Theories of Entanglement Trapping

When cast in terms of strand concentrations, the Bueche-Mullins equation becomes 

The active strand concentration v is obtained from measurements of the initial modulus using Eq.(7.2) with g = 1. Values of v for samples at the same crosslink density but with different primary molecular weights are extrapolated to 1/M=0, giving vc + ve. Values of vc + ve obtained at two or more crosslink densities provide ve, and hence Me, by extrapolation. An advantage of this method is that only relative values of crosslink density must be known absolute values of vc are not required. If absolute values of vc are known, the g factor can be evaluated as well. 

An intuitively more satisfying requirement for trapping was suggested by Ferry and co-workers (285). In order for any entanglement to contribute to the equilibrium modulus, all four strands leading from that entanglement must terminate in crosslinks. To the same approximation as the Bueche-Mullins relation (C N), and with the Flory active strand criterion, the strand concentration becomes 

The Ferry trapping factor is therefore (1 — 2v0/vc)2. Unfortunately, the simple extrapolation procedure of Kraus (286) can no longer be applied, but numerical search techniques should still suffice to determine Me. 

Langley (255) has developed a refined definition of the Ferry trapping factor, and he also accounts theoretically for effects of the primary chain distribution and 

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