Form birefringence

To calculate gt k), we first consider the Langevin equation under the macroscopic velocity field 

Adding this to eqn (5.105), we have the Langevin equation in the presence of the velocity gradient, 

The underlined terms can be calculated in the same way as in eqn (4.141), and gives IkaTL. Thus, using eqn (5.106) we have 

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On the other hand, for dilute solutions one must take care of the refractive indices of polymer and solvent. The refractive index increment dnjdc of the polymer for the chosen solvent must be as small as possible, since otherwise the influence of the shape of the coil (form birefringence) would be noticeable. A more detailed discussion of this problem, however, is postponed to Section 5.1.1. Solvents with a sufficiently small refractive index increment have been called matching solvents. 

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. 

As to this subject a few general remarks should suffice in the present context, as the decision was made at the beginning, to investigate only those systems in which the form birefringence is negligible. From this point of view, only a criterion for the absence of this effect would be needed. However, before this point is discussed, the reasons for the mentioned decision will be explained in more detail. 

The most evident reason is that dilute solution measurements can preferably be compared directly with the unmodified dilute solution theory as reviewed in Chapter 3. As has already been pointed out in Section 2.6.1, the form birefringence in dilute solution can effectively be suppressed by the choice of a solvent of practically the same refractive index as the polymer. In such a "matching solvent the contrast between the coil of the macromolecule and its surrounding practically disappears. This means that, at the same time, the influence of the shape (form) of the coil disappears. Also the comparison with measurements on con-... 

From the foregoing it becomes evident that only a measure of the magnitude of the form birefringence, but not of its influence on the extinction angle, can be given. For this purpose the reader may be reminded that the stress-optical coefficient of an infinitely dilute solution can be expressed by one half of the ratio of Maxwell constant to intrinsic viscosity [eq. (2.33)]. In the absence of the form birefringence the limiting... 

C0Pi6 (137) was the first to give an expression for the contribution of the form birefringence to the Maxwell constant. His theory is based on the elastic dumb-bell model, which has been used in early theories on flow birefringence and viscosity and which is identical with the model used in Sections 2.6.1 and 2.6.2. The ratio of Maxwell constant to intrinsic viscosity is probably unaffected by this simplification, when also the viscosity is calculated with the same model, as Copi6 did. For the absence of the form effect, this has strictly been shown in the mentioned Sections. In fact, in the case of small shear rates the situation is rather simple To a first approximation with respect to shear rate, the chain molecules are only oriented, their intramolecular distances which are needed for the calculation of form birefringence, being unaffected. 

As has been pointed out (63), this is a rather artificial model and, moreover, its application is quite unnecessary. In fact, (a> can be calculated from the refractive index increment (dnjdc), as has extensively been done in the field of light scattering. This procedure is applicable also to the form birefringence effect of coil molecules, as the mean excess polarizability of a coil molecule as a whole is not influenced by the form effect. It is still built up additively of the mean excess polarizabilities of the random links. This reasoning is justified by the low density of links within a coil. In fact, if the coil is replaced by an equivalent ellipsoid consisting of an isotropic material of a refractive index not very much different from that of the solvent, its mean excess polarizability is equal to that of a sphere of equal volume [cf. also Bullough (145)]. 

The Onuki-Doi Theory of Form Birefringence and Dichroism 75 which, when compared with (4.83), gives the result that... 

From the above expressions, the form birefringence and dichioism can be calculated. The form dichroism, in particular, has the simple interpretation of being the anisotropy in the second-moment tensor of the structure factor. It is also evident that the form dichroism appears at a higher order in the wave number than the form birefringence. 

Among the first theories of form birefringence was the calculation of Peterlin and Stuart [58] who solved for the anisotropy in the dielectric tensor of a spheroidal particle with different dielectric constants e j and e2 parallel and perpendicular to its symmetry axis, respectively. If the spheroid is aligned along the z axis, and resides in a fluid of dielectric constant e, the contribution of a single particle to the difference between the principal values of the macroscopic dielectric tensor of the fluid is... 

Theory of Copic for Form Birefringence of a Flexible Chain... 

In section 4.7, the Onuki-Doi theory for form birefringence and dichroism was developed and presented in equations (4.91) and (4.92). It is left to calculate the structure factor, S (q), as a function of flow. This was done in the limit of weak oscillatory flow for the... 

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. 

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. 

Ford equation for the viscosity of a suspension, 602 Form birefringence, 300 Formula of Seitz and Balazs, 450 Fourier number, 59 transform, 361... 

Nitrosonium tetrafluoroborate forms birefringent, orthorhombic, hygroscopic platelets, which decompose by water. It may be stored in glass bottles if dry, and is prepared by reacting sodium tetrafluoroborate with dinitrogen trioxide. Nitrosonium tetrafluoroborate is commercially available. 

The study of the birefringence of the Kraton single crystal and of the effect of stress applied along and perpendicular to the extrusion directfon has shown that the birefringence is entirely due to form birefringence and that the di ersed styrene phase and the butadiene matrix consist of randomly oriented chains Thisresults have been confirmed by infra-red dichroism studies ... 

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