In the constrained-junction model presented in chapter 3, intermolecular correlations were assumed to suppress the fluctuations of junctions. According to this model, the elastic free energy of a network varies between the free energies of the phantom and the affine networks. In a second group of models, to be introduced here, there is a constraining action of entanglements along the chains that may further contribute to the elastic free energy, as if they were additional (albeit temporary) junctions. Consequently, the upper bound of the elastic free energy of such networks may exceed that of an affine network. Since the entanglements along the chain contour are explicitly taken into account in the models, they are referred to as the constrained-chain models. The idea of constrained-chain theories originates from the trapped-entanglement concept of Langley, and Graessley, stating that some fraction of the entanglements which are present in the bulk polymer before cross-linking become permanently trapped by the cross-linking and act as additional cross-links. These trapped entanglements, unlike the chemical cross-links, have some freedom, and the two chains forming the entanglement may slide relative to one other. The two chains may therefore be regarded as being attached to each other by means of a fictitious “slip-link,” as is illustrated schematically in figure 4.1. The entangled system of chains representing the real situation is shown in part (a), and the representation of two entangled chains in this system joined together by a sliplink is shown in part (b). The slip-link may move along the chains by a distance a, which is inversely proportional to the severity of the entanglements. A model based on this picture of slip-links was first proposed by Graessley, and a more rigorous treatment of the slip-link model was given by Ball et al. and subsequently simplified by Edwards and Vilgis; section 4.1 describes this latter treatment in detail. In section 4.2, we present the extension of the Flory constrained junction model to the constrained-chain model by including the effects of constraints along chains, following Erman and Monnerie. One of the newest approaches, the diffused-constraints model, is then described briefly in section 4.3.
Exact variational dynamics of the multimode Bose-Hubbard model based on
SU(M)
coherent states