Theories of the Terminal Relaxation Spectrum

When E is large (M MC) the relaxation times of the affected modes are 

Chompff and Duiser (232) analyzed the viscoelastic properties of an entanglement network somewhat similar to that envisioned by Parry et al. Theirs is the only molecular theory which predicts a spectrum for the plateau as well as the transition and terminal regions. Earlier Duiser and Staverman (233) had examined a system of four identical Rouse chains, each fixed in space at one end and joined together at the other. They showed that the relaxation times of this system are the same as if two of the chains were fixed in space at both ends and the remaining two were joined to form a single chain with fixed ends of twice the original size. 

The network is decoupled by successively replacing junction tetrahedra by two independent strands, each terminated by remaining junctions and one with a slow point in the middle. The order of decoupling and assignment of slow points must be performed in a rather special way the symmetry of remaining junctions must be preserved so that the same decoupling procedure can be applied to them in turn. 

The terminal spectrum is furnished by cooperative motions which extend beyond slow points on chain in the equivalent system. The modulus associated with the terminal relaxations is vEkT, which is smaller by a factor of two than the value from a shifted Rouse spectrum. It is consistent with a front factor g = j given by some recent theories of rubber elasticity (Part 7). The terminal spectrum for E 1 has the Rouse spacings for all practical purposes, shifted along the time axis by an undetermined multiplying factor (essentially the slow point friction coefficient). Thus, the model does not predict the terminal spectrum narrowing which is observed experimentally. 

The Chompff-Duiser procedure of symmetry-preserved decoupling does not appear to be applicable in an easy way to random networks in three dimensions. Junctions which anchor strands of unequal length can be decoupled, but with a considerable increase in difficulty (234). On the other hand, Chompff and Prins (235) have introduced a degree of randomness in the decoupling procedure (the distribution of slow points within each uncoupled chain of the equivalent system was taken to be random) without altering the results appreciably. Random decoupling is much easier to implement, as shown below. 

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