Oscillating reactions computer models

The Runge-Kutta algorithm cannot handle so-called stiff problems. Computation times are astronomical and thus the algorithm is useless, for that class of ordinary differential equations, specialised stiff solvers have been developed. In our context, a system of ODEs sometimes becomes stiff if it comprises very fast and also very slow steps and/or very high and very low concentrations. As a typical example we model an oscillating reaction in The Belousov-Zhabotinsky (BZ) Reaction (p.95). 

Quite different site densities are obtained if these assumptions are changed. Perez et al.13 have calculated the surface site statistics using a computer model which can simulate incomplete layers by removing atoms from complete shells. The atoms removed are those which have the smallest number of first and second nearest neighbours. Many more types of site are considered in the models used by Perez et al. However, one of the most interesting results of their calculations is to demonstrate that for all sites, apart from B2 sites, there are very pronounced oscillations in number as the particle size is increased. Figure 2 shows the variation in the number of B2, B3, and B4 sites, and Figure 3 shows the ratio of B3/B4 sites as a function of particle size. Any reaction which is controlled by this ratio will show activity maxima for particle diameters of 0.8 and 2.0 nm. On the other hand B2 and B2 sites are the ones most likely to catalyse structure insensitive reactions. 

A great deal of more or less detailed computer modelling has been done to predict operational features of chemical lasers since the first studies of this type by Comeil et al. 144>, Cohen et al. m> and Airey It is beyond the scope of this review to account for all the computational approaches that have been made. One paper of this kind was reviewed in Section 7 in connection with power predictions for an H2/F2 laser oscillator. Here the comprehensive work of Igoshin and Oraevskii 109> on the kinetic processes in an HC1 laser may serve as a reference to show the relevant features. The analysis proceeds from the simultaneous solution of chemical kinetics, vibrational relaxation, and radiational processes. The chain reaction model used here is the following... 

Table X compiles the computed gas-phase excitation energies, oscillator strengths, and dipole moments for the valence singlet excited states of the imidazole molecule. Previous MRCI results by Machado and Davidson [137] are also included. Table XI lists the results obtained in the reaction field model. Three valence and two valence n- T7 ...

Chemical reaction network is a typical example of complexity, where the reactants can interact in a variety of ways depending on the nature of interaction (chemical as well as non-chemical). Oscillatory reactions involve a number of steps, including positive and negative feedbacks. The complexity leads to periodic as well as aperiodic oscillations (multi-periodic, bursting/intermittency sequential oscillations separated by a time pause, relaxation and chaotic oscillations). The mechanism is usually determined by non-linear kinetics and computer modelling. Once the reaction mechanism has been postulated, the non-linear time-dependent kinetic equation can be formulated in terms of concentrations of different reactants, which would yield a multi-variable equation. Delay differential equations are sometimes used to characterize oscillatory behaviour as in economics (Chapter 14). 

Analytical and computational methods have proven to be useful mathematical tools for the elucidation of mechanisms of oscillating chemical reactions [1,21. The ordinary or partial differential equations which describe the oscillating system are parameterized by rate constants, initial conditions, boundary conditions, etc. The successful modeling of an oscillating system depends both on 1) an appropriate choice of the form of the differential equations and 2) an appropriate choice of values for each of the parameters. The usual mathematical methods for studying the properties of models of oscillating reactions... 

The pH-dependency of the reaction rate has been proposed as key the feedback mechanism to pH oscillations obtained in a computational model involving... 

Although Eqs. (4-1) and (4-2) have identical expressions as that of the classical rate constant, there is no variational upper bound in the QTST rate constant because the quantum transmission coefficient Yq may be either greater than or less than one. There is no practical procedure to compute the quantum transmission coefficient Yq- For a model reaction with a parabolic barrier along the reaction coordinate coupled to a bath of harmonic oscillators, the quantum transmission... 

Even for purely adiabatic reactions, the inadequacies of classical MD simulations are well known. The inability to keep zero-point energy in all of the oscillators of a molecule leads to unphysical behavior of classical trajectories after more than about a picosecond of their time evolution." It also means that some important physical organic phenomena, such as isotope effects, which are easily explained in a TST model, cannot be reproduced with classical molecular dynamics. So it is clear that there is much room for improvement of both the computational and experimental methods currently employed by those of us interested in reaction dynamics of organic molecules. Perhaps some of the readers of this book will be provide some of the solutions to these problems. 

Compute the fall-off curve using QRRK theory. For this calculation, assume a collision diameter of 4.86 A. Assume that the average energy transfer per N2-C-C5H5 collision is -0.69 kcal/mol (needed to calculate the parameter /5 used in the model). Take the number of oscillators to be, v = actual, with the frequency calculated above. Assume the reaction barrier to be E0, given above. 

Fig. 5. Bursts of oscillation observed in a computer integration of the open BZ reaction model equations (2), for k5 = 5.0 and t = 0.926 hr.

A macrolevel cell can be exemplified by the cellular model [231] and [232] when the cell is incorporated into a system of the strongly bonded Pt crystallites applied to a zeolite. The model allows to describe the complex oscillations of the CO oxidation rate. A heavy dependence of the reaction rates to be computed on the way the coupling rules for the neighboring cells are selected is shown. By varying these rules, it is possible to simulate the various experimental conditions. 

In order to model the oscillatory waveform and to predict the p-T locus for the (Hopf) bifurcation from oscillatory ignition to steady flame accurately, it is in fact necessary to include more reaction steps. Johnson et al. [45] examined the 35 reaction Baldwin-Walker scheme and obtained a number of reduced mechanisms from this in order to identify a minimal model capable of semi-quantitative p-T limit prediction and also of producing the complex, mixed-mode waveforms observed experimentally. The minimal scheme depends on the rate coefficient data used, with an updated set beyond that used by Chinnick et al. allowing reduction to a 10-step scheme. It is of particular interest, however, that not even the 35 reaction mechanism can predict complex oscillations unless the non-isothermal character of the reaction is included explicitly. (In computer integrations it is easy to examine the isothermal system by setting the reaction enthalpies equal to zero this allows us, in effect, to examine the behaviour supported by the chemical feedback processes in this system in isolation... 

Working with the model of B-Z reaction, Sakanoue and Endo (1982) showed by computer simulation the coexistence of a stable and an unstable limit cycle. The existence of an unstable oscillating object between two stable objects had been... 

---