Phantom network different cross-link

Figure 5. The ratio S (x)/S (x) plotted as junction of x for the phantom network at different cross-link functionalities, and for a fixed junction network. A is 2. Key Otfis3 n.f s 4 A, f is 6 V, f is 10 O is junctions fixed.

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... 

The quantities B, and g, are given by Eq. (18a) and (18b). Fitting stress-strain data at different degrees of swelling by combination of Eq, (33), (34) and (35) enables one to determine x, and (presumed to be equal to Mq). Flory and Erman have performed the fitting by assuming = 1 and have determined x, C and the phantom modulus for randomly cross-linked poly(dimethylsiloxane) (PDMS) networks. These results were recently further exploited by Queslel and Mark. 

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 ... 

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