Autowave processes

In this book we summarize the state of the art in the study of peculiarities of chemical processes in dense condensed media its aim is to present the unique formalism for a description of self-organization phenomena in spatially extended systems whose structure elements are coupled via both matter diffusion and nonlocal interactions (chemical reactions and/or Coulomb and elastic forces). It will be shown that these systems could be described in terms of nonlinear partial differential equations and therefore are complex enough for the manifestation of wave processes. Their spatial and temporal characteristics could either depend on the initial conditions or be independent on the initial as well as the boundary conditions (the so-called autowave processes). 

Of special interest is the so-called Belousov-Zhabotinsky class of similar reactions [4, 6-12], This system can serve as an extremely successful example of self-organisation proper mixing of several liquids in a given proportion and at certain temperature demonstrates practically all kinds of the autowave processes just mentioned. 

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. 

Therefore, often main attention in studying chemical oscillations is paid to their formal description on the macroscopic level rather than to an attempt to understand in detail the micromechanism of oscillations. It often results in necessity to make a choice between several alternative models suggested for a particular chemical system. It is difficult to restrict ourselves in theory to a definite universal basic model since it can turn out to be either too complicated for studying a particular kind of the autowave processes or, on the other hand, of a limited use due to its inability to reproduce all types of auto-wave processes. 

The extended Brusselator [2, 5], Oregonator [5, 10] and other similar systems [4, 7] demonstrate other autowave processes whose distinctive spatial and temporal properties are independent on initial concentrations, boundary conditions and often even on geometrical size of a system. As it was noted by Zhabotinsky [4], Vasiliev, Romanovsky and Yakhno [5], a number of well-documented results obtained in the theory of autowave processes is much less than a number of problems to be solved. In fact, mathematical methods for analytical solution of the autowave equations and for analysis of their stability are practically absent so far. 

As it follows from the above-said, nowadays any study of the autowave processes in chemical systems could be done on the level of the basic models only. As a rule, they do not reproduce real systems, like the Belousov-Zhabotinsky reaction in an implicit way but their solutions allow to study experimentally observed general kinetic phenomena. A choice of models is defined practically uniquely by the mathematical formalism of standard chemical kinetics (Section 2.1), generally accepted and based on the law of mass action, i.e., reaction rates are proportional just to products of reactant concentrations. 

Staying within a class of mono- and bimolecular reactions, we thus can apply to them safely the technique of many-point densities developed in Chapter 5. To establish a new criterion insuring the self-organisation, we consider below the autowave processes (if any) occurring in the simplest systems -the Lotka and Lotka-Volterra models [22-24] (Section 2.1.1). It should be reminded only that standard chemical kinetics denies their ability to selforganisation either due to the absence of undamped oscillations (the Lotka model) or since these oscillations are unstable (the Lotka-Volterra model). 

Since the many-point density formalism in its practical applications assumes macroscopically homogeneous system, we will restrict ourselves to a particular class of microscopically self-organized autowave processes. Without investigating in Chapter 8 all possible kinds of autowave processes, we are aimed to answer a principal question - whether these two models under question could be attributed to the basic models useful for the study of autowave processes. 

To treat the stochastic Lotka and Lotka-Volterra models, we have now to extend the formalism presented in Section 2.2.2, where collective variables-numbers of particles iVA and Vg were used to describe reactions. The point is that this approach neglects local density fluctuations in small element volumes. To incorporate both these fluctuations and their correlations due to diffusive conjunction, we are in position now to reformulate these models in terms of the diffusion-controlled processes - in contrast to the rather primitive birth-death formalism used in Section 2.2.2. It permits also to demonstrate in the non-trivial way a role of diffusion in the autowave processes. The main results of this Chapter are published in [21, 25]. 

Lotka-Volterra model reveals different kind of autowave processes with the non-monotonous behaviour of the correlation functions accompanied by their great spatial gradients and rapid change in time. Due to this fact the space increment Ar time increment At was variable to ensure that the relative change of any variable in the kinetic equations does not exceed a given small value. The difference schemes described above were absolutely stable and a choice of coordinate and time mesh was controlled by additional calculations with reduced mesh. 

Therefore, the study of the stochastic Lotka and Lotka-Volterra models carried out in Chapter 8, has demonstrated that the traditional estimates of the complexity of the system necessary for its self-organisation are not correct. Incorporation of the fluctuation effects and thus introduction of a continuous number of degrees of freedom prove their ability for self-organisation and thus put them into a class of the basic models for the study of the autowave processes. 

Now possibilities of the MC simulation allow to consider complex surface processes that include various stages with adsorption and desorption, surface reaction and diffusion, surface reconstruction, and new phase formation, etc. Such investigations become today as natural analysis of the experimental studying. The following papers [282-285] can be referred to as corresponding examples. Authors consider the application of the lattice models to the analysis of oscillatory and autowave processes in the reaction of carbon monoxide oxidation over platinum and palladium surfaces, the turbulent and stripes wave patterns caused by limited COads diffusion during CO oxidation over Pd(110) surface, catalytic processes over supported nanoparticles as well as crystallization during catalytic processes. 

VI. Theoretical Treatment of Autowave Processes in Solid-State Cryochemical Conversions (The Simplest Model)... 

VII. Autowave Processes under Conditions of Uniform Compression of the Sample... 

VIII. Autowave Processes in Film Samples of Reactants... 

IX. The Role of Mechanical Loading Dynamics and Fracture Mode in the Initiation of Autowave Processes... 

To study the wave-front structure (its temperature profile), the autowave process was registered thermographically in a series of experiments. The propagation velocity was measured by the time required for the wave to travel a distance (3-5 cm) between two thermocouples (Fig. 2). 

All the reaction systems considered, despite being greatly different chemically, have been found to have similar dynamic characteristics of the autowave processes occurring therein. Particularly, the linear velocities of the wave-front propagation are in the range of 1 -4 cm/s for all systems. All of them have a certain critical irradiation dose below which the excitation of an autowave process becomes impossible and the system responds to a local disturbance only with local conversion incapable of self-propagating (the situation discussed above and illustrated by Fig. 5). 

The next important step in the elucidation of the role of the thermal factor in the mechanism of the phenomena in question was studying the effect of the sample size on the characteristics of the autowave process. Can the self-sustained wave regime of conversion be made impossible by intensification of heat release at the expense of a decrease in the diameter of a cylindrical sample containing the reactant mixtures By analogy with combustion physics, the question of a critical sample size has been raised. 

VI. THEORETICAL TREATMENT OF AUTOWAVE PROCESSES IN SOLID-STATE CRYOCHEMICAL CONVERSIONS (THE SIMPLEST MODEL)... 

With the above assumptions, the equation describing the autowave process in the systems in question takes a form which looks similar to the fundamental combustion equation32 ... 

The slower autowave process is similar in some respects to classical combustion, despite the differences in their physical nature. The wave velocity shows the same dependence on thermal conductivity as in the case of flame propagation. Analogously to combustion, the reaction zone is near the maximum temperature Tm [it is near Tm that the critical gradient (dT/dx) switching on the reaction is realized], whereas the greater part of the front... 

To test the fit of the theoretical mechanism to the experimentally observed phenomena, it seemed principally important to try to realize experimentally the second mode of the autowave process. Its initiation was performed, in accordance with the theory, not by pulse heating but with the help of a heater whose temperature could be raised slowly. Under such conditions the slower wave could not be excited either in liquid helium or nitrogen, that is, there was only one mode of wave propagation. This was possibly connected with the fact that under conditions of intense heat release into liquid media, high (close to critical) transverse temperature gradients occurred in the samples, which might be a source of severe disturbances impeding the realization of the slower wave mode. 

The realization of the slower wave involved certain difficulties. Not in every experiment could it be initiated by the slow local heating. Frequently the faster wave was initiated as well. There were cases when the well-developed slower wave transformed into the faster one in the course of its propagatioon, so that the first thermocouple in its path registered a front profile similar to that of the solid curve in Figure 11, and the second, a profile similar to the dashed curve. The higher sensitivity of the slower propagation mode to various disturbances has already been noted in the theoretical treatment of the autowave process. 

VII. AUTOWAVE PROCESSES UNDER CONDITIONS OF UNIFORM COMPRESSION OF THE SAMPLE... 

VIII. AUTOWAVE PROCESSES IN FILM SAMPLES OF REACTANTS... 

Comparison of the experiments carried out in glass and metallic cuvettes did not show any appreciable effect of the substrate material on the characteristics of the autowave processes in solid-state conversions. The critical doses required for realization of the self-sustained regimes of conversion in films turned out to be noticeably greater than in massive samples. 

Analogous results have been obtained with thin films of equimolar SOz solution in isoprene. When the surface of such a film immersed in liquid nitrogen was damaged with a thin needle, a copolymerization wave occurred which spread over the sample. The preirradiation dose required to excite the autowave process in this system was 200 kGy, which is also well above the critical values for cylindrical samples. 

The key role of the density mechanism of the autowave process in film samples is also evidenced by the fact that in these systems (as well as in capillaries), under no initiation conditions could the slower wave be excited ... 

IX. THE ROLE OF MECHANICAL LOADING DYNAMICS AND FRACTURE MODE IN THE INTIATION OF AUTOWAVE PROCESSES... 

The series of experiments allows the conclusion that plastic deformation by itself (of which the systems under study are, in principle, capable) cannot initiate the development of the autowave process without the occurrence of brittle fracture. Therefore, creation of high but plastically deforming pressures is not a sufficient condition for the initiation of autowave processes. In order to... 

This conclusion was confirmed also by a different series of experiments elucidating the role of plastic deformations. The experiments were performed in the regime of practically uniform rather than local loading. To this end we employed the procedure developed to study the initiation and development of autowave processes under conditions of uniform compression (see Section VII). But whereas previously what were subjected to y radiation were massive samples under conditions of high static pressure (i.e., the stage of accumulation of active centers in the sample was preceded by plastic deformation during compression), in this work the experimental procedure was modified to fit the task formulated above. 

Therefore, neither the appreciable plastic deformation (both in the case of uniform compression and of local fracture) of the solid reaction systems studied nor their static state of high stress is a factor conditioning the critical phenomena and autowave processes observed during the chemical conversion in the systems. In other words, this series of experiments has provided another telling argument for the decisive role of brittle fracture in the mechanism of the phenomena considered. 

In Section VI we have touched upon the subject of the stability of steady-state wave propagation and pointed out the signs of a monotonic instability in the low-velocity autowave process. Here we shall consider qualitatively another... 

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