Example 3 Belousov-Zhabotinski Reaction

These patterns are an example of what are sometimes called dissipative structures, which arise in many complex systems. Dissipative structures are dynamical patterns that retain their organized state by persistently dissipating matter and energy into an otherwise thermodynamically open environment. 

Lattice gases are micro-level rule-based simulations of macro-level fluid behavior. Lattice-gas models provide a powerful new tool in modeling real fluid behavior ([doolenQO], [doolenQl]). The idea is to reproduce the desired macroscopic behavior of a fluid by modeling the underlying microscopic dynamics. 

While there are mairy variants of the basic, model, one can show that there is a well-defined minimal set of niles that define a lattice-gas system whose macroscopic behavior reproduces that predicted by the Navier-Stokes equations exactly. In other words, there is critical threshold of rule size and type that must be met before the continuum fluid l)cliavior is matched, and onec that threshold is reached the efficacy of the rule-set is no loner appreciably altered by additional rules respecting the required conservation laws and symmetries. 

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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... 

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. 

Another aspect of a very different nature also merits attention. For complex reaction schemes, it can be very cumbersome to write the appropriate set of differential equations and their translation into computer code. As an example, consider the task of coding the set of differential equations for the Belousov-Zhabotinsky reaction (see Section 7.5.2.4). It is too easy to make mistakes and, more importantly, those mistakes can be difficult to detect. For any user-friendly software, it is imperative to have an automatic equation parser that compiles the conventionally written kinetic model into the correct computer code of the appropriate language [37-39],... 

The Belousov-Zhabotinsky reaction system is one example leading to such chemical oscillations. One of the interesting phenomena is the effect of the very narrow range of controlling parameter /x on the stability of the Belousov-Zhabotinsky reaction system. The following reactions represent the Belousov-Zhabotinsky reaction scheme ... 

Example 13.5 The Belousov-Zhabotinsky reaction scheme Field et al. (1972) explained the qualitative behavior of the Belousov-Zhabotinsky reaction, using the principles of kinetics and thermodynamics. A simplified model with three variable concentrations producing all the essential features of the Belousov-Zhabotinsky reaction was published by Field and Noyes (1974). Some new models of Belousov-Zhabotinsky reaction scheme consist of as main as 22 reaction steps. With the defined symbols X = HBr02, Y = Br, Z = Ce4+, B = organic, A = B1O3 (the rate constant contains H+), FKN Model (Field et al., 1972) consists of the following steps summarized by Kondepudi and Priogogine (1999) ... 

An incomparable fascination is the magic of the colorful oscillation reactions which are known today to every chemist, although only a few decades ago they were considered to be completely atypical. Here we shall present an example which, in contrast to the variety of colors often occurring in the course of these processes, demonstrates the principle of a Belousov-Zhabotinsky reaction by means of a single color change. Perhaps we can follow Nietzsche here There are many things which, once and for all, I do not wish to know. Wisdom also sets limits to knowledge. .. ... 

Example 14.3. The Belousov-Zhabotinsky reaction [22,27-29], The reaction is an oxidation of malonic acid by bromate ion in sulfuric acid, catalyzed by a Ce(III)/Ce(IV) redox couple. Many variations with other organic acids and transition-metal ions are possible [22] (Belousov used citric acid, and manganese, ruthenium, or iron can replace cerium). The color of the solution alternates between clear [Ce(III)] and pale yellow [Ce(IV)], and more dramatically between red and blue if ferroin is added as indicator. 

A reaction may be periodic if its network provides for restoration of a reactant or intermediate that has been depleted, while conversion of main reactants to products continues. Periodic behavior often results from competition of two or more contending mechanisms. Predator-prey fluctuations in ecology (Lotka-Volterra mechanisms) provide an easily visualized example. The Belousov-Zhabotinsky reaction—catalyzed oxidation of malonic acid by bromate—involves a similar competition between two pathways. 

Examples include multiple steady states in isothermal CSTRs, predator-prey fluctuations, the Belousov-Zhabotinsky reaction, and a test for stability of quasi-stationary states in reactions with a self-accelerating intermediate steps. 

The spiral or concentric waves observed for the spatial distribution of cAMP (fig. 5.6) present a striking analogy with similar wavelike phenomena found in oscillatory chemical systems, of which the Belousov-Zhabotinsky reaction (fig. 5.7) provides the best-known example (Winfree, 1972a). 

A common example is the Belousov - Zhabotinsky reaction [24], Beautiful patterns of chemical wave propagation can be created in a chemical reaction - diffusion system with a spatiotemporal feedback. The wave behavior can be controlled by feedback-regulated excitability gradients that guide propagation in the specified directions [25, 26]. 

A classical example of reaction-diffusion processes is the Belousov-Zhabotinsky reaction that forms moving Winfree spirals in a two-dimensional reactor. In this reaction the many sequential reaction steps comprise a very complex system, yet the spirals can be simulated with a very simple three-state cellular automaton. This model accounts for the morphology given certain initial conditions but it does not... 

In this volume we also considered the self-organizing systems consisting of the oscillating reactions, of which the Belousov-Zhabotinsky reaction is a well-known example. Attempt was made to demonstrate a generality of the problem on the identification of critical phenomena in branching-chain reactions and in the nonlinear reaction systems as a whole. 

Besides these two regimes, another regime, with a temporally periodic change of the chemical composition (chemical oscillation or self-oscillation), may also be observed. A famous example of this phenomenon is the Belousov-Zhabotinsky reaction. Another example of complex kinetic behavior in open chemical systems is the occurrence of multiple steady states due to the fact that for some components of the reaction mixture the rate of consumption and rate of production can be balanced at more than one point. This type of behavior has become the subject of detailed theoretical and computational analyses (Marin and Yablonsky, 2011 Yablonskii et al., 1991). Bespite the fact that there are many experimental data concerning such complex behavior, the steady-state regime with characteristics that are constant in time still is the most observed phenomenon. 

Example 13.5 The Belousov-Zhabotinsky reaction scheme Field cl al. (1972) explained the qualitative behavior... 

A classical example of an inorganic set of coupled reactions, some of them being autocatalytic, occurs in the Belousov-Zhabotinsky reaction. This reaction, discovered in 1958 by B.P. Belousov and reported to the western world by A.N. Zaikin and A.M. Zhabotinsky in 1970, is the most investigated and best understood... 

The Belousov-Zhabotinsky reaction is an example of a class of processes, in which the chemical compositions exhibit regular periodic variations in time or space. These periodic... 

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