Affinely displaced cross-links

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 ... 

Here N is the number of chains in unit volume of the rubber and T is the absolute temperature. Note that the chain is defined here as part of a long polymer molecule between neighboring cross-linking points. The fundamental assumptions made in deriving Eq. (28) are that all chain ends are displaced to new positions by affine transformation when the rubber is deformed, that there are no intermolecular interactions, and that the end-to-end distance of each chain is much smaller than the contour length of the chain. 

Cross-linked hydrogels can be formed which contain affinity cross-links between polymer-supported hgands and receptors [46]. Such gels can be based only on these biophysical interactions as described by Taylor et al. [28,29]. In these systems, displacement of affinity cross-links by soluble competitors ultimately leads to a gel/sol transition (Figure 16.4) requiring that the responsive phase be constrained between two diffusive membranes to prevent leakage while in the sol phase (Figure 16.5). 

In contrast, the assumption of affine deformation is difficult to remove. The affine network theory assumes that each subchain deforms in proportion to the macroscopic deformation tensor. However, because the external force neither directly works on the chain nor on the cross-links it bridges, the assumption lacks physical justification. In fact, the junctions change their positions by thermal motion around the average position. It is natural to assume that the nature of such thermal fluctuations remains unchanged while the average position is displaced under the effect of strain. 

There is considerable evidence that all the hysteresis effects observed in these materials and most of the viscoelastic behavior can be caused by the time dependent failure of the polymer on a molecular basis and are not due to internal viscosity [1,2]. At near equilibrium rates and small strains filled polymers exhibit the same type of hysteresis that many lowly filled, highly cross-linked rubbers demonstrate at large strains [1-8]. This phenomenon is called the "Mullins Effect" and has been attributed to micro-structural failure. Mullins postulated that a breakdown of particle-particle association and possibly also particle-polymer breakdown could account for the effect [3-5]. Later Bueche [7,8] proposed a molecular model for the Mullins Effect based on the assumption that the centers of the filler particles are displaced in an affine manner during deformation of the composite. Such deformations would cause a highly non-uniform strain and stress gradient in the polymer... 

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