Concentration oscillations

The N20 decomposition, CO oxidation, and H2 oxidation reactions are known to exhibit concentration oscillations over noble metal catalysts. Flytzani-Stephanopoulos et al. (47) have observed oscillations for the oxidation of NH3 over Pt. The effects are dramatic and lead to large temperature cycles for the catalyst wire. Heat and mass transfer effects are important. 

Oscillations are familiar phenomena in mechanical systems and in electric circuits. Noyes and Field discussed the possibilities for concentration oscillations in closed systems and illustrated the principles by means of Oregonator model consists of the following five steps ... 

The mathematical treatment of pulsing illumination has been described [Briers et al., 1926], After a number of cycles, the radical concentration oscillates uniformly with a constant radical concentration [M-] j at the end of each light period of duration t and a constant radical concentration [M-]2 at the end of each dark period of duration t = rt. The two radical concentrations are related by... 

At steady state, by definition the total drug clearance, or loss, is equal to drug input and plasma concentration oscillates around an average figure. However two situations occur which can provide problems in dosing. 

Mukesh, D., Kenney, C. N., and Morton, W. (1983). Concentration oscillations of carbon monoxide, oxygen and 1-butene over a platinum supported catalyst. Chem. Eng. Sci., 38, 69-77. 

Lastly, non-elementary several-stage reactions are considered in Chapters 8 and 9. We start with the Lotka and Lotka-Volterra reactions as simple model systems. An existence of the undamped density oscillations is established here. The complementary reactions treated in Chapter 9 are catalytic surface oxidation of CO and NH3 formation. These reactions also reveal undamped concentration oscillations and kinetic phase transitions. Their adequate treatment need a generalization of the fluctuation-controlled theory for the discrete (lattice) systems in order to take correctly into account the geometry of both lattice and absorbed molecules. As another illustration of the formalism developed by the authors, the kinetics of reactions upon disorded surfaces is considered. 

Obviously, if as t —> oo the stationary solution dnj( )/d = 0 exists, indeed the asymptotic solution rij(oo) of (2.1.1) is one of the solutions n(- of the set (2.1.14). Here we have an example of a simple but very important case of a stable stationary solution. Other stationary points cannot be ascribed to the asymptotic solutions, i.e., n nj(oo), but they are also important for the qualitative treatment of the set of equations. Note that striving of the solutions for stationary values is not the only way of chemical system behaviour as —> oo another example is concentration oscillations [4, 7, 16]. Their appearance in a set (2.1.2) depends essentially on a nature of... 

When pK > 4(32 holds, the singular point remains stable, Reei, 2 < 0, but the roots (2.1.16) have imaginary parts Imei = Im ei. In this case the phase portrait reveals a stable focus - Fig. 2.2. This regime results in damped oscillations around the equilibrium point (2.1.24). The damping parameter pK/(3 is small, for large 3, in which case the concentration oscillation frequency is just ui = y/pK. ... 

This type of a pattern of singular points is called a centre - Fig. 2.3. A centre arises in a conservative system indeed, eliminating time from (2.1.28), (2.1.29), one arrives at an equation on the phase plane with separable variables which can be easily integrated. The relevant phase trajectories are closed the model describes the undamped concentration oscillations. Every trajectory has its own period T > 2-k/ujq defined by the initial conditions. It means that the Lotka-Volterra model is able to describe the continuous frequency spectrum oj < u>o, corresponding to the infinite number of periodical trajectories. Unlike the Lotka model (2.1.21), this model is not rough since... 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

Despite the fact that from a principal point of view a problem of concentration oscillations could be considered as solved [4], satisfactory theoretical descriptions of experimentally well-studied particular reactions are practically absent. Due to very complicated reaction mechanism (in order to describe the Belousov-Zhabotinsky reaction even in terms of standard chemical kinetics several tens of concentration equations for intermediate products should be written down and solved numerically [4, 9, 10]) these equations contain large number of ill-defined parameters - reaction rates [10]. 

A numerical solution of the basic equations demonstrated their ability to reproduce concentration oscillations. At the same time, for the systems possessing three and more intermediate products the standard method to prove existence of periodical solutions, using a phase portrait of a system (Section 2.1.1) fails. An additional reduction in a number of differential equations, e.g., using an idea that one of concentrations, say, [BrOj-], serves as a rapid variable and thus the relevant kinetic equation (8.1.5) could be solved as the stationary [10], cannot be always justified due to uncertainty in the kinetic coefficients hi. 

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). 

Statement 1. Provided K(t) — K — const, i.e., neglecting change in time of the correlation functions, equations (8.2.12) and (8.2.13) of the concentration dynamics describe undamped concentration oscillations with the frequencies u < = y/aj3, dependent on the initial conditions. The de-... 

Fig. 8.1. Phase portraits of the Lotka-Volterra model for d = 3 (a) Unstable focus (re = 0.9) (b) Stable focus (re = 0.2) (c) Concentration oscillations during the steady-state formation (re = 0.1) (d) Chaotic regime (re = 0.05). The values of the distinctive parameter are shown.

Therefore, oscillations of K (t) result in the transition of the concentration motion from one stable trajectory into another, having also another oscillation period. That is, the concentration dynamics in the Lotka-Volterra model acts as a noise. Since along with the particular time dependence K — K(t) related to the standing wave regime, it depends also effectively on the current concentrations (which introduces the damping into the concentration motion), the concentration passages from one trajectory onto another have the deterministic character. It results in the limited amplitudes of concentration oscillations. The phase portrait demonstrates existence of the distinctive range of the allowed periods of the concentration oscillations. 

For a given set of parameters the period of concentration oscillations (or its average for a periodic motion) exceeds greatly the period of the correlation motion. For the slow concentration motion not only the period of the standing wave oscillations but also their amplitudes and, consequently, the amplitude in the K (t) oscillations depend on the current concentrations Na(t) and Nb(t). In other words, the oscillations of the reaction rate are modulated by the concentration motion. Respectively, the influence of the time dependence K K(t) upon the concentration dynamics has irregular, aperiodic character. A noise component modulates the autowave component (the standing waves) but the latter, in its turn, due to back-coupling causes transition to new noise trajectories. What we get as a result is aperiodic motion (chaos). The mutual influence of the concentration and correlation motions and vice versa is illustrated in Fig. 8.2, where time developments of both the concentrations and reaction rates are plotted. 

The conclusion could be drawn from Fig. 8.2 that the peaks in K(t) produce a fine structure in the concentration curves. Despite the fact that these oscillations in K(t) have two orders of magnitude, the fine structure is not of a primary importance. In its turn, the concentration oscillations modulate oscillations of the reaction rate K(t). 

The behaviour of the correlation functions shown in Fig. 8.5 corresponds to the regime of unstable focus whose phase portrait was earlier plotted in Fig. 8.1. For a given choice of the parameter k = 0.9 the correlation dynamics has a stationary solution. Since a complete set of equations for this model has no stationary solution, the concentration oscillations with increasing amplitude arise in its turn, they create the passive standing waves in the correlation dynamics. These latter are characterized by the monotonous behaviour of the correlations functions of similar and dissimilar particles. Since both the amplitude and oscillation period of concentrations increase in time, the standing waves do not reveal a periodical motion. There are two kinds of particle distributions distinctive for these standing waves. Figure 8.5 at t = 295 demonstrates the structure at the maximal concentration... 

Section we show that presence of two such intermediate stages is more than enough for the self-organization manifestation. Lotka [22] was the first to demonstrate theoretically that the concentration oscillations could be in principle described in terms of a simplest kinetic scheme based on the law of mass action [4], Its scheme given by (2.1.21) is similar to that of the Lotka-Volterra model, equation (2.1.27). The only difference is the mechanism of creation of particles A unlike the reproduction by division, E + A - 2A, due to the autocatalysis, a simpler reproduction law E —> A with a constant birth rate of A s holds here. Note that analogous mechanism was studied by us above for the A + B — B and A + B — 0 reactions (Chapter 7). 

In a system with strong damping of the concentration motion the concentration oscillations are constrained they follow oscillations in the correlation motion. As compared to the Lotka-Volterra model, where the concentration motion defines essentially the autowave phenomena, in the Lotka model it is less important being the result of the correlation motion. This is why when plotting the results obtained, we focus our main attention on the correlation motion in particular, we discuss in detail oscillations in the reaction rate K(t). 

The amplitude of the oscillations K (t) is relatively small, the same is true for the amplitudes of the concentration oscillations. The concentration Na(t) oscillates around the value of 6/K where K is a mean value of K (t) whereas jVb(f) oscillates around p/(i respectively. 

Hegedus, L. L., Chang, C. C., McEwen, D. J. and Sloan, E. M., 1980, Response of catalyst surface concentrations to forced concentration oscillations in the gas phase the NO, CO, O2 system over a-alumina. Ind. Engng Chem. Fundam. 19, 367-373. 

I. Jobses, G. Egberts, K. Luyben, J.A. Roels, Fermentation kinetics of zymomonas mo-bilis at high ethanol concentrations oscillations in continuous cultures, Biotechnology and Bioengineering, 28, 868-877, 1986... 

Fig. 3.6 Vanishing oscillations and flux re-routing for increasing ketone concentration, (a) NADH concentration oscillates between two solid curves, the unstable steady state is denoted by the thin dashed curve, (b) L-Carbinol (solid) and D-carbinol (dashed) fluxes, (c) C3 carbon fluxes where time averages are shown in the oscillatory region.

Thus, high fullerene yield at 44 kHz may be explained by the occurrence of electron concentration oscillations in the plasma. That such electron concentration oscillations are highly affecting the fullerene synthesis is known. For details see [8],... 

In real systems, especially in heterogeneous catalytic and biological sys terns, the reactants are often arranged irregularly in space. Therefore, an arising instability may cause simultaneous diffusion of substances from one point to another inside the system to make the reactant concentration oscillations arranged in a certain manner in space during the occurrence of nonlinear chemical transformations. As a result, a new dissipative structure arises with a spatially nonuniform distribution of certain reac tants. This is a consequence of the interaction between the process of diffusion, which tends to create uniformity of the system composition, and local processes of the concentration variations in the course of nonlinear... 

Stationary points, mentioned above. An asymptotic solution of the oscillating type is connected with the concept of the limit cycle. Complicated chemical systems reveal also irregular or chaotic concentration oscillations [8],... 

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