Entanglements and the Tube Model

a plot of either t]q or Tj against molecular weight shows a steep power-law dependence, namely t o shown in Fig. 6.5. The terminal relaxation time may be very difScult 

We have seen that the effect of entanglements on the relaxation of a melt is similar in some respects to the effect of cross-links on the relaxation of a rubber. For example, a bouncing 

We recall that the first of the above definitions arose from Ferry s analogy between entangled melts and crossKnked elastomers, while the second definition was based on the fact that fast Rouse modes allow re-equilibration of tension along the chain, so that one fifth of the initial stress relaxes before the entanglement network interrupts the process. 

Both of the above definitions are controversial, and there is even some justification for including another factor of 1/2 to accoimt for fluctuations of entanglement positions. The relationship between (or ) and is discussed in detail in Section 5.9.3. 

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The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

It became clear in the early development of the tube model that it provided a means of calculating the response of entangled polymers to large deformations as well as small ones [2]. Some predictions, especially in steady shear flow, lead to strange anomaUes as we shall see, but others met with surprising success. In particular the same step-strain experiment used to determine G(t) directly in shear is straightforward to extend to large shear strains y. In many cases of such experiments on polymer melts both Hnear and branched, monodisperse and polydisperse,the experimental strain-dependent relaxation function G(t,Y) may be written... 

Here X ,ax is the single chain limiting extension ratio in the isotropic, unoriented polymer with the same entanglement weight. A complication is that the process of orientation above Tg may result in a loss of entanglement constraint, effectively increasing and One of the ways this loss can take place in the current versions of the tube model is by tube relaxation whereby the process of tube... 

These tube length fluctuation modes (see Section 9.4.5) of the neighbouring chains affect the constraint release modes of a given chain. If entanglements between chains are assumed to be binary, there should be a duality between constraint release events and chain in a tube relaxation events. A release of an entanglement by reptation or tube length fluctuation of one chain in its tube leads to a release of the constraint on the second chain. If this duality is accepted, the distribution of constraint release rates can be determined self-consistently from the stress relaxation modulus of the tube model. 

Fig. 18 Schematic representation of crowded soft systems (a) entangled polymers, (b) repulsive colloidal hard spheres, (c) colloidal star polymers, and (d) attractive hard spheres. The former are described by the tube model for entanglements, whereas the latter three by the general cage model for colloidal glasses...

In order to remove the main differences and confusions mentioned in Sect. 1, the relations between the tube model and other approaches which have been used successfully, such as the model of restricted junction fluctuations and the concept of trapped entanglements, have been discussed in detail and a number of new insights have been achieved. 

Though the tube model is successful, our present understanding of the dynamics in entangled systems is still incomplete. Agreement between theory and experiments is not yet complete as we shall discuss later. More seriously, the tube model does not describe all aspects of the dynamics it describes properties which depend on a single chain... 

The role of positional fluctuations in polymer networks is central to some theories of elasticity, and has been investigated with an MC method based on a modified bond-fluctuation model (265). The simple model used in the simulations gave results close to those calculated from theory for a Bethe lattice (also known as a Cayley tree). More extensive results bearing on the role of fluctuations in polymer networks have been reported by Grest and co-workers (225). They find that entanglements limit fluctuations, giving behavior similar to the description provided by the tube model. 

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