Oscillating reactions travelling waves

Since this early work, interest has been focused mainly on oscillations and traveling waves during chemical reactions 49, 126-129). Butler et al. 130) employed MRI techniques to investigate the extent of reaction in a single crystal of... 

Tyson, J. J. (1976). The Belousov-Zhabotinskii reaction. Springer, Berlin. Field, R. J. and Burger, M. (eds) (1985). Oscillations and travelling waves in chemical systems. Wiley-Interscience, New York. 

Saul, A. and Showalter, K. (1985). Propagating reaction-diffusion fronts. In Oscillations and traveling waves in chemical systems, (ed. R. J. Field and M. Burger), ch. 11, pp. 419-39. Wiley, New York. 

Belousov, B. P. A Periodic Reaction and Its Mechanism in Oscillations and Traveling Waves in Chemical Systems Field, R. J. Burger, M., Ed. Wiley New York, 1985 pp 605-613. 

To simulate the pattern formation observed in our experiments, we employ a model of the BZ reaction (21). We add a global linear feedback term to account for the bromide ion production that results from the actinic illumination, vqf = naxC av ss% where q> is the quantum yield. The results of our simulations mimic those of the experiments. Bulk oscillations and travelling waves are observed in die model for smaller values of g. At higher g values, standing, irregular and localized clusters are observed in the same sequence and with the same patterns of hysteresis as in the experiments... 

Tyson, J. 1985. A Quantitative Account of Oscillations, Bistability, and Traveling Waves in the Belousov Zhabotinskii Reaction, in Oscillations and Traveling Waves in Chemical Systems. (Field, R. J. Burger, M., Eds.). Wiley New York pp. 93-144. Tyson, J. J. 1975. Classification of Instabilities in Chemical Reaction Systems, J. Chem. Phys. 62, 1010-1015. 

In previous chapters we have dealt only with systems which have one or two independent concentrations. This has been sufficient for a wide range of intricate behaviour. Even with just a single independent concentration (one variable), reactions may show multiple stationary states and travelling waves. Oscillations are, however, not possible. To understand the latter point we can think in terms of the phase plane or, more correctly for a one-dimensional system, the phase line (Fig. 13.1(a)). As the concentration varies in time, so the system moves along this line. Stationary-state solutions are points on the line the arrows indicate the direction of motion along the line, as time increases, towards stable states and away from unstable ones. Figure 13.1(b) shows this motion and phase line behaviour represented in terms of some potential, with stable states a minima and an unstable (saddle) solution as a maximum. 

The various steps in this model have all been shown experimentally. The vast difference in the sticking coefficients for O2 on the two forms of the surface has been known for a long time (311-314). The mechanism for the surface-phase transition has been studied in detail by Behm et al. (315,316). During the oscillations, this transition has been observed to be coupled with the respective change in the reaction rate (50,53-55,245). Scanning LEED has also been employed to obtain spatial information, for example, the presence of traveling waves of the surface structure transition (50,53), as shown in Fig. 12. 

J. J. Tyson, A quantitative account of oscillations, bistability, and traveling waves in the Belousov-Zhabotinsky reaction, in Field and Burger (ref. Gl), Chapter 3. 

In this section the whole field of exotic dynamics is considered this term includes not merely oscillating reactions but also oligo-oscillatory reactions, multiple steady states, spatial phenomena such as travelling reaction waves, and chaotic systems. All of these have common roots in autocatalytic processes. This area has continued to expand, and there is a case for treatment in future volumes by a specialist reviewer. An entry into the literature can be gained from a recent series of articles in a chemical education joumal, and in a festschrift issue in honor of Professor R. M. Noyes. Other useful sources are a volume of conference proceedings, and a volume of lecture preprints of a 1989 conference. The present summary is concerned with the chemical rather than the mathematical aspects of the topic. 

Finally, it may be said that the character of the CO -I- O2 oscillating reaction on Pd differs remarkably from that on Pt because (a) different subsurface oxygen (Pd) and (hex) -o- (1 x 1) phase transition (Pt) mechanisms apply and (b) the oxygen front in CO-I-O2 waves travel in reverse directions on Pd it goes from (1 1 0) to (1 0 0) surface, on Pt it travels in the opposite direction. 

We employ a method of numerical continuation which has been earlier developed into a software tool for analysis of spatiotemporal patterns emerging in systems with simultaneous reaction, diffusion and convection. As an example, we take a catalytic cross-flow tubular reactor with first order exothermic reaction kinetics. The analysis begins with determining stability and bifurcations of steady states and periodic oscillations in the corresponding homogeneous system. This information is then used to infer the existence of travelling waves which occur due to reaction and diffusion. We focus on waves with constant velocity and examine in some detail the effects of convection on the fiiont waves which are associated with bistability in the reaction-diffusion system. A numerical method for accurate location and continuation of front and pulse waves via a boundary value problem for homo/heteroclinic orbits is used to determine variation of the front waves with convection velocity and some other system parameters. We find that two different front waves can coexist and move in opposite directions in the reactor. Also, the waves can be reflected and switched on the boundaries which leads to zig-zag spatiotemporal patterns. 

In excitable reaction-diffusion systems, pulses can travel as a periodic wave train. In oscillatory reaction-diffusion systems, too, the existence of plane wave solutions has been theoretically established (Kopell and Howard, 1973 a). In this section we will be concerned with such periodic waves in one space dimension, particularly when the local wavenumber slowly and slightly varies with x. For these systems, the analogy to systems of weakly coupled oscillators might look even weaker. Actually, however, there exists a rather strong formalistic similarity between the two. 

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