Scaling concepts

P. G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY. 1979. 

De Gennes P G 1979 Scaling Concepts In Polymer Physics (Ithaca, NY Cornell University Press) p 113... 

De Gennes P-G (1979) Scaling concepts in polymer physics. Cornell Univ Press, Ithaca, NewYork... 

Power-law scattering features will be discussed in relation to mass-fractal scaling laws. Fractal scaling concepts used to interpret the power-law decay are well published in the literature. 

The realization of complete bench-scale micro reactor set-ups is certainly still in its infancy. Nevertheless, the first investigations and proposals point at different generic concepts. First, this stems from the choice of the constructing elements for such set-ups. Either microfluidic components can be exclusively employed (the so-caUed monolithic concept) or mixed with conventional components (the so-called hybrid or multi-scale concept). Secondly, differences concerning the task of a micro-reactor plant exist. The design can be tailor-made for a specific reaction or process (specialty plant) or be designated for various processing tasks (multi-purpose plant). 

The falling film micro reactor (Figure 5.1) transfers this well-known macro-scale concept to yield films of a few tens of micrometers thickness [1-3]. For this reason, the streams are guided through micro channels. To obtain a reasonable throughput, many micro channels are operated in parallel. 

Since the prediction of the solute diffusion coefficient in a swollen matrix is complex and no quantitative theory is yet possible, Lustig and Peppas [74] made use of the scaling concept, arriving at a functional dependence of the solute diffusion coefficient on structural characteristics of the network. The resulting scaling law thus avoids a detailed description of the polymer structure and yet provides a dependence on the parameters involved. The final form of the scaling law for description of the solute diffusion in gels is... 

Within the scaling concept the answer to the question, how physical properties alter with changes in scale, is of fundamental importance. For example, Fig. 37 shows two polymer chains which result from another altering of the scale (here, number of segments N) by a factor X = 2. Then the following transformation rules are valid for the monomer concentration c and the segment length /[5]... 

Miller, M. A. Reinhardt, W. P., Efficient free energy calculations by variationally optimized metric scaling concepts and applications to the volume dependence of cluster free energies and to solid-solid phase transitions, J. Chem. Phys. 2000,113, 7035-7046... 

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