Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

2.1. Polymer dynamics in fluids 35
Figure 2.3: Single DNA molecule (40µm) relaxing to the coiled state. In this
experiment a latex bead (1µm) is tethered to an end of the molecule. The DNA,
coloured with a fluorescent dye, is stretched by a uniform flow and successively
let free to relax. Images are taken at 5s time intervals, from left to right [34].
where µ is the dynamic viscosity of the solvent. Experiments with DNA molecules
[34] confirm that relaxation can be safely considered linear, provided the elongation
is small compared to the maximum extension R ≪ Rmax (see fig. 2.3).
Actually, the relaxation process can be much more complex than the simple
description of Zimm model. Several microscopic models of polymeric dynamics
have been developed to characterize this process. An introduction to this subject
can be found in the book by Doi and Edwards [35]. Nevertheless, the simple linear
relaxation is able to catch, at least qualitatively, the basic features of polymer
dynamics in fluids.
Finally, let us mention that the description of a polymer as a freely jointed
chain does not consider excluded-volume effects, which are experimentally known
to alter the scaling R0 ∼ M 1/2 of the gyration radius with the degree of polymerization.
The scaling exponent is, indeed, found to be ν = (0.55 ÷0.6). The origin
of these effects is the possibility that segments of the chain have, in the model, to
intersect or superpose to each other, which is not realistic. A better approximation
of the chain dynamics is based on a self-avoiding random walk. While the theory
of the standard random walk is quite simple, the treatment of the self-avoiding
problem is more difficult. A theoretical estimation of ν has been given by Flory
[36] for spatial dimensionality d < 4:
ν = 3
d + 2
(2.11)
which, for d = 3, gives the value ν = 3/5 = 0.6, not far from the experimental
measures; in two dimensions, Flory’s result is exact. When d ≥ 4 the value
ν = 1/2 of the ideal chain is recovered, meaning that the segments have "enough
room" to avoid each other.
35