Rubber stress-optical coefficient

Birefringence. The birefringence of a crosslinked Gaussian rubber subjected to an affine deformation is described by the theories of Kuhn and Grun (1 ) and Treloar (2). These predict a stress-optical coefficient given by... 

Fig. 42. Temperature dependence of the product of stress optical coefficient and temperature CT for natural rubber (NR), poly(ethylene) (PE) and the l.c. elastomer No. 2a from Table 10...

The orientation of the swelling agent (solvent or free chains) has to be taken into account in the analysis of the stress-optical behaviour of swollen networks. Specifically, the segment polarisability (relative to the network chains or to the diluent chains), as currently derived from stress-optical coefficients [33], may not be representative of intrinsic properties of isolated chains. Short-range orientational interactions between the probe molecules and network chains (and between the chains of the matrix itself) must be considered in the interpretation of opticoelastic properties of swollen (and dry) rubbers [67]. 

Ca is called the stress optical coefficient. The value of C depends on the chemical structure of the polymer and is somewhat temperature-dependent. The theory of rubber elasticity leads to the following expression ... 

The validity of the stress optical relationship for flowing liquids has been proved for ideal rubbers and flexible macromolecules at low extension. Many authors have shown that this relationship holds for a broader range of materials, providing the stresses are lower than 10 Pa in shear flow or 1 MPa in elongational flow (see Refs. [6, 13, 14] and part 3 of the present Chapter). Table 1 summarizes values of the stress optical coefficient C for some classical polymers. 

The affine rubber model and the stress-optical coefficient... 

This equation shows that the ratio of the birefringence to the true stress should be independent of stress. The expression on the RHS of equation (11.13) is known as the stress-optical coefficient. A test of equation (11.13) can be made by plotting An against cr, when a straight line should be obtained. Such plots for a vulcanised natural rubber at various temperatures are shown in fig. 11.5. The hysteresis shown in the curves for the lower temperatures is interpreted as being due to stress crystallisation, with the crystallites produced being oriented in the stretching direction and... 

An estimate of the number of monomer units per equivalent random link can be obtained by dividing the value of Aa calculated from the stress-optical coefficient by the anisotropy of the polarisability of the monomer unit calculated from bond polarisabilities. This number can more interestingly be expressed in terms of the number of single bonds in the equivalent random link and is found to be about 5 for natural rubber, about 10 for gutta percha and about 18 for polyethylene. (For the last two the values are extrapolated from measurements at elevated temperature.) The number for polyethylene is considerably higher than the value of 3 suggested by the assumption of totally free rotation around the backbone bonds (see section 3.3.3 and problem 3.7). 

FIGURE 6.16 Optical birefringence for 1,4-polybutadiene and natural rubber networks under tension and compression the stress optical coefficient is given by the slopes = 3.6 and 2.0 GPa, respectively (Mott and Roland, 1996). 

Fig. 6. The variation of the stress optical coefficient with cis content for 1,4-polybutadiene rubbers swollen with a number of solvents. The data include streaming birefringence data of Poddubnyi ct al. and of Phillipoff. (From Ref. 23.)...

We have presented the linear stress-optical rule here as a basic property of polymer melts but, of course, it also holds for rubbers, with unchanged stress-optical coefficients. This must be the case, since stresses arise from a network of chains in both melts and rubbers, so that the arguments presented above apply for both systems equally. Figure 7.25 shows as an example the relation between birefringence and tensile stress as observed for a sample of natural rubber. 

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