Stress optical law

The above argument is crucially supported by the stress optical law, which is obtained by comparing eqn (7.4) with the formula for the intrinsic birefringence (eqn (4.238)) 

In concentrated solutions and melts, the intrinsic birefringence can be regarded as the total birefringence since the form birefringence is negligibly small as was shown in Chapter S. Thus eqn (7.6) is written as 

This is the stress optical law, which has been found experimentally to hold for rubbers and the more general problem of polymeric liquids.  

These results are fully confirmed by experiment. The only explanation for such a general relation is that both the stress and the birefringence have the same physical origin, i.e., the orientational ordering in the polymer segments. 

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Some Experimental Evidence for the Validity of the Stress-Optical Law 178... 

Fig. 1.4. Evidence for the validity of the stress-optical law [Philippoff (8, 9)]. 15 per cent solution of poiyisobutylene B-100 in decalin. (A) extinction angle % and flow birefringence An vs. shear stress pa at 30° C. (o) and An at 50° C. ( ) calculated % according to eq. 1.3 from cone-and-plate measurements at 30° C. ( , An sin2 at 30 and 50° C, respectively...

Fig. 1.4 gives such a plot, which was prepared by Philippoff (8,9) from his early measurements on a 15 per cent solution of polyisobutylene (P-100) in decalin (measurement temperatures 30 and 50°C). From this figure it is clearly seen that An as a function of p21 is non-linear. In contrast to the above mentioned solution of S 111 in methyl 4-bromo-phenyl carbinol, the solution of the poly-isobutylene P-100 in decalin does not form a second order fluid. However, for the product A n sin 2%, one obtains a beautiful straight line. The stress-optical law seems to hold also for this more general type of fluid. ... 

In Chapter 1 the validity of a stress-optical law has been presumed. Furthermore, it has been shown for several polymer systems that this law is, at least approximately, valid and that the second normal stress difference (p22 — p33) must be very small compared with the first normal stress difference (pn — 22). In the present chapter some theoretical considerations of a more general character will be reviewed in order to indicate reasons for this special behaviour of flowing polymer systems. Some additional experimental results will be given. 

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. 

The Behaviour of Single Model Chains 2.6.1. The Stress-Optical Law... 

Of particular interest in connection with the derivation of the stress-optical law is, that an analogous equation exists for the average polarizability tensor of the single chain 62). This expression is derived from the work of Kuhn and Grun (64) and reads ... 

It is immediately noticed that this factor does not depend on the length of the considered chain. [On the other hand, it depends on the chemical nature of the chain as a consequence of the appearance of fa — Oj), cf. Section 5.1.2]. Of particular importance is the elimination of the influence of distribution function ip, as in this way the influence of the state of flow (or strain in a permanent network) is eliminated. In this way, the validity of the stress-optical law, as described by eq. (1.4), seems fairly secured. As will be shown in Section 5.1.3, for flowing systems of uncross-linked chain molecules this result is due to the fact that the interaction of chains with their surroundings is thought to be concentrated in single points like their end-points. A discussion of the quality of this approximation will also be tried in that section. This approximation, however, has tacitly been made by all theories for Gaussain chains known at present. 

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. 

This statement suffices for the present purpose. In fact, a look on eqs. (2.14) and (2.15) which hold for the interesting moment of the memory function, makes the expectation acceptable that only a restricted number of the longest relaxation times will actually be of influence on the final results for slow steady shear flow, provided the g3 s are not too different. For a further discussion of the validity of the stress-optical law see Chapter 5. 

When the stress components are evaluated with the aid of flow birefringence measurements, the following expressions derived from the stress-optical law as formulated in eqs. (1.5) and (1.6), are used ... 

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... 

For the present purpose it should first be stated that the introduction of the subchain model does not change the character of the picture given for the elastic dumb-bell. A much more complicated situation exists, when real chain molecules are considered. It seems, however, that the statistical character of these chains, when they possess a Gaussian distribution of end-points, will suffice for an explanation of the validity of the stress-optical law. 

The flow birefringence pattern of these flows can be obtained through the use of a pair of flat glass walls. Using image enhancement and the stress-optical law (68),... 

The tensor of local anisotropy a is assumed to be determined by orientation of the segments of macromolecules which, according to the stress optical law (see Chapter 10), is proportional to the stress tensor, so that... 

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. 

One can see that, in the case when the intramolecular viscosity is neglected i Pi = 0), the frequency dependence of the components of the dynamo-optical coefficient (10.32) agrees with the analogous dependence of the shear viscosity (see equation (6.20) and Fig. 14). The stress-optical law can be written in the form... 

Validity of the stress optical law and application of birefringence to polymer complex flows... 

The purpose of the present chapter is to introduce the techniques of birefringence, to define the validity of the stress optical law (i.e. the linear relationship between stress and birefringence) in both simple (simple shear and uniaxial elongation) and complex (flow in abrupt contraction) deformations, and to propose some examples of the interest of these techniques for the study of the flow of molten polymers. 

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... 

The stress optical law has been carefully checked by Wales [6] in steady shear conditions for a wide number of molten polymers. However, the experimental set-up limited the range of stresses to a maximum of 10 Pa. In elongational situations, a departure from the linearity was previously observed... 

We presented some applications of the bireftingence techniques for the characterization of orientation and stress field in both simple and complex deformations. The validity of the stress optical law in these particular flow situations was checked and assessed. Birefringence techniques appear to be very powerful for the study of polymer melt processing and for the understanding of polymer flow behaviour. 

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