Rouse Longest relaxation time

This relationship between the relaxation modes and the low shear viscosity is an important one. It indicates that the longest Rouse relaxation time, i.e. the p = 1 mode ... 

Estimate the longest Rouse relaxation time, x, for a polyethylene chain with 154 carbon atoms, C154 at 448 K, making use of the following parameters ... 

Osaki, K., T. Inoue, T. Uematsu, and Y. Yamashita. 2001. Evaluation methods of the longest Rouse relaxation time of an entangled polymer in a semidilute solution.. Polym. Sci. Part B Polymer Phys. 39 1704-1712. 

Hence, the longest Rouse relaxation time, xr, is proportional to N. For t > xr, the motion of the chain becomes essentially diffusive, and the diffusion constant of the center of gravity of the chains is given by the Einstein relation, Eq. (50). 

Chapter 6 treats mean-field theories of melt behavior. We begin with the Rouse model for molecules in dilute solution and its modification by Bueche to deal with unentangled melts. The longest Rouse relaxation time emerges from this treatment and plays an important role in all molecular models. The tube model is introduced, in which the basic relaxation... 

Inoue, T., Uematsu, T., Yamashita, Y., Osaki, K. Significance of the longest Rouse relaxation time in the stress relaxation process at large deformation of entangled polymer solutions. Macromol (2002) 35, pp. 4718-4724... 

The longest Rouse time, corresponding to p = 1, is especially important and is called the longest Rouse relaxation time ... 

These can also be written in terms of the longest Rouse relaxation time. For example, Eq. 6.6 becomes ... 

Thus, by measuring the zero-shear viscosity of the entangled polymer, we can estimate its longest Rouse relaxation time. A similar method has been developed by Pattamaprom and Larson [18], but using an experimentally measured longest relaxation time as the input, and scaling this using the ratio of MJM to the power -1.4 to get the Rouse time (see Eq. 11.41 in Chapter 11) and note that Tj = 2 %. 

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