The Stress-Optical Rule

Equation (9.194) is known as the linear stress-optical rule and is generally valid for polymer melts. The proportionality constant Copt is called the stress-optical coefficient, and its value is a characteristic property for 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. 9.24. 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 triples of principal axes always coincide, as is indicated in the sketches. The inclination angle of the primary axis, c, is 45° for infinitesimally small shear rates and then decreases towards zero on increasing 7. The stress optical rule here reads 

It can be verified by simultaneous measurements of the birefringence ric — ria, the inclination angle 6c and the shear stress Straightforward calculations 4eld the following relation between the principal stresses cr, c and the shear stress a x in the laboratory-fixed coordinate system  

Validity of the linear stress-optical rule is a key observation with regard to the physical nature of the stresses created in flowing pol3mier melts. Generally speaking, stress in a pol rmer fluid arises from all forces acting between 

Indeed, under this assumption the stress-optical rule can be verified and interpreted. We describe a chain in the spirit of the Rouse model, as sketched in Fig. 9.26. Each polymer is subdivided into sequences of equal size, long 

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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. 

The presence of the fourth-rank tensor in (7.127) and its absence in (7.11) suggests that the stress optical rule should not apply for dilute solutions of rigid rods. Unfortunately, because of the difficulty of acquiring truly rigid rods, and the problems of making measurements of stress in dilute systems, there are no data available on dilute rigid rod solutions where the stress optical rule can be investigated on this class of polymer liquids. 

Early work on the use of optical methods on the dynamics of polymeric liquids focused on establishing the validity of the stress-optical rule. A comprehensive account of this research can be found in the books by Janeschitz-Kriegl [29] and Wales [84], The majority of studies have considered simple shear flow, and the rule has been found to hold up to... 

Extensive work investigating the stress-optical rule has also been performed on polymer solutions [101]. Here the rule can be successfully applied if the solvent contributions to the birefringence are properly subtracted. Care must be taken, however, to avoid form birefringence effects if there is a large refractive index contrast between the polymer and the solvent. 

Investigations of the stress-optical rule in extensional flow are fewer in number due to the difficulty in establishing this flow. Using an extrusion device, van Aken and Jan-eschitz-Kriegl [102] produced extensional flows in melts by forcing the material through a... 

As discussed in section 7.1.6.4, semidilute solutions of rodlike polymers can be expected to follow the stress-optical rule as long as the concentration is sufficiently below the onset of the isotropic to nematic transition. Certainly, once such a system becomes nematic and anisotropic, the stress-optical rule cannot be expected to apply. This problem was studied in detail using an instrument capable of combined stress and birefringence measurements by Mead and Larson [109] on solutions of poly(y benzyl L-glutamate) in m-cresol. A pretransitional increase in the stress-optical coefficient was observed as the concentration approached the transition to a nematic state, in agreement of calculations based on the Doi model of polymer liquid crystals [63]. In addition to a dependence on concentration, the stress-optical coefficient was also seen to be dependent on shear rate, and on time for transient shear flows. 

For systems where the stress-optical rule applies, birefringence measurements offer several advantages compared with mechanical methods. For example, transient measurements of the first normal stress difference can be readily obtained optically, whereas this can be problematic using direct mechanical techniques. Osaki and coworkers [26], using a procedure described in section 8.2.1 performed transient measurements of birefringence and the extinction angle on concentrated polystyrene solutions, from which the shear stress and first normal stress difference were calculated. Interestingly, N j was observed to... 

Breakdown of the stress-optic rule can also occur in multiphase or multicomponent mixtures, as well as in melts with crystalline domains. However, as in glassy polymers, for miscible blends, a revised stress-optic law can sometimes be recovered by breaking the stress and birefringence tensors into two components, one for each component in the blend (Zawada et al. 1994 Kannan and Komfield 1994). 

Appendix 18.B — The Stress-Optical Rule in the Case of Simple Shear... 

As Txy is relatively easy to measure, Eq. (18.B.12) is often used to check the validity of the stress-optical rule and determine the value of C. 

Both Arixy and x are measured with a light beam directed perpendicular to the xy plane. Using the third relation of the stress-optical rule,... 

Here, 0 is the nematic coupling constant between the monomeric segments of the components. From Equation (3.57), we note that of the short-chain component does not fully relax even in the range of co where its modulus Gi (co) has fully relaxed but the modulus G2 ((o) of the long-chain component remains unrelaxed. This theoretical prediction is consistent with experiments. We can also confirm that Equations (3.56) and (3.57) give K (co) = Ki (co) + K2 (o) = C G/((o) + G2 (co) and are consistent with the stress-optical rule. Equation (3.53). 

FIGURE 15 Natural rubber sample on left is double network and on right is a conventional NR elastomer (a, b). Both are in a state of mechanical equilibrium (stress = 0). Nevertheless, as seen in (c), the double network transmits light through crossed polarizers, due to its inherent orientation. This is a violation of the stress optical rule, similar to that observed during creep recovery of uncrosslinked rubber. 

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