J. Phys. Chem. 1984, 88, 2851-2857 systematic approach. Work in progress includes the development of mechanisms for the chlorite-iodide and bromate-iodide oscillators and a bifurcation-phase portrait analysis of both the uncoupled and coupled systems. While the study of coupled oscillators in flow reactors may seem a rather esoteric exercise, such systems may be the rule rather than the exception in living organisms. We have already referred to the coupled allosteric enzyme mode1.l' Ross and c o - ~ o r k e r s ~ ~ have carried out extensive studies of the glycolytic enzyme system and have emphasized the possible evolutionary significance of properly matched coupling between the PFK and PK reactions. More recentlyz4 they have studied the effects of periodically ~~~~~~~~~
2851
perturbing an oscillatory thermokinetic reaction. Coupling of two oscillators introduces higher dimensionality and, as pointed out by R O ~ s l e r the , ~ ~possibility of new dynamic phenomena. For this reason and because of the potential significance of these systems in biology and chemistry, the study of coupled chemical oscillators promises to be an area of growing interest. Acknowledgment. We thank Jerzy Maselko, Gyorgy Bazsa, Robert Olsen, and Kenneth Kustin for suggestions, comments, and criticisms. This work was supported by National Science Foundation Grant C H E 8204085. Registry No. Br03-, 15541-45-4; ClOc, 14998-27-7; I-, 20461-54-5; Mn, 7439-96-5.
(23) (a) Termonia, Y.; Ross, J. Proc. Narl. Acad. Sci. U.S.A. 1981, 78, 2952-6. (b) Richter, P. H.; Ross, J. Biophys. Chem. 1980, 12, 285-97. (24) Rehmus, P.; Zimmerman, E. C.; Ross, J.; Frisch, H. L. J . Chem. P h p . 1983, 78, 7241-51.
(25) Rossler, 0. Z . Naturforsch. A 1976, 31, 1664-70.
Phase Response Relationships of the Closed Bromide-Perturbed Belousov-Zhabotinsky Reaction. Evidence of Bromide Control of the Free Oscillating State without Use of a Bromide-Detecting Device Peter Ruofft Department of Chemistry, University of Oslo, Blindern, Oslo 3, Norway (Received: October 19, 1983)
The use of bromide-selective electrodes in the oscillating Belousov-Zhabotinsky reaction (BZR) is still somewhat controversial. In this paper further evidence of bromide control in the oscillating BZR without any use of a bromide-detecting device is given. It is shown that the addition of silver ions, bromide ions, or hypobromous acid to the oscillating BZR results in phase shifts at the platinum electrode which behave as predicted by models simulating the Field-Koros-Noyes mechanism. Additional complexities due to higher initial Ce(1V) concentrations are reported and discussed.
'
simulate the FKN mechanism. This constitutes evidence of Brcontrol without the use of a Br--detecting device. Finally, we report and discuss additional complexities due to an increased initial Ce(1V) concentration when AgNO, is used as the perturbant .
Introduction In its now classical form, the Belousov-Zhabotinsky reaction (BZR) is the metal-ion-catalyzed bromination of an organic material in aqueous acidic media. Under suitable conditions it may result in many repeated nearly unchanged concentration oscillations of the catalyst and other intermediates. Oscillations were first reported by Belousov' using citric acid as the organic substrate and the study was initiated by Zhabotinsky,* showing that different organic substrates may be used. The basic mechanism of the oscillations is now well established, notably by the work of Field, Koros, and Noyes3 (FKN mechanism). In this mechanism, Br- ions play the role of a control i~~termediate.~ On the basis of the FKN mechanism several mathematical models5-10 have been developed which qualitatively or semiquantitatively simulate most of the observed phenomena. However, in a series of papers, Noszticzius and co-workers question the importance of Br- ions in the oscillating BZR. The argument is based mainly on the experimental fact that Br--selective electrodes are also sensitive to hypobromous acid"J2 and other compound^'^ which may contribute considerably to the electrode's potential. Like many oscillating systems BZ oscillations are structurally stable and exhibit a limit cycle, Le., periodicities in concentration are stable to perturbations in amplitude and frequency but not in phase.I4 In this paper we report an investigation on the phase shifts observed at a platinum electrode when the Br- concentration of an oscillating BZR is perturbed by solutions of KBr, A g N 0 3 , or HOBr. It will be shown that the phase shifts observed at the platinum electrode behave exactly as predicted by models which
Experimental Section All experiments were performed in a closed stirred thermostat-regulated glass container at 25 OC. The stirring rate was kept constant at ca. 500 rpm with a magnetic stirrer. The oscillations were followed with a platinum electrode against a double junction Belousov, B. P. ReJ Radiats. Med. 1958, 145. Zhabotinsky, A. M. Dokl. Akad. Nauk SSSR 1964, 157, 392. Field, R. J.; Koros, E.; Noyes, R. M. J . Am. Chem. Soc. 1972, 94, (4) (5) (6) (7) 417. (8)
I
Edelson, D.; Noyes, R. M.; Field, R. J. Inr. J. Chem. Kinet. 1979, 1 1 ,
155
(9) Showalter, K.; Noyes, R.M.; Bar-Eli, K. J . Chem. Phys. 1978, 69, 25 14. (lo! Ganapathisubramanian, N.; Ramaswamy, R.; Kuriacose, J. C. In "Kinetics of Physicochemical Oscillations"; Franck, U. F.; Wicke, E. Eds.; Deutsche Bunsengesellschaft fur Physikalische Chemie: Aachen, 1979. (11) Noszticzius, Z. Acra Chim. Acad. Sci. Hung. 1981, 106, 347. (12) Noszticzius, Z.; Noszticzius, E.; Schelly, Z. A. J . Am. Chem. SOC. 1982, 104, 6194. (13) Noszticzius, Z.; Noszticzius, E.; Schelly, Z. A. J . Phys. Chem. 1983, 87, 510. (14),See papers of Busse, H.-G.; Vavilin, V. A.; et al., Zaikin, A. N.; Zhabotinsky, A. M.; Zhabotinsky, A. M. In "Biological and Biochemical Oscillators", Chance, B.; Pye, E. K.; Gosh, A. K.; Hess, B.; Eds.; Academic Press: New York, 1973.
Permanent address: Rogaland Regional College, P.O. Box 2540 Ullandhaug, N-4001 Stavanger, Norway.
0022-3654 ,/84 ,/2088-28 5 1 $01.50/ O
Noyes, R. M. J . Am. Chem. SOC.1980, 202, 4644. Fpld, R. J.; Noyes, R. M. J . Chem. Phys. 1974,60, 1877. Field, R. J. J. Chem. Phys. 1975, 63, 2289. Edelson, D.; Field, R. J.; Noyes, R. M. Znt. J . Chem. Kinei. 1975, 7,
0 1984 American Chemical Societv -
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The Journal of Physical Chemistry, Vol. 88, No. 13, I984
Ruoff
Ag/AgCl reference electrode (Metrohm, sleeve type) with a Z+fy (05) saturated KC1 solution as the inner electrolyte, and 1 M H2SO4 where X = HBr02, Y = Br-, Z = 2Ce(IV), A = Br03-, and P as the outer electrolyte. All chemicals were of analytical grade = HOBr. The concentrations of A and P are constant. As in and used without further purification. As organic substrate the original work: the resulting differential equations were first malonic acid (MA) was used. The reaction volume was 150 cm3 written in dimensionless form. Original rate constants were used, and oscillators were started by rapidly mixing the reagents in the only corrected for the experimental values of Br0; and H+ ions. following order: 1 M H2S04-MA-(NH4)2Ce(N03)6-KBr03. The parameters kos andfwere fixed at 1.0. We also considered The initial compositions were CMA= 0.28 M, CCe(Iv) = 5.0 X IO4 a model which explicitly includes HOBr as an intermediate: or 2.1 x M, and CKB~O, = 0.1 M. When the experiments were run under anaerobic conditions, Br03- + Br- 2H+ G HBrOz + HOBr (Rl) nitrogen was first bubbled vigorously for 20 min into the sulfuric kR1 2.1 M-3 S-l k-R1 = 1 X lo4 M-’ S-’ acid. Then the reagents were added and bubbling of nitrogen continued. So that the dry nitrogen gas would not carry off solvent, HBrOz + Br- + H+ ~t 2HOBr (R2) the gas was passed through two washing bottles filled with water before going into the reaction vessel. kR2 = 2 X lo9 M-2 S-l k 4 2 = 5 X lo-’ M-’ S-’ After the induction period, the oscillator ran several free cycles. Br03- HBrOz + H+ s 2Br02- H 2 0 Then perturbation experiments were performed by adding a single (R3) drop of perturbant from a conventional Pasteur pipet to the rek ~ 3= 1 X lo4 M-’ S-’ k 4 3 = 2 X lo7 M-’ S-’ action medium. After perturbation the system was allowed to resettle until limit cycle oscillations again appeared. Under our Ce” Br02- + H+ F? Ce4+ + H B r 0 2 (R4) experimental conditions, the BZR evolves toward an excitable steady state. Near the border between the excitable and oscillating k ~ 4= 6.5 X 10’ M-’S-’ k - ~ 4= 2.4 X 10’ M-’ S-’ regime the oscillations may become irregular with a considerable 2HBr02 + HOBr B r 0 < H+ (R5) increase of period length.s Therefore, in the experiments only the first 30-35 oscillations were used. Thus, several runs had to be kR5 = 4 x lo7 M-’ S-’ k-R5 = 2 X lo-’’ M-’ S-’ performed until a complete phase response curve was obtained. Because of the slight increase in period lengths the experimentally Ce4+ + BrMA gBr- + Ce3+ product (R6) determined phase shifts (see below) are expressed in fractions of k * ~ 6= kR6CBrMA = 1.0 S-’ g = 0.9 the period length Po of the preceding unperturbed cycle. As perturbants, aqueous solutions of KBr, HOBr, and AgN03 HOBr Br- H+ s Br2 H 2 0 (R7) were used. Hypobromous acid, HOBr, was prepared by adding 8.0 g of bromine to 100 cm3 of a 0.5 M A g N 0 3 solution which k ~ 7= 8 X lo9 M-’ s - ~ k-R7 = 2.0 M-’ S-’ was stirred magnetically. After 10 min, the AgBr formed was removed by filtration and the solution stored at 0 OC. After 1 Br2 + MAenol BrMA Br- product (R8) h, the solution was again filtered and freed from bromine by three k*RS = kRgCMAen,, = 1.0 s-’ successive extractions using ice-cold CC14. Then, if necessary, the HOBr solution was filtered yet again and perturbation exThis model is similar to a mechanism previously used by periments were started. During the experiments the HOBr soShowalter et a1.: extended by the inclusion of bromine hydrolysis lution was stored at 0 OC. The HOBr solution was analyzed by (R7) and the bromination of MAeno~ (enol form of malonic acid) potentiometric titration in 1 M H 2 S 0 4by using a 0.01 M KBr (R8), as treated for irreversible reaction steps by Ganapathisusolution as titrant and a Br--selective electrode (Philips, IS 550) bramanian et a1.I0 Concentrations of Br03-, H+, H 2 0 , BrMA for determination of the end point. The yield of HOBr formed (bromomalonic acid), and MAenolare treated as constants. As was 82% of the theoretical value. in the Oregonator, kR6and g are variable parameters. Their values The average drop size of the Pasteur pipet was estimated (by are still somewhat uncertain. weighing 2 X 50 drops of water) to be 31 wL. In all model calculations, the phase shifts are first expressed in the time scale of the respective models, Le., in the Oregonator Simulation Models and Method of Calculation we are using the dimensionless 7 scale,4 while in the reaction Calculations were performed on a CDC CYBER digital comscheme Rl-R8 the time is given in seconds. However, in order puter using the program package DARE P with a version of Gear’s to compare the theoretical results with experiments, we have, as algorithrn.l5 In the models treated numerically, phase response in the case of the experimentally determined phase shifts, also curves were calculated by the following procedure. The model expressed the calculated phase shifts in fractions of the unperoscillator was first allowed to converge to the limit cycle by running turbed period length Po of the model. several cycles freely. Thereafter one period of the limit cycle was recorded for all variable intermediates. Then for various positions Theoretical Predictions and Comparison with Experiments in the cycle, instantaneous perturbations were applied (“phase of The Initial Phase Shift o f t h e Ce(IV) Spike. The essential stimulation”) and the phase shift was calculated. Because of the feature of the FKN mechanism is that Br- ions act as a control relationship between the redox potential of the platinum electrode i ~ ~ t e r m e d i a t eAbove . ~ . ~ a certain critical bromide concentration, and the logarithm of the Ce(IV)/Ce(III) concentration ratio, CBFt,Br- is removed by reactions with bromate and other incalculations of phase shifts (see next section) were performed for termediates until CBFtis reached. The sum of these reactions only the Ce(1V) variable. is denoted as process A. Then an autocatalytic process sets in The simplest model treated numerically was the Oregonator so that a new which drives the Br- concentration far above CBrRit developed by Field and no ye^.^ It consists of the following five cycle can start. irreversible reaction steps: According to the FKN mechanism, the addition of Br- ions during process A should lengthen the period since more Br- ions A+Y-+X+P (01) have to be removed in order to reach C B ~On ~ the ~ ~other . hand, X+Y-2P (02) removing Br- ions (but not forcing the Br- concentration below CBrCarit) should lead to a shortened period. More specifically, the A X 2X Z (03) addition of Br- ions should lead to a delay in the appearance of 2X-+A+P (04) the Ce(1V) spike (compared with the unperturbed oscillations) while irreversible removal of Br- ions should lead to a corresponding advance. In the following it will be shown that this initial (15) Wait, J. V.; Clarke, D. DARE P Users Manual, Version 4, Report 299, University of Arizona, 1976. phase shift of the experimental Ce(1V) spike (in the paper simply
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The Journal of Physical Chemistry, Vol. 88, No. 13, 1984 2853
Phase Response in BZ Reactions
+0.4/ i0.31
A
-0.14
PHASE OF STIMULATION
/
i1.0-
time
B
io.9-
i0.8 i0.7.o-
k
i0.6- 3
6
l
- 0.11 Figure 1. (A) Partition of the unperturbed period Poin A and the slow Br- removing process A. (B) The initial phase shift At$ at the platinum electrode is defined as the time difference between the appearance of the perturbed Ce(IV) maximum (at t = tPrJ and the unperturbed Ce(1V) top (at t = Po).Addition of Br- ions at t = t’(”phase of stimulation”) causes a delay in the Ce(1V) spike (positive phase shifts), while an irreversible removal of Br- ions causes an advance in the appearance of the Ce(IV) top (negative phase shifts). denoted as the “phase shift”) behave axactly as predicted by the F K N mechanism. Addition of Br- Zons. A simple model of the FKN mechanism may be constructed by using for process A the following irreversible reaction
A
+Y
-
product
”
”
l
1.0 PHASE OF STIMULATION
Figure 2. Experimental and calculated phase response curves (phase shift vs. the phase of stimulation) using Br- ions as perturbant. (A) The M of unreacted perturbation strength in the experiment is 2.1 X added KBr. The solid line represents eq 5a and 5b with AC,, = 6.0 X IO” M, A = 15 s, Po = 40 s, ksl = 2.1 M-I s-I, and CBrmax = M. (B) The perturbation strength in the experiment is 2.1 X lo-’ M of unreacted added KBr. The solid line represents the calculation using the Oregonator (curve 1 in Figure 3A). In both (A) and (B) the initial Ce(1V) concentration was 2.1 X IO-) M and the reaction medium was in contact with air.
From Figure 1B we see at once that A$ L 0, as expected. Inserting eq 2 and 3 in eq 4 we obtain the retarded phase shift, A$rct:
(SI)
ksl = 2.1 M-I sdl where A = Br0,- and Y = Br-, while “product” is here an uninteresting reaction product. The autocatalytic process is regarded as an indifferent reaction which lasts a certain time A and finally drives the Br- concentration to CErmax (Figure 1A). The rate equation for Y during process A is simply k*sl = kslCA= constant d(ln [ Y ] )= -k*sl dt (1) We will now consider an instantaneous addition of Br- ions, ACE,, at t = t’ (the “phase of stimulation”). The new Br- concentration at t = t’is C;, ACE,. Integrating eq 1 from C’,, + ACB, to CB,crit
+
Ca”‘
d(ln [ Y ] )= - k * s , l , f m d t and from C’,, to CErCrit L:*’d(ln
[ Y ] )= -k*sl J,%t
(3)
we get an explicit expression of the phase shift A$ which is given by (Figure 1B)
Ad = (tpcrt - t 9 - (PO- t 9 =
tprt
- Po
(4)
Integrating the unperturbed system from t = A to t = t’, we obtain the following expression for PEr: C’,, =
exp(-k*sl(t’- A))
(5b)
Despite the simplicity of the model it is surprising how good the agreement with the experiment actually is. Figure 2A shows the experimental data for a low concentration of added Br-. The solid line on Figure 2A represents eq 5a and 5b. The jump back to the zero line occurs at 0.94 in the experiment and not at 1.O as in the model. An improved model but still a skeleton is the Oregonator model (01-05). In Figure 3A we see the calculated phase shifts for three different concentrations of added Br- ions. Although the Oregonator gives a better description of the jump to the zero line, the behavior of low perturbant concentrations is essentially that of our simple model. Using a high added Br- concentration in the model (curve 1 in Figure 3A), we observe a characteristic top at the end of the cycle. This is actually found in the experiment and Figure 2B shows the good semiquantitative agreement. Irreuersible Removal of BY- Zons. The F K N mechanism predicts an advance in the appearance of the Ce(1V) spike (A$ < 0) when Br- ions are irreversibly removed from the reaction medium. With our simple model (Sl, Figure l), similar calcu-
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The Journal of Physical Chemistry, Vol. 88, No. 13, 1984
2854
PHASE 0F STIMULATION
5.0
0 +
15.0
10.0
20.0
300-
-5.0+ k
A
Ln
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PHASE OF STIMULATION
Figure 4. Calculated phase shifts using the model R1-RE (Po= 26.4 s) where AgN03 irreversibly and instantaneously removes Br- ions. Note the buffer effect of the solubility product of AgBr just before the sudden jump to the excitable branch. The perturbation strengths (M of added unreacted Agt ions) are as follows: curve 1, 3.3 X curve 2, 3.3 X curve 3, 3.3 x 1oP.
g
TABLE I: Average Experimental Period Length in the Dependence on Initial Ce(IV) Concentration CCe(IV),M PO? s 2.1 x 10-3 45-47 5.0 x 10-4 28-30 I
Anaerobic condition; however, no significant difference between anaerobic and aerobic conditions was observed.
-VI
Figure 3. Calculated phase shifts using the Oregonator (Po= 314.4 9 21.7 s). (A) Addition of Br-ions (M of added unreacted Br- ions): curve (B) Irreversible removal curve 2, 5 X 10”; curve 3,5 X 1,5 X of Br- ions (M of instantaneously removed Br- ions): curve 1, 5 X IO? curve 2, 5 x lo+; curve 3, 5 x 10”. lations, as performed in the previous section, lead to the following expressions for the advanced phase shifts, A$adv:
=
A@ = -(Po- 2’)
if C’,, - ACB, 5 CB?
(6b)
Interesting is the linear relationship in eq 6b. It states that when the Br- coflcentration is forced below CBFita Ce(IV) spike is immediately induced. This is the excitability property of the BZR recently found experimentally in nonoscillatory state^.'^,'^ We therefore call eq 6b the “excitable branch“.ls Figure 3B shows the calculated phase shifts from the Oregonator when Br- ions are instantaneously removed, for example by the addition of AgN03. In the calculationS of Figure 3B, the solubility product of AgBrI9 was not cohsidered, Le., we used the approximation that in situations when the Br- concentration was lower than the amount of Br- ions removed, the resulting Brconcentration was made equal to zero. However, we also performed calculations with the inclusion of the solubility product of AgBr19 using the model Rl-R8. The results are shown in (16) (a) Ruoff, P. Chem.Phys. Leu. 1982,9476.(b) Ibid. 1982,92,239.
(c) Natunvissenschaften 1983,70,306. (17)Ruoff, P. Chem. Phys.Leu. 1983.96, 374. (18) In this context, it is interesting to note that certain biological oscillators also have such an excitable branch. Thus, the record of phase response
curves of these oscillators immediately leads us to the existence of their excitable kinetics. (19)“Handbook of Chemistry and Physics”;CRC Press: Cleveland, 1980, 60th ed, p B-220.
Figure 4. The experimental data using different AgN03 solutions as perturbants are presented in Figure 5 . We note that, in the model calculations, the excitable branch (solid lines in Figure 5 , A and B) lies parallel beneath the experimental curve (dashed line). The reason for this is probably due to the neglect of various intermediate reaction steps in the models which cause an additional delay in the appearance of the Ce(1V) spike. Nevertheless, for low perturbant concentrations (Figure 5 , A and B) we obtain a semiquantitative agreement between computations and experiments. Figure 2 together with Figure 5 clearly shows that the free oscillating state is conttolled by Br- ions. Addition of HOBr. We found it interesting to use hypobromous acid as an additional check of the bromide control. By R7, HOBr reacts rapidly with Br- ions and forms bromine which subsquently reacts with malonic acid to produce Br- ions and bromomalonic acid (R8). Thus the removed and produced Br- ions balance and we therefore expect either a zero phase shift or, due to the delay inherent in the reaction sequence R7-R8, a slight delay in the appearance of the Ce(1V) pulse. Only in the case when the HOBr concentration is so high that the Br- concentration is forced below CBtntshould we immediately induce a Ce(1V) spike, Le., observe the excitable branch. This predicted behavior is indeed found by calculations using the FKN model Rl-R8 (Figure 6). Characteristic is the continuation of the excitable branch into the domain of positive phase shifts at the end of the cycle. This is due to the delay of R7-R8 mentiohed above. Figure 7 shows the experimental resblts using two different initial Ce(1V) concentrations. It is interesting to note that, at higher initial Ce(1V) concentrations (Figure 7B), the jump to the excitable branch occurs much later in the cycle than when the lower initial Ce(IV) concentration is used (Figure 7A). This effect is not unexpected, since Ce(1V) itself initiates radical reactions which finally liberate Br- ionsZo Ce(1V)
+ CH,(COOH)2
-
Ce(II1)
-
+ H+ + .CH(COOH)2 (AI)
C H ( C O O H ) 2 + BrCH(COOH)2 CHz(COOH)2 + CBr(COOH)2 (A2) .CBr(COOH)z
+ H20
-
Br-
+ H + + C(OH)(COOH), (A31
Phase Response in BZ Reactions
The Journal of Physical Chemistry, Vol. 88, No. 13. 1984 2855
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