Hierarchy of length scales

Figure 12.4. Hierarchy of length scales of structure and time scales of motion in polymers. Tg denotes the glass transition temperature. After Uhlherr and Theodorou (1998) (courtesy Elsevier...

In conclusion, this chapter has attempted to bring together the measurement capabilities that can be united under the heading of MR techniques . Each of these can be a research field in their own right, but from the perspective of the chemical engineer it is the combination of these methods—particularly in the hierarchy of length-scales they probe—that makes MR methods so powerful. 

If the one-equation model of diffusion and reaction in a micropore-macropore system is not valid, one needs to proceed dovm the hierarchy of length scales to develop an analysis of the transport process in both the macropore region and the micropore region. This leads to yet another averaging volume that is illustrated as level III in Figure 1.4. Analysis at this level leads to a micropore effectiveness factor that is discussed by Carberry (1976, Sec. 9.2) and by Froment and Bischoff (1979, Sec. 3.9). 

The development of the model for the wave phenomena is based on a hierarchy of length scales [61 On the microscale this is the bubble radius R, on the mezoscale it is the cell radius A, and on the macroscale we deal with a characteristic wave length L. We make the assumption that the bubble distribution is monodisperse, that bubbles keep their spherical shape and do not fragment. The void fraction (p is then related to the bubble number density n and the radius R by... 

The ultimate goal is a reversible bottom-up, top-down approach, based on first principles QM, to characterize properties of materials and processes at a hierarchy of length and timescales. This will improve our ability to design, analyze, and interpret experimental results, perform model-based prediction of phenomena, and to control precisely the multi-scale nature of material systems for multiple applications. Such an approach is now enabling us to study problems once thought to be intractable, including reactive turbulent flows, composite material instabilities,... 

As for all disordered complex macromolecular materials, polymers can be characterized by a hierarchy of different length and time scales, and these scales span an extremely wide range, as outlined in Fig. 1.1 [15]. The diffusion... 

Similarly to catalysis, the properties of these composite materials are also determined by a hierarchy of structures on very different length/time scales. Therefore, linking mesoscale molecular models and continuum descriptions is relevant for their understanding and optimization. Together with advanced synthesis methods and functional testing, it is thus necessary also to develop new improved computational methods to provide an understanding of materials properties and to assist in the development of new functional materials. 

In fact, due to an inevitable nonuniformity of the distribution of conductive spots over the membrane surface, a whole hierarchy of circulation on different length scales sets in the diffusion layer. Namely, this complex multiscale convection is expected to cause the mixing of the entire diffusion layer and the resulting overlimiting CP behaviour of the C-membranes. 

A hierarchy of models can often be derived from a more detailed model under certain assumptions. This approach was discussed above in the case of deterministic, continuum models (see Fig. 3a). Such hierarchical models can be valuable in multiscale modeling. Let us just mention two cases. First, one could use different models from a hierarchy of models for different situations or length scales. This approach plays a key role in hybrid multiscale simulation discussed extensively below. Second, one could easily apply systems tasks to a simpler model to obtain an approximate solution that is then refined by employing a more sophisticated, accurate, and expensive model from the hierarchy. 

---