Entanglement strand volume

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)]. 

Me molecular mass of these entanglement strands) and the volume spanned by the entanglement distance is approximately a constant... 

With the number of entanglement strands ne/volume, the desired number of binary contacts per entanglement strand becomes... 

Pc> Pb> Pf cumulative number fraction of grid squares that exhibit craze formation, craze fibril breakdown, and catastrophic fracture, respectively probability that a given entangled strand survives craze fibril formation disentanglement time of i strands in a fibril that survive fibril formation craze interface velocity volume fraction of polymer within craze... 

In view of the fact that the cross-tie fibrils contain some of the entangled strands that were imagined to either break or disentangle in the development of Eqs. (15) and (20), one can ask how accurate these formulae are given the cross-tie fibril microstructure of a typical craze. From the meridonal LAED reflections, Miller estimated that the cross-tie fibrils comprised only at most about 15% of the volume of the main fibrils and therefore that the corrections required to Eqs. (15) and (20) for the cross-tie fibrils are negligible. 

The plateau modulus is given by the usual formula for entangled polymers, Gq vksT, where v is the number of entanglement strands per unit volume of melt that is,... 

Table 9.1 shows and calculated from the measured plateau modulus. All flexible polymers have Pg —20 overlapping entanglement strands defining the entanglement volume a, which is the overlap criterion for entanglement in polymer melts. 

Owing to the fact that the tube diameter is always larger than the correlation length (fl > 0, the entanglement strand is a random walk of correlation volumes in any solvent ... 

On length scales larger than the correlation length but smaller than the tube diameter a, hydrodynamic interactions are screened, and topological interactions are unimportant. Polymer motion on these length scales is described by the Rouse model. The relaxation time Tg of an entanglement strand of monomers is that of a Rouse chain of N jg correlation volumes [Eg. (8.76)] ... 

The number of entanglement strands within the confinement volume is determined as the ratio of confinement... 

The plateau modulus in this regime is G VefeeT, where Ve is the number density of entanglement strands. The volume per entanglement strand is =fa. The scaling model... 

Since, in contrast to experiment, the simulation knows in detail what the connectivity looks hke, how long the strands are, and how the network loops are distributed, one can attribute this behavior to the non-crossability of the chains. Actually, one can even go further by allowing the chains to cross each other but still keep the excluded volume. Such a technical trick, which is only possible in simulations, allows one to isolate the effect of entanglement and non-crossability in such a case. As one would expect, if one allows chains to cross through each other one recovers the so-called phantom network result. 

In this relation, 2C2 provides a correction for departure of the polymeric network from ideality, which results from chain entanglements and from the restricted extensibility of the elastomer strands. For filled vulcanizates, this equation can still be applied if it can be assumed that the major function of the dispersed phase is to increase the effective strain of the rubber matrix. In other words, because of the rigidity of the filler, the strain locally applied to the matrix may be larger than the measured overall strain. Various strain amplification functions have been proposed. Mullins and Tobin33), among others, suggested the use of the volume concentration factor of the Guth equation to estimate the effective strain U in the rubber matrix ... 

A determination of how Gat depends on the chemical structure of a polymer requires a definition of an entanglement. Various ideas have been proposed a fixed number of binary contacts per entanglement (Brochard and De Gennes, 1977) or per entanglement volume (Colby et al., 1992) a fixed number of strands per entanglement volume (Ronca, 1983 Lin, 1987 KavassaUs and... 

As the stress of the freed chain strands increases, slippage through entanglements is assumed to set in at a rate that has the same functional form as that for vdW breakings (Eq. 1) but, with different values for the activation energy U and volume fi. In Eq. 1, a now denotes the difference in stress in two consecutive chain strands that are separated by an entanglement. This stress difference is calculated using the classical treatment of... 

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