Linear stress-optical rule

Equation (7.159) is known as the linear stress-optical rule and is generally valid for all polymer melts. The proportionality constant Copt is called the stress-optical coefficient , and its value describes a characteristic property of each polymer. 

The linear stress-optical rule also holds under the conditions of simple shear flow. Observed data comply with the scenario depicted schematically in Fig. 7.22. The drawings show the principal axes of the stress tensor and of the optical indicatrix, for different shear rates. Data evaluation proves that the orientations of the two tripels of principal axes always coincide, as is indicated in the sketches. The inclination angle of the primary axis, 0c is 45° for infinitesimally small shear rates, and then decreases towards zero on increasing 7. The stress optical rule here reads... 

Indeed, validity of the linear stress-optical rule is a key observation with regard to the physical nature of the stresses created in flowing polymer melts. Generally speaking, stress in a polymer fluid arises from all forces acting between monomers on alternate sides of a reference plane. In polymers, we may divide them into two parts. We have first the strong valence bond forces... 

Using this representation of the chains in a melt, they may be entangled or not, we can deduce the linear stress-optical rule. Previously, in Eq. (6.60), we formulated an expression for the contribution of stretched springs to the shear stress azx... 

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. 

Validity of the linear stress-optical rule points at the dominant role of the network forces in pol3mier melts. The Lodge equation of state can be interpreted on this basis. We introduced the equation empirically, as an ap>-propriate combination of properties of rubbers with those of viscous liquids. It is possible to associate the equation with a microscopic model. Since the entanglement network, although temporary in its microscopic structure, leads under steady state conditions to stationary viscoelastic properties, we have to assume a continuous destruction and creation of stress-bearing chain sequences. This implies that at any time the network will consist of sequences of different ages. As long as a sequence exists, it can follow all imposed deformations. 

Scattering or form birefringence contributions will cause a deviation in the stress optical rule. As seen in equation (7.36), these effects do not depend on the second-moment tensor, but increase linearly with chain extension. 

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