Director elastic properties

Figure 5.48 Geometry discussed in Section 6.3 for tubule formation based on chiral elastic properties. Here, r is tubule radius, l is tubule length, n is molecular director, m is projection of n into local tangent plane (normalized to unit magnitude), <(> is angle in tangent plane between m and curvature direction (equator running around cylinder), and N is local normal vector. Adapted from Ref. 132 with permission of the author. Copyright 1996 by the American Physical Society.

The viscous properties of a smectic A are characterized by the same five independent viscosities that characterize the nematic. As we shall see, however, the elastic properties of the smectic are very different from those of a nematic, and some flows permitted to the nematic are effectively blocked for the smectic. For smectic C, for which the director is tilted with respect to the layers, there are some 20 viscosities needed to characterize the viscous properties (Leslie 1993). Formulas for these, derived using a method analogous to that used for nematics by Kuzuu and Doi (1983, 1984) can be found in Osipov et al. (1995). The smectic phase for which rheological properties are most commonly measured is smectic A, however, and hereafter we will limit our discussion to it. 

The static continuum theory of elasticity for nematic liquid crystals has been developed by Oseen, Ericksen, Frank and others [4]. It was Oseen who introduced the concept of the vector field of the director into the physics of liquid crystals and found that a nematic is completely described by four moduli of elasticity Kn, K22, K33, and K24 [4,5] that will be discussed below. Ericksen was the first who understood the importance of asymmetry of the stress tensor for the hydrostatics of nematic liquid crystals [6] and developed the theoretical basis for the general continuum theory of liquid crystals based on conservation equations for mass, linear and angular momentum. Later the dynamic approach was further developed by Leslie (Chapter 9) and nowadays the continuum theory of liquid crystal is called Ericksen-Leslie theory. As to Frank, he presented a very clear description of the hydrostatic part of the problem and made a great contribution to the theory of defects. In this Chapter we shall discuss elastic properties of nematics based on the most popular version of Frank [7]. 

As already mentioned, for the fixed direction of the nematic director n the shear modulus is absent because the shear distortion is not coupled to stress due to the material slippage upon a translation. The compressibility modulus B is the same as for the isotropic liquid. New feature in the elastic properties originates from the spatial dependence of the orientational part of the order parameter tensor, i.e. director n(r). It is assumed that the modulus S of the order parameter Qij r) is unchanged. In Fig. 8.4 we can see the difference between the translation and rotation distortion of a nematic. 

The SmA phase has the same symmetry and the same dielectric ellipsoid as in nematics, therefore, everything said above about the birefringence and dichroism is valid for the SmA phase. However, due to specific elastic properties of the layered structure, the director fluctuations are strongly quenched, and the SmA preparations are much more transparent than the nematic ones. This is related to specific elastic properties of the lamellar SmA phase [14]. 

In the smectic A phase the director is always perpendicular to the plane of the smectic layers. Thus, only the splay distortion leaves the interlayer distance unchanged [7], and only the elastic modulus K i is finite while K22 and Kzz diverge when approaching the smectic A phase from the nematic phase. On the other hand, the compressibility of the layered structure and the corresponding elastic modulus B is taken into account when discussing the elastic properties of smectic phases. The free energy density for the smectic A phase, subjected to the action of an external electric field, is... 

The described fluorinated compound shows interesting elastic properties [159]. The sample remains fully transparent when stretched either parallel or perpendicular to the director (Sect. 5.1). These results have been correlated with high-resolution X-ray scattering [135]. An increase in the FWHM of the smectic peak found during stretching corresponds to a decrease of the average domain size from the original 180 nm down to about 45 nm at the threshold to plastic deformation. At this level, at... 

These results are in good agreement with X-ray data d 4.3 nm, lb k Id k 18.5 nm, A/q = 2nlb/p 13° [6]). In addition to electron microscopy, atomic force microscopy (AFM) was also used to study a TGBa phase. The latter study indicates a surface modulation which corresponds to the pitch of the director field and was attributed to the elastic properties [106]. 

The elastic properties of liquid crystal phases can be modelled using continuum theory. As its name su ests, this involves treating the medium as a continuum at the level of the director, neglecting the structure at the molecular scale. 

The intense light scattering of nematics is due to thermally induced orientational fluctuations of the nematic director field. These orientational fluctuation modes are related to the viscous and elastic properties of the nematic (see, for example, Litster [27]). 

In general, the LC alignment originates from a symmetry breaking at the surface of the functionalized substrate. Every kind of siuface causes a particular orientation of the molecular director n close to it the alignment present at the surface propagates in the material over macroscopic distances, due to the elastic properties of LCs. 

The angular momentum conservation equation couples the viscous and the elastic effects. The angular profiles of the director and the effective viscosity data are computed for one set of material parameters based on published data in literature. The velocity profiles are also attained from the same dataset. The results show that the alignment of molecules has a strong influence on the lubrication properties. 

Physical properties of liquid crystals are generally anisotropic (see, for example, du Jeu, 1980). The anisotropic physical properties that are relevant to display devices are refractive index, dielectric permittivity and orientational elasticity (Raynes, 1983). A nematic LC has two principal refractive indices, Un and measured parallel and perpendicular to the nematic director respectively. The birefringence An = ny — rij is positive, typically around 0.25. The anisotropy in the dielectric permittivity which is given by As = II — Sj is the driving force for most electrooptic effects in LCs. The electric contribution to the free energy contains a term that depends on the angle between the director n and the electric field E and is given by... 

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