Polymer Physics

reptation chain diffuses even slower than the Rouse chain. The overall scaling
relationships of the chain dynamics for the reptation chain can be summarized
below, as also demonstrated in Fig.5.6.

<[r(t)-r(0)]^2 >rep~t,ift<t 0 ; (within the scale of b^2 )
~t1/2,ift 0 <t<te; (within the scale of a^2 )
~t1/4,ifte<t<tR; (within the scale ofn1/2ab)
~t1/2,iftR<t<tt; (within the scale ofnb^2 )
~t,iftt<t.

Similar to the scaling analysis for the short chains in concentrated solutions, we
can assume the screening length of hydrodynamic interactions for the long chains,
corresponding to the screening length of the volume repulsive interactions, as

xhx (5.51)

Applying the blob model, one can obtain

xg^3 =^5 C^3 =^4 (5.52)

At the motion length scale smaller thanxh, the Zimm-chain model is valid. In
contrast, at the motion length scale abovexh, there occur two different scenarios:
the Rouse-chain model is applicable at the scale smaller than the critical entangle-
ment lengthne; and the reptation-chain model is applicable at the scale larger than
ne. The results are similar with the semi-dilute solutions of short chains, withtx
betweent 0 andte, i.e. inserting 2/3 scaling exponent before 1/2 scaling exponent.
In the following, we discuss two examples to demonstrate how the results of
scaling analysis facilitate our better understanding to the deformation and flow
behaviors of polymer chains.
A very small stresssworking on the polymer melt will reveal the elastic
response of the fluid, and the strain

eðtÞ¼sJðtÞ (5.53)

τe

Log

10

<[

r(t

)-

r(0)]

2 >

Log 10 t

1

1/2

1

τ 0 τt

b^2

nb^2

τR

1/2 n1/2ab

1/4

a^2

Rouse chain

Reptation chain

Fig. 5.6 Double logarithmic
plot of the mean-square
displacements of monomers
versus the time for the scaling
law of a long chain in the bulk
polymer phase. Reptation
chains are slower (half-down
indexes) than Rouse chains
due to the tube confinement

5.3 Long Chains 87