Topology of entanglements

The process of chain retraction can be applied to more complex topologies of entangled polymers under the same assumptions discussed for linear polymers... 

Everaers R, Sukumaran SK, Grest GS, Svaneborg C, Sivasubramanian A, Kremer K (2004) Rheology and microscopic topology of entangled polymeric liquids. Science 303 823-826 Ewen B, Richter D (1995) The dynamics of polymer melts as seen by neutron spin echo spectroscopy. Macromol Symp 90 131-149... 

A similar anomalous behavior has been detected also in 3d polymer melts but only for rather short chains [41] for longer chains, several regimes occur because of the onset of entanglement (reptation ) effects. In two dimensions, of course, the topological constraints experienced by a chain from... 

The presented scheme offers several extensions. For example, the model gives a clear route for an additional inclusion of entanglement constraints and packing effects [15]. Again, this can be realized with the successful mean field models based on the conformational tube picture [7,9] where the chains do not have free access to the total space between the cross-links but are trapped in a cage due to the additional topological restrictions, as visualized in the cartoon. 

Equation (29) shows that the modulus is proportional to the cycle rank , and that no other structural parameters contribute to the modulus. The number of entanglements trapped in the network structure does not change the cycle rank. Possible contributions of these trapped entanglements to the modulus therefore cannot originate from the topology of the phantom network. 

But the topological nature has not been confirmed more directly to date. It is expected that the topological restriction increases with an increase in molecular weight (M) and the number density of entanglement (ve). Therefore, the studies of the M or ve dependence of crystallization behavior should be important in confirming directly the important role of topological nature in polymer crystallization. 

As we conjectured in the introduction, the fundamental role of topology in this approach to entangled polymer dynamics would indicate that changes to the topology of the molecules themselves would radically affect the dynamic response of the melts. In fact rheological data on monodisperse star-branched polymers, in which a number of anionically-polymerised arms are coupled by a multifunctional core molecule, pre-dated the first application of tube theory in the presence of branching [22]. Just the addition of one branch point per molecule has a remarkable effect, as may be seen by comparing the dissipative moduli of comparable linear and star polymer melts in Fig. 5. 

A feature of theories for tree-like polymers is the disentanglement transition , which occurs when the tube dilation becomes faster than the arm-retraction within it. In fact this will happen even for simple star polymers, but very close to the terminal time itself when very little orientation remains in the polymers. In tree-like polymers, it is possible that several levels of molecule near the core are not effectively entangled, and instead relax via renormalised Rouse dynamics (in other words the criterion for dynamic dilution of Sect. 3.2.5 occurs before the topology of the tree becomes trivial). In extreme cases the cores may relax by Zimm dynamics, when the surroundings fail to screen even the hydro-dynamic interactions between the slowest sections of the molecules. 

As we consider the mechanical properties of SPs, it is often useful to consider them in the context of entanglements, which are intermolecular interactions that transfer mechanical forces from one molecule to the next. In this chapter we use the term entanglement in a very general way, so that it includes topological entanglements (one polymer chain is physically wrapped around another), chemical entanglements (attractive intermolecular interactions between polymer chains), and surface adsorption (attractive intermolecular interactions between polymer chains and a particle surface, e.g., from a filler). The important, fundamental characteristic is that it is an interaction that allows a mechanical stress on one molecule to be distributed or transferred to another molecule with which it is entangled. 

In response to the above characterization problems and an interest in understanding the topology of intramolecular entanglement a membrane viscometer was developed.(A) In the membrane viscometer a solution is passed through a thin ( 10 )im) membrane with well-defined pores of fixed diameter that are nearly perpendicular to the membrane surface. The important feature is... 

The first two terms on the right of Eq. (7.30) will be recognized as the independent strand contribution to the entropy. The topological or entanglement contribution is then... 

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