,+I is the time between spike n + 2 and spike n + 1, starting with spike n = 1, 2, 3, ....
Stirring Effects. The system we discuss is a classical metal ion catalyzed BZ reaction, characterized by the switchingbehavior between an oxidized and a reduced steady state. The R4cz system:9a where stirring effects have recently been observed and discussed by Noszticzius et al.>9bis not covered here, because the Rdcz system is not a relaxation oscillator and does not show excitability.33 In fact, stirring effects in Rdcz systems differ considerably from those observed in claarsical BZ systems: a decrease in stirring rate does not lead to a decrease in period length but leaves period length almost unchanged (or slightly increased).29b Bromine hydrolysis controlled (BHC) systems34show "opposite" stirring behaviors compared to classical BZ systems; Le., in these systems an increased stirring gas purging,35or an organicsubatratewhichreacts with bromine (Br2)36maybeneeded to remove Br2, which otherwise, due to the bromine hydrolysis step
chaos is observed when the system is near the excitable steady state. In the computations, the Occurrence of chaos can be achieved by "balancing" the influences of K and G,. High K and q, values correspond to a high 'noise level" of the stirrer. However, K and q, act in opposite directions. An increase of K increases the average time between two successive excitations. Contrary, an increase of &x decreases the average period length. Thus, chaos in BZ systems can be understood either as an oscillatory system which is very near a nonoscillatory (here excitable) steady state, and where fluctuations (due to K ) drive the system occasionally into the nonoscillatory region, or as a nonoscillatory excitable system where fluctuations (due to hX) occasionally drive the average bromide ion below [BrImt. Figure 9 shows calculated return maps of the simulated spikes in Figure 8. They are in agreement with the experimental return maps (Figure 7) and clearly reflect the stochastic nature of our model. Table I1shows &xbrdcr values for given values of K and$ These n,values define the border between the oscillatory and nonoscillatory state. Close to this border (at the oscillatory side), the system behaves irregularly. The supplementarymaterial contains the calculated concentration time plots of these transitions for f values equal to 0.8, 1.2, and 1.6 and for K values equal to 0.05, 0.1,0.2,0.4, and 0.8. From these results, we find that chaotic
would buffer the bromide ion concentration above [Br]dt and leave the BZ reaction in a reduced (but excitable) n o n d a t o r y steady state.34 In BHC systems, higher stirring speedswill lower Brz levels and induce oscillationsor increase the frequency as the stirring rate is increased. Very high stirring rates leave a BHC system in a nonoscillatory oxidized steady state. In the (classical) BZ systemstudied here, an increase in stirring rate leads generally to a nonoscillatory reduced steady state. In our view, the key to understanding stirring effects in classical BZ systems is the fragmentation of the reaction volume into individual fluid packages, the presence of locally excited fluid packages, and their dynamics due to stirring. Menzinger and co-workersz8have discussed stirring effects both for BZ systems and for the chlorite-iodide reaction, and they have stressed the importance of local fluctuations. During process A, bromine (Brz) is generated and consumed by bromination of malonic acid, with the result that bromide ion is depleted. Because of the random fluctuations of reaction intermediates, in some places the bromide ion concentration will be below [Br]dt, and local centers (nuclei) of oxidation form. It is the bromide ion concentration of the remaining part of the solution that determines whether these oxidation nuclei survive
-
1
I
0
4
I
I
8 12 Xn, Min
I
I
16
20
F i e 7.
Br,
+ H,O
H++ Br- + HOBr
(C)
The Journal of Physical Chemistry, Vol. 97, No. 24, 1993 6409
Excitations Induced by Fluctuations
0 -1.8
-am
& -2.3
-esB
A
-2.8
-k
-3.3 f 0
I
I
6000
I
12000 Time, s
18000
24000
-1-1.8
B
& -2.3 e B
E 0)
- -3.3 1 0
I
I
6000
0
I
12000 Time, s
24000
18000
Figwe 8. Simulation of chaotic spike generation using the Oregonator and the cell model. In both calculations,f = 1.0, N = 600, n, = 30, and ‘T = 1.0 s. (A) K = 0.85; (B) K = 0.92. 12,
I
TABLE Ik Determined 4xblaer Values for Given f and I( f = 0.8 f = 1.2 f = 1.6 K
0.05 0.1 0.2 0.4 0.8
n,b*
16 15 15 12 10
K
0.05 0.1 0.2 0.4 0.8
n,40 41 42 39 41
K
0.05 0.1 0.2 0.4 0.8
n,61 65 66 64 74
Also, the stirring-induced chaotic excitation spikes observed by
0
10
20
30
40
50
60
Xn, Min
Figure 9. Return map of calculated spikes shown in Figure 8A,B. X, is the time between spike n 1 and spike n,whileX*l is the time between spike n + 2 and spike n + 1, starting with spike n = 1, 2, 3, ....
+
and “ignite” the whole solution or whether they get quenched by the bromide ion from other parts of the solution. ch.os. Interestingly, the same BZ system can exhibit both stirring effects and chaos (Figures 2 and 5). Both effects can be explained using the originalzzOregonator together with the cell model, although it is generally assumed39 that the original Oregonator is not able to show chaos. The approach of generating chaos by using the cell model appears applicable to other excitable systems. In fact, an analogous observation of chaos in a closed excitable enzymatic model system with noise was previously reported by Hahn et a1.&
Menzinger and GiraudiZBnin the chlorite-iodide system appear very similar to the chaotic spikes generated in the closed3O BZ reaction and can probably be modeled by using the cell model. Chaos in the Belousov-Zhabotinsky reaction is exclusively studied in open flow reactors. This technique has the advantage that true stationary states can be obtained. Although this is not true for closed systems (which evolve toward equilibrium,and all phenomena observed in closed systems are therefore transitory), the study of chaos in closed systems30may have the advantage that mixing problems of in-flow reagents are avoided. The work by Gykgyi and Field4Ihas shown that BZ systemsin flow reactors may show a chaotic response when the oscillator inside the reactor bulk phase couples dynamically with the in-flow reagents. Therefore, contrary to the general assumptionthat deterministic chaos observed in CSTRs reflects the determiniim of the chemical rate equationsof the BZ system, two-cycle coupling4*leads to the possibility that certain hydrodynamicconditions in a CSTR may lead to deterministic chaos. There is a striking similarity between chaos observed in closed and open systems: chaos is only observed when the BZ reaction is near a bifurcation point. In flow BZ systems, this is a Hopf while in a closed system the analogous bifurcation separatesan oscillatorystate froma nonoscillatoryexcitablestcady state. On the other hand, we note that the dynamical behavior in flow reactors when various return maps have been constructed can be different. In these ~ystems,ll.~~ return maps have been constructedon the different local maxima or minima of electrode potentials (i.e., the oscillatory potentials are not constant from one spike to the other), and it therefore appears that t h e s ~ * l * ~ ~ deterministic chaotic responses are not connected to the appearance of chaotic excitation spikes in closed systems. Our interpretation using the cell model alongwith experimental return maps (Figure 7) suggests that the origin of the chaotic
6410 The Journal of Physical Chemistry, Vol. 97, No. 24, 1993
Ruoff
=9 0)
-1.8
B
a
E -2.3
-Et
A
-2.0
a
-
t
6000
12000 Time, s
18000
24000
B -3.3
1 0
I
6000
I
I
12000 Time, s
18000
24000
Figure 10. Chaotic spike generation by increase of theffactor in the Oregonator. For both calculations, N = 600, T = 1.0 a, K = 0.05, and n, = 16. (i)f= 0.8100; ( B ) j = 0.8125.
excitation spikes is random noise. The system is very sensitive to small variations in the bromide ion concentration, where fluctuations decide when the next excitation of the reacting solution appears. However, once local excitations survive, a fully developed oxidation spike with constant height is generated. This behavior is found both in the experiments (Figure 6) and in our computations (Figure 8). The importance of sensitivity of excitable systems in relation to chaos was earlier discussed by Noyes.44 Nature of Fluctuations. It is important to realize that the chaotic response is only observed when the system is sufficiently near the border (bifurcation) between the oscillatory and the nonoscillatory (but excitable) regions. As pointed out by Prigogine:’ near a nonequilibrium bifurcation point the law of large numbers breaks down: fluctuations become as important as average values. As indicated by our calculations, fluctuations in B r ion concentration are an obvious explanation of the observed stirring effects and the (stochastic) chaotic response. Fluctuations in [ B r ] occur because fluctuations of reaction rates take place near the bifurcation between the oscillatory and the nonoscillatory excitable states and because the stirrer is not able to eliminate these fluctuations completely. There are additional possible mechanisms of how [ B r ] fluctuations can occur. By performing the experiment under an argon blanket, one can exclude the involvement of oxygen, but a stirring-dependentloss of Br2 may still take place. Fluctuations could arise because of local variations in the transport rates when Brz crosses the liquid-gas interphase. This mechanism is of importance for bromine hydrolysis controlled (BHC) systems where there is a major buildup of elementary bromine. BHC systemswill show an opposite stirring effect compared to a classical BZ reaction; Le., in BHC systems an increase of stirring will lower the dissolved Br2 concentration (which buffers the bromide ion concentration) and therefore increase the frequency of the oscillations with the increase of stirring rate. A nice example of such stirring effects in a BHC system together with the influence of adsorption of Br2 on a Teflon stirring bar was shown by Pojman and co-workers in the manganese/hypophosphite/acetone BZ system.38 For a classical BZ system, however (as studied here), a Brz-removal mechanism seems to play a negligible role because of the presence of a high malonic acid concentration (0.29 M) which keeps the Br2 concentration
Although thermal fluctuations can probably be excluded as the primary inducer of local changes in [Br],4’ the opposite behavior, i.e., local thermal variationsdue to fluctuationsin [B r ] , cannot be ruled out. Nagy-Ungvarai and K6r6s4*showed that most of the heat output in a classical BZ system occurs while process B is occurring, i.e., when the system is in an oxidized state. Thus, local variations in the bromide ion concentration may lead to the appearance of oxidized reaction centers with locally high heat releases. This local heat release may affect (increase) the rate of the local excitation process, thus causing a complex interplay between concentration fluctuations and thermal fluctuations. At present, the simulation of such effects appears beyond the scope of feasibility. Conclusion We have demonstrated that inclusion of random concentration fluctuations in the original Oregonator can explain both stirring effects and chaos in closed classical BZ systems. For closed BZ systems, we explain chaos as random noise where fluctuations occasionally exceed the excitation threshold of the BZ reaction. The prerequisite for chaos in closed systems is that the system is near the bifurcation between the oscillatory and nonoscillatory steady states. Acknowledgment. I thank Prof. Richard M. Noyes and Prof. Patricia Harris for hospitality during a stay at the University of Oregon (UoO) in 1989. Dr. John Guslander is thanked for help in using MS-DOSat UoO. This work was supported by grants from the Norwegian Research Council NAVF and the Nansen Foundation. NAVF is thanked for providing computing time at the SINTEF Supercomputing Center in Trondheim. Supplementary Material Available: Concentration-time plots of chaotic systems near the border of the nonoscillatory excitable steady state forfvalues equal to 0.8, 1.2, and 1.6; return map of the random number generator used in the computations; differences in calculatedconcentration-time plots due to different numerical precision when using Sun SPARC, HP9000, and Cray computers (23 pages). Ordering information is given on any current masthead page. References and Notes (1) Bclousov, B. P. A Periodic Reaction and Its Mechanism. In Oscillations and Traveling Wwes in Chemical Systems; Field, R.J., Burger, M.,Eds.;Wilcy: New York, 1985.
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