Flow models reptation/tube model

The pom-pom polymer reptation model was developed by McLeish and Larson (60) to represent long chain-branched LDPE chains, which exhibit pronounced strain hardening in elongational flows. This idealized pom-pom molecule has a single backbone confined in a reptation tube, with multiple arms and branches protruding from each tube end, as shown in Fig. 3.12(a). Mb is the molecular weight of the backbone and Ma, that of the arms. 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

Polymer melts and semidilute and concentrated solutions of polymer are highly viscous. Even at a concentration of 1 wt %, solutions of polymer with a molecular weight greater than several million g/mol can flow only slowly. Their behaviors are even elastic like rubber at accessible time and frequency ranges. These exquisite properties had eluded researchers for decades until the tube model and the reptation theory elegantly solved the mystery. The tube model and the reptation theory were introduced by de Gennes." They were refined and applied to the viscoelasticity of semidilute solutions of polymers and polymer melts in the late 1970s by Doi and Edwards." Until then, there had been no molecular theory to explain these phenomena. We will leam the tube model and the reptation theory here. 

Tube models have been used to predict this material function for linear, monodisperse polymers, and a so-called standard molecular theory [159] gives the prediction shovm in Fig. 10.17. This theory takes into account reptation, chain-end fluctuations, and thermal constraint release, which contribute to linear viscoelasticity, as well as the three sources of nonlinearity, namely orientation, retraction after chain stretch and convective constraint release, which is not very important in extensional flows. At strain rates less than the reciprocal of the disengagement (or reptation) time, molecules have time to maintain their equilibrium state, and the Trouton ratio is one, i.e., % = 3 7o (zone I in Fig. 10.17). For rates larger than this, but smaller than the reciprocal of the Rouse time, the tubes reach their maximum orientation, but there is no stretch, and CCR has little effect, with the result that the stress is predicted to be constant so that the viscosity decreases with the inverse of the strain rate, as shown in zone II of Fig. 10.17. When the strain rate becomes comparable to the inverse of the Rouse time, chain stretch occurs, leading to an increase in the viscosity until maximum stretch is obtained, and the viscosity becomes constant again. Deviations from this prediction are described in Section 10.10.1, and possible reasons for them are presented in Chapter 11. 

The earliest tube models included only the simplest nonlinearities, that is, convective constraint release was neglected (since its importance was not clearly recognized), and the retraction was assumed to occur so fast relative to the rate of flow that the chains were assumed to remain imstretched. The linear relaxation processes of constraint release and primitive path fluctuations were also ignored, so that the model contained only one linear relaxation mechanism, namely reptation, and only the nonlinearity associated with large orientation of tube segments, but no stretch. Subsequent models added the omitted relaxation phenomena, one at a time, and in what follows we present the most important constitutive models that included these effects, starting with models for monodisperse linear polymers. 

The beauty of the reptation model is that it is able to make predictions about molecular flow both in solution and at fracture by assuming that the molecules undergo the same kind of motions in each case. For both self-diffusion in concentrated solutions and at fracture, the force to overcome in pulling the polymer molecule through the tube is assumed to be frictional. 

In the subsequent 20 years (1960-80), the main principles of modern polymer physics were developed. These include the Edwards model of the polymer chain and its confining tube (Chapters 7 and 9), the modern view of semidilute solutions established by des Cloizeaux and de Gennes (Chapter 5), and the reptation theory of chain diffusion developed by de Gennes (Chapter 9) that led to the Doi-Edwards theory for the flow properties of polymer melts. 

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