Stress relaxation and viscosity

The reptation ideas discussed above will now be combined with the relaxation ideas discussed in Chapter 8 to describe the stress relaxation modiihis G t) for an entangled polymer melt. On length scales smaller than the tube diameter a, topological interactions are unimportant and the dynamics are similar to those in unentangled polymer melts and are described by the Rouse model. The entanglement strand of monomers relaxes by Rouse motion with relaxation time Tg [Eq. (9.10)]  

The Rouse model predicts that the stress relaxation modulus on these short time scales decays inversely proportional to the square root of time [Eq. (8.47)]  

The relaxation time of the Kuhn monomer tq is the shortest stress relaxation time in the Rouse model, given by Eq. (8.56) with p N  

The stress relaxation modulus at tq is the Kuhn modulus (kT per Kuhn monomer)  

Consider tor example, a melt of 1,4-poly butadiene linear chains with M—130000gmol The molar mass of a polybutadiene Kuhn monomer is Mo= I05gmol (see Table 2.1) so this chain has N = MjMo 1240 Kuhn monomers. At 25 °C, this polymer is 124 K above its glass transition and its oscillatory shear master curve is shown in Fig. 9.3. The time scale for monomer motion is tq = 0.3 ns, An entanglement strand of 1,4-polybutadiene has molar mass M — I900gmol (see Table 9.1) and therefore contains N — M jMQ= % Kuhn monomers. The whole chain has N/Ne = MjMs 68 entanglements. The Rouse time of the entanglement strand Te = 0.1 ps [Eq. (9,16)]. 

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Experimental signatures of this peculiar transition has been found essentially in chalcogenide glasses fi om Raman scattering [18], stress relaxation and viscosity measurements [19], vibrational density of states [17], Brillouin scattering [20], Lamb-Mossbauer factors [21], resistivity [22], and Kohlrausch fractional exponents [23]. 

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