Nonlinear chemical dynamics reaction

I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics Oscillations, Waves, Patterns, and Chaos (New York Oxford University Press, 1998) I. R. Epstein, K. Kustin, P. De Kepper, and M. Orban, Scientific American, March 1983, p. 112 and H. Degn, Oscillating Chemical Reactions in Homogeneous Phase, J. Chem. Ed. 1972,49. 302. 

Oscillating reactions, a common feature of biological systems, are best understood within the context of nonlinear chemical dynamics and chaos theory based models that are used to predict the overall behavior of complex systems. A chaotic system is unpredictable, but not random. A key feature is that such systems are so sensitive to their initial conditions that future behavior is inherently unpredictable beyond some relatively short period of time. Accordingly, one of the goals of scientists studying oscillating reactions is to determine mathematical patterns or repeatable features that establish relationships to observable phenomena related to the oscillating reaction. 

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 study of nonlinear chemical dynamics begins with chemical oscillators - systems in which the concentrations of one or more species increase and decrease periodically, or nearly periodically. While descriptions of chemical oscillators can be found at least as far back as the nineteenth century (and chemical oscillation is, of course, ubiquitous in living systems), systematic study of chemical periodicity begins with two accidentally discovered systems associated with the names of Bray (2) and of Belousov and Zhabotinsky (BZ) 3,4), These initial discoveries were met with skepticism by chemists who believed that such behavior would violate the Second Law of Thermodynamics, but the development of a general theory of nonequilibrium thermodynamics (5) and of a detailed mechanism 6) for the BZ reaction brought credibility to the field by the mid-1970 s. Oscillations in the prototypical BZ reaction are shown in Figure 1. 

Given the importance of the BZ reaction in nonlinear chemical dynamics, it is not surprising that polymers and polymerizations would be coupled to it. Pojman et al. studied the BZ reaction to which acrylonitrile was added and showed that the polyacrylonitrile was produced periodically in phase with the oscillations (41). Given that radicals are produced periodically from the oxidation of malonic acid by ceric ion, it seemed reasonable to assume the periodic appearance of polymer was caused by periodic initiation. However, Washington et al. showed that periodic termination by bromine dioxide caused the periodic polymerization (42). 

We summarize our findings. It is suggested the criterion of critical condition, that is, the extremal behavior of the reaction species concentration, may be applied to reveal the critical conditions of nonlinear chemical dynamic systems. This is with the changeover of different dynamic modes of the reactions, such as the quasi-periodic and chaotic oscillations of the intermediate concentrations, as well as the steady-state mode. At the same time the Hamiltonian formalism makes it possible not only to have a successful numerical identification of the critical reaction conditions, but also to specify the role of individual steps of the reaction mechanism under different conditions. 

Various types of oscillating behaviors such as emergence of chemical waves, chaotic patterns, and a rich variety of spatiotemporal structures are investigated in oscillatory chemical reactions in association with nonlinear chemical dynamics [1-3]. In non-equilibrium condition, the characteristic dynamics of such chemically reacting systems are capable to self-organize into diverse kinds of assembly patterns. With the help of nonlinear chemical dynamics, the complexity and orderliness of those chemical processes can be explained properly. Various biological processes which exhibited very time-based flucmations especially when they are away from equilibrium have also been described by mechanistic considerations and theoretical techniques of nonlinear chemical dynamics [4-7]. 

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). 

Oscillations of chemical origin have been present as long as life itself. Every living system contains scores, perhaps hundreds, of chemical oscillators. The systematic study of oscillating chemical reactions and of the broader field of nonlinear chemical dynamics is of considerably more recent origin, however. In this chapter, we present a brief and extremely idiosyncratic overview of some of the history of nonlinear chemical dynamics. 

In the previous chapter, we developed a set of conceptual and mathematical tools for analyzing the models and experimental data that form the subject matter of nonlinear chemical dynamics. Here, we describe some of the key items of experimental apparatus used to obtain these data so that the reader can better appreciate the results discussed in the following chapters and can learn how to begin his or her own investigations. The first several sections are devoted to measurements of temporal behavior, with emphasis on the techniques used to monitor reactions in time and on the reactors in which these reactions are studied. The final section focuses on the study of spatial patterns and waves in chemical systems. 

The most commonly employed and convenient methods for studying nonlinear chemical dynamics employ potentiometric techniques. Electrochemical methods offer speed, low cost and, in many cases, excellent selectivity and sensitivity. A platinum electrode and a reference electrode are all that is required for any system with a species that changes its oxidation state during the reaction. The potential E is given by the Nemst equation ... 

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... 

Only the BZ reaction has played a more central role in the development of nonlinear chemical dynamics than the chlorite-iodide reaction (De Kepper et al., 1990). This latter system displays oscillations, bistability, stirring and mixing effects, and spatial pattern formation. With the addition of malonic acid, it provides the reaction system used in the first experimental demonstration of Turing patterns (Chapter 14). Efforts were made in the late 1980s to model the reaction (Epstein and Kustin, 1985 Citri and Epstein, 1987 Rabai and Beck, 1987), but each of these attempts focused on a different subset of the experimental data, and none was totally successful. Since each model contains a different set of reactions fitted to a different set of data, individual rate constants vary widely among the different models. For example, the rate constant for the reaction between HOCl and HOI has been given as zero (Citri and Epstein, 1987), 2 x10 s (Rabai... 

One way to think of a chemical reaction mechanism is as a network that connects the various reactants, intermediates, and products. This point of view has led to important general results about the properties of reaction mechanisms. In nonlinear chemical dynamics, one is interested in the question of network stability (Clarke, 1980), that is, whether or not all the steady states of a given network or mechanism are stable, A related question, called by Clarke the stability diagram problem, is to find for a given mechanism the boundary (in a space whose coordinates are the rate constants) that separates unstable from stable networks. 

The most frequently encountered numerical problem in nonlinear chemical dynamics is that of solving a set of ordinary, nonlinear, first-order, coupled, autonomous differential equations, such as those describing the BZ reaction. We hope you understand by now what nonlinear means, but let us comment on the other modifiers. The equations are ordinary because they do not contain partial derivatives (we consider partial differential equations in the next section), first order because the highest derivative is the first derivative, and coupled because the time derivative of one species depends on the concentrations of other species. In the absence of time-dependent external forcing, rate equations are autonomous, meaning that time does not appear explicitly on the right-hand side. 

Just as the BZ reaction has become the experimental prototype for nonlinear chemical dynamics, the Oregonator model (Field and Noyes, 1974b), is easily the most familiar and thoroughly studied model system in nonlinear chemical dynamics. We recall from Chapter 5 that the model equations are... 

One of the great appeals of nonlinear chemical dynamics lies in the striking visual demonstrations that can be created. It is a rare person who is not impressed the first time he or she sees a clear solution repeatedly turn brown, then blue, then back to clear In this appendix, we explain how to perform demonstrations of oscillating reactions, fronts, and waves. In the next section, we provide information on how some of these systems can be systematically studied in the upper-level undergraduate laboratory. 

Chemical Reactions in Clusters, E.R. Bernstein Quantum Mechanics in Chemistry, J. Simons and J. Nichols An Introduction to Hydrogen Bonding, G.A. Jeffrey Hydrogen Bonding A Theoretical Perspective, S. Scheiner Fractals in Molecular Biophysics, T.G. Dewey Molecular Orbital Calculations for Biological Systems, A.-M. Sapse An Introduction to Nonlinear Chemical Dynamics Oscillations, Waves. Patterns, and Chaos, I.R. Epstein and J.A. Pojman... 

The models for chemically reacting media discussed above described the evolution of the system on macroscopic scales. In some instances, especially when one considers applications of nonlinear chemical dynamics to biological systems or materials on nanoscales, a mesoscopic description will be more appropriate or even essential. In this section, we show how one can construct mesoscopic models for reaction-diffusion systems and how these more fundamental descriptions relate to the macroscopic models considered previously. 

Over the past years enormous progress has been made in making experiments on macroscopic spatial or temporal structures formed in chemical reactions and interpreting them theoretically. In this way chemistry has given an outstanding contribution to the study of systems driven far from thermal equilibrium. It is a particular pleasure for me to present this talk in Bordeaux where important contributions to this new field were given which is also witnessed by the two volumes on nonlinear chemical dynamics edited by A. Pacault ana C. Vidal [l]. 

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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