Polymer Physics

We know the Einstein relationship

Dp¼

kT

zp

(5.27)

and consider the free-draining mode,

zp¼

zn

p

(5.28)

We substitute the three equations above into (5.25), and obtain

tpð

n

p

Þ^2 (5.29)

Whenp¼1, the characteristic time for each chain diffusing through its coil size
is often referred as theRouse relaxation time, i.e.

tRn^2 (5.30)

Therefore, the diffusion coefficient for the whole Rouse chain is

D

R^21

tR

n^1 (5.31)

which can also be derived directly from (5.27) and (5.28).
By definition, the mean-square displacement of monomers in the time period
oftpis comparable to the mean-square end-to-end distances of sub-molecules
(see (5.2)). From (5.29), we have

<½rðtpÞrð 0 ފ^2 >R^2 p

nb^2

p

t^1 p=^2 (5.32)

Therefore, within the time window between the monomer characteristic time
t 0 and the whole-chain characteristic time tR, there exists following scaling
relationship.

<½rðtÞrð 0 ފ^2 >t^1 =^2 (5.33)

Such a smaller scaling exponent (1/2) compared with the simple fluids (1) can be
attributed to the fact that the motions of monomers are slowed down due to their
chain connection. Below or above this time window, the monomers or the whole
chain exhibit the characteristics of simple fluids, following the scaling law

<½rðtÞrð 0 ފ^2 >t (5.34)

82 5 Scaling Analysis of Polymer Dynamics