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Chemical Physics 313 (2005) 101–106

Effects of the electrolytes in a closed unstirredBelousov–Zhabotinsky medium

Federico Rossi a, Federico Pulselli a, Enzo Tiezzi a, Simone Bastianoni a, Mauro Rustici b,*

a Dipartimento di Scienze e Tecnologie Chimiche e dei Biosistemi, Universita di Siena, Via della Diana 2/a, Siena, Italyb Dipartimento di Chimica, Universita di Sassari, Via Vienna 2, 07100 Sassari, Italy

Received 7 July 2004; accepted 3 January 2005Available online 20 January 2005

Abstract

Complex periodic and aperiodic behaviors are reported in an unstirred Belousov–Zhabotinsky reaction at different concentra-tions of added electrolytes. A route to chaos following a Ruelle–Takens–Newhouse (RTN) scenario is identified. A 3-fold torus

was also found in phase space. In this paper, we prove that the concentration of inert added electrolytes is a bifurcation parameterfor the sequence period-1 ! period-2 ! period-3 ! chaos.� 2005 Elsevier B.V. All rights reserved.

Keywords: Belousov–Zhabotinsky; Nonlinear dynamics; RTN scenario; Transient chaos; Electrolytes; Sulphate ions

1. Introduction

The Belousov–Zhabotinsky (BZ) reaction is themost famous oscillatory chemical reaction in a homo-geneous liquid phase for studying temporal, spatialand spatiotemporal nonlinear dynamics in nonequilib-rium systems [1–3]. Significant research has beenundertaken to understand chemical chaos that havebeen usually observed in a continuously flow stirredtank reactor (CSTR) [1,3,4]. It has been also reportedthat transient chaotic oscillations are observed in theBZ oscillating chemical reaction in a stirred batchreactor experimentally and numerically [5–8].

Recently, we have observed aperiodic oscillationsduring the chemical evolution of a closed unstirredcerium catalyzed BZ system [9]. These aperiodic oscil-lations are an example of transient chaos because theyare sensitive to the initial conditions, the major dis-

0301-0104/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2005.01.003

* Corresponding author. Tel.: +79 229561; fax: +79 212069.E-mail address: rustici@uniss.it (M. Rustici).

tinctive features of chaos [9,10]. The chaotic regimeis bounded by two periodic zones. The onset of chaosspontaneously starts by a Ruelle–Takens–Newhouse(RTN) scenario [11,12] as soon as convection motioncouples to diffusion and local kinetics. The chaoticmotion continues for about 2 h and ends by an in-verse RTN scenario [13]. Differently from the firsttransition (RTN scenario), the last seems related tothe consumptions of the reactants [12,13]. In previouspapers, we studied the effect of different experimentalconditions on the system dynamics. We showed thatviscosity, temperature and reactor geometry areimportant control parameters for the transition tochaos [14–17]. They play an important role in the cou-pling of chemical kinetics, diffusion and convectionallowing or preventing the onset of chaos. A commonfeature of these systems is that the transition to chaosalways takes place through an RTN scenario.

In this paper, we will show that the added electrolyteconcentration is another bifurcation parameter respon-sible for the transition chaos to periodicity of a closedunstirred BZ reaction.

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Fig. 1. (a) A typical spectrophotometric recording in absence ofstirring. (b) Time series of the region 3058–5656. (c) Typical broadband spectrum of a chaotic regime in the Fourier transform of theregion 3058–5656.

102 F. Rossi et al. / Chemical Physics 313 (2005) 101–106

2. Experimental

All experiments were performed isothermically at�20 �C in a batch reactor (spectrophotometer cuvette,1 · 1 · 4 cm3). The dynamics were monitored by thesolution absorbance at 320 nm using quartz UV gradespectrophotometer cuvettes. A double beam spectro-photometer (Varian, series 634) was used. All chemi-cals were of analytical quality and were usedwithout further purification. The following concentra-tions of reactants stock solutions were used: Ce(SO4)20.004 M, malonic acid 0.30 M, KBrO3 0.09 M; eachstock solution was 1 M H2SO4. The oscillator wasstarted by mixing equal quantities of reactants in aflask. Several experiments were performed by addingNa2SO4 Æ 10H2O, MgSO4 or Al2(SO4)3 to the solutionat different concentrations. The solution was stirredfor 10 min with a 1-cm length teflon-coated magneticstirrer, at a constant high stirring rate; then electro-lytes were added and the solution was stirred for an-other 2 min, the total volume was 12 ml.

The solution was then poured into the cuvette untilthe sample reached the top, and measurement of the sig-nal began. The cross-sectional area of the spectropho-tometer light beam was 30 mm2. The volume spannedby the beam was 300 mm3 (7.5% of the total volume)and was located 2 cm away from the liquid-air interface,1 cm away from the bottom of the cuvette and about0.4 cm away from the sides.

The spectrophotometer was connected to an IBMcompatible PC for data acquisition by an analog to dig-ital board converter with a 16-bit resolution. The absor-bance was recorded with a ss = 1 s sampling time. Timeseries points were recorded and stored in the computerfor data analysis.

3. Results

In this section, we will illustrate in detail the resultsobtained adding Na2SO4 to the BZ solution; we ob-tained analogous results for the others electrolytes.Fig. 1(a) shows the typical spectrophotometric recordingin absence of stirring. Two transitions: periodic ! aperi-odic ! periodic can be observed. The Fourier transformof a significant interval of the aperiodic region (Fig.1(b)), shows a broad band spectrum (Fig. 1(c)); it is acommon feature of all aperiodic time series. Our previ-ous experiments [9,12,13] show that these aperiodicoscillations of the BZ system are an example of chaotictransient. For this reason we can refer to this behaviorby the following scheme: periodicity ! chaos !periodicity.

Fig. 2(a) shows a typical spectrophotometric timeseries recording after 0.2707 g of Na2SO4 (0.070 M)was added to the BZ solution. This spectrum is not

distinguishable from that obtained in absence ofNa2SO4. Fig. 2(b) shows an aperiodic pattern like thatobtained in absence of Na2SO4 (Fig. 1(b)). In fact theFourier transform (Fig. 2(c)) of the signal shown onFig. 2(b) (range 3058–5656 s) presents a typical broadband spectrum like that of Fig. 1(c). The Fouriertransform is not able to distinguish between chaosor random motion as both give broad band spectra,nevertheless there is no reason to believe that Na2SO4

can induce a transition from chaos to random motion.Therefore the behavior that appears on Fig. 2(a) is amanifestation of chaos. This type of behavior does notchange until the Na2SO4 concentration reaches a crit-ical concentration of about 0.087 M.

When Na2SO4 concentration reaches 0.087 M theaperiodic pattern disappears and only the periodicbehavior occurs as showed in (Fig. 3(a)). In fact theFourier transform (Fig. 3(b)) of this region is nearlysinusoidal with only harmonics of the fundamentalfrequency (f1 � 0.00579 Hz) above the noise level.

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Fig. 3. (a) Spectrophotometric time series recording of BZ solutionwith Na2SO4 0.087 M. (b) Spectrum of the region 3058–5656 with onlyone fundamental frequency typical of a periodic regime. (c) Limit cyclereconstructed applying the delay vector method to the region 3058–5656.

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Fig. 2. (a) Spectrophotometric time series recording of BZ solutionwith Na2SO4 0.07 M. (b) Time series of the region 3058–5656.(c) Broad band spectrum of the region 3058–5656.

F. Rossi et al. / Chemical Physics 313 (2005) 101–106 103

The same result is demonstrated when the delay vectormethod is used to reconstruct the attractor. A limitcycle is shown in (Fig. 3(c)).

The increase of Na2SO4 concentration induces thedisappearance of chaotic behavior, making the salt�sconcentration an important control parameter for theonset of chaos in this system.

The transition scenario was further investigated per-forming a series of experiments over a wide range of val-ues of the controlling parameter: 0.070 M < [Na2SO4]< 0.087 M.

When the Na2SO4 concentration reaches the valueof 0.080 M (Fig. 4(a)), three irrationally related fre-quencies and their combinations are observed (Fig.4(b)): f1 = 0.0082397 Hz, f2 = 0.0097650 Hz and f3 =0.0106800 Hz. The attractor should correspond to a3-fold torus in the phase space. It is almost rare to

find very convincing examples of quasiperiodic systemwith a 3-fold torus, because the nonlinear coupling be-tween the modes corresponding to the different fre-quencies tend to destroy quasiperiodicity and replaceit by chaos. Nevertheless in Figs. 4(a) and (b) we havea very good example of this type of system. Broad-band spectrum, namely a chaotic regime, is observedwhen the Na2SO4 concentration is under 0.080. Fina-lly, when the Na2SO4 concentration reaches the valueof 0.085 M (Fig. 5(a)), two irrationally related fre-quencies and their combinations are observed (Fig.5(b)): f1 = 0.009765625 Hz, f2 = 0.004902812 Hz. Theattractor should correspond to a 2-fold torus in thephase space. These results allow us to identify anRTN transition as reported below:

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Fig. 4. (a) Spectrophotometric time series recording of the region 3058–5656 of BZ solution with Na2SO4 0.080 M. (b) Spectrum of the region3058–5656 with three fundamental frequencies and their combinationsthat should correspond to a 3-fold torus in the phase space.

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Fig. 5. (a) Spectrophotometric time series recording of the region 725–1935 of BZ solution with Na2SO4 0.085 M. (b) Spectrum of the region725–1935 with two fundamental frequencies and their combinationsthat should correspond to a 2-fold torus in the phase space.

104 F. Rossi et al. / Chemical Physics 313 (2005) 101–106

periodic oscillationðlimit cycleÞ½Na2SO4� P 0:087 M

#quasiperiodic oscillationð2-fold torusÞ

½Na2SO4� ’ 0:085 M

#quasiperiodic oscillationð3-fold torusÞ

½Na2SO4� ’ 0:080 M

#chaotic oscillationðstrange attractorÞ

½Na2SO4� 6 0:080 M

4. Discussion and concluding remarks

Recently, it has been showed that the BZ reactionin a closed reactor presents a lot of complex behaviorswhich depend sensibly on the experimental conditions.Many clues led us to claim that the route to chaosfollowing an RTN scenario is due to a coupling be-tween nonlinear chemical kinetics and transport phe-nomena (diffusion and convection). Unfortunately,we are not able to completely decouple these threephenomena and to tune independently one of them.On the other hand, we can choose suitable experimen-tal conditions in order to enhance some effect and/ordump others at the time. The addition of electrolytesis not an obvious issue, increasing the concentrationof salts in solution will cause an increment in the ionicstrength, so that we can influence diffusion phenomenabut the species ðSO2�

4 Þ could affect the reaction kinet-ics through the modification of pH or cerium complexequilibria.

F. Rossi et al. / Chemical Physics 313 (2005) 101–106 105

It is well known that the characteristics of the BZoscillatory regime that are deeply connected with thereaction kinetics are the induction period (IP) and theoscillation frequency (OF) [18]. In order to evaluatethe influence of electrolytes on the kinetics of the BZwe conducted some experiments in well stirred systems.In fact, without transport phenomena, the reaction is

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Fig. 6. Potentiometric time series recordings of BZ solution with(b) [Na2SO4] = 0.087 mol/l. (c) [Na2SO4] = 0.100 mol/l. (d) [Na2SO4] = 1.000

only under kinetic control. These experiments were per-formed at room temperature with potentiometric sys-tems: as reference we used mercury sulphate electrodeAMEL Hg/HgSO4 and as measure an AMEL platinumelectrode with glass corp. The electrodes were connectedwith high precision pH-meter/potentiometer AMELmod.338.

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Fig. 7. Superposition of the first oscillations of BZ solution withdifferent concentrations of Na2SO4.

Table 1Mean value of the induction period (IP) and the oscillation frequency(OF) refers to BZ systems without added electrolyte and with[Na2SO4] = 0.1 mol/l

Without salt ½SO2�4 � ¼ 0:1 mol=l D%

IP (s) 348.6 360 +3.16OF (Hz) 0.018229 0.017252 �5.35

106 F. Rossi et al. / Chemical Physics 313 (2005) 101–106

In Fig. 6 are shown four potentiometric recordings ofBZ reaction with increasing salt concentration; as wecan see the time series are quite similar until½SO2�

4 � < 0:1 mol=l, and, instead of a little phase delay,they are almost superimposable (Fig. 7). At highconcentration of sulphate ð½SO2�

4 � ¼ 1 mol=lÞ, we cansee a drastic change in the reaction�s kinetic, inductionperiod is more than doubled (5 m 48 s ! 11 m 28 s),and the oscillation frequency is drastically decreased(0.017578 Hz ! 0.005859 Hz). In Table 1 are shownthe mean values of IP and OF deduced from several

potentiometric measures for BZ reactions performedwithout salt and with ½SO2�

4 � ¼ 0:1 mol=l.These last results does not allow to neglect the kinet-

ics effect of sulphate ions on the global dynamic of reac-tion. Although it is not possible to evaluate the effect ofsulphate on kinetics and diffusion, nevertheless theadded electrolytes ðSO2�

4 Þ play an important role as con-trolling parameter in the transition to chaos. In particu-lar the concentration of sulphate is a bifurcationparameter for RTN scenario.

Acknowledgement

This work was supported by MIUR (Italy).

References

[1] S.K. Scott, Oscillations, Waves and Chaos in Chemical Kinetics,Oxford Chemistry Primers, Oxford, 1994.

[2] A.M. Zhabotinsky, Chaos 1 (1991) 379.[3] R.J. Field, M. Burger (Eds.), Oscillations and Travelling Waves in

Chemical Systems, Wiley, New York, 1985.[4] I.R. Epstein, J.A. Pojman, An Introduction to Nonlinear Chem-

ical Dynamics, Oxford University Press, 1998.[5] J. Wang, P.G. Sørensen, F. Hynne, J. Phys. Chem. 98 (1994) 725.[6] S.K. Scott, B. Peng, A.S. Tomlin, K. Showalter, J. Chem. Phys.

94 (1991) 1134.[7] P.E. Strizhak, A.L. Kawczynsky, J. Phys. Chem. 99 (1995) 10830.[8] A. Goryachev, P. Strizhak, R. Kapral, J. Chem. Phys. 107 (1997)

2881.[9] M. Rustici, M. Branca, C. Caravati, N. Marchettini, Chem. Phys.

Lett. 263 (1996) 429.[10] M. Masia, S. Bastianoni, M. Rustici, Phys. Chem. Chem. Phys. 3

(2001) 5516.[11] J.P. Eckmann, Rev. Mod. Phys. 53 (1981) 643.[12] M. Rustici, M. Branca, A. Brunetti, C. Caravati, N. Marchettini,

Chem. Phys. Lett. 293 (1998) 145.[13] M. Rustici, M. Branca, C. Caravati, E. Petretto, N. Marchettini,

J. Phys. Chem. 103 (1999) 6564.[14] N. Marchettini, M. Rustici, Chem. Phys. Lett. 317 (2000) 647.[15] M. Rustici, R. Lombardo, M. Mangone, C. Sbriziolo, V.

Zambrano, M.L. Turco, Liveri Faraday Discuss. 120 (2001) 39.[16] M. Masia, N. Marchettini, V. Zambrano, M. Rustici, Chem.

Phys. Lett. 341 (2001) 285.[17] M.L. Turco Liveri, R. Lombardo, M. Masia, C. Calvaruso, M.

Rustici, J. Phys. Chem 107 (2003) 4834.[18] I.R. Epstein, J.A. Pojman, J. Chem. Educ. 77 (2000) 450.