Modelling of oscillatory reactions

Before we discuss the mechanism of the B-Z reaction, it would be worthwhile to review the earlier hypothetical reaction schemes, which generate oscillations by taking recourse to computer solutions of differential equations. These were pioneering efforts since earlier it was difficult to conceive how a reaction system could generate oscillations. In fact, the principles and paradigms emanating from such analyses paved the way for formulating the reaction mechanism, as we shall see in the next section. 

The first attempt in understanding the chemistry of isothermal oscillatory reaction was made by Lotka [20, 21] who proposed a model based on preypredator model, i.e. 

The major weakness of the above model was that both amplitude and time period depend on the initial values of the oscillating species X and Y. Hence, it could generate infinite number of trajectories in X-Y plan depending on initial conditions. 

Next major attempt was made by Lefever and Prigogine [22] which results in a simple model called Brusselator as given below 

This model yields sustained oscillations even if back reactions are considered in the above scheme, provided some assumptions are made regarding the values of the rate constants. 

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In chapters 2-5 two models of oscillatory reaction in closed vessels were considered one based on chemical feedback (autocatalysis), the other on thermal coupling under non-isothermal reaction conditions. To begin this chapter, we again return to non-isothermal systems, now in a well-stirred flow reactor (CSTR) such as that considered in chapter 6. 

Until the 1950s, the rare periodic phenomena known in chemistry, such as the reaction of Bray [1], represented laboratory curiosities. Some oscillatory reactions were also known in electrochemistry. The link was made between the cardiac rhythm and electrical oscillators [2]. New examples of oscillatory chemical reactions were later discovered [3, 4]. From a theoretical point of view, the first kinetic model for oscillatory reactions was analyzed by Lotka [5], while similar equations were proposed soon after by Volterra [6] to account for oscillations in predator-prey systems in ecology. The next important advance on biological oscillations came from the experimental and theoretical studies of Hodgkin and Huxley [7], which clarified the physicochemical bases of the action potential in electrically excitable cells. The theory that they developed was later applied [8] to account for sustained oscillations of the membrane potential in these cells. Remarkably, the classic study by Hodgkin and Huxley appeared in the same year as Turing s pioneering analysis of spatial patterns in chemical systems [9]. 

We are thus, in many instances, more interested in the transient behaviour early in a reaction than we are in the more easily studied final or equilibrium state. With this in mind, we shall be concerned in our early chapters with simple models of chemical reaction that can satisfy all thermodynamic requirements and yet still show oscillatory behaviour of the kind described above in a well-stirred closed system under isothermal or non-isothermal conditions. 

Though reduced to the barest of essentials, the scheme shows many features observed in real examples of oscillatory reactions a pre-oscillatory period, a period of oscillatory behaviour, and then a final monotonic decay of reactant and intermediate concentrations to their equilibrium values. We can identify from the model such features as the dependence of the length of the pre-oscillatory period on the initial reactant concentration and the rate constants, an estimate for the number of oscillations, and the length of the oscillatory phase. By tuning the parameters we can obtain as many oscillations as we wish. 

Bykov, V. I., Ivanova, A. N. Yablonskii, G. S. 1979 On one class of kinetics models of oscillatory catalytic reactions. Kinet Phys. Chem. OscilL 2,468-476. 

In the literature there is a small number of reactions exhibiting oscillations, observed experimentally, which motivated a vast number of studies either devising a model for the reaction scheme or analyzing the small variations thereof. Although oscillatory behavior has been recognized in the past by a handful of chemists, it is recently that oscillatory behavior of chemical systems attracted considerable attention. As a result, studies carried out by various groups of researchers have been reviewed and summarized in review articles. Some of these reviews are more comprehensive than others and cover multiple examples of oscillatory reactions. A partial list of these articles is given in Table II with some annotations. 

While Belousov was describing his e)q)eriments into oscillatory chemical reactions, Ilya Prigogine in Brussels was developing theoretical models of nonequilibrium thermodynamics and ended with the notion of "structure dissipative" for which he was awarded the 1977 Nobel Prize in Chemistry. The concept of "Dissipative Structure" is ejq)licitly mentioned in the Nobel quotation "The 1977 Nobel Prize in Chemistry has been awarded to Professor Ilya Prigogine, Brussels, for his contributions to non-equilibrium thermodynamics, particularly the theory of dissipative structures". In the first half of the 1950s, Glansdorff and Balescu defined with Prigogine the thermodynamic criteria necessary for oscillatory behavior in dissipative systems [7]. Nicohs and Lefever then applied these to models of autocatalytic reactions [8]. 

The differential equations are stiff that is, several processes are going on at the same time, but at widely differing rates. This is a common feature of chemical kinetic equations and makes the numerical solution of the differential equations difficult. A steady state is never reached, so the equations cannot be solved analytically. Traditional methods, such as the Euler method and the Runge-Kutta method, use a time step, which must be scaled to fit the fastest process that is occurring. This can lead to large number of iterations even for small time scales. Hence, the use of Stella to model this oscillatory reaction would lead to an impossible situation. 

In the case of oscillatory reaction under discussion, reactions are ionic in nature and oscillating species are ions. The oscillating species Br and Ce +/Ce + are detected by bromide and platinum sensitive electrodes in conjunction with standard calomel electrode. The essential challenging task of developing a reaction mechanism is to postulate how the concentration of Ce + and Br builds-up in the course of time and how it is periodically inhibited. In the light of Brusselator model discovered by... 

As stated earlier in the section, for a comprehensive investigation of mechanism of oscillatory reactions, detailed study of kinetics (determination of rate constants) and mechanism of component reactions is also needed as a supporting study to provide information relevant for computer modelling of modified FKN mechanism. 

It may also happen that the concentration versus time curves tend to a periodic trajectory after an initial transient. This behaviour of differential equations is used when modelling the extremely important phenomena of oscillatory reactions. These reactions and their models will be treated in Section 4.5. 

Theoretical discussion of oscillatory reaction began since the discovery of the most famous Lotka s model in 1921. After the non-equilibrium theories of Prigogine (1968), many theories to explain the oscillatory chemical reactions have been proposed. A brief summary of some important oscillator models have been described below. 

It is the most basic and properly explained model of oscillating reactions. In 1921, Lotka proposed a model to explain some oscillatory phenomena in biological systems [29]. This model composed numbers of sequential steps which are presented in Table 1.1. As suggested, the each step referred to a specific mechanism in which the reactant molecules come together to form some useful intermediates and products. 

One approach to understanding the general aspects of oscillatory reactions is to seek key features that are shared by a family of oscillators and to incorporate these features into a general model for the family. The pH oscillators (Rabai et ah, 1990) consist of a dozen or more reactions in which there is a large amplitude change in the pH and in which the pH change is the driving force for, rather than a consequence or an indicator of, the oscillation. Luo and Epstein (1991) identified a number of common features in pH oscillators and abstracted from this analysis a scheme that captures the essential elements of these systems. Their... 

Particularly useful applications of the Monte Carlo method include modelling complex oscillatory reactions and studying enzyme catalysis [8,9]. As an example of the latter treatment, we will consider a system involving an initial reversible complex formation between the enzyme and the substrate, accompanied by a reversible step of inhibition of the catalyst... 

Recently, Vigil and Willmore [67] have reported mean field and lattice gas studies of the oscillatory dynamics of a variant of the ZGB model. In this example oscillations are also introduced, allowing the reversible adsorption of inert species. Furthermore, Sander and Ghaisas [69] have very recently reported simulations for the oxidation of CO on Pt in the presence of two forms of oxygen, namely chemisorbed atomic O and oxidized metal surface. These species, which are expected to be present for reaction under atmospheric pressure, are relevant for the onset of oscillatory behavior [69]. 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

Recently there has been an increasing interest in self-oscillatory phenomena and also in formation of spatio-temporal structure, accompanied by the rapid development of theory concerning dynamics of such systems under nonlinear, nonequilibrium conditions. The discovery of model chemical reactions to produce self-oscillations and spatio-temporal structures has accelerated the studies on nonlinear dynamics in chemistry. The Belousov-Zhabotinskii(B-Z) reaction is the most famous among such types of oscillatory chemical reactions, and has been studied most frequently during the past couple of decades [1,2]. The B-Z reaction has attracted much interest from scientists with various discipline, because in this reaction, the rhythmic change between oxidation and reduction states can be easily observed in a test tube. As the reproducibility of the amplitude, period and some other experimental measures is rather high under a found condition, the mechanism of the B-Z reaction has been almost fully understood until now. The most important step in the induction of oscillations is the existence of auto-catalytic process in the reaction network. 

In addition to bistability and hysteresis, the minimal model of glycolysis also allows nonstationary solutions. Indeed, as noted above, one of the main rationales for the construction of kinetic models of yeast glycolysis is to account for metabolic oscillations observed experimentally for several decades [297, 305] and probably the model system for metabolic rhythms. In the minimal model considered here, oscillations arise due to the inhibition of the first reaction by its substrate ATP (a negative feedback). Figure 24 shows the time courses of oscillatory solutions for the minimal model of glycolysis. Note that for a large... 

NMR properties, 33 213, 274 in nutation-NMR spectroscopy, 33 333 in sheer silicate smdies of, 33 340-341 layer structure, 32 184-186 on metal surfaces, 32 194-197 model, oscillatory reactions, 39 97-98 number... 

The outline of this paper is as follows. First, a theoretical model of unsteady motions in a combustion chamber with feedback control is constructed. The formulation is based on a generalized wave equation which accommodates all influences of acoustic wave motions and combustion responses. Control actions are achieved by injecting secondary fuel into the chamber, with its instantaneous mass flow rate determined by a robust controller. Physically, the reaction of the injected fuel with the primary combustion flow produces a modulated distribution of external forcing to the oscillatory flowfield, and it can be modeled conveniently by an assembly of point actuators. After a procedure equivalent to the Galerkin method, the governing wave equation reduces to a system of ordinary differential equations with time-delayed inputs for the amplitude of each acoustic mode, serving as the basis for the controller design. 

One particular pattern of behaviour which can be shown by systems far from equilibrium and with which we will be much concerned is that of oscillations. Some preliminary comments about the thermodynamics of oscillatory processes can be made and are particularly important. In closed systems, the only concentrations which vary in an oscillatory way are those of the intermediates there is generally a monotonic decrease in reactant concentrations and a monotonic, but not necessarily smooth, increase in those of the products. The free energy even of oscillatory systems decreases continuously during the course of the reaction AG does not oscillate. Nor are there specific individual reactions which proceed forwards at some stages and backwards at others in fact our simplest models will comprise reactions in which the reverse reactions are neglected completely. 

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