Polymer Physics

According to thefluctuation-dissipation theoremin Brownian motions (Nyquist
1928 ), both the driving forces and the frictional forces on a particle are initiated by
the collisions of the surrounding particles with the thermal energykT. Accordingly,
we have the Einstein relationship (Einstein 1905 )

kT¼Dz (5.4)

On the other hand, the frictional forces are induced by the viscosity of fluids. The
Stokes lawreveals the relationship between the friction coefficientz and the
viscosity(Stokes 1851 ), as given by

z¼ 6 pR (5.5)

Therefore, one can obtain the Stokes-Einstein relationship as

D¼

kT

6 pR

(5.6)

One can measure the viscosity and the diffusion coefficient to determine the
so-calledhydrodynamic radiusas

Rh

kT

6 pD

(5.7)

This quantity of sizes reflects the effective volume-exclusion range of the moving
particle interacting with its surrounding particles. For a single polymer chain in a
good solvent, the theoretical hydrodynamic radiusRhtheocan be defined as

1

Rtheoh



1

n^2

<

X

i 6 ¼j

1

rij

> (5.8)

where<...>is an ensemble average, andnis the number of monomers in the
polymer (Des Cloizeaux and Jannink 1990 ). Such a theoretical definition makes the
hydrodynamic radius close to the radius of gyration of the polymer coil. However,
as we have introduced for polymer solutions in the previous chapter, the hydro-
dynamic radius of an anisotropic coil could be larger than its static radius of
gyration. Thus from the dynamics point of view, the actual critical overlap concen-
tration appears smaller than the theoretical prediction.
In dilute solutions of hard spheres, Einstein has found that the viscosity

¼sð 1 þ 2 : 5 cÞ (5.9)

wheresis the solvent viscosity, andcis the volume fraction of hard spheres
(Einstein 1911 ).

78 5 Scaling Analysis of Polymer Dynamics