Stress-optical relation

Instead of checking the second stress-optical relation, viz. eq. (1.6), Philippoff preferred to use eq. (1.3), assuming % = %. That this assumption is justified, can be seen in the same Fig. 1.4, In this figure also the extinction angle is plotted against the shear stress. Orientation angles calculated with the aid of eq. (1.3), fit rather well on the extinction angle curve. The normal stress difference (pn — p22) has been measured in the way explained in the previous section. Similar results were published later by the same author for solution of carboxy methyl celluose in water (33) and for S 111 in Aroclor (34) a chlorinated biphenyl. 

From the derivation of the stress-optical relation, which has been sketched in Section 1.1 and which will be given in somewhat more detail in Chapter 2, it follows that only small deviations from the equilibrium state of chain conformations are permitted. As a consequence, the validity of eq. (1.22) becomes doubtful, when the shear stress is increased too much. 

Limits and Background of the Reviewed Theory 5.1. Stress-Optical Relation... 

Perhaps the most intriguing feature of the described theory is the validity of the stress-optical relation also for flowing systems. In fact, the validity of this relation forms the central point of the present concept, viz. the usefulness of flow birefringence measurements for the investigation of elastico-viscous properties. 

In the derivation of the stress-optical relation, as sketched in Section 2.6.1, several points of interest have only scarcely been touched The form birefringence in dilute solution, the nature of the anisotropy of the random link and the background of the quasistatic treatment. 

Fig. 7.22. Stress-optical relation as observed in polymer melts under simple shear flow Optical indicatrix (ellipsoids drawn with continuous lines) and stress tensor (ellipsoids depicted with broken lines) show equal orientations of the principal axes and proportionality between the birefringence nc — na and the principal stress difference o c — o-a- The inclination angle 6c decreases with increasing shear rate...

A serious problem in the measurement of stresses in glass is the fact that the stress field is rarely uniaxial. One usually needs to consider the effect of a biaxial stress field perpendicular to the direction of light propagation (a stress component along the direction of light propagation has no effect). The stress-optic relation has to be modified to read... 

Additional characterization is often required. Although the direct measurement of a modulus or a viscosity is often of immediate utility, in the optical case some additional measurements may be required to establish the relationship between the optical properties and the mechanical ones. For example, Ae stress-optical relation, to be discussed in Section 9.4, predicts that the shear stress and first normal stress difference may be obtained from birefringence measurements, but only after a quantity called the stress-optic coefficient is determined. 

The stress-optical relation (SOR) lies at the very heart of the use of flow birefringence in rheology (Janeschitz-Kriegl, 1969, 1983 A.S. Lodge, 1955 Tsvetkov, 1964 Fuller, 1990). Given a polymer liquid undergoing flow, both a stress tensor r and an index of refraction tensor n can be defined, The SOR comprises two statements about these tensors ... 

In an incompressible medium, the rheological state at a point is, as far as the stress is concerned, completely described by the shear stresses (three in a general flow) and in the differences of the normal or direct stresses. We shall denote components of the stress tensor by (TijiiJ= 1,2, 3) and suppose the stress tensor is symmetric, so that Gij=Oji. (If Cij is a shear stress if i=j, Gij is a direct or normal stress.) We can change axes by rotation to reduce Gij to principal form—in these (mutually orthogonal) principal axes the principal stresses are g, G2 and 0-3 and all shear stresses vanish. The simplest stress-optical relation is to suppose that the dielectric tensor (Kij) and the stress tensor are coaxial, i.e. have the same principal axes, and that the differences in principal stresses are proportional to the corresponding differences in (principal) refractive indices. Hence if 2( 3) is a principal axis, and g and G2 lie in the xy plane, we have the simplest stress-optic relation as... 

From a theoretical viewpoint, the stress-optical relation is found to be true for flexible macromolecules at low extensions and for rigid particles where Brownian motion dominates. Leal and Hinch have shown that the stress and optical ellipses are not coaxial for rigid particles undergoing weak Brownian motion. We shall return to this discussion of the range of validity of equation (8) later. If we accept this relation, then the principal stress differences can be found if A is given everywhere. 

The birefringence and isoclinic angles may be determined using the above methods. However, the rheologist usually desires the stresses. To do this, the stress-optic relation must be applied, as discussed in Section 20.2. The above discussion deals with steady flow situations. Transient flows (except those with very slow changes) need other methods and recently a number of new ideas have emerged. 

Peterlin plotted his optic and shearing stress data from the two apparatuses according to equation (21) and obtained a linear relation indicating a constant stress-optic coefficient. He also used the normal stress data to calculate Xia compared this to the measured Xo and found the two equal at equivalent shear rates. Therefore, the stress-optic relation was confirmed, at least for this system and level of principal stresses (di —(T2 2 x 10 Pa). 

To overcome the problem of shear field inhomogeneity a small angle (truncated) cone and plate geometry has been used by Janeschitz-Kriegl and co-workers. The stress-optic relation was found to be true for polymer melts in both transient and steady shear conditions up to shear stresses of the order of Pa (some deviation occurred in the transient measurement and was attributed to... 

Elongational flows have been produced using several apparatuses — fibre spinning, tensile experiments and the four-roll mill. What is immediately obvious is that the stress-optic relation appears to be invalid in a number of instances, but there are some cases where the simple stress-optic relation holds true in elongational flow these are mainly when stresses are small enough. 

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