Entanglements topology

From this rough outline of some examples of current problems in the physics of rubber elasticity, it is clear that it is important to have a well-founded statistical-mechanical theory of equilibrium properties of rubber-elastic networks. Consequently, first junction and entanglement topology are described and discussed. Then a section briefly reviews the theory of the phantom network. In the following two sections, theories of equilibrium properties of networks and a comparison of theoretical results with experimental data are presented. 

The crystallisation of polymers from the melt has proved even more controversial, as a single molecule is unlikely to be laid down on a crystalline substrate without interference from its neighbours, and it might be expected that the highly entangled topology of the chains that exists in the melt would be substantially retained in the crystalline state. These... 

In a homopolymer blend, where the i and j species have identical chemical structure but dissimilar size, and where reptation is the prevailing mode of macromolecular relaxation, it is to be expected that and are equal. This is not the case in heteropolymer blends due to modifications of (a) the entanglement topology and (b) the frictional resistance encountered by an i chain after mixing. 

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. 

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. 

This section presents results of the space-time analysis of the above-mentioned motional processes as obtained by the neutron spin echo technique. First, the entropically determined relaxation processes, as described by the Rouse model, will be discussed. We will then examine how topological restrictions are noticed if the chain length is increased. Subsequently, we address the dynamics of highly entangled systems and, finally, we consider the origin of the entanglements. 

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. 

The purpose of this review is to solve these two unresolved problems by confirming the nucleation during the induction period of nucleation and the important role of the topological nature with experimental facts regarding the molecular weight (M)- or number density of the entanglement (independence of nucleation and growth rates. 

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. 

The molecular weight (M) dependence of the steady (stationary) primary nucleation rate (I) of polymers has been an important unresolved problem. The purpose of this section is to present a power law of molecular weight of I of PE, I oc M-H, where H is a constant which depends on materials and phases [20,33,34]. It will be shown that the self-diffusion process of chain molecules controls the Mn dependence of I, while the critical nucleation process does not. It will be concluded that a topological process, such as chain sliding diffusion and entanglement, assumes the most important role in nucleation mechanisms of polymers, as was predicted in the chain sliding diffusion theory of Hikosaka [14,15]. 

Fig. 33 Schematic illustration of the model of two-stage melt relaxation. When ECSCs are melted, the chains within ECSCs are rapidly changed to a random coiled conformation. Then, chains are gradually entangled with each other. Cross-mark denotes the entanglement. rconf and tent are the conformational and topological relaxation time, respectively. At is the melt annealing time (see text)...

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

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