Oscillatory kinetics and nonlinear dynamics

Reactions at Solid Surfaces. By Gerhard Ertl Copyright 2009 John Wiley Sons, Inc. 

Oscillatory kinetics with a surface reaction had been observed as early as in 1828 by Fechner [4] with an electrochemical system. As an example for these t) es of reactions. Fig. 7.1 shows the variation of the potential at a Pt electrode with time for the electrochemical oxidation of H2 in the presence of copper ions [5]. While the potential at low-current density j is constant (a), at higher j kinetic oscillations occur because of periodic poisoning and activation transitions of the electrode by underpotential deposition and dissolution of a passivating Cu overlayer. With further increase of , at first period doubling and then transition to an irregular situation (chaos) take place. 

The rich variety of this type of temporal self-organization was mainly explored in detail with the famous Belousov-Zhabotinsky reaction [6,7], but with heterogeneously catalyzed reactions the oscillatory kinetics were first reported only aroimd 1970 for the oxidation of CO on Pt catalysts [8,9]. Since then oscillatory kinetics have been found with more than a dozen catalytic reactions, and this field has also been extensively reviewed [10,17]. 

Chemical oscillators are described on the basis of nonlinear dynamics, in that the underlying kinetic equations under steady-state conditions are nonlinear. If one assumes that the spatial distribution of the reaction species is uniform, then these variables will only depend on time, and mathematical description in the mean field approximation for the concentration variables c,- is achieved by a set of coupled (nonlinear) ordinary differential equations (ODEs). This will be the approach applied in this chapter. 

FIGURE7.1. Time series of the potential ofaPt electrode during electrochemical oxidation of H2 in the presence of copper ions at different current densities [5]. 

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Nonlinear Dynamics Oscillatory Kinetics and Spatio-Temporal Pattern Formation... 

The occurrence of kinetic instabilities as well as oscillatory and even chaotic temporal behavior of a catalytic reaction under steady-state flow conditions can be traced back to the nonlinear character of the differential equations describing the kinetics coupled to transport processes (diffusion and heat conductance). Studies with single crystal surfaces revealed the formation of a large wealth of concentration patterns of the adsorbates on mesoscopic (say pm) length scales which can be studied experimentally by suitable tools and theoretically within the framework of nonlinear dynamics. [31]... 

Chapter 3 is an overview of chemical and biological nonlinear dynamics. The kinetics of several types of reactions -first order, binary, catalytic, oscillatory, etc - and of ecological interactions -predation, competition, birth and death, etc - is described, nearly always within the framework of differential equations. The aim of this Chapter is to show that, despite the great variety of mechanisms and processes occurring, a few mathematical structures appear recurrently, and archetypical simplified models can be analyzed to understand whole classes of chemical or biological phenomena. The presence of very different timescales and the associated methodology of adiabatic elimination is instrumental in recognizing that. 

With their combination of complex kinetics and thermal, convective, and viscosity effects, polymerizing systems would seem to be fertile ground for generating oscillatory behavior. Teymour and Ray reported both laboratory-scale Continuous Stirred-Tank Reactor (CSTR) experiments and modeling studies on vinyl acetate polymerization [63-66]. The period of oscillation was long, about 200min, which is typical for polymerization in a CSTR (Figure 2.8). Papavasiliou and Teymour [67] reviewed nonlinear dynamics in CSTR polymerizations. 

The study of the response of nonlinear systems to external periodic perturbations leads to interesting information.Cool-flame, 9 oscillations occur in a number of combustion reactions, and we discuss an experimental study of the effect of external periodic perturbations on such systems. The application of perturbations to a chemical reaction can reveal important information about the stability, kinetics, and dynamics of the reaction. This technique is well known in the field of relaxation kinetics, in which perturbations are applied to a chemical system at equilibrium. In our work, periodic perturbations are first applied to the input rates of acetaldehyde and oxygen, one at a time, in the combustion of acetaldehyde in a CSTR. We measure periodic responses in five entrainment bands as we vary the frequency and amplitude of the external periodic perturbation. Outside of entrainment bands we find quasi-periodic responses. Next-phase rnapslO, of the experimental results are constructed in real time and used in the observation and interpretation of entrainment and quasi-periodic behavior. Within the fundamental entrainment band, we measure critical slowing down and enhancement of the response amplitude. As the bath temperature is increased, so that the oscillatory system approaches a Hopf bifurcation, we observe an increase in the amplitude enhancement. The predictions of a five-variable thermokinetic model agree well with the experimental results. 

Chemical reactions in the atmosphere form a large system of interacting chemical species described by nonlinear kinetics. It has been shown that certain components of this system can also exhibit oscillatory dynamics in some range of the parameters (see e.g. Poppe and Lustfeld (1996)). When the typical period of these oscillations... 

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