Delays and Differential Delay Equations

Mathematically speaking, the most important tools used by the chemical kineticist to study chemical reactions like the ones we have been considering are sets of coupled, first-order, ordinary differential equations that describe the changes in time of the concentrations of species in the system, that is, the rate laws derived from the Law of Mass Action. In order to obtain equations of this type, one must make a number of key assumptions, some of which are usually explicit, others more hidden. We have treated only isothermal systems, thereby obtaining polynomial rate laws instead of the transcendental expressions that would result if the temperature were taken as a variable, a step that would be necessary if we were to consider thermochemical oscillators (Gray and Scott, 1990), for example, combustion reactions at metal surfaces. What is perhaps less obvious is that our equations constitute an average over quantum mechanical microstates, allowing us to employ a relatively small number of bulk concentrations as our dependent variables, rather than having to keep track of the populations of different states that react at different rates. Our treatment ignores fluctuations, so that we may utilize deterministic equations rather than a stochastic or a master equation formulation (Gardiner, 1990). Whenever we employ ordinary differential equations, we are making the approximation that the medium is well mixed, with all species uniformly distributed; any spatial gradients (and we see in several other chapters that these can play a key role) require the inclusion of diffusion terms and the use of partial differential equations. All of these assumptions or approximations are well known, and in all cases chemists have more elaborate techniques at their disposal for treating these effects more exactly, should that be desirable. Another, less widely appreciated idealization in chemical kinetics is that phenomena take place instantaneously—that a change in [A] at time t generates a change in [B] time t and not at some later time t + τ. On a microscopic level, it is clear that this state of affairs cannot hold.

Turing Patterns

In the first chapter of this book, we noted the “dark age” of nearly forty years separating the work of Bray and Lotka in the early 1920s and the discovery of the BZ reaction in the late 1950s. Remarkably, the history of nonlinear chemical dynamics contains another gap of almost the same length. In 1952, the British mathematician Alan Turing wrote a paper in which he suggested that chemical reactions with appropriate nonlinear kinetics coupled to diffusion could lead to the formation of stationary patterns of the type encountered in living organisms. It took until 1990 for the first conclusive experimental evidence of Turing patterns to appear (Castets et al., 1990). Turing was a formidable figure (Hodges, 1983). He was responsible for much of the fundamental work that underlies the formal theory of computation, and the notion of a “Turing machine” is essential for anyone who wishes to understand computing and computers. During World War II, Turing was a key figure in the successful effort to break the Axis “Enigma” code, an accomplishment that almost certainly saved many lives and shortened the war in Europe. His 1952 paper, entitled “The Chemical Basis of Morphogenesis” was his only published venture into chemistry, but its impact has been enormous. Recently, this classic paper has been reprinted along with some of Turing's unpublished notes on the origins of phyllotaxis, the arrangement of leaves on the stems of plants (Saunders, 1992). In this chapter, we shall describe the nature of Turing patterns and some of the systems in which they may play a role, explore why they have been so elusive, examine the experimental systems in which they have been demonstrated, and consider other systems and other methods for generating them. Much of our discussion will focus on the chlorite-iodide-malonic acid (CIMA) reaction in which the patterns were first seen. In the study of Turing patterns, the CIMA system and its relatives play much the same role today that the BZ reaction played during the 1960s and 1970s in the study of chemical oscillation.

Computational Tools

It is fair to say that the field of nonlinear chemical dynamics would not be where it is today, and perhaps it would not exist at all, without fast digital computers. As we saw in Chapter 1, the numerical simulation of the essential behavior of the BZ reaction (Edelson et al., 1975) did much both to support the FKN mechanism and to make credible the idea that chemical oscillators could be understood without invoking any new principles of chemical kinetics. In 1975, solving those differential equations challenged the most advanced machines of the day, yet the computers used then were less powerful than many of today’s home computers! Despite the present widespread availability of computing power, there remain many challenging computational problems in nonlinear dynamics, and even seemingly simple equations can be difficult to solve or maybe even lead to spurious results. In this chapter, we will look at some of the most widely used computational techniques, try to provide a rudimentary understanding of how the methods work (and how they can fail!), and list some of the tools that are available. There are several reasons for utilizing the techniques described in this chapter: 1. For a complicated system, it is generally not possible to measure all of the rate constants in a proposed mechanism. One way to estimate the remaining parameters is to simulate numerically the behavior of the system, varying the unknown rate constants until the model satisfactorily reproduces the experimental behavior. 2. If a mechanism, which may consist of dozens of elementary chemical reactions, is valid, then it should reproduce the observed dynamical behavior. Proposed mechanisms are most commonly tested by integrating the corresponding rate equations numerically and comparing the results with the experimental time series, or by comparing the results of many such simulations with different initial conditions (or of a numerical continuation study) to the experimental phase diagram. 3. Numerical results can act as a guide to further experiments. The real reason for developing models is not to interpolate between our experimental observations but to extrapolate into unknown realms.

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. A persuasive argument can be made that it was progress in unraveling the mechanism of the prototype BZ reaction in the 1970s that gave the study of chemical oscillators the scientific respectability that had been denied it since the discovery of the earliest oscillating reactions. The formulation by Field, Körös, and Noyes (Field et al., 1972) of a set of chemically and thermodynamically plausible elementary steps consistent with the observed “exotic” behavior of an acidic solution of bromate and cerium ions and malonic acid was a major breakthrough. Numerical integration (Edelson et al., 1975) of the differential equations corresponding to the FKN mechanism demonstrated beyond a doubt that chemical oscillations in a real system were consistent with, and could be explained by, the same physicochemical principles that govern "normal" chemical reactions. No special rules, no dust particles, and no vitalism need be invoked to generate oscillations in chemical reactions. All we need is an appropriate set of uni- and bimolecular steps with mass action kinetics to produce a sufficiently nonlinear set of rate equations. Just as the study of molecular structure has benefited from new experimental and theoretical developments, mechanistic studies of complex chemical reactions, including oscillating reactions, have advanced because of new techniques. Just as any structural method has its limitations (e.g., x-ray diffraction cannot achieve a resolution that is better than the wavelength of the x-rays employed), mechanistic studies, too, have their limitations. The development of a mechanism, however, has an even more fundamental and more frustrating limitation, sometimes referred to as the fundamental dogma of chemical kinetics. It is not possible to prove that a reaction mechanism is correct. We can only disprove mechanisms.

Introduction—A Bit of History

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 1828, Fechner described an electrochemical cell that produced an oscillating current, this being the first published report of oscillations in a chemical system. Ostwald observed in 1899 that the rate of chromium dissolution in acid periodically increased and decreased. Because both systems were inhomogeneous, it was believed then, and through much of our own century, that homogeneous oscillating reactions were impossible. Degn wrote in 1972 (p. 302): “It is hard to think of any other question which already occupied chemists in the nineteenth century and still has not received a satisfactory answer.” In that same year, though, answers were coming. How it took so long for the nature of oscillating chemical reactions to be understood and how that understanding eventually came about will be the major focus of this chapter. Although oscillatory behavior can be seen in many chemical systems, we shall concentrate primarily on homogeneous, isothermal reactions in aqueous solution. In later chapters, we shall broaden our horizons a bit. While the study of oscillating reactions did not become well established until the mid-1970s, theoretical discussions go back to at least 1910. We consider here some of the early theoretical and experimental work that led up to the ideas of Prigogine on nonequilibrium thermodynamics and to the experimental and theoretical work of Belousov, Zhabotinsky, Field, Körös, and Noyes, all of whom did much to persuade chemists that chemical oscillations, traveling fronts, and other phenomena that now comprise the repertoire of nonlinear chemical dynamics were deserving of serious study. Alfred Lotka was one of the more interesting characters in the history of science. He wrote a handful of theoretical papers on chemical oscillation during the early decades of this century and authored a monograph (1925) on theoretical biology that is filled with insights that still seem fresh today.

Stirring and Mixing Effects

In almost everything that we have discussed so far, we have assumed, explicitly or implicitly, either that the systems we are looking at are perfectly mixed or that they are not mixed at all. In the former case, concentrations are the same everywhere in the system, so that ordinary differential equations for the evolution of the concentrations in time provide an appropriate description for the system. There are no spatial variables; in terms of geometry, the system is effectively zero-dimensional. At the other extreme, we have unstirred systems. Here, concentrations can vary throughout the system, position is a key independent variable, and diffusion plays an essential role, leading to the development of waves and patterns. Geometrically, the system is three-dimensional, though for mathematical convenience, or because one length is very different from the other two, we may be able to approximate it as one- or two-dimensional. In reality, we hardly ever find either extreme—that of perfect mixing or that of pure, unmixed diffusion. In the laboratory, where experiments in beakers or CSTRs are typically stirred at hundreds of revolutions per minute, we shall see that there is overwhelming evidence that, even if efforts are made to improve the mixing efficiency, significant concentration gradients arise and persist. Increasing the stirring rate helps somewhat, but beyond about 2000 rpm, cavitation (the formation of stirring-induced bubbles in the solution) begins to set in. Even close to this limit, mixing is not perfect. In unstirred aqueous systems, as we have seen in Chapter 9, it is difficult to avoid convective mixing. Preventing small amounts of mechanically induced mixing requires considerable effort in isolating the system from external vibrations, even those caused by the occasional truck making a delivery to the laboratory stockroom. It is possible to suppress the effects of convection and mechanical motion in highly viscous media, such as the gels used in the experiments on Turing patterns as discussed in the previous chapter. There, we can finally study a pure reaction-diffusion system. Systems in nature—the oceans, the atmosphere, a living cell—are important examples in which chemical reactions with nonlinear kinetics occur under conditions of imperfect mixing.

Coupled Oscillators

We have thus far learned a great deal about chemical oscillators, but, except in Chapter 9, where we looked at the effects of external fields, our oscillatory systems have been treated as isolated. In fact, mathematicians, physicists, and biologists are much more likely than are chemists to have encountered and thought about oscillators that interact with one another and with their environment. Forced and coupled oscillators, both linear and nonlinear, are classic problems in mathematics and physics. The key notions of resonance and damping that arise from studies of these systems have found their way into several areas of chemistry as well. Although biologists rarely consider oscillators in a formal sense, the vast variety of interdependent oscillatory processes in living systems makes the representation of an organism by a system of coupled oscillators a less absurd caricature than one might at first think. In this chapter, we will examine some of the rich variety of behaviors that coupled chemical oscillators can display. We will consider two approaches to coupling oscillatory chemical reactions, and then we will look at the phenomenology of coupled systems. We begin with some general considerations about forced oscillators, which constitute a limiting case of asymmetric coupling, in which the linkage between two oscillators is infinitely stronger in one direction than in the other. As an aid to intuition, picture a child on a swing or a pendulum moving periodically. The forcing consists of an impulse that is applied, either once or periodically, generally with a frequency different from that of the unforced oscillator. In a chemical oscillator, the forcing might occur through pulsed addition of a reactive species or variation of the flow rate in a CSTR. Mathematically, we can write the equations describing such a system as . . . dX/dt = f(x) + ε g(X, t) (12.1) . . . where the vector x contains the concentrations, the vector function f(x) contains all the rate and flow terms in the absence of forcing, g(x) represents the appropriately scaled temporal dependence of the forcing, and the scalar parameter e specifies the strength of the forcing.

Fundamentals

Before plunging into the meat of our discussions, we will review some basic but necessary ideas. Much of this material will be familiar to many readers, and we encourage you to move quickly through it or to skip it completely if appropriate. If you have not encountered these concepts before, you will find it worthwhile to invest some time here and perhaps to take a look at some of the more detailed references that we shall mention. We begin with a review of chemical kinetics. We then consider how to determine the stability of steady states in an open system using analytical and graphical techniques. Finally, we look at some of the methods used to represent data in nonlinear dynamics. The problems that we are interested in involve the rates of chemical reactions, the study of which forms the basis of chemical kinetics. This is a rich and beautiful subject, worthy of whole volumes. For those interested in a less superficial view than we have room to present here, we recommend several excellent texts on kinetics (Jordan, 1979; Cox, 1994; Espenson, 1995). We review here a minimal set of fundamentals necessary for what comes later.

Polymer Systems

In the classic 1967 film “The Graduate” the protagonist, Benjamin (Dustin Hoffman), is attempting to plan his postcollege path. His neighbor provides one word of advice, “Plastics.” This counsel has become part of American culture and is often parodied. But, it is good advice, because not since the transformations from stone to bronze and then to iron have new materials so completely transformed a society. Plastics made from synthetic polymers are ubiquitous, from Tupperware to artificial hearts. About half the world’s chemists work in polymer-related industries. In this chapter, we will survey some of the work that has been done in applying nonlinear dynamics to polymerization processes. These systems differ from those we have considered so far because they do not involve redox reactions. We will consider polymerization reactions in a CSTR that exhibit oscillations through the coupling of temperature-dependent viscosity and viscosity-dependent rate constants. Emulsion polymerization, which produces small polymer particles dispersed in water, can also oscillate in a CSTR. Both types of systems are important industrially, and their stabilities have been studied by engineers with the goal of eliminating their time-dependent behavior. Our favorite oscillating system, the Belousov-Zhabotinsky reaction, can be used to create an isothermal periodic polymerization reaction in either a batch or continuous system. This, however, is not a practical system because of the cost of the reagents. In most industrial processes, nonlinear behavior is seen not as an advantage but as something to be avoided. However, we will look at several reaction-diffusion systems that have desirable properties precisely because of their nonlinear behavior. Replication of RNA is autocatalytic and can occur as a traveling front. Since not all RNA molecules replicate equally well, faster mutants gradually take over. At each mutation, the front propagates faster. Evolution can be directly observed in a test tube. Propagating polymerization fronts of synthetic polymers may be useful for making new materials, and they are interesting because of the rich array of nonlinear phenomena they show, with pulsations, convection, and spinning fronts. Finally, we will consider photopolymerization systems that exhibit spatial pattern formation on the micron scale, which can be used to control the macroscopic properties.

Complex Oscillations and Chaos

After studying the first seven chapters of this book, the reader may have come to the conclusion that a chemical reaction that exhibits periodic oscillation with a single maximum and a single minimum must be at or near the apex of the pyramid of dynamical complexity. In the words of the song that is sung at the Jewish Passover celebration, the Seder, “Dayenu” (It would have been enough). But nature always has more to offer, and simple periodic oscillation is only the beginning of the story. In this chapter, we will investigate more complex modes of temporal oscillation, including both periodic behavior (in which each cycle can have several maxima and minima in the concentrations) and aperiodic behavior, or chaos (in which no set of concentrations is ever exactly repeated, but the system nonetheless behaves deterministically). Most people who study periodic behavior deal with linear oscillators and therefore tend to think of oscillations as sinusoidal. Chemical oscillators are, as we have seen, decidedly nonlinear, and their waveforms can depart quite drastically from being sinusoidal. Even after accepting that chemical oscillations can look as nonsinusoidal as the relaxation oscillations shown in Figure 4.4, our intuition may still resist the notion that a single period of oscillation might contain two, three, or perhaps twenty-three, maxima and minima. As an example, consider the behavior shown in Figure 8.1, where the potential of a bromide-selective electrode in the BZ reaction in a CSTR shows one large and two small extrema in each cycle of oscillation. The oscillations shown in Figure 8.1 are of the mixed-mode type, in which each period contains a mixture of large-amplitude and small-amplitude peaks. Mixedmode oscillations are perhaps the most commonly occurring form of complex oscillations in chemical systems. In order to develop some intuitive feel for how such behavior might arise, we employ a picture based on slow manifolds and utilized by a variety of authors (Boissonade, 1976; Rössler, 1976; Rinzel, 1987; Barkley, 1988) to analyze mixed-mode oscillations and other forms of complex dynamical behavior.