Coherent transport diffusivities

Up to now, no direct measurements of diffusion coefficients have been reported for any system that show the low-temperature upturn just described, and it may well be that for most systems involving hydrogen such effects would occur only at ultra-low temperatures and minuscule diffusion rates. Also, the impurities and imperfections always present in real materials might well trap nearly all the diffusant atoms at the low temperatures at which coherent transport might be expected in ideal material. However, a recent measurement by Kiefl et al. (1989) of the (electronic) spin relaxation rate of muonium in potassium chloride over a range of temperatures gives spectacular support to the concept of coherent tunneling at low temperatures. (See Fig. 6 of Chapter 15 and the associated discussion.)... 

In separation or catalytic applications, it is the transport diffusivity, Dt, which matters (this quantity is also named Fickian or chemical diffusivity). Transport diffusivities are traditionally obtained under non-equilibrium conditions [2], but they can be measured at equilibrium by coherent QENS [5]. Coherent neutron scattering is in principle more comphcated than incoherent scattering, but under certain conditions transport diffusivities can be extracted from the neutron data. 

In short, incoherent scattering allows one to determine the self-diffusivity, Ds, whereas coherent scattering gives access to the transport diffusivity, Du from experiments performed at equihbrium. When the scattering is both incoherent and coherent, then both diffusivities can in principle be determined simultaneously. 

Coherent QENS measurements and MD simulations have been performed for N2 and CO2 in silicalite [30,31]. It has been found that the self-diffusivities of the two gases decrease with increasing occupancy, while the transport diffusivities increase. For a comparison with other systems, it is appropriate to remove the influence of the thermodynamic correction factor and to discuss the collective mobility in terms of the corrected diffusivity (also called Maxwell-Stephan diffusivity). Dq(c) is directly obtained from the Simula-... 

The characteristic feature of solid—solid reactions which controls, to some extent, the methods which can be applied to the investigation of their kinetics, is that the continuation of product formation requires the transportation of one or both reactants to a zone of interaction, perhaps through a coherent barrier layer of the product phase or as a monomolec-ular layer across surfaces. Since diffusion at phase boundaries may occur at temperatures appreciably below those required for bulk diffusion, the initial step in product formation may be rapidly completed on the attainment of reaction temperature. In such systems, there is no initial delay during nucleation and the initial processes, perhaps involving monomolec-ular films, are not readily identified. The subsequent growth of the product phase, the main reaction, is thereafter controlled by the diffusion of one or more species through the barrier layer. Microscopic observation is of little value where the phases present cannot be unambiguously identified and X-ray diffraction techniques are more fruitful. More recently, the considerable potential of electron microprobe analyses has been developed and exploited. 

The exponent turned out to be x 1. This finding demonstrates that coherent flow determines transport in the mechanical dispersion regime and that diffusion is negligible under such conditions. For a discussion also see Ref. [43]. 

Out of this concept grew the cardinal idea of carrier mediated transport. Necessary for this was the development of a more coherent theoretical analysis built upon the general notion of facilitated diffusion. The major insight here came from Widdas who proposed in 1952 that carrier mediated transport would explain earlier data such as the transport of glucose across the sheep placenta, as well as his own observations on glucose entry into the erythrocyte. There were three assumptions made in developing this quantitative hypothesis ... 

The ratio Vo/B determines the transition from coherent diffusive propagation of wavefunctions (delocalized states) to the trapping of wavefunctions in random potential fluctuations (localized states). If I > Vo, then the electronic states are extended with large mean free path. By tuning the ratio Vq/B, it is possible to have a continuous transition from extended to localized states in 3D systems, with a critical value for Vq/B. Above this critical value, wave-functions fall off exponentially from site to site and the delocalized states cannot exist any more in the system. The states in band tails are the first to get localized, since these rapidly lose the ability for resonant tunnel transport as the randomness of the disorder potential increases. If Vq/B is just below the critical value, then delocalized states at the band center and localized states in the band tails could coexist. 

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