The Brusselator

Another hypothetical mechanism of a chemical reaction is the model called Brusselator, investigated by the Brussels school of Prigogine 

The following system of kinetic equations without diffusion corresponds to the mechanism (6.79)  

We shall introduce dimensionless variables, making substitutions in (6.80) 

An analysis of the system (6.82) will begin with the determination of its stationary states. The requirement x = 0, y = 0 results in the determination 

Sensitive states correspond to the condition Re(At 2) = 0. The analysis of equation (6.86) leads to a conclusion that for a 0, b 0 the only sensitive state is given by 

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Once the door was opened to these new perspectives, the works multiplied rapidly. In 1968 an important paper by Prigogine and Rene Lefever was published On symmetry-breaking instabilities in dissipative systems (TNC.19). Clearly, not any nolinear mechanism can produce the phenomena described above. In the case of chemical reactions, it can be shown that an autocatalytic step must be present in the reaction scheme in order to produce the necessary instability. Prigogine and Lefever invented a very simple model of reactions which contains all the necessary ingerdients for a detailed study of the bifurcations. This model, later called the Brusselator, provided the basis of many subsequent studies. 

GEN.278.1. Prigogine, L apport de TEcole de Thermodynamique et de Mecanique statistique de Bruxelles (The contribution of the Brussels school of thermodynamics and statistical mechanics), article redige dans le cadre d un ouvrage sur les activites de TULB a I approche de fan 2000 (non public). 

Classical dynamics and orthodox quantum mechanics are constructed along the model of integrable systems in the sense of Poincare. Our aim is to construct dynamics for nonintegrable systems. As far as we know, this is a new attempt, which has its roots in the early work of the Brussels School [1-9]. The main result is that we have to replace the unitary transformation f/ by a nonunitary... 

The buds and the leaves (less often) of the Brussels sprout plant (Brasska olera-cea var. gemmifera) are eaten cooked with the main meal. In Brussels sprouts, breakdown products from glucosinolates are dominant and represent about 80-90% of the volatiles in headspace samples [176]. The residual volatiles are mostly sulfur compounds [176]. Compounds likely to be associated with the aroma of Brussels sprouts are 2-propenyl isothiocyanate, dimethyl sulfide, dimethyl disulfide and dimethyl trisulfide (Table 7.6) [35,176]. 

Let us go back to the Brusselator (1). In a one-dimensional medium of unit length the reaction-diffusion equations (12) become... 

In this simplified version of the Brusselator model, the trimolecular autocatalytic step, which is a necessary condition for the existence of instabilities, is, of course, retained. However, the linear source-sink reaction steps A—>X—>E are suppressed. A continuous flow of X inside the system may still be ensured through the values maintained at the boundaries. The price of this simplification is that (36) can never lead to a homogeneous time-periodic solution. The homogeneous steady states are... 

Three model kinetic schemes have been studied relatively intensively with periodic forcing the first-order non-isothermal CSTR of chapter 7 the Brusselator model, which is closely related to the cubic autocatalysis of chapters 2 and 3 and the surface reaction model discussed in 12.6. We will use the last of these to introduce some of the general features. 

An imagined chemical reaction with two compounds X, Y exhibiting a limit cycle is the Brusselator ... 

Finally, the basic model could be also constructed ad hoc just to reproduce the kinetic phenomena observed experimentally in time and in space the well-known examples are the Brusselator or Prigogine-Lefever model (see [2]) and the model by Smoes [7]). Practically any basic model is oriented for a simplest and transparent description of a particular kind of the autowave processes. 

Reprint F is an example of analyzing a reaction in formal kinetics. Gray and Scott introduced the autocatalytic A + 2B = 3B as a simple model reaction that proved to have a rich behavior, much richer than the Brusselator for example. However, A + 2B smacks of a three-body interaction, which is a sufficiently rare occurrence as to be avoided. I had done a pseudo-steady-state analysis before I visited Leeds at Gray s invitation, and the chance of working with the fons et origo of this reaction, so to speak, was an opportunity to make sure that the limiting behavior was not lost when certain parameters were small, but not actually zero. For another analysis of autocatalytic behavior, see [107]. 

Gray, B. F. and Morley-Buchanan, T., 1985, Some criticism concerning the Brusselator model of an oscillating chemical reaction. J. Chem. Soc. Faraday Trans. 2 81, 77. 

In figure 2b, there are clearly folds in the left-hand side of the 3/2 and 2/1 resonance horns. This phenomenon had not (when we observed it) been seen in other forced oscillators such as the Brusselator model (Kai Tomita 1979) and the non-isothermal cstr (Kevrekidis et al. 1986), although it may have been missed in previous numerical studies that did not use arc-length continuation. It is however also to be found in unpublished work of Marek s group. The cusp points at M and L are quite different from the apparent cusp ... 

A number of reference materials for vitamins in foods are under development. Extensive analysis and stability testing have been conducted to assess the potential of vitamin-enriched milk powder, wholemeal flour, and margarine as well as lyophilized brussels sprouts, mixed vegetables, and pork liver for use as reference materials (193-196). The certification study for vitamin C in the brussels sprouts reference material has been completed. However, methodology problems continue to have a significant negative effect on the development of reference materials. 

Before addressing the main topic of chemical evolution, I would like to discuss briefly the rather curious story of the Belgian school of thermodynamics, often called the Brussels school. It took shape at the end of the 1920s and during the 1930s. At a time when the great schools of thermodynamics, such as the Californian school founded by Lewis and the British school with Guggenheim, directed their efforts almost exclusively to the study of equilibrium systems, the point of view presented by the Brussels school appeared as quite unorthodox and somewhat controversial. Indeed, the Brussels school tried to approach equilibrium as a special case of nonequilibrium and concentrated its efforts on the presentation of thermodynamics in a form that would be applicable also to nonequilibrium situations. This story is rather curious from the point of view of the history of science, so let me go into a little more detail. 

The introduction of affinity by De Donder marks the birth of the Brussels school the first publication appeared around 1922, but it took some years to make these concepts more precise.4 What was the reaction of the scientific community When we go through the proceedings of the Belgian Royal Academy, we see that De Donder s work indeed aroused much local interest. Verschaffel from Ghent and Mund from Louvain were among the people who became active in this newborn nonequilibrium chemical thermodynamics. However, one has to say that elsewhere De Donder s approach met with skepticism and even with hostility. His introduction of affinity was thought of as merely a different notation. 

To confirm the above conjectures we have performed a numerical simulation of equation (29) on the Brusselator model chemical reaction.46 The results are shown in Fig. 7. We start with an initial condition corresponding to a clockwise wave. Under the effect of the counterclockwise field this wave is deformed and eventually its sense of rotation is reversed. In other words, the system shows a clear-cut preference for one chirality. As a matter of fact we are witnessing an entrainment phenomenon of a new kind, whereby not only the frequency but also the sense of rotation of the system are adjusted to those of the external field. More complex situations, including chaotic behavior, are likely to arise when the resonance condition w = fl, is not satisfied, but we do not address ourselves to this problem here. 

Fig. 7. Numerical simulation of equation (29) for the Brusselator model on a ring. At t = 0 a clockwise wave propagates along the ring at t = 4.49 the effect of the counterclockwise external field deforms this wave appreciably after a while the sense of rotation is reversed and for t > 22.49 one obtains a stable wave solution in the counterclockwise direction. The period of both the external field and of the linearized solution of the unperturbed system is 4.28 time units.

Recently, a whole "zoo of models has been investigated. Its most known inhabitants, the "brussellator and "oregonator Ref. [2], contain the steps... 

The Brussels Roundtable discussions also point out the interactions between the different stakeholders, e.g. the industry and consumers. As Terence Koh of the Singapore Chemicals Industry Council argued, the chemicals industry demands that it should not be reduced to a role of polluter and in fact, it has recognised that it does well in accommodating public concerns. Daniel Verbist of the European Chemicals Industry Council further argued that We need to learn from scientific progress as well as from public perception to update our knowledge and adapt our practice. ... 

These considerations are supported by the experience made within the Montreal Protocol. Part of the success, as presented by Rajendra Shende, Head of the United Nations Environment Programme (UNEP) s OzonAction Unit, at the Brussels Roundtable, was due to the fact that the protocol supports its NFPs in networking with other regional groups. Further the protocol successfully cooperates with NGOs, e.g. the EIA. 

The range of practical recommendations from the Brussels Roundtable discussions indicates the wide field in which stakeholders operate and may help to tap the full potential of a strengthened co-operation to ... 

Paul Kaye is the Brussels Editor with ENDS Europe Daily. 

Jan. 25,1917, Moscow, Russia - May 28, 2003, Brussels, Belgium) Ilya Prigogine studied chemistry and physics at the Universite Libre de Bruxelles, where he completed his PhD in 1941, and became a professor in 1947. He joined the Brussels -> thermodynamics school founded by Theophile De Donder (1873-1957) and Jean Timmermans (1882-1971). He received the Nobel Prize in Chemistry in 1977 for his contributions to nonequilibrium thermodynamics, particularly the theory of dissipative structure. His work established a thermodynamic basis of -> transport phenomena in electrolyte solutions and -> electrochemical oscillations. 

The Brussels Commission also gave its approval to three large-scale operations the ICI and BP Chemicals exchange of polyethyl-ene/PVC, the vertical integration of vinyl chloride involving AKZO and Shell Chemicals,... 

In those countries that had rescinded the tradition by which a commanding officer s orders exempted a subordinate from punishment, a return to the earlier principles was soon demanded. For example, in its verdict of June 29, 1951, against Lippert and others,126 the Belgian court-martial in Ltittich rejected the criminal liability of the accused because these had acted under orders. The Brussels court-martial came to a similar decision on March 9, 1951, in its verdict against General von Falkenhausen.127... 

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