Static permittivity, liquid crystals

Thus the number of independent components of the permittivity tensor will depend on the symmetry of the liquid crystal phase. The frequency dependence of the permittivity is described in terms of real and imaginary parts, and these also will be tensor quantities. Apart from complications of anisotropic internal fields, the static or low frequency part of the permittivity tensor can be related to the molecular polarizability and dipole moment averaged over the appropriate orientational distribution functions. 

In this section we wish to consider all the possible contributions to the electric permittivity of liquid crystals, regardless of the time-scale of the observation. Conventionally this permittivity is the static dielectric constant (i.e. it measures the response of a system to a d.c. electric field) in practice experiments are usually conducted with low frequency a.c. fields to avoid conduction and space charge effects. For isotropic dipolar fluids of small molecules, the permittivity is effectively independent of frequency below 100 MHz, but for liquid crystals it may be necessary to go below 1 kHz to measure the static permittivity polymer liquid crystals can have relaxation processes at very low frequencies. 

As was pointed out in Sec. 1 of this chapter the symmetry of the phases will determine the number of independent components of the second rank electric permittivity furthermore the point group symmetry of the phase and the constituent molecules will fix the orientational order parameters that contribute to a microscopic expression for the permittivity. In order to complete the description of the low frequency or static electric permittivity of liquid crystals, it is necessary to consider the additional effects of chirality, and the translational order associated with smectic phases. 

The dielectric theory for such a material has been given by Attard and is a development of the theories of Nordio et aV and Maier and Meier for the dielectric permittivity of a monodomain of LC material to include the case of partially aligned material. This theory involves the Wigner rotation matrix method for coordinate transformations. A simpler treatment has been given recently by Attard, Araki and Williams and uses a cartesian rotation matrix method for coordinate transformations. Maier and Meiershowed that the principal static permittivities for a monodomain of a uniaxial liquid crystal are given by... 

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