Chemical Physics Letters 480 (2009) 322–326

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Chemical Physics Letters

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Chaotic dynamics in an unstirred ferroin catalyzed Belousov–Zhabotinsky reaction

Federico Rossi a,*, Marcello Antonio Budroni b, Nadia Marchettini b, Luisa Cutietta a, Mauro Rustici c,Maria Liria Turco Liveri a

a Dipartimento di Chimica Fisica ‘F. Accascina’, Università degli Studi di Palermo, Viale delle Scienze Ed. 17, 90128 Palermo, Italyb Dipartimento di Chimica, Università degli Studi di Siena, Via della Diana 2/a, 53100 Siena, Italyc Dipartimento di Chimica, Università degli Studi di Sassari, Via Vienna 2, 07100 Sassari, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 July 2009In final form 3 September 2009Available online 8 September 2009

0009-2614/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cplett.2009.09.018

* Corresponding author. Fax: +39 091590015.E-mail address: f.rossi@unipa.it (F. Rossi).

The Belousov–Zhabotinsky (BZ) reaction is the best known example of far from equilibrium self-organiz-ing chemical reaction. Among the many dynamical behaviors that this reaction can exhibit, spatio-temporal chaos attracted particular interest, both for the ferroin and cerium catalyzed systems. In recentyears transient chaos was found in the cerium catalyzed BZ reaction, when conducted in batch andunstirred reactors. It was established that the chaotic oscillations, originated by the coupling amongchemical kinetics and transport phenomena, appeared and disappeared following a Ruelle–Takens–Newhouse scenario. In this Letter, we show results about the ferroin catalyzed system conducted in batchand unstirred reactors. An aperiodic transient regime was detected and we show that this is a genuinemanifestation of spatio-temporal chaos.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Chaotic dynamics have been found in many oscillating phenom-ena, in chemical science in particular, the well known Belousov–Zhabotinsky (BZ) reaction [1,2] can exhibit complex, aperiodicand chaotic oscillating regimes in many experimental conditions[3]. Complex dynamics in batch stirred reactors were found bothin cerium [4–6] and ferroin [7,8] catalyzed BZ systems.

In recent years, our group detected a chaotic transient in theoscillations of a cerium catalyzed BZ reaction [9] in closed and un-stirred reactors, generally spectrophotometric cuvettes. In Fig. 1 isreported a typical spectrophotometric time-series where the oscilla-tions of the Ce(IV) absorbance show the transient chaotic behavior.

The transient was found to appear by following a Ruelle–Takens–Newhouse (RTN) scenario [11,12] and disappear by follow-ing an inverse RTN [13]. The origin of the chaotic oscillations wasattributed to the coupling among the BZ nonlinear kinetics andthe main transport phenomena present in the system: diffusionand natural convection. The unstirred cerium catalyzed systemwas studied in several experimental conditions [14–17] and partic-ularly by varying the viscosity of the solution with non-ionicmicelle-forming surfactants, it was inferred the role of the convec-tion in the onset of the aperiodic behavior. This result was con-firmed by using reactors with special geometry [18] and, finally,theoretically proved by numerical simulations [19] where theGrashof number was the bifurcation parameter in the transitionfrom the periodic regime to chaos.

ll rights reserved.

In this Letter, we present the first results about the unstirredferroin catalyzed systems. With this work we are going to under-take an extensive study on the relationship between transientchaos and spatial dissipative structures. In the past, several col-leagues drawn our attention to the possible role of 3-dimensionalstructures which are likely to form in the bulk solution when leftunstirred (see for example the 2002 Faraday discussion on nonlin-ear chemical kinetics [20]). Recently, we also showed how naturalconvection can induce spatio-temporal chaos in the motion of aspiral tip [21]. Following these inputs we decided to investigatethe ferroin catalyzed BZ system which is more suitable for studyingpattern formation. The substitution of the catalyst is not an obvi-ous issue and the new system needs to be characterized. In fact,cerium and ferroin catalyzed BZ systems show different behaviorsboth in temporal and spatial evolution, mainly because of the re-dox potential of the catalysts [22–24]. In particular, the ferroin cat-alyzed system does not show the initial single lengthy oscillation,known as induction period (IP), typical of the cerium catalyzed sys-tems. Moreover, a different reactivity between the two catalysts,with respect to the organic substrates [25] and the oxybrominespecies was found [26].

In this Letter, we show that also the ferroin catalyzed BZ reactioncan exhibit a transient chaotic regime. We also characterized theroute which lead the system to chaos and back to the periodic regime.

2. Experimental methods

Malonic acid (MA), sodium bromate, sulfuric acid and ferroinðFeðo-phenÞ3SO4Þ were commercial grade reactants (Fluka). All

A

B C D

Fig. 1. (A) Temporal evolution of a cerium catalyzed BZ reaction in a batch and unstirred reactor. Time-series was recorded following [Ce(IV)] at k ¼ 320 nm. (B), (C) and (D)Phase portraits of the first periodic – chaotic – second periodic region, respectively. The phase portraits were reconstructed by using the delay vector method [10].

F. Rossi et al. / Chemical Physics Letters 480 (2009) 322–326 323

chemicals were used without further purification. Stock solutionwere prepared by weight before use. Deionized water from reverseosmosis (Elga, model Option 3), having a resistivity higher than1 MX cm, was used to prepare all solutions. The followingconcentration of reactants stock solutions were used:MA 2 mol dm�3, NaBrO3 1:2 mol dm�3, H2SO4 5 mol dm�3 andferroin 0:025 mol dm�3. Samples for the kinetics runs were pre-pared in a flask by diluting and stirring for 15 min a solution con-taining the BZ components, then 2 mL of solution were poured intothe spectrophotometric quartz cuvettes ð1� 1� 5 cmÞ and placedin a UV–Vis spectrophotometer (Analytik Jena Specord S 600equipped with a magnetic stirring apparatus) for data acquisition.All experiments were performed at room temperature. Time serieswere recorded by following the absorbance of ferroin atwavelength k ¼ 510 nm and ferriin at k ¼ 630 nm where the molarextinction coefficients are � ¼ 1:1� 104 mol�1 dm3 cm�1 and� ¼ 620 mol�1 dm3 cm�1, respectively.

3. Results and discussion

Because of the batch conditions of our experiments, the onlystable attractor of the system is the final thermodynamic equilib-rium, therefore, the observed chemical oscillations have a transientcharacter. Nevertheless the relatively slow kinetics of the reactionallows to consider stationary some portions of the signal for a gi-ven interval of time, so that the pool chemical approximation canbe applied [27,28].

Fig. 2 shows a typical spectrophotometric time series recordedat k ¼ 630 nm, where the oscillations in the concentration of theferriin, i.e. the oxidized form of the catalyst, were followed. Perusalof Fig. 2 reveals that, as in the case of the cerium catalyzed reactionreported in Fig. 1, three dynamical regimes can be identified inthree different regions of the time series (panels A–C). Even at afirst glance, it is evident that regions A and C appear to oscillatewith a regular period, while the oscillations in the region B havean aperiodic character. The regular oscillations of region C lastfor about 1 h more (total lifespan is 15800 s) until the reaction

reaches the thermodynamic equilibrium. Nevertheless, the instru-mental noise overlaps the small amplitude peaks, typical of the endof the reaction, making them unusable for data analysis.

The time series reported in the panel A of Fig. 2 shows an expo-nential decay trend due to the decomposition of the catalystaccording to the reaction [22,29]

FeðphenÞnþ3 þ 3Hþ !kd Fenþ þ 3ðphenÞHþ n ¼ 2;3 ð1Þ

being the kinetics a first order respect to the iron–phenanthrolincomplexes. The phase portraits reconstructed from the original timeseries roll up on the exponential trend in the phase space, makingtheir interpretation not clear. In order to overcome this problemwe fitted the analytical concentration trend of the complexes([Fe(II)] + [Fe(III)]) to get a rough estimation of the decay constant.The portraits reported in panel D–F were then reconstructed fromthe time series once the fitted signal has been subtracted. Thedecomposition of the catalyst certainly affects the global kineticsof the oscillations yielding the aforementioned subtraction proce-dure not perfectly correct. Nevertheless the characteristic time ofreaction (1) ðkd ’ 10�5—10�4 s�1Þ is on a slower time scale respectto the reactions governing the oscillation frequency, and process(1) becomes important when initial concentrations of reactantsare very high and concentration of ferroin is low [26,29].

A detailed characterization of the oscillatory regimes will begiven later in the text.

The dynamical nature of the aperiodic region was confirmed tobe related with the spatial inhomogeneities originated by transportphenomena, as in the case of cerium catalyzed BZ systems. In fact,in Fig. 3 it is showed the effect of the homogenization due to thestirring. At the beginning the solution is left unstirred and, afterthe initial periodic oscillations, it evolves towards the aperiodicregime. If during the aperiodic transient, stirring is started, the sys-tem immediately shifts to a periodic regime. Again, if stirring isturned off, the system slowly goes back to an aperiodic phase.

Fig. 4 shows the characterization of the aperiodic region in theinterval 840–4434 s. Two distinctive features of deterministicchaos were detected, namely a broad band spectrum in the fre-

Fig. 2. Temporal evolution of a ferroin catalyzed BZ reaction in a batch and unstirred reactor. ½MA� ¼ 0:74 mol dm�3, ½BrO�3 � ¼ 0:126 mol dm�3, ½H2SO4� ¼ 0:35 mol dm�3,½ferroin� ¼ 9:3� 10�4 mol dm�3. Panels A–C in the lower part show the enlargement of the A–C areas in the temporal time series. Panels D–F show the phase portraits ofregions A–C.

Fig. 3. Stirring effect on the dynamic evolution of a ferroin catalyzed BZ reaction.½MA� ¼ 0:74 mol dm�3, ½BrO�3 � ¼ 0:126 mol dm�3, ½H2SO4� ¼ 0:35 mol dm�3, ½ferroin� ¼9:3�10�4 mol dm�3.

a

Fig. 4. Characterization of the region 840–4434 s of the time series reported in Fig. 2. (a)

324 F. Rossi et al. / Chemical Physics Letters 480 (2009) 322–326

quencies domain and positive maximal Lyapunov exponents.Fig. 4a shows the decomposition of the signal in its sinusoidal com-ponents operated by a Fast Fourier Transform (FFT) algorithm. Thebroad band spectrum does not permit to extract any fundamentalfrequency, a typical behavior of chaos and noise. In order to distin-guish between the two, further analysis are required. In particular,the largest Lyapunov exponent, k, has been calculated by using theKantz algorithm from TISEAN package [30]. The positive trend corre-sponding to the embedding dimension m ¼ 3;4;5 is reported inFig. 4b. The linear regression of the curves Sð�;m; tÞ, in the zone be-tween 0 and 40 iterations, gave a value of k ¼ 0:00532� 0:00002.The previous analysis, together with the phase portrait reportedin the panel E of Fig. 2, shows that the aperiodic regime can be clas-sified as a genuine manifestation of deterministic chaos and here-after we can refer to the region 840–4434 s in the time series as achaotic transient.

Once the nature of the transient has been established we alsocharacterized the transition scenarios. Fig. 5 shows the FFTs ofthe regions A and C of Fig. 2. In particular, we identified four inter-vals with different dynamical regimes: P1 (0–377 s, Fig. 5a), QP1

b

Broad band FFT spectrum and (b) computation of the largest Lyapunov exponent, k.

a

d

b

c

Fig. 5. Characterization of the direct (a, b) and inverse (c, d) RTN transition scenarios to and from the chaotic regime. The insets report the interval of the time series wherethe FFTs were analyzed. The values of the frequencies are given along the text.

F. Rossi et al. / Chemical Physics Letters 480 (2009) 322–326 325

(378–839 s, Fig. 5b), QP2 (4435–5072 s, Fig. 5c), P2 (5072–5276 s,Fig. 5d). P1 and P2 are periodic regimes with 1 fundamental fre-quency and its harmonics: f1 ¼ 2:73� 10�2 Hz for P1 andf6 ¼ 1:95� 10�2 Hz for P2. P2 represents only a small portion ofthe region C because of the non-stationarity of the signal. Never-theless, the analysis of subsequent intervals starting from 5277 s,reveals a succession of periodic areas with decreasing oscillationfrequencies.

A more complex behavior has been found for QP1 and QP2. Inpanels b and c of Fig. 5 two fundamental frequencies and theirlinear combinations can be easily detected: f2 ¼ 2:53� 10�2 Hzand f3 ¼ 2:93� 10�2 Hz for QP1, f4 ¼ 2:05� 10�2 Hz andf5 ¼ 3:13� 10�2 Hz for QP2. The dynamic regimes P1 and P2

correspond to a limit cycle in the phase space, while QP1 and QP2

correspond to a 2-dimensional Torus. The phase space behaviorsare the fingerprint of two RTN scenarios, one direct, P1 ! QP1

! chaos, the other inverse, chaos! QP2 ! P2.

4. Conclusion

In this Letter, we showed that the ferroin catalyzed BZ reactioncan exhibit transient chaotic oscillations for a broad spectrum ofthe reagents initial concentrations. Similarly to the cerium cata-lyzed reaction, the chaotic regime appears by following a RTN tran-sition (periodic ? quasi periodic ? chaos) and disappears byfollowing an inverse RTN scenario (chaos ? quasi periodic ? peri-odic). In this respect, the ferroin and cerium catalyzed systemswere found to follow the same qualitative evolution. This confirmsthat the main role in the onset of chaotic oscillations is played bytransport phenomena. Numerical studies performed with a generic

Oregonator model [19], are therefore suitable to simulate the sys-tem presented in this Letter.

Acknowledgements

The authors thank the Università degli Studi di Palermo forfinancial support (Research Funds ex quota 60%). Thanks are dueto Casimiro Caruso (University of Palermo) for experimental help.

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