Hydrodynamic scaling model

The scaling dependence of the diffusion coefficient on N and Cobs Iso poses a number of questions. While the original scaling predictions, based on reptation dynamics [26,38], oc N, have been verified by some measurements [91,98], significant discrepancies have been reported too [95,96]. Attempts to interpret existing data in terms of alternative models, e.g., by the so-called hydrodynamic scaling model [96], fail to describe observations [100,101]. 

Phillies, GDJ, The Hydrodynamic Scaling Model for Polymer Self-Diffusion, Journal of Physical Chemistry 93, 5029, 1989. 

Phillies, G., The hydrodynamic scaling model for polymer self-diffusion. Journal of Physical Chemistry, 1989, 93, 5029-5039. 

On the other hand, the existence of a concentration-independent crossover probe diameter d (R, R is consistent with polymer solution models based on an assumed dominance of hydt ynatnic interactions in nondilute solution. Models such as the hydrodynamic scaling model (6.7) identify the chain radius as the primary solution length scale at all concentrations at which the model applies. With this identification, a crossover from small-probe to large-probe behavior, perhaps correlated with differential ability to interact with internal chain modes, would at all concentrations occur over the same range of d/R, or d/R, precisely as observed experimentally. 

G. D. J. Phillies. Derivation of the universal scaling equation of the hydrodynamic scaling model via renormalization group analysis. Macromolecules, 31 (1998), 2317-2327. 

Figure 8.36 Scaling pre factora as a function of M using results from Refs. (O) (4) for polystyrene in CCI4, ( ) (16), ( ) (25,24), (A) (9), (A) (21,23) with linear chains, (+) (21,22) with/ = 3, (X) (21) with / = 8, (El) (23) with /=18, (1), ( ) (2), (V) (4) for polymers in C6D6, ( ) (10), (0) (27), ( ) (26), ( ) (22) for linear polybutadiene, ( ) (22) for / = 3 polybutadiene, (>) (2) for PEO in water, and (<) (3) for xanthan in water. Dashed line indicates best-fit line with a jj O.98 Solid line is the no-free-parameter prediction of a from the hydrodynamic scaling model, Chapter 17. Other details as in Figure 8.34.

Here the scaling exponent v and scaling prefactor a are independent of polymer concentration, but may depend on polymer molecular weight. Phillies hydrodynamic scaling model predicts not only the molecular weight dependences of v and a but also approximate numerical values for both parameters(14). 

The five primary steps of the derivation are (1) extension of the renormalization group derivation of the hydrodynamic scaling model for Ds to treat the zero-shear viscosity t], (2) description of the experimental phenomenology of r] c),... 

Hydrodynamic scaling model for self-diffusion and viscosity As shown in Chapter 8 on polymer self-diffusion, the concentration dependence of the polymer self-diffusion coefficient Ds is uniformly given by... 

Afterword hydrodynamic scaling model for polymer dynamics... 

This very short chapter sketches a theoretical scheme - the hydrodynamic scaling model - that is consistent with the results in the previous chapter, and that predicts aspects of the observed behavior of polymers in nondilute solution. The model is incomplete it does not predict everything. However, where it has been applied, its predictions agree with experiment. Here the model and its developments as of date of writing are described qualitatively, the reader being referred to the literature for extended calculations. 

The hydrodynamic scaling model is an extension of the Kirkwood-Riseman model for polymer dynamics(l). The original model considered a single polymer molecule. It effectively treats a polymer coil as a bag of beads. For their collective coordinates, the beads have three center-of-mass translations, three rotations around the center of mass, and unspecified other coordinates. The use of rotation coordinates causes the Kirkwood-Riseman model to differ from the Rouse and Zimm models(2,3). The other collective coordinates of the Kirkwood-Riseman model are lumped as internal coordinates whose fluctuations are in first approximation ignored. The beads are linked end-to-end, the links serving to estabhsh and maintain the coil s bead density and radius of gyration. However, the spring constant of the finks only affects the time evolution of the internal coordinates it has no effect on translation or rotation of the coil as a whole. 

And that is the current state of the hydrodynamic scaling model. There are clearly very large gaps, entire categories of phenomena that have not been treated, so certainly there is no obligation to believe more than that the model happens to be correct in a few particular cases. However, in the cases that have been treated, the model works rather well. 

With respect to the previous chapter, the hydrodynamic scaling model is consistent with many of its conclusions. Hydrodynamic scaling correctly predicts... 

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