Chapter 5
Scaling Analysis of Polymer Dynamics
5.1 Simple Fluids
In this part, we start with the basic laws of molecular motions in simple fluids, to
learn the scaling analysis of polymer dynamics, followed with polymer deformation
and polymer flow.
The trajectories of Brownian motions of hard spherical molecules can be analogous
to random walks. As we have leant in Chap. 2 , the mean square end-to-end distance
of a random walk is proportional to the number of steps, i.e.<R^2 >~n. The three-
dimensional mean-square displacement of particles in Brownian motions is also
proportional to the motion timet,as
<½rðtÞrð 0 Þ^2 >¼ 6 Dt (5.1)whereDis thediffusion coefficient. The discovery of such a law in the Brownian
motion of the molecular particles is actually one of Einstein’s milestone
contributions in 1905 (Einstein 1905 ). Accordingly, thecharacteristic timeis
defined as the moving time of a particle through a distance of its own size, as
tR^2
D
(5.2)
In simple fluids, the external driving forces on a small particle are equilibrated
with the friction due to the collisions with its surrounding medium. Therefore, the
total frictional forcefis proportional to the activated constant velocityvof the
moving particle with respect to its surrounding medium, as
f¼zv (5.3)wherezis thefriction coefficient. This law of fluid dynamics is similar to the
Newton’s second law for the external force proportional to the acceleration.
W. Hu,Polymer Physics, DOI 10.1007/978-3-7091-0670-9_5,
#Springer-Verlag Wien 2013
77