Polymer Physics

5.3 Long Chains

When a long chain diffuses within its bulk melt phase, it remains a coil-like shape
containing many other interpenetrated chains. The other chains are not allowed to
diffuse together with this chain; rather, they block local vertical diffusion along the
contour length of this chain due to the volume exclusion effect, which is often
called theentanglementeffect. Such entanglements force the chain to diffuse
mostly along its contour length, behaving like a snake. Edwards first proposed the
tube model(Edwards 1967 ). He regarded such entanglements around the primitive
path of the long chain to form an effectively dynamic network, and to trap the long
chain in a tube, as demonstrated in Fig.5.5. De Gennes later-on proposed the
reptation model(De Gennes 1971 ). He assumed the tube diametera(about 10 nm)
andn/neblobs, each with the size ofa, forming the tube lengthZ~an/ne, wherene
is the critical chain length for the effect of chain entanglement.

a^2 neb^2 (5.44)

The characteristic time for the chain to diffuse out of the tube is

tt¼

Z^2

Dt

(5.45)

Log

10

<[

r(t

)-r

(0)]

2 >

Log 10 t

1

1/2

1

τ 0 τR

b^2

nb^2 C-1/4

τξ

2/3

g6/5b^2

Zimm Rouse

Fig. 5.4 Double logarithmic
plot of the mean-square
displacements of monomers
versus the time for the scaling
laws for a short chain in semi-
dilute solutions. The Zimm
chains are slightly faster than
the Rouse chains due to their
less frictional barrier in the
non-draining mode

Fig. 5.5 Illustration of the
tube model for a long chain
diffusing in the bulk phase

5.3 Long Chains 85