Length scales phenomenological model

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

Classical surface and colloid chemistry generally treats systems experimentally in a statistical fashion, with phenomenological theories that are applicable only to building simplified microstructural models. In recent years scientists have learned not only to observe individual atoms or molecules but also to manipulate them with subangstrom precision. The characterization of surfaces and interfaces on nanoscopic and mesoscopic length scales is important both for a basic understanding of colloidal phenomena and for the creation and mastery of a multitude of industrial applications. 

As seen in Chapter 2 for turbulent flow, the length-scale information needed to describe a homogeneous scalar field is contained in the scalar energy spectrum E k, t), which we will look at in some detail in Section 3.2. However, in order to gain valuable intuition into the essential physics of scalar mixing, we will look first at the relevant length scales of a turbulent scalar field, and we develop a simple phenomenological model valid for fully developed, statistically stationary turbulent flow. Readers interested in the detailed structure of the scalar fields in turbulent flow should have a look at the remarkable experimental data reported in Dahm et al. (1991), Buch and Dahm (1996) and Buch and Dahm (1998). 

The total mixing time can be computed by inserting the mixing rate y(Jp,) into the following simple phenomenological model for the scalar length scale ... 

There are several ways that dispersion (and the corresponding length scale) can be brought at the phenomenological level into the conservative part of the model. The two most well known examples of such theories are gradient (or van der Waals) models with energy... 

Simulation can also be applied to longer length-scale phenomena. Examples include attempts to model the structural and mechanical properties of catalyst pellets, the mesopore structure of particle aggregates and phenomenological studies of crystallization. Here I mention just two examples, studies of crystallite morphology and quantitation of the effect of pore blockages on effective sorption capacities. 

Models have been developed to describe the SAXS from microemulsions, which contain polar, nonpolar, and amphiphilic constituents [70]. There are a few different ways that one can analyze the SAXS from these systems. These approaches are reviewed here. Qualitatively, the Teubner-Strey phenomenological model accounts for a bi-continuous repeat distance, d, as well as the average length scale of the... 

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