Complex oscillations in the Belousov–Zhabotinsky batch reaction with methylmalonic acid and manganese(ii)

The Belousov–Zhabotinsky (BZ) oscillating chemical reaction involves the oxidation of an organic compound by the bromate ion in the presence of a metal ion catalyst such as cerium(iv), manganese(ii), or ferroin. Simple periodic oscillations are generally obtained for the BZ reaction in a batch (closed) system. However, complex oscillations have been observed for the BZ reaction in batch with malonic acid and either cerium or ferroin ions as the catalyst. We report here that fascinating complex oscillations in the potential of a Pt electrode have been found in the batch BZ reaction with methylmalonic acid (MeMA) and manganese(ii). Relatively high initial concentrations of NaBrO3 and MeMA are required, and the [NaBrO3]0/[MeMA]0 ratio is the main factor determining the type of oscillations obtained. Other relevant factors are [NaBr]0, [MnSO4]0, [H2SO4]0 or [NaOH]0, temperature, and stirring rate. Complex phenomena observed include mixed mode oscillations, birhythmicity, quasiperiodicity, bursting, and possible chaos. A mechanism is proposed involving the reversible formation of a manganese(iii) complex with bromomethylmalonic acid followed by two-electron oxidation to methyltartronic acid and Mn2+.


Introduction
The oscillation patterns exhibited in the rhythmic beating of cardiac pacemaker cells, the central pattern generators that control rhythmic actions, and the bursting patterns found in various types of neurons due to changing membrane potential are similar to those observed under certain conditions in the Belousov-Zhabotinsky (BZ) reaction system. In the well-studied BZ oscillating chemical reaction an organic compound is oxidized by the bromate ion in the presence of a metal ion catalyst such as cerium(IV), manganese(II), ruthenium(II), or ferroin. The classical BZ system consists of malonic acid (MA), BrO 3 À , and Ce(IV) in H 2 SO 4 . 1 Chemical oscillations have been observed for the BZ reaction both in a closed (batch) reactor and in a continuous-ow stirred tank reactor (CSTR). Among the organic substrates found to give oscillations with the BZ system is methylmalonic acid (MeMA).
Most commonly, the BZ reaction in batch gives simple periodic oscillations in absorbance at a xed wavelength and/or in potential of a Pt or specic-ion electrode versus a reference electrode. However, in a few BZ systems with MA in batch, complex oscillations have been observed. Ruoff 2 reported chaotic behavior in the BZ reaction catalyzed by cerium ion.
Wang, Sorensen and Hynne 3 found period doubling, intermittency, mixed mode and quasiperiodic oscillations in the BZ reaction also with cerium ion. Kawczinski et al. 4,5 obtained mixed mode oscillations in the BZ reaction catalyzed by ferroin. Johnson, Scott and Thompson 6 conrmed the complex oscillations of Wang et al. and modeled this behavior in the ceriumcatalyzed BZ reaction. Experimental and numerical evidence for complex oscillations in the BZ reaction catalyzed by cerium in batch was reported subsequently by Kolar-Anic et al. 7 and Blagojevic et al. 8 Manganese(II)-catalyzed BZ systems have been less studied than those catalyzed by cerium(IV) or ferroin. Hansen and Ruoff 9 used NMR to observe oscillations in a closed BZ system using MeMA and catalyzed by Mn(II). Subsequently, Jwo et al. 10 reported on a thorough study of the Mn(II)-catalyzed BZ reaction in batch with MeMA as well as with other related acids of MA. The reaction was followed potentiometrically with a bromide ion selective electrode and periodic oscillations were obtained. In both of these studies using Mn(II) as catalyst only simple periodic oscillations were obtained.
Recently, Frerichs et al. 11 reported the surprising result that the closed BZ system containing BrO 3 À , MA, and Mn(II) gave signicant pH oscillations under a variety of conditions. The conclusion was that the pH oscillations were due primarily to the reversible formation of a manganese(III) complex with bromomalonic acid. Because of the close similarity in their structures, it was decided to search for pH oscillations with the BZ reaction using MeMA in place of MA. Although signicant pH oscillations were not found, a variety of fascinating complex oscillations, to our knowledge never reported before in this system, were observed. Our goal in this study is to uncover the wide range of oscillation patterns exhibited by the Mn(II)-catalyzed BZ reaction with MeMA as the organic substrate in a batch reactor. The oscillation patterns found include simple periodic oscillations, mixed mode oscillations (MMO), birhythmicity (BI), bursting (BR), quasiperiodicity (Q), and possible chaotic oscillations. We rst describe the experimental procedure used, present the results in relation to how different conditions of reaction mixture composition, temperature, and stirring rate affect the oscillation patterns, then discuss observed oscillations in the context of a possible reaction mechanism for the BZ system, and nally provide a conclusion.

Experimental section
Experiments were carried out in a glass reactor covered by a reactor cap and having a total solution volume of about 50 mL. The cap had a small opening covered by ParaFilm having a very small perforation. A Pt electrode and a combination pH electrode, each with a Ag/AgCl reference electrode, were inserted in the reactor cap. Both potential and pH were recorded concurrently using a Servogor 124 chart recorder. Stirring was done at various rates from 308 to 886 rpm using a Teon-coated stir bar and either a Fisher Scientic or Heidolph Hei-Standard stirrer. Runs were carried out at room temperature (21 AE 1) C, as well as at 25, 30, and 40 C using a constant-temperature circulator. The reagents NaBrO 3 (Fisher), MeMA (TCI), NaBr (Fisher) and MnSO 4 $H 2 O (Fisher) were ACS-certied. Ultrapure water was bubbled with N 2 for about an hour before use.
Initially, the procedure for experiments was to add the solid reagents directly to the water in the reactor. Later, some runs were done using pre-dissolved NaBrO 3 . Eventually, the procedure involved pre-dissolving both NaBrO 3 and MeMA before adding MnSO 4 $H 2 O and NaBr in that order. In most runs, H 2 SO 4 was used and was added before the other reagents. A few experiments were done using NaOH, and some were carried out without either acid or base. Since the reaction vessel used was a CSTR reactor with closed ports, a few experiments were carried out at room temperature using a 100 mL beaker and comparable results were obtained.  Fig. 1(a). Immediately aer addition of all reactants the solution color was dark orange. During the induction period, the color went through the transition to lighter orange, yellow, and then nearly colorless before oscillations began. As each peak was formed, the solution color became pink to light tan.

Results
In between oscillations, the solution again became nearly colorless, suggesting a transition from Mn(III) to Mn(II) ions. Oscillations were periodic, but not symmetric, as typically a "shoulder" appeared as the peak returned to the baseline. These oscillations continued for an extended period of time, sometimes for hours.  These were of the mixed mode types 2 1 and 3 1 , using the notation L S where L is the number of large-amplitude oscillations followed by S small-amplitude oscillations. One such result is shown in Fig. 1 Subsequently, a series of runs was done with varying [MeMA] 0 , not only for 1.00 M NaBrO 3 , but also for 1.60 M and 2.00 M NaBrO 3 . An overall summary of the results is given in Table 1. Reactant concentrations used for NaBr, MnSO 4 , and H 2 SO 4 are given in the table. A variety of mixed mode oscillations (MMO) was obtained, ranging from 2 1 to 10 2 .
Perhaps the principal conclusions to be drawn from Table 1 are that the [NaBrO 3 ] 0 /[MeMA] 0 ratio is the main determinant of whether complex oscillations are obtained, and of what type. For runs done with 1.00 M NaBrO 3 , MMO were observed over a reactant ratio range of 2.50 : 1 to 1.00 : 1. With 1.60 M NaBrO 3 , MMO were found over a range of 4.44 : 1 to 1.25 : 1, and with 2.00 M NaBrO 3 the reactant ratio range giving complex oscillations was 3.70 : 1 to 1.00 : 1. Also, the greater the initial concentration of NaBrO 3 the more complex the oscillations tended to be. Further, for a given [NaBrO 3 ] 0 the complexity of oscillations generally increased with decreasing [NaBrO 3 ] 0 / [MeMA] 0 ratio (i.e., with increasing [MeMA] 0 ). Finally, for a given [NaBrO 3 ] 0 both the cap times and induction times tended to decrease with decreasing [NaBrO 3 ] 0 /[MeMA] 0 ratio. The "cap" refers to the broad maximum that generally occurs during  Fig. 2(a). Also, bursting appears to be present for 2.00 M NaBrO 3 , especially at values of [NaBrO 3 ] 0 / [MeMA] 0 from 1.50 : 1 to 1.00 : 1. This phenomenon, which occurs when periods of quiescent behavior alternate with periods of relatively fast large-amplitude oscillations, is illustrated in Fig. 2(b).
Using the procedure described above, with [NaBrO 3 ] 0 ¼ 1.60 M and [MeMA] 0 ¼ 0.500 M, the dependence of oscillations on temperature, [NaBr] 0 , [MnSO 4 ] 0 , and presence of acid or base was investigated. The results are summarized in Table 2. Regarding the effect of temperature, similar MMO were observed at room temperature, at 20 C, and at 25 C, the main difference being the number of 2 1 oscillations. Not surprisingly, the cap time, induction time, and period of oscillations all decreased with increasing temperature. As to the dependence of oscillations on [NaBr] 0 , good MMO were found with 0.160 M and 0.240 M NaBr, but the former had many more 2 1 oscillations. Interestingly, both the cap time and induction time were reduced as [NaBr] 0 was lowered. Also, similar MMO were obtained with 0.0100 M and 0.0120 M MnSO 4 , although the former gave signicantly more 2 1 oscillations. Regarding the impact of acid or base, good MMO were obtained with 0.0188 M H 2 SO 4 , with 0.010 M and 0.020 M NaOH, as well as with no acid or base added. One difference was that 2 1 oscillations were found with H 2 SO 4 , whereas 3 1 oscillations were the main type observed with NaOH or with no added acid or base.
Because the dissolution of NaBrO 3 in water is an endothermic process, and also to be sure the solid is completely dissolved before any reaction begins, it was decided to predissolve NaBrO 3 before adding it to the reactor. Also, the initial reaction of NaBrO 3 with NaBr is quite exothermic. Thus, in order to eliminate any possible temperature effect, a constant-temperature circulator was used to maintain the reactor at 25 C. A number of experiments were done using this procedure, including many runs with the same reactant concentrations as used previously. Although solid MeMA generally dissolved very readily aer being added to the reactor, eventually it was decided to pre-dissolve this reactant as well. All subsequent experiments were done using the above method.
Since Ruoff and Schwitters 12 had reported a stirring effect on the number and shape of oscillations in their study of the closed BZ system with MeMA and Ce(IV), it was thought important to investigate this effect also in the present system. Previously, all runs were carried out at the rather slow stirring rate of 308 rpm.
To determine the effect of stirring rate, the present system was explored over a wide range from 308 to 886 rpm. Several previously used sets of concentrations were studied. The results are summarized in Table 3. It is clear that stirring rate can have a signicant effect on the type of complex oscillatory behavior exhibited.
In addition to MMO, birhythmicity (BI), bursting (BR), and quasiperiodicity (Q), along with possible chaos, were observed. An example of a run giving MMO and BI is shown in Fig. 3(a), and one showing a transition from MMO to Q and BR is given in Fig. 3(b). In the case of birhythmicity, a lower oscillatory state rst appeared with peak amplitudes in the 1-40 mV range. Later in the run, there was a very large increase in potential to a higher oscillatory state and these peaks typically had amplitudes of 200-300 mV. Quasiperiodicity involves the appearance  3 1 2 1 (3 1 ) 2 (2 1 ) 2 (3 1 ) 2 2 1 3  of two incommensurate frequencies of oscillations, one much lower than the other. With 1.00 M NaBrO 3 and 0.667 M MeMA, BI was observed at a stirring rate of 635 rpm or less, while both BI and MMO occurred at 379 and 512 rpm. Interestingly, at 765 rpm period-halving (P-1/2) took place, in which the system switched to a new behavior with half the period of the original oscillations. For the series of runs with 1.60 M NaBrO 3 and 0.500 M MeMA, MMO were found at all stirring rates, but BI only at 400 rpm or lower.
With 2.00 M NaBrO 3 and 1.33 M MeMA, BI, Q, and BR were all present except at the highest stir rate (886 rpm), which did not give BI. In the case of 2.00 M NaBrO 3 and 1.60 M MeMA only the higher oscillatory state was present, with Q and BR being observed at all stirring rates but MMO only at 635 and 886 rpm. The results in Table 3, including experiments done with 2.00 M NaBrO 3 and 1.60 M MeMA, suggest that Q and BR are more likely to be observed when [NaBrO 3 ] 0 is near the upper limit of concentrations giving complex oscillations, as long as [MeMA] 0 is sufficiently high. It is to be noted from this table that possible chaos also is more likely to be found under these conditions.

Discussion
Based on the similarity of the structures of MeMA and MA, it would be reasonable to expect the mechanisms of their reactions in the BZ system to be rather similar. A proposed model is given in Table 4. As with the model used for the BZ reaction with MA, 11 we include the minimal bromate subsystem which generates Br 2 and Mn 3+ . This is followed by the bromine-methylmalonic acid subsystem in which MeMA is converted to the enol, which then reacts with Br 2 to produce bromomethylmalonic acid (BrMeMA). Finally, we include the Higher state: (5 1 ) 2 3 1 (4 1 ) 6

0 886
Lower state: none MMO,Q,BR Higher state: (2 1 ) 2 8 1 5 1 4 1 (6 1 ) 4 7 1 (5 1 ) 2 4 0 manganese-bromomethylmalonic acid subsystem which features the reversible formation of a manganese(III) complex with BrMeMA, followed by two-electron oxidation to methyltartronic acid (MeTA) and Mn 2+ . The fact that the solution color turns brown during the formation of a peak similarly to what was observed in the BZ reaction with MA lends support to the role of [Mn(III)BrMeMA] + . For a typical run not showing birhythmicity, one can generally identify four rather distinct regions leading up to the appearance of oscillations. The reaction of BrO 3 À with Br À and H + in step 1 of the proposed model takes place very rapidly, as evidenced by the nearly immediate appearance of a reddish-orange color apparently due to Br 2 formation. Simultaneously, a very large increase in potential occurs. This is followed by a much slower reaction involving the reversible conversion of MeMA to the enol form (step 8). During this process a more gradual increase in potential takes place and generally a "cap" is formed. As this is occurring, the orange color turns yellow orange, indicating partial reaction of the newly formed enol with Br 2 . The next phase of the reaction corresponds to the somewhat faster reaction of enol with Br 2 to HBrO 2 + Br À + H + 4 2HOBr form BrMeMA in step 9 of the model. There is a noticeable drop in potential during this step and the solution becomes yellow. Finally, there is another rather slow process (step 11) which is thought to be the reversible reaction of Mn 3+ with BrMeMA to form the [Mn(III)BrMeMA] + complex. During this step, the potential drops further and the solution becomes light yellow. Eventually, oscillations begin, as indicated both by the increase in potential and the appearance of the brown color of the complex. During oscillations, the solution color then oscillates between amber and brown, likely due to the equilibrium between Mn 3+ and [Mn(III)BrMeMA] + in step 11. Finally, oscillations stop when [Mn(III)BrMeMA] + is sufficiently depleted by conversion to MeTA and Mn 2+ in step 12.
Computer simulations based on the model in Table 4 were carried out with Berkeley Madonna soware using the Rosenbrock method for integrating stiff differential equations. The rate constants used for the minimal bromate subsystem are based on previously published values. 11 Over many years, the kinetics of the enolization of MeMA as in step (8) has been studied by several investigators, including Furrow, 13 Ruoff et al., 9,12,14,15 Williams and Graham, 16 and Yoshimoto et al. 17 Their results as summarized in Table 5 reect a disparity of nearly an order of magnitude in k 8 . Data from the latter two references mentioned above allow an estimate of the value of k À8 for the reverse of the enolization reaction, but they do not agree. However, there is agreement that k À8 [ k 8 .
The rate constant values for reactions in the brominemethylmalonic acid and manganese-bromomethylmalonic acid subsystems were obtained by treating them as variable parameters in computer simulations using the model in Table  4.
Before attempting to reproduce the complex oscillations obtained in most of the experiments, we rst tried to simulate the simpler periodic oscillations observed using lower concentrations of reactants as in Fig. 1(a). To determine the importance of [Mn(III)BrMeMA] + , simulations were attempted without including the manganese-bromomethylmalonic acid subsystem (i.e., letting k 11 ¼ k À11 ¼ k 12 ¼ 0). Interestingly, it was not possible to produce oscillations under these conditions.
Once it was determined the manganese-bromomethylmalonic acid subsystem needed to be part of the model, it was found that the set of rate constants giving the best t under the experimental conditions for Fig. 1(a) was: k 8 ¼ 4.5 Â 10 À4 s À1 ; k À8 ¼ 100 s À1 ; k 9 ¼ 5 Â 10 4 M À1 s À1 ; k 10 ¼ 80 M À1 s À1 ; k 11 ¼ 5 Â 10 5 M À1 s À1 ; k À11 ¼ 1 Â 10 5 M À2 s À1 ; k 12 ¼ 600 M À1 s À1 . The calculated oscillations under these conditions are shown in Fig. 4(a). Note that these oscillations give good agreement with experiment in terms of the induction period, period of oscillations, and total number of peaks observed. Also, as the period increases toward the latter part of the experiment, the peaks begin to show a broadening or formation of a "shoulder," as observed experimentally. This is especially evident in Fig. 4(b) where a k 10 value of 40 M À1 s À1 is used in the plot of [BrO 2 ] and [HBrO 2 ] vs.time. This suggests that the intermediates, BrO 2 and HBrO 2 , may play a signicant role in determining the shape of the oscillatory peaks.   Table 4; (b) [BrO 2 ] 0 (blue) and [HBrO 2 ] 0 (green) using rate constants given in Table 4 except k 10 ¼ 40 M À1 s À1 .
complex may be formed in all Mn(II)-catalyzed BZ reactions with MA or MeMA, but in some cases may not be present at a high enough concentration that it can be detected.) Although simulation of periodic oscillations with the proposed model was successful, this was not the case for complex oscillations. It is suggested that one or more steps involving formation of H + may need to be added to the model in order to simulate complex oscillations. This could involve formation of further oxidation products, such as Pyr and/or HOAc. It is interesting that bursting behavior similar to that described above was originally reported for biological systems. For example, bursting due to a changing membrane potential is common in various types of neurons, including pacemaker neurons. The latter have been shown to control rhythmic tasks such as breathing, locomotion, eating, and sleeping. 21 Bursting also has been studied in various electrochemical systems, including H 2 O 2 reduction on platinum, iron dissolution in sulfuric acid with halogen additives, and with dichromate ions coupled with graphite or zinc electrodes. 22 In addition, the chlorine dioxide-iodide reaction in a system consisting of two coupled CSTRs (continuous-ow stir tank reactors) has been found to give neuron-like bursting behavior. 23 The insight into the mechanisms that generate the observed oscillation patterns in the present work possibly could elucidate processes that lead to similar oscillation patterns in biological systems and other natural phenomena.
It is our hope that this study of the Mn(II)-catalyzed BZ reaction using MeMA as the organic substrate presents new investigation opportunities in nonlinear chemical systems. For example, it would be informative to study this reaction in a ow system, especially to observe how the specic type of complex oscillations might vary with the reciprocal residence time k 0 .

Conflicts of interest
There are no conicts of interest to declare.