Statistical theory for real networks

Abstract

As already mentioned, the basic challenge in the molecular theory of rubber elasticity is to relate the state of deformation at the molecular level to the externally applied macroscopic deformation. The two models described in the previous chapter, the affine network and phantom network models, are the two simplest models adopted for this purpose. In the affine network model, the junctions are assumed to be embedded securely in the network structure, showing no fluctuations over time. As a consequence of being embedded in the network, the junctions translate affinely with macroscopic strain. No assumption is made with regard to the parts of a chain between its junctions. The junctions in the phantom network model, on the other hand, reflect the full mobility of the chains subject only to the effects of the connectivity of the network. The position of each junction may be defined in terms of a time-averaged mean location and an instantaneous fluctuation from it. According to this other extreme case (James and Guth, 1947), the mean locations of junctions transform affinely with macroscopic deformation, whereas the instantaneous fluctuations are not affected. The independence of the instantaneous fluctuations from the macroscopically applied state of deformation is a consequence of the phantomlike nature of the chains. During the course of their fluctuations the chains may pass freely through each other, being unaffected by the volume exclusion effects of neighboring chains and therefore by the macroscopically applied deformation.