The Belousov-Zhabotinsky BZ Reaction

Chemical mechanisms for real oscillating reactions are very complex and presently not understood in every detail. Nevertheless, there are approximate mechanisms which correctly model several crucial aspects of real oscillating reactions. In these simplified systems, often not all physical laws are strictly obeyed, e.g. the law of conservation of mass. 

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 

2BrOf+ 3CH2(COOH)2 +j2H — 2BrCH(COOHf2 +-3C02 + 4H20 (3.94) 

A short induction period is typically followed by an oscillatory phase, visible by the alternating colour of the aqueous solution due to the different oxidation states of the metal catalyst. Addition of a coloured redox indicator, such as the Fe,llZ,hl) phen)n couple, results in more dramatic colour changes. Typically, several hundred oscillations with a periodicity of approximately one minute, gradually die out within a couple of hours and the system slowly drifts towards its equilibrium state. 

In order to understand the BZ system Field, Koros and Noyes developed the so-called FKN mechanism. From this, Field and Noyes later derived the Oregonator model, an especially convenient kinetic model to match individual experimental observations and predict experimental conditions under which oscillations might arise. 

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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). 

In this section, we consider the breakdown of the condition of normal hyperbolicity. First, we explain a simple example where breakdown of normal hyperbolicity leads to a bifurcation in reaction processes. In the Belousov-Zhabotinsky (BZ) reaction [40], the bifurcation from the stable fixed point to the limit cycle takes place through the breakdown of normal hyperbolicity. This is the simplest case where mathematical analyses are in progress [41]. 

Despite the importance of the chlorite-iodide systems in the development of nonlinear chemical dynamics in the 1980s, the Belousov-Zhabotinsky(BZ) reaction remains as the most intensively studied nonlinear chemical system, and one displaying a surprising variety of behavior. Oscillations here were discovered by Belousov (1951) but largely unnoticed until the works of Zhabotinsky (1964). Extensive description of the reaction and its behavior can be found in Tyson (1985), Murray (1993), Scott (1991), or Epstein and Pojman (1998). There are several versions of the reaction, but the most common involves the oxidation of malonic acid by bromate ions BrOj in acid medium and catalyzed by cerium, which during the reaction oscillates between the Ce3+ and the Ce4+ state. Another possibility is to use as catalyst iron (Fe2+ and Fe3+). The essentials of the mechanisms were elucidated by Field et al. (1972), and lead to the three-species model known as the Oregonator (Field and Noyes, 1974). In this... 

The most studied and cited example of complex dynamic behavior in homogeneous catalysis is the Belousov - Zhabotinsky (BZ) reaction named after B. P. Belousov who discovered the reaction and A. M. Zhabotinsky who continued Belousov s early work. The reaction is theoretically important in that it shows that chemical reactions do not have to be dominated by equilibrium. These reactions are far from equilibrium and remain so for a length of time. In this sense, they provide an interesting chemical model of nonequilibrium biological phenomena, and the mathematical model of the BZ reactions themselves are of theoretical interest. 

We now turn to our application of MSIMPC to examine the behavior of an oscillatory reaction. To compare experimental kinetic results to theoretical chemical mechanisms, the differential equations derived from the mechanism must be solved. The Oregonator model, which is a simple model proposed to explain the oscillatory behavior of the Belousov-Zhabotinsky (BZ) reaction, is a typical case. It involves five coupled differential equations and five unknown concentrations. We do not discuss details of this mechanism or the overall BZ reaction here, since it has received considerable attention in the chemical literature. 

The best known oscillating reaction is without a doubt the Belousov-Zhabotinsky (BZ) reaction, the oxidation of an organic substrate, typically malonic acid, CH2(C00H)2, by bromate, Br03, in an acidic medium in the presence of a metalion catalyst. It was discovered by Belousov in the early 1950s [32], and modified by Zhabotinsky [497]. The mechanism of the BZ reaction was elucidated by Field, Koros, and Noyes in 1972 [326, 130, 325] and reduced to five essential steps by Field and Noyes [131]. This model is called the Oregonator and in the version presented by Tyson and Fife [442] it is given by... 

Studies carried out by Yoshida and coworkers have coupled this phenomena with oscillating chemical reactions (such as the Belousov-Zhabotinsky, BZ, reaction) to create conditions where pseudo non-equilibrium systems which maintain rhythmical oscillations can demonstrated, in both quiescent (4) and continuously stirred reactors (5). The ruthenium complex of the BZ reaction was introduced as a functional group into poly(N-isopropyl acrylamide), which is a temperature-sensitive polymer. The ruthenium group plays it s part in the BZ reaction, and the oxidation state of the catalyst changes the collapse temperature of the gel. The result is, at intermediate temperature, a gel whose shape oscillated (by a factor of 2 in volume) in a BZ reaction, providing an elegant demonstration of oscillation in a polymer gel. This system, however, is limited by the concentration of the catalyst which has to remain relatively small, and hence the volume change is small. 

It is well known that the Belousov-Zhabotinsky (BZ) reaction can initiate free radical polymerisation (2) while it is less known that polymers can also affect the dynamics of the BZ reaction. Recently, we have performed preliminary experiments perturbing the BZ reaction with two different water-soluble nonionic polymers containing alcoholic end-groups, namely polypropylene glycol and polyethylene glycol (PEG) (i). It was realized that the Belousov-Zhabotinsky reaction responded to the perturbation in an unexpected way. Thus, a systematic study was undertaken to inquire whether the perturbation effect can be attributed exclusively to PEG reactive endgroups (here primary alcoholic groups) or the chemical nature of polymeric backbone plays also a relevant role. 

Although some of the fimdamental discoveries in nonlinear chemical dynamics were made at the beginning of the twentieth century and arguably even earlier, the field itself did not emerge until the mid-1960 s, when Zhabotinsky s development (1) of the oscillatory reaction discovered by Belousov (2) finally convinced a skeptical chemical community that periodic reactions were indeed compatible with the Second Law of Thermodynamics as well as all other known rules of chemistry and physics. Since the discovery of the Belousov-Zhabotinsky (BZ) reaction, nonlinear chemical dynamics has grown rapidly in both breadth and depth (3). 

When ionized acrylamide polymer gel undergoes extensive swelling, an extremely fine pattern appears on the surface, and evolves with time[201]. On the other hand, when cylindrical gels of acrylamide shrink, they exhibit bubble pattern and bamboo-like pattern[202]. Under certain conditions, alternate swollen and shrunken portions appear (bubble pattern). In other cases, cross-sectional planes made of the collapsed gel membrane, whose thickness comparable to the wavelength of light, appear in the cylinder (bamboo-like pattern). Gel has also played an important role in the study of the Belousov-Zhabotinsky (BZ) reaction [168, 203], which induces spatiotemporal patterns like the Turing pattern [204, 205] and spiral wave [206]. In the reaction, gel works as supporting... 

The most studied and cited example of complex dynamic behavior in homogeneous catalysis is the Belousov-Zhabotinsky (BZ) reaction named after B.P. Belousov, who discovered the reaction, and A.M. Zhabotinsky, who continued the early work of... 

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