Polymer Physics

For the short chains in the melt phase, the whole chain could not relax within the
time scale of the Rouse time, but rather forms a dynamic network to display an
elastic response to the external stress. According to (5.56), we have

E 0 ¼

ckT

n

(5.60)

Therefore,

RE 0 tR

1

n

n^2 ¼n (5.61)

For the reptation model,

E 0 ¼

ckT

ne

(5.62)

Therefore,

repE 0 ttn^3 (5.63)

The experimental observations on the zero-shear melt viscosity reveal that, for the
chain length belowne(more exactly 2ne), the scaling relation indeed follows the
prediction of the Rouse-chain model, while abovene, it shows an index of 3.4, quite
close to the 3.0 predicted by the reptation-chain model, as demonstrated in Fig.5.8.
Doi proposed to consider thecontour length fluctuations(CLF), which theoretically
corrects the scaling exponent to 3.5 (Doi 1983 ). The molecular-weight index of the
self-diffusioncoefficient can be further corrected into2.25, close to the experimental
observation2.3 (Frischknecht and Milner 2000 ). Recently, Liu et al. reported that
the deviation of the exponent from 3.0 might be attributed to cooperative motion of
multiple chains (Liu et al. 2006 ). For flexible chains, experimental studies found that
when the molecular weight is higher than a critical valueMr, the scaling exponent
returns to 3.0 (Colby et al. 1987 ). The tube length is sufficiently long so that its length
fluctuations might not be important anymore.
Presently, the influences of polydispersity and chain branching on the dynamics
of polymers have been analyzed theoretically. The disentanglement along the
reptation of a long chain is accelerated by the surrounding short chains due to the

Fig. 5.8 Illustration of the
scaling relationship between
the zero-shear viscosity and
the molecular weight. Herene
should be more precise as 2ne

5.3 Long Chains 89