Spatially coupled

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 ... 

As outlined above, for the regular oscillations on Pt( 100) at high temperatures, spatial coupling will predominantly be mediated through the gas phase, which process is occurring so rapidly that no phase difference between different parts of the surface are to be expected. However, with this system wave phenomena were also observed by means of PEEM under conditions for which the temporal oscillations were less regular... 

A rapid lowering of the temperature will trigger the onset of oscillations simultaneously at various parts of the surface, which, however, will soon drift out of phase due to slight frequency differences if spatial coupling is insufficient. This was the case in our experiments, where the temperature... 

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. 

The Evolution Law of the Potential Drop Across the Double Layer and the Nature of Spatial Coupling in Electrochemical Systems... 

Fig. 4. Top Potential distribution at the working electrode (z — WF,j. Bottom Corresponding spatial coupling term (Eq. 17) for the 2-d cylinder geometry (cf. Fig. 2.2d) and three different values of fi Solid line, fi = 4 long-dashed line fl — 0.5 dashed line fl — 0.1.

If, on the other hand, the aspect ratio is very large, the spatial coupling converges toward a characteristic function with a maximum range... 

Fig. 5. Coupling functions H in Eq. (24) for ring working electrodes with circumference L for two different values of / . Solid line large aspect ratio jl resulting in nonlocal spatial coupling. Dashed line very small aspect ratio / resulting in local (diffusive) spatial coupling. (Courtesy of J. Christoph.)...

In the preceding section we discussed the spatial coupling in electrochemical systems, which possesses several peculiar features. Its interplay with the homogeneous dy-... 

Diffusion of the inhibitor parallel to the electrode is much smaller than the characteristic rate of spatial coupling of the activator (migration), and can therefore be ignored [26]. 

Concepts developed in nonlinear dynamics facilitated the classification of nonlinear phenomena in electrochemical systems and revealed the origins of the diversity of temporal and spatial patterns in electrochemical systems. The diversity results on the one hand from the fact that the electrode potential might act as a positive or as a negative feedback variable. On the other hand, it is a consequence of the different kinds of spatial coupling present in an electrochemical cell and of the unique property that the extent of the spatial couplings is influenced by parameters that can be easily manipulated in an experiment. 

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