The Stress-Optical Relation

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  

The terms T12 and Th — t22 correspond to the shear stress and the first normal stress difference, respectively. The SOR may now be rewritten 

It is important to emphasize that the SOR is not the inevitable consequence of fundamental physical principles rather, it is a very plausible hypothesis, which has extensive experimental support for polymer solutions and melts. In other words, there is no reason to assume that the SOR is valid under all possible flow conditions or for all possible polymer liquids. Some situations under which the SOR is expected to fail are mentioned in the next section. Many constitutive relations for solutions and melts predict that the SOR will hold, but even this apparent generality is somewhat misleading. The derivation of an SOR starts at a measurable molecular property, the optical polarizability of an isolated molecule a, and leads to a macroscopic refractive index tensor n, in a nontrivial way several substantial assumptions are necessary. Most rheological models (for flexible chains) that proceed to an SOR assume the derivation of Kuhn and Gritn (1942) for the polarizability anisotropy of a Gaussian subchain and thus in a sense make the same assumptions for the optical half of the SOR (Larson, 1988). Therefore differences between constitutive relations and their predictions for an SOR usually stem from differences in the calculation of t. 

Steady shear viscosity (open symbols) and (absolute value of the) stress-optical coefficient (solid symbols) (a) For melts of polystyrenes of narrow molecular mass distribution as functions of shear rate. 

x and T 2 reached their steady state values after about 100-200 seconds. From Janeschitz-Kriegl and Gortemaker (1974). 

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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. 

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. 

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... 

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. 

Birefringence induced by applied stress is caused by the two components of the refracted light traveling at different velocities. This generates interference which is characteristic of the material. The change in refractive index, An, produced by a stress S is often related by a factor C called the stress-optical coefficient as follows ... 

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. 

This relation has first been proposed by Philippoff (9). It becomes particularly suitable on condition that the stress-optical law, eq. (1.4), is valid. In this case dynamic measurements can be compared with the extinction angle of flow birefringence. 

As a consequence, the given proof for the validity of the stress-optical law remains formally true. The same holds for the relation between the diagonal components of the macroscopic stress tensor and the stored free energy per unit of volume. In fact, it does not make any difference, whether this energy is thought to be built up of the contributions of all complete chains or all subchains contained in the unit of volume. Only one statement will be revized, viz. that with respect to the coil expansion of the entire chain. A detailed discussion of this point will be given in Section 3.3. 

It is clearly seen that the validity of the stress-optical law is more general than that of the said relation for second order fluids. As the... 

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

In this situation, which is also discussed in Section 7.5, we refer to experimental evidence according to which components of the relative permittivity tensor are strongly related to components of the stress tensor. It is usually stated (Doi and Edwards 1986) that the stress-optical law, that is proportionality between the tensor of relative permittivity and the stress tensor, is valid for an entangled polymer system, though one can see (for example, in some plots of the paper by Kannon and Kornfield (1994)) deviations from the stress-optical law in the region of very low frequencies for some samples. In linear approximation for the region of low frequencies, one can write the following relation... 

Of course, these relations are trivial consequences of the stress-optical law (equation (10.12)). However, it is important that these relations would be tested to confirm whether or not there is any deviations in the low-frequency region for a polymer system with different lengths of macromolecules and to estimate the dependence of the largest relaxation time on the length of the macromolecule. In fact, this is the most important thing to understand the details of the slow relaxation behaviour of macromolecules in concentrated solutions and melts. 

The stress-optical coefficient C is defined by equation (10.27) and the relaxation times t,1 and t][ are defined by relations (2.30). One can see that the dynamo-optical coefficient of dilute polymer solutions depends on the non-dimensional frequency t w, the measure of internal viscosity ip and indices zv and 6... 

In solid photoelasticimetry, birefringence is related to local stresses through the stress optical law, which expresses that the principal axes of stress and refractive index tensors are parallel and that the deviatoric parts of the refractive index and stress tensors are proportional ... 

As will be shown below, the experiment in the sliding plate rheometer does not allow one to determine Nl, since the normal force is in fact related to the second normal stress difference. For this reason, we studied the stress-optical law in shear by assuming that the principal directions of shear and refractive index are close to each other in the x-y plane. It is then straightforward to express the difference of principal stresses in the x-y plane... 

LDPE is a highly branched materied, whose flow behavioTir exhibits some peculiarities, which constitute a good test for a numerical simulation. As presented in Section lll-l, birefringence patterns are perturbed around the re-entrant comer, which leads to the appearance of W-shaped fringes at the entry of the die land. It can be seen in Fig. 39 that the mPTT model ilows on to obtain these characteristic shapes (the computation is related to experiments carried out at 175 °C and 21 s, for an estimated value of the stress optical coefficient of... 

P and Q are also called as Neumann s constants and in turn they are related to Pockel s coefficients, which can be similarly defined. Stress optic coefficients are thus related to differences in refractive indices of e-and o- rays. Glasses containing PbO and such highly polarisable ions have been known to exhibit very low values of stress-optic coefficients. The stress optic coefficients have the units of inverse pressure and are of the order of lO Pa" (known as Brewster) or 10 cm dyne" and generally vary between 2.5 - 4.0 Brewsters in glasses. 

While we have thus been able to correlate certain conformation-related properties with connectivity indices, we have not been able to do so for many other conformation-related properties. For example, the very complex nature of the stress-optic coefficient defies a simple treatment. Such complicated conformation-related properties are still best predicted via more sophisticated calculations [6-9], such as those using rotational isomeric state theory. 

FIGURE 6.17 (Bottom) Birefringence for uncrosslinkedPIB after imposition of a shear strain = 2.5. Linear fitting yields 3.07 GPa - for the stress optical coefficient. (Top) Ratio of refractive index components (1 is the flow direction and 2 the direction of the gradient), which equals the value of the shear strain, in accord with the Lodge-Meissner relation (Eq. (6.66)) (Balasubramanian et al., 2005). 

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,... 

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