Relaxation time, molecular-weight dependence

In connection with a discussion of the Eyring theory, we remarked that Newtonian viscosity is proportional to the relaxation time [Eqs. (2.29) and (2.31)]. What is needed, therefore, is an examination of the nature of the proportionality between the two. At least the molecular weight dependence of that proportionality must be examined to reach a conclusion as to the prediction of the reptation model of the molecular weight dependence of viscosity. 

Evaluate the relaxation time associated with each of these molecular weights and verify that the molecular weight dependence of r corresponds to the value given in Sec. 2.13. 

Fig. 12. Molecular weight dependences of the normalized chain relaxation time, tJTs, for linear polymers ( ), branched fractions ( ), and branched feed polymer (+). (Reproduced with permission from [88]. Copyright 2001 American Chemical Society.)...

The exponent b is a matter of some current dispute, being 2 in nearly all studies but appearing to range as high as 3 in a few instances (Section 5.4.4). In any case, the explicit molecular weight dependence is lost, and concomitantly the concentration dependence is increased (173). If all relaxation times had been scaled up by the same factor (the relative spacings and intensities of the Rouse spectrum being retained), the Rouse forms would have been preserved for M > (Mc)soln. 

C.J. Farrell, A. Keller, M.J. Miles, and D.P. Pope, Conformational relaxation time in polymer solutions by elongational flow experiments 1. Determination of exten-sional relaxation time and its molecular weight dependence, Polymer, 21,1292 (1980). 

The difference in the molecular-weight dependence of the terminal relaxation time can be attributed to the change of the mechanisms (diffusive and repta-tion, correspondingly) of conformational relaxation in these systems. Further on in this section, we shall calculate dynamic modulus and discuss characteristic quantities both for weakly and strongly entangled systems. 

A frequency dependence of complex dielectric permittivity of polar polymer reveals two sets or two branches of relaxation processes (Adachi and Kotaka 1993), which correspond to the two branches of conformational relaxation, described in Section 4.2.4. The available empirical data on the molecular-weight dependencies are consistent with formulae (4.41) and (4.42). It was revealed for undiluted polyisoprene and poly(d, /-lactic acid) that the terminal (slow) dielectric relaxation time depends strongly on molecular weight of polymers (Adachi and Kotaka 1993 Ren et al. 2003). Two relaxation branches were discovered for i.s-polyisoprene melts in experiments by Imanishi et al. (1988) and Fodor and Hill (1994). The fast relaxation times do not depend on the length of the macromolecule, while the slow relaxation times do. For the latter, Imanishi et al. (1988) have found... 

This is exactly the molecular-weight dependence of conformational relaxation times of polymer in non-entangled state and for the region of diffusive mobility (see equation (4.41), weakly-entangled system). 

A rigorous theoretical approach to the characterization of the molecular-weight dependence of relaxation times in PL has been used only for the simplest models of the polymer chain for Gaussian subchains without hydrodynamic interactions and by taking into account the effect of hydrodynamic interactions on the PL The two-time approximation (see Eq. (4.5.1))... 

For Ni>Ns> Ng Ng is the critical entanglement molecular weight), r,- is the reptation time, while for N Ng it represents the Rouse time [60,69]. Because Xi has a strong molecular weight dependence, long chains keep then-orientation while short already relax to the isotropic state. For Nt 2> N one finds [70]... 

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