Polymer Physics

whereJ(t)is called the time-dependentcreep compliance.Whent<tt,polymer
chains are trapped in an elastic network of entanglements, and the compliance shows
a plateau as the time increases, which is characteristic for the long-chain melt.

JðtÞ!J 0 (5.54)

where J 0 is the steady-state compliance, as demonstrated in Fig. 5.7. The
corresponding plateau modulus,

E 0 ¼

1

J 0

(5.55)

and the corresponding equation of state for the transient rubber state is

E 0 ¼

ckT

Ne

(5.56)

wherec¼1/a^3 , andais the size of the monomers. The critical molecular weight of
polystyrene 1.13 104 g/mol corresponds to a plateau modulus of 2.24 105 Pa,
in agreement with the experimental observations, 2 105 Pa (Sperling 2006 ).
When t>tt, the chain eventually “reptates” out of the tube. Under the stress,
polymer melt can perform the steady-state flow with a permanent deformation, thus

JðtÞ¼J 0

t

tt

(5.57)

From a Newtonian fluid

s¼

deðtÞ

dt

(5.58)

Then we can obtain the melt viscosity

¼

dt

dJðtÞ

¼

tt

J 0

¼E 0 tt (5.59)

Fig. 5.7 Illustration of the
rubber plateau for the creep
compliance within the time
scale shorter than the
characteristic time of the
reptation chain in the melt
phase at a given temperature

88 5 Scaling Analysis of Polymer Dynamics