Calculation by reptation model

Next we consider a polymer melt of high molecular weight in which entanglement is very important. To calculate G(t) it is convenient to consider the stress relaxation after a step strain. Suppose at t = 0 a shear strain y is applied to the system in equilibrium. The strain causes the deformation of the molecular conformation, and creates the stress, which relaxes with time as the conformation of polymers goes back to 

At t Xg, the whole polymer is confined in a deformed tube. As time passes, the polymer reptates, and at time t the parts of the polymer near the ends have disengaged from the deformed tube, while the part in the middle is still confined in the tube. Since only the segments in the deformed tube are oriented and contribute to the stress, the stress is proportional to the fraction (t) of the polymers still confined in the 

The average of the contour length a(t) of this part is equal to Lip f). 

To obtain G, we utilize the fact that at I t, the Rouse-Uke behaviour (eqn (7.38)) smoothly crosses over to die reptation behaviour (eqn 

Equations (7.39) and (7.44) give the relaxation modulus in the highly entangled state. We shall now compare this result with experiments. 

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