Waves Propagating Reaction-Diffusion Fronts

Chemical Waves Propagating Reaction-Diffusion Fronts 

The great physical chemist Walther Nernst happened to be in the audience and voiced his skepticism about Luther s velocity equation, asking, Who has derived this formula  

When Luther said he did, Nernst asked, But this is not published yet  

Luther replied, No, but it is a simple consequence of the corresponding differential equation.  

Nernst said that he looked forward to seeing the complete publication. It is easy to imagine that this exchange was not exactly congenial—and Luther s glib reply may not have been completely up front because the result is not a simple consequence of the differential equation.,  

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Chemical Waves Propagating Reaction-Diffusion Fronts... 

Saul, A. and Showalter, K. (1985). Propagating reaction-diffusion fronts. In Oscillations and traveling waves in chemical systems, (ed. R. J. Field and M. Burger), ch. 11, pp. 419-39. Wiley, New York. 

We have restricted our discussion in this section to bistability in well-stirred, homogeneous systems. Multiple steady states may also occur in unstirred systems, where domains of the system in one steady state coexist with domains in the other steady state. In addition to the obvious application to nondhemical systems, chemical systems (in fact the iodate-arsenite system considered here) sometimes exhibit domains that are connected by propagating reaction-diffusion fronts. We will return to this system in our discussion of chemical waves, which will include a description of these fronts. 

Propagating reaction-diffusion fronts were first studied around the turn of the century as models for wave behavior in biological systems [1, 2]. However, as recently as 10 years ago they fell into the category of exotic phenomena , as only a few experimental examples were known and their mechanisms were poorly understood. Today, many autocatalytic reactions are known to support propagating fronts [3], and more complex wave behavior is of widespread interest for modeling excitable media in biological systems [4]. The theoretical treatment of reaction-diffusion fronts, while first addressed over a half-century ago [5-7] has also advanced in recent years. Many features are now well understood and, in addition, new theoretical challenges are apparent. 

Even in systems which are of more than one dimension, such as a tube, one can envision the propagation of a planar front with the properties described in the preceding sections. Such waves may be solutions to the governing reaction-diffusion or reaction-conduction equations, but if they are to be realized and observed in practice they must also be stable to the inevitable small fluctuations in local concentration and temperature. There is a long history of stability analysis for nonisothermal flame propagation [30-32], although the absence of exact analytical solutions to even the 1-D flame front equation makes these rather difficult. The same questions about the stability of isothermal reaction-diffusion fronts seem not to have been addressed until only recently [12]. 

Nettesheim S, von Oertzen A, Rotermund FI FI and ErtI G 1993 Reaction diffusion patterns in the catalytic CO-oxidation on Pt(110) front propagation and spiral waves J. Chem. Rhys. 98 9977-85... 

This reaction can oscillate in a well-mixed system. In a quiescent system, diffusion-limited spatial patterns can develop, but these violate the assumption of perfect mixing that is made in this chapter. A well-known chemical oscillator that also develops complex spatial patterns is the Belousov-Zhabotinsky or BZ reaction. Flame fronts and detonations are other batch reactions that violate the assumption of perfect mixing. Their analysis requires treatment of mass or thermal diffusion or the propagation of shock waves. Such reactions are briefly touched upon in Chapter 11 but, by and large, are beyond the scope of this book. 

Again the scenario we envisage is similar to that shown qualitatively in Fig. 11.7 we expect our best chance of such behaviour if the decay rate is small, i.e. k2 1. The reaction wave has a leading front moving with a steady velocity c1, through which most of the conversion of A to B occurs. After this front, the dimensionless concentration of A is almost zero and that of B is almost unity. At some distance, the first front is followed by a recovery wave, possibly more diffuse, in which A is completely removed and the autocatalyst also decays. The velocity of the recovery wave is c2. If ct exceeds c2, the first front will move away from the second, so the pulse will increase in width if c, = c2, the pulse will move with a constant shape if, however, c2 exceeds c, we can expect the second wave to catch the first, in which case propagation may fail. 

This type of equation is also encountered in other areas, such as nonlinear waves, nucleation theory, and phase field models of phase transitions, where it is known as the damped nonlinear Klein-Gordon equation, see for example [165, 355, 366]. In the (singular) limit r 0, (2.15) goes to the reaction-diffusion equation (2.3). Front propagation in HRDEs has been studied analytically and numerically in [149, 150, 152, 151, 374]. The use of HRDEs in applications is problematic. Such equations are obtained indeed very much in an ad hoc manner for reacting and dispersing particle systems, and they can be derived neither from phenomenological thermodynamic equations nor from more microscopic equations, see below. 

Reaction-diffusion systems are expected to show spatio-temporal chaos in various circumstances. A few specific cases will be discussed. They include the turbuhzation of uniform oscillations, of propagating wave fronts and of rotating spiral waves. 

Fig. 7. Chemical front propagation in the period-1 domain ( 2 = 1.4). (Top panel) Deterministic reaction-diffusion equation simulation of a circulating trigger wave. The diffusion coefficient in reduced units is AtD/ Ax) = 1 /8. The times reported in the figure are in units of 10 At with At = 1.783 x 10 . The concentration of species X is plotted as a function of space and time. (Middle panel) Automaton simulation for the same parameter values as in the top panel. (Bottom panel) Automaton simulation with D = /2.

Diffusion in solids does not ensure the experimentally observed velocity of combustion wave propagation in the systems which are traditionally considered as gasless and burned in the mode of solid flames (gasless solid-state combustion). The phenomenology of indirect interactions, the thermochemistp and dynamics of the gas-phase carriers formation, as well as their participation in the reactants transport are studied in the systems Mo-B and Ta-C. The distributions of the main species in the gas phase of the combustion wave are measured in situ with the use of a dynamic mass-spectrometry (DMS) technique which allows for high temporal and spatial resolution. The detailed chemical pathways of the processes were established. It was shown that the actual mechanism of combustion in the systems under study is neither solid state nor gasless and the reactions are fiilly accomplished in a narrow front. 

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