Optical Properties of Uniaxial Phases

We begin with the electric displacement vector Dj = ZyEi where i, j = x , y, / are Cartesian coordinates and the summation over repeated indices is inferred. The tensor of dielectric permittivity is symmetric Sy = Ej,and generally (even for biaxial medium) has six independent components. If an insulator is placed in the electric field, the stored electric energy density is given by 

This is an equation of an ellipsoid arbitrary oriented with respect to any Cartesian frame [1]. The frame may be chosen in such a way that the eUipsoid will be oriented with its principal axes along the co-ordinate axes, fri the new frame x, y, z, the tensor is diagonal that is all the off-diagonal terms vanish  

The same energy may be expressed in terms of the electric displacement vector components  

11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals 

Finally we go back to the x, y, z space replacing vector by vector r. 

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One of the most common LC phases is the nematic, where the molecules have no positional order, but they have long-range orientational order. Thus, the molecules flow and their center of mass positions are randomly distributed as in a liquid. Most nematics are uniaxial the molecules are orientationally ordered about a common axis defined as the director and represented by the unit vector n (Table 2). It is very important to stress that nematics can be easily aligned by an external magnetic or electric field. Aligned nematics have the optical properties of uniaxial crystals and this makes them extremely useful in liquid crystal display technology. 

At temperatures above 232°C, we observe the colours of birefringent layers as previously, together with the threads characteristic of nematic phases. Below 213.5°C, the threads disappear and we observe the focal conic texture which typifies the smectic in Fig. 9.0c. In fact, certain regions of our preparation appear dark when viewed under crossed polarisers, whatever their disposition. Such an observation could only be possible if the optical properties of our smectic phase were the same as those of the uniaxial nematic described in the last section, and if this optical axis were parallel to the microscope axis. This implies that the smectic layers are parallel to the microscope slide in these regions of the preparation, and also that molecules are on average normal to the layers. Cooling further, at temperatures below 182.5°C, the dark regions become coloured and reveal a threadlike texture very similar to that of the nematic phases (see Fig. 9.10). 

Birefringence is the difference between two principal refractive indices of the polymer. Where it is optically uniaxial, the principal axes are normal to and perpendicular to the director. Of course, some smectic phases are optically biaxial as are some interesting biaxial nematic polymers. It is important to appreciate that birefringence is an optical property of a representative volume of the polymer with dimensions at least those of the wavelength of light. It is not a molecular property but is directly proportional to (P2(cos or)), at least... 

Twisting a nematic structure around an axis perpendicular to the average orientation of the preferred molecular axes, one arrives at the molecular arrangement commonly called cholesteric (Kelker and Hatz, 1980). The twisted nematic phase is optically uniaxial, however with the axis perpendicular to the (rotating) director. Such a mesophase combines the basic properties of nematics with the implications of chirality The structure itself is chiral and as a consequence, a non-identical mirror image exists as it is shown schematically in Fig. 4.6-7. Besides the order parameters mentioned before, the essential characteristics of a cholesteric mesophase are the pitch, i.e., the period of the helical structure as measured along the twist axis, and its handedness, i.e., whether the phase is twisted clockwise or anticlockwise. 

The local symmetry of smectic A phase is the same as that of the nematics, be., its point group is D h, while the symmetry of the smectic C phase is ( b/, (a ( 2 symmetry axis plus a reflection plane perpendicular to the axis). In addition, both smectic phases exhibit a one-dimension translational order. Owing to the difference in symmetry, the smectic phases show different optical properties. The smectic A phase is optically uniaxial, but the smectic C phase is optically biaxial. 

The dielectric tensor of an FLC at optical frequencies could be regarded as uniaxial [60-62]. Then the FLC possesses only two refractive indices, n along the director and n perpendicular to it. This approximation is confirmed by direct measurements of the FLC optical properties. As a rule, the biaxiality does not exceed 10 and tends to zero at the phase transition point Tc [60-62]. 

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