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)... 

Zhabotinsky A M, Buohholtz F, Kiyatin A B and Epstein I R 1993 Csoillations and waves in metal-ion oatalysed bromate osoillating reaotion in highly oxidised states J. Phys. Chem. 97 7578-84... 

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

Zaikin A N and Zhabotinsky A M 1970 Concentration wave propagation in two-dimensional liquid-phase self-oscillating system Nature 225 535-7... 

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. 

Belouzov-Zhabotinsky reaction [12, 13] This chemical reaction is a classical example of non-equilibrium thermodynamics, forming a nonlinear chemical oscillator [14]. Redox-active metal ions with more than one stable oxidation state (e.g., cerium, ruthenium) are reduced by an organic acid (e.g., malonic acid) and re-oxidized by bromate forming temporal or spatial patterns of metal ion concentration in either oxidation state. This is a self-organized structure, because the reaction is not dominated by equilibrium thermodynamic behavior. The reaction is far from equilibrium and remains so for a significant length of time. Finally,... 

Zhabotinsky A. M. (1964) Periodic processes of malonic add oxidation in a bquid phase. Biofizika, 9, 306-311... 

In an interesting variant of the CA model, cells can adopt "excited states." This has been used to model the spatial waves observed in the Zaikin-Zhabotinsky reaction. 

Reaction-diffusion systems can readily be modeled in thin layers using CA. Since the transition rules are simple, increases in computational power allow one to add another dimension and run simulations at a speed that should permit the simulation of meaningful behavior in three dimensions. The Zaikin-Zhabotinsky reaction is normally followed in the laboratory by studying thin films. It is difficult to determine experimentally the processes occurring in all regions of a three-dimensional segment of excitable media, but three-dimensional simulations will offer an interesting window into the behavior of such systems in the bulk. 

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... 

A.M. Zhabotinsky and A.B. Rovinskii. Self-Organization, Autowaves and Structures Far from Equilibrium. Springer, Berlin, 1984. 

V.K. Vanag, L.F. Yang, M. Dolnik, A.M. Zhabotinsky, and l.R. Epstein. Oscillatory cluster patterns in a homogeneous chemical system with global feedback. Nature, 406(6794) 389-391, 2000. 

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. 

A. M. Zhabotinsky, Periodic process of the oxidation of malonic acid in solution. Study of the kinetics of Belousov s reaction. Biofizika 9, 1306 (1964). 

The kinetics of the manganese(II)- and cerium(III)-catalysed Belusov-Zhabotinsky (BZ) oscillatory reactions were studied with mixed organic acid-ketone substrates. 

The Belusov-Zhabotinsky (BZ) reaction is catalyzed by a different mechanism when low-reduction-potential couples such as [Fe(phen)3] +/[Fe(phen)3] + are employed. Experimental results for the BZ reaction with this couple in aerated conditions are compared with satisfactory agreement to a model calculation based on an 18-step skeleton mechanism, which includes reactions of organic radicals and molecular oxygen. 

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