The Relative Permittivity Tensor

In order to examine the optical anisotropy, we begin with the relative permittivity tensor for the system , which is defined (see, for example, Born and Wolf 1970 Landau et al. 1987) by the relation 

One can make use of the heuristic model mentioned previously, in Section 1.1 each macromolecule consists of 2 segments and is surrounded by solvent molecules. It is not essential now to know whether the segments in the chain are connected or independent the results of this section are applicable in both cases. 

When considering the system consisting of solvent molecules and segments, the simple old-fashion (Vleck 1932 Frohlich 1958) speculations allow us to 

The time of relaxation of the mean orientation of the lateral vector ex is considered to be much less than the time of relaxation of the mean orientation of the axial vector e11, so that the last term in (10.3) can be neglected for rather low frequencies and one can continue with the simpler case (10.2). 

The true molecular field F acting both on the segment and on molecules of solvent differs from the average field E because the scale of the dimensions of the segments is molecular. Each solvent molecule makes an isotropic contribution to the polarisability vector the contribution of each segment of the macromolecule is anisotropic and is expressed by the formula 

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The written set of equations has a simple solution for the components of the polarisation vector. We use them to write, in accordance to equation (10.1), the relative permittivity tensor... 

The written relations define the relative permittivity tensor for the system, which is formulated below to within second-order terms in the orientation tensor... 

In conformity with the significance of the terms employed by investigators of anisotropy (Tsvetkov et al. 1964), the effects associated with the first-order terms in equation (10.6) may be called the effects of intrinsic anisotropy, while the second-order effects may be referred to as the effects of mutual interaction. In the second approximation, the principal axes of the relative permittivity tensor do not coincide, generally speaking, with the principal axes of the orientation tensor. It is readily seen that interesting situations may arise when Aa < 0 in this case, the coefficients of the first- and second-order terms have different signs. 

Equation (10.6), formulated in the previous section, defines the relative permittivity tensor in terms of the mean orientation of certain uniformly distributed anisotropic elements, which we shall interpret here as the Kuhn segments of the model of the macromolecule described in Section 1.1. We shall now discuss the characteristic features of a polymer systems, in which the segments of the macromolecule are not independently distributed but are concentrated in macromolecular coils. 

As before, we shall consider each macromolecule to be divided into N subchains and assume that every subchain of the macromolecule is in the equilibrium. So, using the above formula relating the tensor of the mean orientation of the segments of the macromolecules (ejek) to the distance between the ends of the subchains, we arrive from relation (10.6), taken in the first approximation, at Zimm s (1956) expression for the relative permittivity tensor... 

Expression (10.8) for the relative permittivity tensor in terms of the normal co-ordinates introduced by means of equations (1.13), assumes the form... 

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... 

One admits that the relative permittivity tensor of the system is determined by the mean orientation of the segments, so that we consider expression (10.13) to be equivalent to the first-order terms of relation (10.6) and, at comparison, obtain the expression for the mean orientation of segments of macromolecules in an entangled system... 

In the simplest cases, the optical anisotropy of polymer systems is studied under the conditions of simple elongation, when the elongation velocity gradient i/ii is given. The system investigated then becomes, generally speaking, a triaxial dielectric crystal with components of the relative permittivity tensor... 

For a system undergoing simple shear, when the velocity gradient v 2 7 0, the relative permittivity tensor is non-diagonal... 

Note that a frequency-dependent stress-optical coefficient C w) can be introduced by comparing the stress tensor and the relative permittivity tensor... 

Let us consider the anisotropy of polymer system undergoing simple steady-state shear. This situation can be realised experimentally in a simple way (Tsvetkov et al. 1964). The quantity measured in experiment are the birefringence An and the extinction angle x which are defined by formulae (10.19) and (10.20), correspondingly, through components of the relative permittivity tensor. 

One can turn to equation (10.10) to find the components of the relative permittivity tensor. Using expressions for the moments (2.42), one determines the gradient dependence of the quantities for dilute polymer solutions to within second-order terms... 

Now we refer to formula (10.13) for the relative permittivity tensor to determine the characteristic quantities in this case of strongly entangled linear polymers. We use expansions (7.32) and (7.43) for the internal variables to obtain the expression for the components of the tensor through velocity gradients... 

One can turn to discussion of the dynamo-optical coefficient, defined by equation (10.22). The expression for the relative permittivity tensor (10.10) and equation (2.41) for the moments allow one to write... 

The number of components in the relative permittivity tensor is limited by the sample symmetry. In axially oriented samples two independent components are found, parallel and perpendicular to the alignment direction (ey and e L or j and e2), since the sample is isotropic in the plane normal to the direction of orientation, i.e. ej = e2. 

The principal components of the relative permittivity tensor, measured parallel to the crystal axes, a, b and c, at 1 kHz and room temperature, are (Orczyk, 1990) ... 

To consider the relative permittivity tensor of polymeric system, one makes use of the heuristic model of a macromolecule as freely-jointed segments each macromolecule consists of z segments and is surrounded by solvent molecules (Sect. 2.1). Then, the simple old-fashion [111, 112] speculations allow us to determine the relative permittivity tensor of polymeric system in terms of the mean orientation of anisotropic segments of the macromolecules (6,6 ). The relative permittivity tensor is formulated below to within first-order terms in the orientation tensor... 

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