Theoretical schemes for orientation in polymers

The discussion in Section 1.5.1 above outlines the complexity of determining orientation in a polymer in the most general situation. There is therefore considerable merit in examining theoretical schemes for predicting the orientation from the macroscopic deformation which occurs during the drawing or forming process. 

One of the best-known of such schemes is the AFFINE deformation model for rubbers. The rubber is considered to be a network of flexible chains, and the macroscopic strain is imagined to be transmitted to the network such that lines joining the network junction points rotate and translate exactly as lines joining corresponding points marked on the bulk material. If we assume that the flexible chains consist of rotatable segments called random links , and that some statistical model can describe the configurational situation, it is then possible to obtain explicit expressions which relate the segmental orientation to the macroscopic deformation. 

Such ideas formed the basis of the Kuhn and Grun model for the stress-optical behaviour of rubbers. The birefringence of a uniaxially stretched rubber is given by 

Pi and P2 are the polarisabilities of the random link along and perpendicular its length respectively and A is the extension ratio or draw ratio. 

This expression has been shown to provide a quantitative understanding of the behaviour of rubbers. Because of the many configurational possibilities for the chains between the network junction points, the segmental orientation for strains 1(K)-2(X)% is much less than the orientation of the lines joining the junction points. The development of birefringence which reflects this segmental orientation is therefore slow, particularly at low extensions where the chains are still close to the 

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