Aggregation frequencies, modeling

The chief phenomenological instrument of the population balance model of an aggregation process is the aggregation frequency. It represents the probability per unit time of a pair of particles of specified states aggregating. Alternatively, it represents the fraction of particle pairs of specified states aggregating per unit time. This interpretation must, however, be modified for the aggregation frequency commonly used in population balance models in which the population is regarded as well-mixed and external coordinates do not appear explicitly in the population density. We will subsequently derive this modified frequency from the quantity that we have just defined. 

It is essential to consider only one of the above order for a given pair of particles. The explicit time dependence in (3.3.1) in the aggregation frequency is generally not a desirable feature in models and is eliminated in the remaining treatment. 

Most aggregation frequencies in population balance models are such that they have the property ... 

After a-helices, P-sheets are the most prominent secondary structure feature in globular proteins. However, the most widely used tool for secondary analysis, ultraviolet ECD, is so dominated by its sensitivity to the a-helix that, at best, it can only poorly characterize ( -sheet content. IR on the other hand has a good differential sensitivity to the p-sheet because of the large frequency shift ( 30 cm-1) from the helical band and because the extinction coefficient for model extended sheets is often higher than for other conformations. However, study of the p-sheet conformation with peptide models has long been hindered by their natural tendency to aggregate. Furthermore, no (or very few) peptide models of parallel sheets are available, to our knowledge. 

The molecule is often represented as a polarizable point dipole. A few attempts have been performed with finite size models, such as dielectric spheres [64], To the best of our knowledge, the first model that joined a quantum mechanical description of the molecule with a continuum description of the metal was that by Hilton and Oxtoby [72], They considered an hydrogen atom in front of a perfect conductor plate, and they calculated the static polarizability aeff to demonstrate that the effect of the image potential on aeff could not justify SERS enhancement. In recent years, PCM has been extended to systems composed of a molecule, a metal specimen and possibly a solvent or a matrix embedding the metal-molecule system in a molecularly shaped cavity [62,73-78], In particular, the molecule was treated at the Hartree-Fock, DFT or ZINDO level, while for the metal different models have been explored for SERS and luminescence calculations, metal aggregates composed of several spherical particles, characterized by the experimental frequency-dependent dielectric constant. For luminescence, the effects of the surface roughness and the nonlocal response of the metal (at the Lindhard level) for planar metal surfaces have been also explored. The calculation of static and dynamic electrostatic interactions between the molecule, the complex shaped metal body and the solvent or matrix was done by using a BEM coupled, in some versions of the model, with an IEF approach. 

A similar approach has also been developed by Susteric [108], who compares the behavior during low-amplitude deformation of rubbers, loaded with aggregated carbon black, with the visco-elastic behavior of macromolecules undergoing high-frequency deformation. The specific features of the breaking of carbon black aggregates defined by the deformation amplitude of loaded rubbers are described by the above author by a mathematical model developed for the description of the dynamic, visco-elastic behavior of polymer molecules. This approach revealed... 

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