Analysis of Chemical Oscillations

Many of the most remarkable achievements of chemical science involve either synthesis (the design and construction of molecules) or analysis (the identification and structural characterization of molecules). We have organized our discussion of oscillating reactions along similar lines. In the previous chapter, we described how chemists have learned to build chemical oscillators. Now, we will consider how to dissect an oscillatory reaction into its component parts—the question of mechanism. 

Just as the study of molecular structure has benefited from new experimental and theoretical developments, mechanistic studies of complex chemical reactions. 

To date, mechanisms have been proposed for about half of the known oscillators. These mechanisms vary in the level of detail that they attempt and in the quality of the agreement between their predictions and the experimental data. In this chapter, we describe how one goes about constructing a mechanism for a chemical oscillator. We then examine a few examples of actual mechanisms. We conclude with a brief look at the possibility of using mechanistic ideas to categorize chemical oscillators in a fashion complementary to the approach described in Chapter 4. 

To be precise, the term mechanism or reaction mechanism should only be used to designate a molecular level description that is, a complete set of elementary steps that specifies how a chenucal reaction takes place. However, the word is often employed in reference to less detailed characterizations of reactions. These descriptions, which can involve several levels of simplification or assumption, are more properly referred to as models, rather than mechanisms. Models are not necessarily inferior to, or less useful than, mechanisms. They simply offer a different type of description. For certain purposes, one may gain more insight into how a system works from a three-step model than from a thirty-step mechanism. 

One class of models that has played a key role in the development of nonlinear chemical dynamics consists of abstract models. Typically, these have a small number of variables, perhaps two or three, and are meant to elucidate a particular phenomenon, reaction, or class of reactions. Models of this type can be derived 

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Field, R.J., Noyes, R.M., and Koros, E., Oscillations in chemical systems. 2. Thorough analysis of temporal oscillation in bromate-cerium-malonic acid system, JACS, 94,8649,1972. 

Bigeleisen, J. and Ishida, T. Application of finite orthogonal polynomials to the thermal functions of harmonic oscillators. I. Reduced partition function of isotopic molecules, J. Chem. Phys. 48, 1311 (1968). Ishida, T., Spindel, W. and Bigeleisen, J. Theoretical analysis of chemical isotope fractionation by orthogonal polynomial methods, in Spindel, W., ed. Isotope Effects on Chemical Processes. Adv. Chem. Ser. 89, 192 (1969). 

R.J.Field, E.Koros and R.M.Noyes, Oscillations in Chemical Systems. II. Through Analysis of Temporal Oscillations in the Bromate-Cerium-Malonic Acid System, Journal of the American Chemical Society, 94, 8649-8664(1972). 

The mechanisms of chemical oscillations can be very complex. The BZ reaction is thought to involve more than twenty elementary reaction steps, but luckily many of them equilibrate rapidly—this allows the kinetics to be reduced to as few as three differential equations. See Tyson (1985) foi this reduced system and its analysis. 

R. J. Field, E. Koros and R. M. Noyes, Oscillations in chemical systems, Part 2. Thorough analysis of temporal oscillations in the Ce-Br03 — malonic acid system , J. Am. Chem. Soc., 94, 8649 (1972). 

Luo, Y. Epstein, I. R. Systematic design of chemical oscillators. 58. Feedback analysis of mechanisms for chemical oscillators. Adv. Chem. Phys. 1990, 79, 269-299. 

Grace, J.R. In Handbook of Multiphase Systems, (Eds., Hestroni, G.), Section F., McGraw-Hill, New York, 1982 In Recent Advances in Engineering Analysis of Chemical Reaction Systems (Ed., Doraiswamy, L.K.), Wiley Eastern, New Delhi, 1984 In Gas Fluidization Technology, (Ed. Geldart, D.), Wiley, New York, 1986. Gray, P, and Scott, S.K, Chemical Oscillations and Instabilities, Oxford Science, New York, 1990. 

Field, R.J., Koros, E., Noyes, R.M. Oscillations in chemical systems. II. Thorough analysis of temporal oscillation in the bromate-cerium-malonic acid system. J. Am. Chem. Soc. 94(25), 8649-8664 (1972). http //dx.doi.org/10.1021/ja00780a001 Field, R.J., Noyes, R.M. Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. J. Chem. Phys. 60(5), 1877-1884 (1974). http //dx. doi.org/10.1063/1.1681288... 

To our knowledge, Sal nikov [21] was the first who developed such type of model and applied it to bifurcation analysis of chemical kinetics in oscillating systems. Further on, the thermokinetic models were successfully applied to the stability problems in the operation of chemical reactors [22]. Lately similar approach was applied to investigation of electrochemical processes [23]. 

One of the few successful attempts, by investigators other than Clarke, to utilize stochastic network analysis was carried out by Eiswirth et al. (1991), who divide the known mechanisms of chemical oscillators into four categories two based on the positive feedback loop, and a threefold division of one of these according to the negative feedback loop. The structure of the classification scheme resembles somewhat the results of the approach described in section 5.4.2, but the analysis is considerably more detailed and rigorous. 

Hwang, J.-T. Dougherty, E. P. Rabitz, S. Rabitz, H. 1978. The Green s Function Method of Sensitivity Analysis in Chemical Kinetics, J. Chem. Phys. 69, 5180-5191. Hynne, F, Sorensen, P. G. 1987. "Quenching of Chemical Oscillations, J. Phys. Chem. 

Field, R., K5r5s, E., Noyes, M. Oscillations in chemical systems. II. thorough analysis of themporal oscillation in the bromate-cerium-malonic acid system. Journal of the Americal Chemical Society 94, 8649-8666 (1972)... 

Furthermore, the existing sources of new oscillators did not provide encouragement for those wishing to test general theories of chemical oscillation. Serendipity, in many ways the most fruitful source, had furnished only two examples and was too unreliable to depend upon for more. Oscillators derived from living systems, though plentiful, appeared to be too complex for the kind of detailed mechanistic analysis needed. Variants and hybrids of known reactions were too similar to their parent systems to provide additional insights into the nature of chemical oscillation. 

The solution of the initial value problem described by Eq. (2.9) can be visualised so that the calculated concentrations are plotted as a function of time as shown in Fig. 2.1a. Another possibility is to explore the solution in the space of concentra-tirais as in Fig. 2.1b. In this case, the axes are the concentrations and the time dependence is not indicated. The actual concentration set is a point in the space of concentrations. The movement of this point during the simulation outlines a curve in the space of cOTicentrations, which is called the trajectory of the solution. This type of visuaUsation is often referred to as visualisation in phase space. In a closed system, the trajectory starts from the point that corresponds to the initial value and after a long time ends up at the equilibrium point. In an open system where the reactants are continuously fed into the system and the products are continuously removed, the trajectory may end up at a stationary point, approach a closed curve (a limit cycle in an oscillating system) or follow a strange attractor in a chaotic system. It is not the purpose of this book to discuss dynamical systems analysis of chemical models in detail, and the reader is referred to the book of Scott for an excellent treatment of this topic (Scott 1990). 

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