Polymer Physics

as illustrated in Fig.5.3. Both experiments and computer simulations on short
polymer chains in the melt phase have verified such a scaling relationship of the
Rouse chain.
Since polymer chains in the melt are in a highly interpenetrating state, the
volume exclusion effect between chain units has been well screened, and at
the meantime the hydrodynamic interactions have been screened as well. Thus,
the free-draining mode considered in the Rouse model is roughly applicable in the
melt of short polymer chains. In dilute solutions, however, the hydrodynamic
interactions could not be screened. In 1956, Zimm considered further the non-
draining mode for the dynamics of single chains in dilute solutions on the basis of
the Rouse model (Zimm 1956 ). He started with the definition of the characteristic
timet¼R^2 /D, and inserted it into the Einstein relationD¼kT/z. For the non-
draining mode, the friction coefficient of the single coil follows the Stocks law
(5.5). Then he obtained

tZR^3 (5.35)

In a good solvent, the expanded coil sizes of single chains areR¼bn3/5, then the
Zimm relaxation time is

tZn^9 =^5 (5.36)

The scaling exponent (9/5) appears smaller than that (2) for the characteristic
relaxation time of the Rouse chain. This implies that the Zimm chain diffuses faster
than the Rouse chain, because the non-draining mode of the single coil incurs
less frictional hindrance than the free-draining mode. Accordingly, the diffusion
coefficient of the Zimm chain is

DZ

R^2

tZ

n^3 =^5 (5.37)

Similar to the derivation of the scaling law of the Rouse chain, the mean-square
displacement of monomers within the time period of the characteristic timetpZfor
p-mode sub-molecules is

Log

10

<[

r(

t)-r

(0)]

2 >

Log 10 t

1

1/2

1

τ 0 τR

nb^2

b^2

Rouse chain

Fig. 5.3 Double logarithmic
plot of the mean-square
displacement of monomers
versus the time, illustrating
the scaling laws of the Rouse
chain. Monomers are moving
slower than simple fluids due
to their chain connection

5.2 Short Chains 83