Reaction-Diffusion Systems with Two Intermediates

We now consider reaction-diffusion systems with two intermediates and multiple stationary states, which may be nodes or foci. For a real eigenvalue that approach is monotonic for a complex eigenvalue with negative real part that approach is one of damped oscillations. In the absence of cross diffusion the deterministic rate equations in one dimension, 2, are 

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In a reaction-diffusion (R-D) system with two intermediates X and Y where X is an activator and Y is an inhibitor, criterion for Turing symmetry breaking instability is that... 

Then we address these same questions in Chap. 3 for multivariable systems, with two or more intermediates. Now our approach takes inherent fluctuations fully into account and we find a state function (analogous to AG) that satisfies the stated requirements. We also present a deterministic analysis of multivariable systems in Chap. 4 and compare the approach and the results with the fluctuational analysis. In Chap. 5 we turn to the study of reaction-diffusion systems and the issue of relative stability of multiple stationary states. The same issue is addressed in Chap. 6 on the basis of fluctuations, and in Chap. 7 we present experiments on relative stability. 

The Hanusse theorem [23] discussed in Section 2.1.1 was later generalized for the case of diffusion by Tyson and Light [32], Therefore, the mono- and bimolecular reactions with one or two intermediate products are expected to strive asymptotically, as t —> oo, for the stationary spatially-homogeneous solution Ci(r, oo) = nt(oo) corresponding to equations (2.1.2) for a system with the complete particle mixing. 

The problem of obtaining and suppressing intermediate phases is of particular interest when solid-phase reactions in powder mixtures are analyzed. Let us consider the initial stage of sintering of the simplest binary mixture when the particles of the initial components stiU preserve independence and one can speak of two connected surfaces with different total squares Sa, Sg, brought into contact via fast surface diffusion and diffusion through the gas phase. Eor simplicity, let us take a system of two almost mutually insoluble components, giving two... 

The investigation of the initial stages of reaction-diffusion in multilayers, carried out during recent years, by differential scanning calorimetry (DSC), proved that the stage of intermediate phase nucleation at solid-state reaction does take place. DSC experiments [6-8] have shown that the formation of a new phase in multilayers can involve two stages. For example, the curve illustrating the dependence of heat flux on time at formation of NbAls in multilayers Nb/Al (obtained by deposition) has two maxima. X-ray analysis and electron microscopy confirmed that both peaks correspond to the formation of the phase NbAls. Similar curves with two peaks are obtained for such systems as Co/Al, Ni/Al, Ti/Al, Ni/amorphous Si, and V/amorphous Si. 

It is therefore possible that the initial fate of ZnCH3 in the static system work is dimerization. Under the conditions used diffusion to the surface can compete successfully with reaction (2) so formation of an intermediate dimer could be either a homogeneous or a heterogeneous process. By elimination, the two hydrogen atoms required to convert the dimer to 2 Zn+2 CH4 must come from dimethyl zinc itself and must leave a product that does not undergo subsequent decomposition. It is possible that this occurs via a cyclic intermediate, with adsorbed dimethyl zinc leaving surface-adsorbed radicals which may undergo polymerization, viz. 

Two other crucial factors are mass transfer and heat transfer. In Chapter 3 we assumed that the reactions were homogeneous and well stirred, so that every substrate molecule had an equal chance of getting to the catalytic intermediates. Here the situation is different. When a molecule reaches the macroscopic catalyst particle, there is no guarantee that it will react further. In porous materials, the reactant must first diffuse into the pores. Once adsorbed, the molecule may need to travel on the surface, in order to reach the active site. The same holds for the exit of the product molecule, as well as for the transfer of heat to and from the reaction site. In many gas/solid systems, the product is hot as it leaves the catalyst, and carries the excess energy out with it. This energy must dissipate through the catalyst particles and the reactor wall. Uneven heat transfer can lead to hotspots, sintering, and runaway reactions. 

The molecular size pore system of zeolites in which the catalytic reactions occur. Therefore, zeolite catalysts can be considered as a succession of nano or molecular reactors (their channels, cages or channel intersections). The consequence is that the rate, selectivity and stability of all zeolite catalysed reactions are affected by the shape and size of their nanoreactors and of their apertures. This effect has two main origins spatial constraints on the diffusion of reactant/ product molecules or on the formation of intermediates or transition states (shape selective catalysis14,51), reactant confinement with a positive effect on the rate of the reactions, especially of the bimolecular ones.16 x ... 

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