Rouse time entanglement strand

The first part of the final relation of Eq. (9.8) is the Rouse time of an entanglement strand containing Ae monomers ... 

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)] ... 

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)]. 

At the Rouse time of an entanglement strand Te, the chain finds out that its motion is topologically hindered by surrounding chains. Free Rouse motion of the chain is no longer possible on time scales t > Te. The value of the stress relaxation modulus at le is the plateau modulus G, which is kT per entanglement strand [Eq. (9.5)] ... 

On length scales larger than the correlation length but smaller than the tube diameter a, hydrodynamic interactions are screened, and topological interactions are unimportant. Polymer motion on these length scales is described by the Rouse model. The relaxation time Tg of an entanglement strand of monomers is that of a Rouse chain of N jg correlation volumes [Eg. (8.76)] ... 

Tnteractions are not important. The dynamics on these intermediate scales (for r < t< Te) are described by the Rouse model with stress relaxation modulus similar to the Rouse result for unentangled solutions [Eq. (8.90) with the long time limit the Rouse time of an entanglement strand Tg]. At Te, the stress relaxation modulus has decayed to the plateau modulus Gg[kT per entanglement strand, Eq. [(9.37), see Fig. 9.9)]. The ratio of osmotic pressure and plateau modulus at any concentration in semidilute solution -in athermal solvents is proportional to the number of Kuhn monomers in ... 

At the Rouse time of an entanglement strand tg, the chain in semidilute solution finds out that it is trapped in the confining tube. The stress relaxation modulus between and the reptation time r ep is almost con-... 

The constraint release process for the P-mer can be modelled by Rouse motion of its tube, consisting of P/A e segments, where is the average number of monomers in an entanglement strand. The average lifetime of a topological constraint imposed on a probe P-mer by surrounding A -mers is the reptation time of the A -mers Trep(A ). The relaxation time of the tube... 

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