Tube model constraints release

A second appealing feature of tube model theories is that they provide a natural hierarchy of effects which one can incorporate or ignore at will in a calculation, depending on the accuracy desired. We will see how, in the case of linear polymers, bare reptation in a fixed tube provides a first-order calculation more accurate levels of the theory may incorporate the co-operative effects of constraint release and further refinements such as path-length fluctuation via the Rouse modes of the chains. 

The approximate treatment described above accounts rather well for the linear rheology of star polymer melts. In fact it has been remarked that the case for the tube model draws its real strength from the results for star polymers rather than for linear chains, where the problems of constraint release and breathing modes are harder to account for (but see Sect. 3.2.4.). However, there are still some outstanding issues and questions ... 

Fig. 14. Data (points) for G (co) and G (co) for a range of compositions of a blend of two polyisoprene stars of molecular weights 28 and 144 kg mol The fractions of the bigger star are in order 0.0,0.2,0.5,0.8 and 1.0. Curves are theoretical predictions of the tube model with co-operative constraint release treated by dynamic dilution [56]. The choice of 2.0 rather than 7/3 for the dilution exponent p is compensated for by taking M = 5500 kg mol" ...

Newer rubber elasticity theories based on the tube model (35) consider special constraint release mechanisms which allow a physi-... 

This combination is equivalent to saying that the overall diffusion coefficient is the sum of the coefficients of the two processes. Watanabe and Tirrell [18] suggested that the rate constant of constraint release depends on the tube configuration, but the comparison between the two models [19] does not allow us to conclude in favor of either one of the models. Thus we wiQ use the simplest one given by relation (5-2). 

The Doi-Edwards model has been extended to allow processes of primitive-path fluctuations, constraint release, and tube stretching. These extensions of the theory allow accurate prediction of many steady-state and time-dependent phenomena, including shear thinning, stress overshoots, and so on. Predictions of strain localization and slip at walls... 

In interpreting the relaxation behavior of polydisperse systems by means of the tube model, one must consider that renewal of the tube occurs because the chain inside it moves thermally, either by reptation mode, by fluctuation of the tube length in time (breathing motion), or in both ways (13,14). Moreover, the tube wall can be renewed independently of the motion of the chain inside the tube because the segments of the chains of the wall are themselves moving. The relaxation mechanism associated with the renewal of the tube is called constraint release. 

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... 

Constraint release has a limited effect on the diffusion coefficient it is important only for the diffusion of very long chains in a matrix of much shorter chains and can be neglected in monodisperse solutions and melts. The effect of constraint release on stress relaxation is much more important than on the diffusion and cannot be neglected even for monodisperse systems. Constraint release can be described by Rouse motion of the tube. The stress relaxation modulus for the Rouse model decays as the reciprocal square root of time [Eq. (8.47)] ... 

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. 

Consider an isolated long probe P-mer entangled in a melt of shorter Wmers. Tube dilation assumes that as soon as short chains relax, stress in the long P-mer drops to zero. In particular, a version of tube dilation called double reptation imposes an exact symmetry between single chains in a tube and multi-chain processes. As one chain reptates away, stress at a common entanglement (stress point) is relaxed completely. In constraint release models, this stress relaxes only partially due to connectivity of the P-mer. 

Comparing the advantages and shortcomings of the various concepts, the tube model may be considered as the most flexible and the one most suited to successive improvement. The main task remaining is the development of a quantitative theory of the topological constraints for the whole range of network parameters of interest. In particular, this theory should contain a theory or at least a more refined model of the constaint release processes proposed in polymer networks. 

The molecular theory of extensional viscosity of polymer melts is again based oti the standard tube model. It considers the linear viscoelastic factors such as reptation, tube length fluctuations, and thermal constraint release, as well as the nonlinear viscoelastic factors such as segment orientations, elastic contractimi along the tube, and convective constraint release (Marrucci and lannirubertok 2004). Thus, it predicts the extensional stress-strain curve of monodispersed linear polymers, as illustrated in Fig. 7.12. At the first stage, the extensional viscosity of polymer melts exhibits the Newtonian-fluid behavior, following Trouton s ratio... 

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