Density of entanglements

The different experimental systems all yield a similar pattern of variation of toughness with interface width. The toughness initially increases slowly with width at low interface width, and then increases rapidly with width and saturates at high width at a value close to the bulk toughness. If the density of entangled strands controlled the toughness, then the interface width at which the toughness... 

Entanglement dependence of the nucleation rate I is qualitatively obtained for the first time by changing the number density of entanglement (ve) within the melt. An experimental formula of I as a function of ve was obtained on PE, I(ve) a exp(- yve) where y is a constant. 

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. 

Finally we require an expression for the relaxation modulus consistent with the dilution hypothesis in which each tube segment relaxing its stress typically at a time r(x) does so in a background whose effective density of entangled strands is 0 The appropriate general form is... 

Force for chain breakage Surface density of entangled chains... 

The toughness Gc (Eq. 20) is related to material features such as the surface density of entangled chains in the fibrils vs and the force for chain scission /b. The value of vs depends on the amount of disentanglement induced by the fibrillation together with the initial entanglement density of the bulk. 

To a first approximation, ktc is inversely proportional to the density of entangled coils, resulting in the functional dependence... 

Figure 3.19 shows the dependence of the elastic modulus of craze fibrils Efibril of PMMA, SAN and PS on the entanglement density (v ). It is known that the increasing density of entanglement in a polymer induces an increase of the bulk... 

Since monomers are space-filling in the melt, the number density of entanglement strands is just the reciprocal of the entanglement strand volume, leading to a simple expression for the plateau modulus of an entangled polymer melt [Eq. (7.47)]. 

One can determine the density of entanglement cluster network from the value of Poisson s ratio p by means of the following approximation [21] ... 

Further evidence bearing on the physical nature of the discrepancy is provided by two other observations C2 does not appear to be strongly dependent on temperature and therefore does not appear to be associated with the energetics of chain conformations and it is closely correlated with the tendency of the polymer chains to form molecular entanglements. For example, those polymers that have a high density of entanglements in the bulk state (Table I) yield rubbery networks with a relatively high C2 stress component... 

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