Fluctuating cross-link model

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

These measurements for the first time allowed experimental access to the microscopic extent of cross-link fluctuations. The observed range of fluctuation is smaller than predicted by the phantom network model, for which... 

Whereas k = 1.3 is derived from the above-presented NSE data, k = 2.75 is expected for a four-functional PDMS network of Ms = 5500 g/mol on the basis of Eq. (67). Similar discrepancies were observed for a PDMS network under uniaxial deformation [88]. Elowever, in reality this discrepancy may be smaller, since Eq. (67) provides the upper limit for k, calculated under the assumption that the network is not swollen during the cross-linking process due to unreacted, extractable material. Regardless of this uncertainty, the NSE data indicate that the experimentally observed fluctuation range of the cross-links is underestimated by the junction constraint and overestimated by the phantom network model [89],... 

A dynamical cross-linking simulation using the bond fluctuation model has been developed by Gilra and co-workers (241,242). They find a strong dependence of network properties on equilibration times, leading them to conclude that a 20% excess of cross-linker results in optimum properties. This conclusion is different from the assertion that balanced stoichiometry yields the best networks, as obtained both with statistical methods as well as the static MC method (226). As the authors point out, this may reflect slow relaxation of network systems beyond the gel point. The more perfect the network, the slower it will appear to relax because there are fewer fast-moving defect structures. 

The classical affinity model assumes that the doss-links are immobile with respect to the whole network. The fluctuations of the positions of cross-links induced by thermal motion are taken into account in the phantom model proposed by James and Guth. It suggests that the fluctuations of a given CTOss-link proceed independently of the presence of subchains linked to it, and during such fluctuations the subchains can pass freely through each other like phantoms. The classic phantom theory predicts the shear modulus G as ... 

In this model, the chains are viewed as having zero cross-sectional area, and can pass through one another as phantoms [2, 9, 28, 29], The cross-links undergo considerable fluctuations in space, and in the deformed state these fluctuations occur in an asymmetric manner so as to reduce the strain below that imposed macroscopically. The deformation thus viewed is very non-affine. Because of this reduction in the strain sensed by the network chains, the modulus is predicted to be diminished relative to that in Eq. (1.16) by incorporation of the factor < 1 ... 

The two models postulate an affine displacement of the positions occupied by the cross-links of the network resulting from a deformation, but differ about the movements undergone by these cross-links. For the Flory-Rehner affine model, cross-links move proportionally to the macroscopic deformation and remain in a given position of space at constant deformation. In the James-Guth phantom model, cross-links are assumed to freely move or fluctuate around an average position corresponding to the affine deformation. The amplitude of such fluctuations is independent of the deformation but depends on the valence of the cross-links and the length of elastic chains ... 

Actually these two models correspond to two extreme situations. The affine model is well appropriate to describe the case of networks made of short elastic chains—in this case, the fluctuation of cross-links is hindered by the presence of adjacent chains—whereas the phantom model is better suited to networks comprising... 

---