Entangled melt

Several forms for the prefactor of D have been discussed in the literature. The effects of the entanglements are also clearly seen in the viscosity, which for large N, has the scaling form  

Direct evidence for these intermediate time regimes has been seen by pulsed gradient spin-echo field cycling and rotating-frame NMR and 

The reduced mobility of the chains can also be seen in the relaxation of the long-wavelength modes of the long chain. In the reptation model, the relaxation time of a Rouse mode p, with N/p Ne is enlarged by a factor of NjNe, giving  

Hess derived a similar expression from his microscopic model by explicitly considering the effective entanglement as a dynamic effect. Hess included the important many chain cooperative effects of constraint release and tube renewal, which are necessary in order to get quantitative predictions for the stress relaxation functions. Ultimately this does not affect the N dependence of the relaxation time. He found that after an initial fast Rouselike decay up to time r, Tp Hess = / irp Rep- Both models describe essentially the same physical picture. For the generalized Rouse model, Kavassalis and Noolandi found that Tp rm N /p. MD simulation results of Kremer and Grest could not distinguish between the standard reptation and Hess models but could rule out the generalized Rouse model. 

In the reptation model, the viscosity t) scales with N. The exponent 3 is slightly smaller than that found experimentally, eq. (4.19). There have been numerous models to explain this difference between the reptation model and experiment. Though the question is not completely settled, it is believed by many that the higher exponent is a finite size crossover effect due to the tube length fluctuations. DoP was the first to suggest this difference was a crossover effect and that for finite chain length, contour length fluctuations enhance the relaxation of the stress. Doi sug- 

---

On the other hand, the viscosity q of a non-entangled melt is expressed by... 

In a mode coupling approach, a microscopic theory describing the polymer motion in entangled melts has recently been developed. While these theories describe well the different time regimes for segmental motion, unfortunately as a consequence of the necessary approximations a dynamic structure factor has not yet been derived [67,68]. 

Immediately following a step strain deformation, all of the segments of a fully entangled melt can be assumed to have the same degree of orientation. In other words, both the short and the long chains will be characterized by identical functions (uu)f Q+, where... 

Pierre-Gilles deGennes (47) utilized this concept and coined the term in his work to explain why the relaxation times of entangled melts have a X A/34 dependence. Earlier, the lateral confinement of melt chains to a tubelike region had been postulated by Edwards (54). Since these early days of the reptation theory, a very significant volume of work has been dedicated to incorporating features that are physically reasonable and warranted in... 

The semi-crystalline structures often formed by crystallizable polymers are known to consist of thin crystalline lamellae separated by amorphous regions [1-3]. For crystallization from the melt, where the conditions are far from equilibrium, the polymer chains must achieve a regular conformation from the highly entangled melt and align parallel to each other to form thin... 

Slow crystallization promotes disentangling since the process of reeling-in of chains from the entangled melt onto the crystal surface is promoted and this in turn enhances the maximum drawability, despite the fact that this increases the crystallinity. However, there is a critical lower limit for the number of entanglements between crystals in order to achieve high drawability and in the extreme limit of very slow isothermal crystallization, linear standard polyethylenes (not the UHMW-PE grades) can lose their drawability completely and become brittle materials. 

Fig. 1. Typical flow curve of commercial LPE. There are five characteristic flow regimes (i) Newtonian (ii) shear thinning (iii) sharkskin (iv) flow discontinuity or stick-slip transition in controlled stress, and oscillating flow in controlled rate (v) slip flow. There are three leading types of extrudate distortion (a) sharkskin like, (b) alternating bamboo like in the shaded region, and (c) spiral like on the slip branch. Industrial extrusion of polyethylenes is most concerned with flow instabilities occurring in regimes (iii) to (v) where the three kinds of extrudate distortion must be dealt with. The unit shows the approximate levels of stress where the sharkskin and flow discontinuity occur respectively. There is appreciable molecular weight and temperature dependence of the critical stress for the discontinuity. Other highly entangled melts such as 1,4 polybutadienes also exhibit most of the features illustrated herein...

Molecular Behavior of Entangled Melts at Solid Surfaces... 

Fig. 3. Interfacial slip of an entangled melt at a non-adsorbing perfectly smooth surface, where the dots represent an organic surface (e.g., obtained by a fluoropolymer coating), which invites little chain adsorption. Lack of polymer adsorption produces an enormous shear rate jiat the entanglement-free interface between the dots and the first layer of (thick) chains. y-x=vs/a is much greater than the shear rate y present in the entangled bulk. This yields an extrapolation length b, which is too large in comparison to the chain dimensions to be depicted here...

Fig.4.a A perfectly flat wall in presence of sufficient polymer adsorption and interfacial chain entanglements, b An entangling melt under high stresses (o>oc) in contact with a molecularly smooth wall, where the adsorbed (thick) chains undergo a coil-stretch transition and the unbound chains are no longer in entanglement with the tethered chains. Here the first layer of adsorbed chains is stagnant, as the unbound chains flow by. 

Evidently, desorption and coil-stretch transition are two interfacial molecular processes, both of which may produce massive interfacial slip between a highly entangled melt and a solid surface. The former can be viewed as a true adhesive failure whereas the latter may not be regarded as such and can be viewed as cohesive. However, it is misleading and inappropriate to link the coil-stretch transition of adsorbed or entrapped chains to any bulk behavior in shear because these chains are either confined or trapped at the surface and should be distinguished... 

In a large range of concentrations (< )S0.05) where the high component is assumed to behave as an entangled melt, the variations of the terminal relaxation time (Oni in the iso-free volume state (Fig. 29) confirms relation (6-8). As the steady-state compUance J l scales as ( )" (Fig. 30), the zero-shear viscosity tiol varies as expected and the plateau modulus Gn which reveals the entanglement network is proportional to 0 . ... 

All the masses lie below Mg in the literatru e, the Rouse description of the relaxation modes of non-entangled melts or solutions is also used for polydisperse samples by means of a linear blending law. In order to consider the free voltune variations of each mass in the blend, the relaxation times have to be shifted by a factor X which is the ratio of the monomeric fiiction coefficients of the blend Cob and of each species (Co)-... 

Despite the complications produced by entanglements, melts and concentrated solu-... 

Another important relaxation process in entangled melts is constraint release, depicted in Fig. 3-27. When an end of a surrounding chain moves past a test chain, an entanglement constraint restricting the motion of the test chain is released, and a portion of the latter is freed to reorient (Graessley 1982 Montfort et al. 1986 Pearson 1987 Viovy et al. 1991). Constraint release can only be completely neglected for the case of an isolated chain... 

Theories for material instabilities or slip in highly entangled melts and solutions are still under active development. 

Problem 3.7 Consider an entangled melt of linear flexible polymer chains of molecular weight M = 100,000 and zero-shear viscosity jo = 10" P- If is increased to 300,000, what would... 

The viscosity of a polymer melt is predicted to be proportional to molar mass for unentangled melts (the Rouse model) and proportional to the cube of molar mass for entangled melts (the reptation model). 

What is the mean-square curvilinear displacement of monomers in an entangled melt along the confining tube on time scales between the... 

---