Reptation scaling

Liu, C. Y., Keunings, R., and Bailly, C., 2006. Do deviations from reptation scaling of entangled polymer melts result from single- or many[Pg.229]

There are three basic time scales in the reptation model [49]. The first time Te Ml, describes the Rouse relaxation time between entanglements of molecular weight Me and is a local characteristic of the wriggling motion. The second time Tro M, describes the propagation of wriggle motions along the contour of the chain and is related to the Rouse relaxation time of the whole chain. The important... 

W. Paul, K. Binder, D. Heermann, K. Kremer. Dynamics of polymer solutions and melts. Reptation prediction and scaling of relaxation times. J Chem Phys 95 7726-7740, 1991. 

The scaling dependence of the diffusion coefficient on N and Cobs Iso poses a number of questions. While the original scaling predictions, based on reptation dynamics [26,38], oc N, have been verified by some measurements [91,98], significant discrepancies have been reported too [95,96]. Attempts to interpret existing data in terms of alternative models, e.g., by the so-called hydrodynamic scaling model [96], fail to describe observations [100,101]. 

The scaling results above all pertain to local segmental relaxation, with the exception of the viscosity data in Figure 24.5. Higher temperature and lower times involve the chain dynamics, described, for example, by Rouse and reptation models [22,89]. These chain modes, as discussed above, have different T- and P-dependences than local segmental relaxation. 

Fig. 25. Rouse-scaling representation of the PEP homopolymer data at 492 K. Above solid lines represents the Ronca model [50] below solid lines display the predictions of local reptation [53]. The solid lines from below tooabove correspond to Q = 0.135 A-1 Q = 0.116 A-1 Q = 0.097 A-1 Q = 0.078 A- Q = 0.068 A-1 Q = 0.058 A-1). The symbols along the lines are data points corresponding to the respective Q-value. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

So far we have invoked reptation theory to describe the behaviour in the melt state. We can use scaling theory in the form of Equations (5.122) and (5.123) to express the concentration dependence of the modulus and viscosity. By inspection of Equations (5.128) and (5.130) ... 

Combining the predictions for the reptation time of the H cross-bar and the effective modulus when the arms are acting as solvent gives a prediction for the scaling of the viscosity of a melt of H-polymers on their structural parameters ... 

The final chapter develops the most modern insights in the relation between the rheological properties and the large scale architecture of polymers. Indeed, the largest effects of branching are encountered in their melt relaxation properties. In the absence of reptation, which dominates relaxation processes in Hnear polymers, a rich variety of other relaxation processes becomes apparent. The control ot the melt properties of polymers by means of their long-chain branch architecture will continue to lead to new industrial applications. 

The reptation model predicts that the viscosity of a melt scales with the chain length to the third power while the diffusion coefficient decreases with the second power of the chain length. 

Fig. 3.16 Scaling presentation of the dynamic structure factor from a M =36,000 PE melt at 509 K as a function of the Rouse scaling variable. The solid lines are a fit with the reptation model (Eq. 3.39). The Q-values are from above Q=0.05,0.077,0.115,0.145 A The horizontal dashed lines display the prediction of the Debye-Waller factor estimate for the confinement size (see text)...

In order to calculate the effects of CLF we have to ask how the fraction of monomers that is released through CLF at the chain ends grows with time. It has been recently shown that for Ktr the effect of reptation on escaping from the tube is negligible in comparison to CLF [90]. It is the first passage of a chain end that is assumed to relax the constraint of a tube segment on a chain. From the scale invariance of the Rouse equation (Eq. 3.7) an exact asymptotic result... 

The large scale molecular motions which take place in the rubber plateau and terminal zones of an uncross-linked linear polymer give rise to stress relaxation and thereby energy dissipation. For narrow molecular weight distribution elastomers non-catastrophic rupture of the material is caused by the disentanglement processes which occur in the terminal zone, e.g., by the reptation process. In practical terms it means that the green strength of the elastomer is poor. 

It is not clear how improvements can be made without real progress on the difficult fundamental problems of diffusion in media with obstacles and cooperation of large-scale motions between interpenetrating chains which do not violate chain connectivity. The DeGennes reptation model (225) makes a significant contribution to the first problem, although in an admittedly simplified system. Rigorous calculations or computer simulations on well-defined models which relate to the second problem would be extremely valuable, even if the models themselves were not completely faithful representations of the assumed physical situation. It is not obvious how even to pose solvable problems, simplified or not, which relate to interchain cooperation. 

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