Chaos, chemical

At the end of this chapter we will now briefly discuss a theoretical approach to the description of chaotic processes encountered in chemical kinetics, see Section 1.3 and Sections 6.2.2.4, 6.3.2.4. In Section 6.2.2.4 we described the method of generation and physical meaning of a chaotic state of the Belousov-Zhabotinskii reaction carried out in a flow reactor. 

The chaotic dynamics can be conveniently followed by monitoring a concentration of just one reagent, for example thr Br ion (using a suitable ion-selective electrode or colorimetrically). The measurement of the Br concentration at constant time intervals t is most straightforward. The consecutively measured concentrations will be 

This type of transition to a chaotic state is called the Feigenbaum cascade. Such a scenario of generation of chaos involves consecutive losses of stability by successive orbits, see Fig. 103. 

The measurements of x, reveal that the observed phenomena can be modelled by means of the recurrent equation 

The experimental phenomena observed in the Belousov-Zhabotinskii reaction, such a doubling of the oscillation period, chaotic oscillations or alternate periodical and chaotic oscillations, can be modelled still more exactly by the recurrent equation 

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Swinney H L, Argoul F, Arneodo A, Richetti P and Roux J-C 1987 Chemical chaos from hints to confirmation Acc. Chem. Res. 20 436-42... 

Graduate-level introduction mainly to theoretical modelling of nonlinear reactions Scott S K 1993 Chemical Chaos (Oxford Oxford University Press)... 

In tills chapter we shall examine how such temporal and spatial stmctures arise in far-from-equilibrium chemical systems. We first examine spatially unifonn systems and develop tlie tlieoretical tools needed to analyse tlie behaviour of systems driven far from chemical equilibrium. We focus especially on tlie nature of chemical chaos, its characterization and the mechanisms for its onset. We tlien turn to spatially distributed systems and describe how regular and chaotic chemical patterns can fonn as a result of tlie interjilay between reaction and diffusion. 

The existence of chaotic oscillations has been documented in a variety of chemical systems. Some of tire earliest observations of chemical chaos have been on biochemical systems like tire peroxidase-oxidase reaction [12] and on tire well known Belousov-Zhabotinskii (BZ) [13] reaction. The BZ reaction is tire Ce-ion-catalyzed oxidation of citric or malonic acid by bromate ion. Early investigations of the BZ reaction used tire teclmiques of dynamical systems tlieory outlined above to document tire existence of chaos in tliis reaction. Apparent chaos in tire BZ reaction was found by Hudson et a] [14] aiid tire data were analysed by Tomita and Tsuda [15] using a return-map metliod. Chaos was confinned in tire BZ reaction carried out in a CSTR by Roux et a] [16, E7] and by Hudson and... 

In addition to tire period-doubling route to chaos tliere are otlier routes tliat are chemically important mixed-mode oscillations (MMOs), intennittency and quasi-periodicity. Their signature is easily recognized in chemical experiments, so tliat tliey were seen early in the history of chemical chaos. 

Scott, S. K. (1991) Chemical Chaos, Clarendon Press, Oxford. 

M. Dolnik and E.M. Bollt. Communications with chemical chaos in the presence of noise. Chaos, 8(3) 702-710, 1998. 

One phenomenon that is drawing increasing attention, not only from chemists, but from mathematicians and physicists as well, has been dubbed chemical chaos", the existence of sustained aperiodic oscillation of species concentrations in a flow sys-... 

Ganapathisubramanian, N. and Noyes, Richard M., "A discrepancy between experimental and computational evidence for chemical chaos", to be published J. Chem. Phys. 

Refs. [i] Scott SK (1991) Chemical chaos. Clarendon Press, Oxford [ii] Fechner GT (1828) Schweigg / / Chem Phys 53 129 [Hi] Wojtow-icz I (1972) Oscillatory behaviour in electrochemical systems. In Bock-ris JO M, Conway BE (eds) Electrochemical oscillations, vot 8. Plenum Press, New York [iv] Hudson JL, Tsotsis TT (1994) Chem Eng Sci 49 1493 [v] Hudson JL, Bassett MR (1991) Oscillatory electrodissolution of metals. In Luss D, Amundson NR (eds) Reviews in chemical engineering. Freund, London [vi] Albahadily IN, Schell M (1991) J Electroanal Chem 308 151 [vii] Inzelt G (1993) J Electroanal Chem 348 465 [viii] Buck RP, Griffith LR (1962) J Electrochem Soc 109 1005 ... 

S.K. Scott, Chemical Chaos (Oxford University Press, 1993). 

In 1933, M. S. Kharasch and F. W. Mayo at the University of Chicago brought order to this chemical chaos by discovering that the orientation of addition of hydrogen bromide to the carbon-carbon double bond is determined solely by the presence or absence of peroxides. 

In this section we describe some beautiful experiments on the Belousov-Zhabotin-sky chemical reaction. The results show that strange attractors really do occur in nature, not just in mathematics. For more about chemical chaos, see Argoul et al. (1987). 

In the BZ reaction, malonic acid is oxidized in an acidic medium by bromate ions, with or without a catalyst (usually cerous or ferrous ions). It has been known since the 1950s that this reaction can exhibit limit-cycle oscillations, as discussed in Section 8,3. By the 1970s, it became natural to inquire whether the BZ reaction could also become chaotic under appropriate conditions. Chemical chaos was first reported by Schmitz, Graziani, and Hudson (1977), but their results left room for skepticism—some chemists suspected that the observed complex dynamics might be due instead to uncontrolled fluctuations in experimental control parameters. What was needed was some demonstration that the dynamics obeyed the newly emerging laws of chaos. 

The elegant work of Roux, Simoyi, Wolf, and Swinney established the reality of chemical chaos (Simoyi et al. 1982, Roux et al, 1983). They conducted an experiment on the BZ reaction in a continuous flow stirred tank reactor. In this standard set-up, fresh chemicals are pumped through the reactor at a constant rate to replenish the reactants and to keep the system far from equilibrium. The flow rate acts as a control parameter. The reaction is also stirred continuously to mix the chemicals. This enforces spatial homogeneity, thereby reducing the effective number of degrees of freedom. The behavior of the reaction is monitored by measuring S( ), the concentration of bromide ions. 

IIIC) Ganapathisubramanian, N., Noyes, R. M. A Discrepency between Experimental and 1982-4 Computational Evidence for Chemical Chaos. J. Chem. Phys. 76, 1770-1774. 

Fig. 104. Chemical chaos in the Belousov- Zhabotinskii reaction. Reprinted with permission from C. Vidal, page 49, Springer Series in Synergetics, H. Haken (Ed.), Vol. 12. Nonlinear Phenomena in Chemical Dynamics.

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