Spatially inhomogeneous distribution

OS 92] [R 32] [P 72/The iodate-arsenous acid reachon proceeds to one of two stationary states in different parts of the capillary when an electrical field of specific strength is applied [68]. Accordingly, a spatially inhomogeneous distribution of reaction products is generated along the capillary. 

The microenvironment generates a spatially inhomogeneous distribution of reagents within the reaction volume. The local properties of a solubilization site can strongly affect the energetics of solvent-solute interactions and therefore the intrinsic reac-... 

In addition to changes due to chemical reactions we must account also for changes caused by transport processes. Since we are not interested in spatially inhomogeneous distributions (cf. the example of a stirred flow reactor), we introduce a general dilution flux term d) = (t) that removes material in proportion to the amount produced. 

As can be deduced from numerous experimental studies of superconducting cuprates, the hole excitations in the CUO2 layers are characterized by a spatially inhomogeneous distribution with a tendency to quasi one-dimensional structurization in the form of stripes [1,2]. This nanoscale segregation is found to be more pronounced in underdoped curates [3] with npcritical temperature Tcifip) on the threshold line enclosing the protectorate of percolation superconductivity. 

As the amplitude and scale of Q fluctuations increase, their correlation arises, which is in no way represented by an equation of the type 1.6-1. Moreover, correlated, extended and long-living fluctuations of Q lead to a spatially inhomogeneous distribution of Q f) in the range T < I c, which also is not taken into account in Equation 1.6 1. 

Illustration Short-time behavior in well mixed systems. Consider the initial evolution of the size distribution of an aggregation process for small deviations from monodisperse initial conditions. Assume, as well, that the system is well-mixed so that spatial inhomogeneities may be ignored. Of particular interest is the growth rate of the average cluster size and how the polydispersity scales with the average cluster size. 

An example Hollander et al. (2001a) nicely demonstrated how the strong inhomogeneities in stirred-tank flow result in unpredictable scale-up behaviour and that the impact of the detailed hydrodynamics and of the non-uniform spatial particle distribution on agglomeration rate is larger and more complex than usually assumed their study once more illustrated the risks of scale-up on the basis of keeping a single non-dimensional number. Sophisticated CFD, especially on the basis of LES, offers an attractive alternative indeed. 

We can think of the reactant concentration and some initial spatial distribution of the intermediate concentration and temperature profiles specifying a point on Fig. 10.9. If we choose a point above the neutral stability curve, then the first response of the system will be for spatial inhomogeneity to disappear. If the value of /r lies outside the range given by (10.79), then the system adjusts to a stable spatially uniform stationary state. If ji lies between H and n, we may find uniform oscillations. 

We shall call this a quasilinear Fokker-Planck equation, to indicate that it has the form (1.1) with constant B but nonlinear It is clear that this equation can only be correct if F(X) varies so slowly that it is practically constant over a distance in which the velocity is damped. On the other hand, the Rayleigh equation (4.6) involves only the velocity and cannot accommodate a spatial inhomogeneity. It is therefore necessary, if F does not vary sufficiently slowly for (7.1) to hold, to describe the particle by the joint probability distribution P(X, V, t). We construct the bivariate Fokker-Planck equation for it. 

The density of a liquid is hundreds of times greater than that of a gas. The spatial inhomogeneity in the distribution of reagents is more pronounced here, the remains for quite a long time, therefore, the diffusion processes in condensed media proceed more slowly. As a result, the kinetics of reactions in a substance in the condensed phase is different than it is in gases, and gives a different final radiation effect. 

However, the spatial inhomogeneity in the distribution of reagents is not the only reason why the radiolysis of substances in the condensed state is different from that of gases. As we have already mentioned in Section VIII, as we pass from the gaseous state to the condensed one, at the primary stage of radiolysis we already observe a redistribution of yields of primary active particles (resulting in the increase of the yield of ionized states). Also different are the subsequent relaxation processes, as well as the processes of decay of excited and ionized states.354 Another specific feature of processes in a condensed medium is the cage effect, which slows down the decay of a molecule into radicals.355 Finally, the formation of solvated electrons is also a characteristic feature of radiation-chemical processes in liquids.356... 

Fig. 33. Schematic representation of the potential distribution in the electrolyte as a result of an inhomogeneous distribution of the electrode potential, DL, and the effect of migration currents induced by the inhomogeneous potential distribution on the local temporal evolution of the potential (a) for the case that the length of the WE is much smaller than the distance between the WE and the CE and (b) for the case that the length of the WE is much larger than the distance between the WE and the CE. The length of the arrows in the representations below the potential distributions indicate how the contribution of the migration couplings to the temporal evolution of DL changes as a function of distance from the disturbance. (x, z spatial coordinates parallel and perpendicular to the WE, respectively. The electrode is assumed to be one-dimensional and the electrolyte two-dimensional.)...

The latter result shows that to interpret the mechanical properties of networks we do not need to take into account the spatial inhomogeneities of crosslink distribution in a sample, at least in the rubbery state. The analysis of epoxy networks performed under the framework of a tree-like model and experiments 7,10-26) brought the... 

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