Spatiotemporal coupling

The search for Turing patterns led to the introduction of several new types of chemical reactor for studying reaction-diffusion events in feedback systems. Coupled with huge advances in imaging and data analysis capabilities, it is now possible to make detailed quantitative measurements on complex spatiotemporal behaviour. A few of the reactor configurations of interest will be mentioned here. 

An altogether different behavior emerges for slightly larger values of 7 ( 40 -tSee also our discus.sion of spatiotemporal chaos in coupled-map lattices in section 8.2. 

The possibility for the formation of spatiotemporal patterns with heterogeneously catalyzed reaction systems was pointed out early (114-116). Here the following transport mechanisms may be responsible for spatial coupling ... 

Coupling between surface diffusion and reaction. If this mode is dominating [such as with the CO oxidation on Pt(100)], chemical waves propagating across the surface will give rise to spatiotemporal pattern formation. 

Spatiotemporal pattern formation at the electrode electrolyte interface is described by equations that belong in a wider sense to the class of reaction-diffusion (RD) systems. In this type of coupled partial differential equations, any sustained spatial structure comes about owing to the interplay of the homogeneous dynamics or reaction dynamics and spatial transport processes. Therefore, the evolution of each variable, such as the concentration of a reacting species, can be separated into two parts the reaction part , which depends only on the values of the other variables at one particular location, and another part accounting for transport processes that are induced by spatial variations in the variables. These latter processes constitute a spatial coupling among different locations. 

Table 1. Summary of typical spatiotemporal behaviors in (H)N-NDR and S-NDR systems under different experimental conditions corresponding to different types of spatial coupling, (n is the wave number of the resulting pattern.)...

Oscillations connected with adsorbate-induced surface restructuring were studied also in [29]. The model used was aimed at mimicking oscillations in NO reduction by H2 on a mesoscopic Pt particle containing two catalytically active (100) areas connected by an inactive (111) area that only adsorbed NO reversibly. NO diffusion on and between facets was much faster than other steps. The results obtained show that the coupling of the catalytically active sublattices may synchronize nearly harmonic oscillations observed on these sublattices and also may result in the appearance of aperiodic partially synchronized oscillations. The spatiotemporal patterns corresponding to these regimes are nontrivial. In particular, the model predicts that, due to phase separation, the reaction may be accompanied by the formation of narrow NO-covered zones on the (100) sublattices near the (lOO)-(lll) boundaries. These zones partly prevent NO supply from the (111) sublattice to the (100) sublattices. 

This method has recently been utilized by the Konstanz quantum optics group, who has prepared the signal photon in a well-defined, highly pure spatiotemporal mode which can be matched with, coupled into, or caused to interfere with, a classical laser mode. Based on this achievement, we have demonstrated homodyne tomography of the single-photon Fock state [Lvovsky 2001]... 

Theoretical studies of the properties of the individual components of nanocat-alytic systems (including metal nanoclusters, finite or extended supporting substrates, and molecular reactants and products), and of their assemblies (that is, a metal cluster anchored to the surface of a solid support material with molecular reactants adsorbed on either the cluster, the support surface, or both), employ an arsenal of diverse theoretical methodologies and techniques for a recent perspective article about computations in materials science and condensed matter studies [254], These theoretical tools include quantum mechanical electronic structure calculations coupled with structural optimizations (that is, determination of equilibrium, ground state nuclear configurations), searches for reaction pathways and microscopic reaction mechanisms, ab initio investigations of the dynamics of adsorption and reactive processes, statistical mechanical techniques (quantum, semiclassical, and classical) for determination of reaction rates, and evaluation of probabilities for reactive encounters between adsorbed reactants using kinetic equation for multiparticle adsorption, surface diffusion, and collisions between mobile adsorbed species, as well as explorations of spatiotemporal distributions of reactants and products. 

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