Chemical reaction Brusselator

Participants in the second conference (April 1925) of the Institut International de Chimie Solvay in Brussels. The topic was "Structure and Activity," and four papers were devoted to activation or mechanism in chemical reactions. Henry Armstrong and Jean Perrin are seated at the center section of the adjoining tables. Andre Job and Thomas Martin Lowry are to Perrin s left. Courtesy of the Instituts Intemationaux de Physique et Chimie (Solvay), Brussels. 

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. 

Szepe, S., Ph.D. Thesis, Illinois Institute of Technology, 1966 also see Szepe, S., and O. Levenspiel, Chem. Eng. ScL, 23, 881 (1968) Catalyst Deactivation, p. 265, Fourth European Symposium on Chemical Reaction Engineering, Brussels, September 1968, Pergamon, London, 1971. 

Szepe, S, Levenspiel, O., Proceedings of the Fourth European Symposium on Chemical Reaction Engineering, Pergamon Press, Brussels (1971)... 

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

As it was mentioned in Section 2.1.1, the concentration oscillations could be simulated quite well by a set of even two ordinary differential equations of the first order but paying the price of giving up the rigid condition imposed on interpretation of mechanisms of chemical reactions namely that they are based on mono- and bimolecular stages only (remember the Hanusse theorem [19]) An example of what Smoes [7] called the heuristic-topological model is the well-known Brusselator [2], Its scheme was discussed in Section 2.1.1 see equations (2.1.33) to (2.1.35). 

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. 

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. 

Some autocatalytic chemical reactions such as the Brusselator and the Belousov-Zhabotinsky reaction schemes can produce temporal oscillations in a stirred homogeneous solution. In the presence of even a small initial concentration inhomogeneity, autocatalytic processes can couple with diffusion to produce organized systems in time and space. 

Example 12.7 Linear stability analysis Brusselator scheme This example is from Kondepudi and Prigogine (1999). Consider the chemical reaction scheme in Eq. (12.86). Assume that the concentration values of A, B, and E, F are maintained at uniform values, and X and Y are the only remaining variables. Then the kinetic equations are... 

Slesser. C. G. M W. T. Allen, A. R. Cuming. V. Faivlowsky. and J. Shields, Chemical Reaction Engineering, Proceedings of the Fourth European Symposium, Brussels 1968, supplement to Chemical Engineering Science, pp. 41-45. 

A. G. Buekens, G. F. Froment, Fundamental aspects of the design of tubular cracking units. Proceedings of the 4th European Symposium on Chemical Reaction Engineering, Brussels, 9-11 September 1968. 

As a student and collaborator of Professor de Bonder, Professor R. Defay has contributed to the work of the Brussels School omthe Thermodynamics of chemical reactions. His own researches are chiefly related to azeotropism, to surface tension, to the extension of the phase rule, to capillary systems and to the study of surfaces not in adsorption equilibrium. 

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

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 "Brusselator model contains only two variables (X and Y). It corresponds formally to a system of four coupled chemical reactions in a complete non-ecpiilibrium situation ... 

Another model of oscillating chemical reactions, the so-called Brusselator model was proposed by I. Prigogine and his collaborators at the Free University of Brussels. 

Example 5.3. Consider the model called the brusselator, to which case (d) applies. The model describes a chemical reaction with two intermediate substances [19]. The nondimensional concentrations u and v of these substances satisfy the system... 

As shown in Appendix B, a can happen to be negative for the Brusselator, at least in the vicinity of the Hopf bifurcation point. Moreover, it is seen from (B.19) that I a lean be made arbitrarily small by suitably choosing y4, Dx, andZ7y, while p and y can remain of ordinary magnitude. Thus, we have at least one chemical reaction model showing phase turbulence. 

Paris, J.R. and Stevens, W.F. Fourth European Symposiiin on Chemical Reaction Engineering, Brussels (1968), 73,... 

Abstract chemical models exhibiting nonlinear phenomena were proposed more than a decade ago. The Brusselator of PRIGOGINE and LEFEVER [54] has oscillatory (limit cycle) solutions, and the SCHLOGL [55] model exhibits bistability, but these models have only two variables and hence cannot have chaotic solutions. At least 3 variables are required for chaos in a continuous system, simply because phase space trajectories cannot cross for a deterministic system. As mentioned in the Introduction, the possibility of chemical chaos was suggested by RUELLE [1] in 1973. In 1976 ROSSLER [56], inspired by LORENZ s [57] study of chaos in a 3 variable model of convection, constructed an abstract 3 variable chemical reaction model that exhibited chaos. This model used as an autocatalytic step a Michaelis-Menten type kinetics, which is a nonlinear approximation discovered in enzymatic studies. Recently more realistic biochemical models [58,59] have also been found to exhibit low dimensional chaos. 

Influence of external noise speed on the direction of shift of a Hopf bifurcation We consider the Brusselator chemical reaction scheme... 

Fig. 14.30 Some of the participants in the 20th Solvay Congress on Chemistry, Chemical Reactions and their Control on the Femtosecond Time Scale (November 1995, Brussels). From left to right the author. Professor J. Troe, Professor R. Marcus, Professor A. Zewail, and Professor R. Levine.

Prigogine s teacher De Bonder was the founder of The Brussels School of Thermodynamics that focused on the thermodynamic treatment of irreversible processes, including chemical reactions. Much of the work of The Brussels School was connected with rates of entropy-production of systems that are not at equilibrium. One of Prigogine s early achievements, after he was appointed to the ULB physical chemistry faculty, was to demonstrate that non-equilibrium systems that are close to equilibrium necessarily will evolve so as to approach the equilibrium state in such a way that the rate of entropy production is as low as is possible. 

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