Belousov-Zhabotinsky

Winfree A T 1984 The prehistory of the Belousov-Zhabotinsky osoillator J. Chem. Eduo. 61 661-3 [11 ] Zhabotinsky AM 1991 A history of ohemioal osoillations and waves Chaos 1 379-86... 

Tyson J J 1976 The Belousov-Zhabotinsky Reaotion (Leoture Notes in Biomathematios vol 10) (Berlin Springer)... 

Gyorgyi L and Field R J 1992 A three-variable model of deterministic chaos in the Belousov-Zhabotinsky reaction Nature 355 808-10... 

An example of the application of J2-weighted imaging is afforded by the imaging of the dynamics of chemical waves in the Belousov-Zhabotinsky reaction shown in figure B 1.14.5 [16]. In these images, bright... 

Figure Bl.14.5. J2-weighted images of the propagation of chemical waves in an Mn catalysed Belousov-Zhabotinsky reaction. The images were acquired in 40 s intervals (a) to (1) using a standard spin echo pulse sequence. The slice thickness is 2 nun. The diameter of the imaged pill box is 39 nun. The bright bands...

This subject has been reviewed by Noyes and Field,8 who give reference to the original formulation as well as a more explicit treatment. The presentation here will be given not in general terms but by means of one striking example, the oxidation of malonic acid by bromate ions catalyzed by cerium(IV). It is called the Belousov-Zhabotinsky (or BZ) reaction, after its discoverers.9 The stoichiometry of the reaction with excess malonic acid is... 

This reaction can oscillate in a well-mixed system. In a quiescent system, diffusion-limited spatial patterns can develop, but these violate the assumption of perfect mixing that is made in this chapter. A well-known chemical oscillator that also develops complex spatial patterns is the Belousov-Zhabotinsky or BZ reaction. Flame fronts and detonations are other batch reactions that violate the assumption of perfect mixing. Their analysis requires treatment of mass or thermal diffusion or the propagation of shock waves. Such reactions are briefly touched upon in Chapter 11 but, by and large, are beyond the scope of this book. 

How relevant are these phenomena First, many oscillating reactions exist and play an important role in living matter. Biochemical oscillations and also the inorganic oscillatory Belousov-Zhabotinsky system are very complex reaction networks. Oscillating surface reactions though are much simpler and so offer convenient model systems to investigate the realm of non-equilibrium reactions on a fundamental level. Secondly, as mentioned above, the conditions under which nonlinear effects such as those caused by autocatalytic steps lead to uncontrollable situations, which should be avoided in practice. Hence, some knowledge about the subject is desired. Finally, the application of forced oscillations in some reactions may lead to better performance in favorable situations for example, when a catalytic system alternates between conditions where the catalyst deactivates due to carbon deposition and conditions where this deposit is reacted away. 

Fig. 5 MR images of traveling (reaction-diffusion)waves in the manganese-catalysed Belousov-Zhabotinsky reaction, taken from the centre of a bed packed with 1 mm diameter glass spheres (22). Waves are formed both inside the bed and above it in the liquid phase. Images (a-d) are shown at time intervals of 16 s. 

The Runge-Kutta algorithm cannot handle so-called stiff problems. Computation times are astronomical and thus the algorithm is useless, for that class of ordinary differential equations, specialised stiff solvers have been developed. In our context, a system of ODEs sometimes becomes stiff if it comprises very fast and also very slow steps and/or very high and very low concentrations. As a typical example we model an oscillating reaction in The Belousov-Zhabotinsky (BZ) Reaction (p.95). 

The Belousov-Zhabotinsky (BZ) reaction involves the oxidation of an organic species such as malonic acid (MA) by an acidified aqueous bromate solution in the presence of a metal ion catalyst such as the Ce(m)/Ce(IV) couple. At excess [MA] the stoichiometiy of the net reaction is... 

The 1970s saw an explosion of theoretical and experimental studies devoted to oscillating reactions. This domain continues to expand as more and more complex phenomena are observed in the experiments or predicted theoretically. The initial impetus for the smdy of oscillations owes much to the concomitance of several factors. The discovery of temporal and spatiotemporal organization in the Belousov-Zhabotinsky reaction [22], which has remained the most important example of a chemical reaction giving rise to oscillations and waves. 

At the same time as the Belousov-Zhabotinsky reaction provided a chemical prototype for oscillatory behavior, the first experimental studies on the reaction catalyzed by peroxidase [24] and on the glycolytic system in yeast (to be discussed in Section 111) demonstrated the occurrence of biochemical oscillations in vitro. These advances opened the way to the study of the molecular bases of oscillations in biological systems. 

The next two important steps in this narrative are considered to be the following (i) the description of the bruxellator by Prigogine and Lefever, who, following on from Turing s work, analyzed theoretically the ingredients that should be present in a model of chemical reactions in order to produce spatial self-organization (Prigogine and Lefever, 1968) (ii) the description of the Belousov-Zhabotinsky (B-Z) reaction. 

Winfree, A. T. (1984). The prehistory of the Belousov-Zhabotinsky oscillator. J. Chem. 

S. Vajda and T. Tur3nyi, Principal component analysis for reducing the Edelsan-Field-Noyes model of Belousov-Zhabotinsky reaction, J. Phys. Chem. 

The famous Oregonator model (ref. 9) is a highly simplified (but very successful) description of the Belousov - Zhabotinsky oscillating reaction ... 

Of special interest is the so-called Belousov-Zhabotinsky class of similar reactions [4, 6-12], This system can serve as an extremely successful example of self-organisation proper mixing of several liquids in a given proportion and at certain temperature demonstrates practically all kinds of the autowave processes just mentioned. 

Numerous versions of the Belousov-Zhabotinsky system differ by chemical compounds used. The typical reaction involves oxidation of some organic compound by bromate ion (BrOj ) occurring in acid medium with metal catalyst (Ce3+, Mn2+, as well as complexes of Fe2+, Ru2+). As an example, a particular reaction [4] could be mentioned, where an organic reductor is malonic acid CH2(COOH)2 and Ce3+ ions serve as a catalyst. In this reaction a solution changes periodically its colour due to oscillations in Ce3+ concentration. Generally speaking, the reaction consists of two stages. At the first one metal is oxidized... 

Despite the fact that from a principal point of view a problem of concentration oscillations could be considered as solved [4], satisfactory theoretical descriptions of experimentally well-studied particular reactions are practically absent. Due to very complicated reaction mechanism (in order to describe the Belousov-Zhabotinsky reaction even in terms of standard chemical kinetics several tens of concentration equations for intermediate products should be written down and solved numerically [4, 9, 10]) these equations contain large number of ill-defined parameters - reaction rates [10]. 

Field, K6ros and Noyes [13] suggested to use as the basic model for the Belousov-Zhabotinsky system a rather complicated set of chemical reactions with seven intermediate products. Its more global analysis based on macrokinetic stages and retaining still the principal features of this reaction [14] has led to the simplified scheme with three intermediate products only. This model called Oregonator [9, 15] is described by the following equations ... 

As it follows from the above-said, nowadays any study of the autowave processes in chemical systems could be done on the level of the basic models only. As a rule, they do not reproduce real systems, like the Belousov-Zhabotinsky reaction in an implicit way but their solutions allow to study experimentally observed general kinetic phenomena. A choice of models is defined practically uniquely by the mathematical formalism of standard chemical kinetics (Section 2.1), generally accepted and based on the law of mass action, i.e., reaction rates are proportional just to products of reactant concentrations. 

A role of other parameters of the model is investigated by Kuzovkov [26], It is demonstrated that an increase of the ratio a//3 for a fixed lj0 = (cr/3)1/2 and the control parameter k acts to accelerate a change of the focal regime for chaotic. Simultaneously, the amplitudes of oscillations in concentration for particles of different kinds are no longer close. A study of the stochastic Lotka-Volterra model performed here shows that irregular concentration motion observed experimentally in the Belousov-Zhabotinsky systems [8] indeed could take place in a system with mono- and bimolecular stages and two intermediate products only. 

Hudson, J. L., Hart, M. and Marinko, D., 1979, An experimental study of multiple peak periodic and nonperiodic oscillations in the Belousov-Zhabotinski reaction. J. Chem. Phys. 71,1601-1606. 

One of the well-studied systems that illustrates this successive-bifurcation behavior is the Belousov-Zhabotinski reaction. Let me briefly show you the results of some experiments done at the University of Texas at Austin,8 referring for further details to the discussion by J. S. Turner in this volume. The experimental setup of the continuously stirred reactor... 

Fig. 5. Experimental arrangement of the continuously stirred Belousov-Zhabotinski reaction.

Fig. 7. Mixed mode oscillations in the Belousov-Zhabotinski reaction when it is farther from equilibrium than it is in Fig. 6.

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