Distortions director-field interactions

As mentioned earlier, most studies of field interactions with liquid crystals are done using thin films with a well-defined initial state, usually a monodomain or a thin film with a simple distortion induced by incommensurate surface anchoring. These conditions simplify observation and theoretical analysis. However, most liquid crystal materials that are not specially prepared contain topological defects that are very important to their response to external fields. One class of defect commonly observed in nematics is the disclinalion line. At a disclination line the director field is ill defined. The director field turns around the disclination line a multiple of half-integer times. Several disclination lines are shown in Fig. 8. 

For materials with > 0 the interaction free-energy minimum corresponds to a parallel alignment of the director with respect to the field. In nematic cells -such as the ones shown in Fig. 2.- a competition takes place between the orienting action of the substrates and that of the external field (unless the initial alignment of the director coincides with the direction of the applied field). As a result the initial director pattern becomes distorted. 

Associated with the distortion of the director field there is an elastic free-energy which was discussed by Ossen, Frank and others. (For details see the monographs. ) The equilibrium director configuration can be determined by minimizing the total free-energy, i.e. the sum of the interaction and the elastic contributions. An equivalent method is to determine the balance between the volume torques arising from the interaction with the field and from the distortion respectively, The former torque can be given as... 

Chiral liquid crystals belong to a wide class of soft condensed phases. The director field in the ground state of chiral phases is nonuniform because molecular interactions lack inversion symmetry. Among the broad variety of spatially distorted structures the simplest one is the cholesteric phase in which the director n is twisted into a helix. The spatial scale of background deformations, e.g., the pitch p of the helix, is normally much larger than the molecular size ( > 0.1 pm) since the interactions that break the inversion symmetry are weak. 

The first rheological theory for Uquid crystals was developed by LesUe and Ericksen, building on Ericksen s earUer transversely isotropic fluid. The theory is formulated in terms of a director field n, and it is similar to the fiber theory in the preceding section, except that it includes a contribution to the free energy from an interactive potential that causes the molecules to align at rest. The usual form of the free energy F resulting from distortions of the director field is... 

The first term is related to the bulk distortion energy, determined by the elastic deformation and external bulk field interactions. The second term is the surface contribution of the limiting surface D. The actual director field can be obtained by minimizing the energy. In a simple sandwich cell geometry (see Figure 4.27), the director (n(r)) can be expressed with the tilt angle as Q = cos (n k)> where k is the unit vector of the surface normal. 

There are four types of terms in Eq. [30], The first four terms concern only the value of the orientational order S(r). The next two terms account for spatial variation of S(r). Next there is a term concerned with the spatial variation of h(r) we have expressed this term in the familiar form of splay, twist, and bend distortions of the director field n(r). It should be noted that to second order in the Landau expansion there are only two independent elastic constants, L and L2, whereas in the nematic phase there are known to be three independent elastic constants. The last two terms in Eq. [30] represent the interaction between spatial variations of S(f) and spatial variations of n(r). Clearly, the mathematics can be quite complicated if S(r) and n(r) are allowed to vary simultaneously. 

The spatial and temporal response of a nematic phase to a distorting force, such as an electric (or magnetic) field is determined in part by three elastic constants, kii, k22 and associated with splay, twist and bend deformations, respectively, see Figure 2.9. The elastic constants describe the restoring forces on a molecule within the nematic phase on removal of some external force which had distorted the nematic medium from its equilibrium, i.e. lowest energy conformation. The configuration of the nematic director within an LCD in the absence of an applied field is determined by the interaction of very thin layers of molecules with an orientation layer coating the surface of the substrates above the electrodes. The direction imposed on the director at the surface is then... 

Here, the first term describes the nematic-like elastic energy in raie crmstant approximation (K in 9). This term allows a discussion of distortions below the AF-F threshold (a kind of the Frederiks transition as in nematics in a sample of a finite size). In fact, the most important specific properties of the antiferroelectric are taken into account by the interaction potential W between molecules in neighbour layers the second term in the equation corresponds to interaction of only the nearest layers (/) and (/ + 1). Let count layers from the top of our sketch (a) then for odd layers i, i + 2, etc. the director azimuth is 0, and for even layers / + 1, / + 3, etc. the director azimuth is n. The third term describes interactimi of the external field with the layer polarization Pq of the layer / as in the case of ferroelectric cells. Although for substances with high Pq the dielectric anisotropy can be neglected, the quadratic-in-field effects are implicitly accounted for by the highest order terms proportiOTial to P. ... 

In this section we will consider the orientational deformations of a nematic director which are flexoelectric in nature, i.e., induced due to the interaction of the flexoelectric polarization (see (4.2)) with the external electric field. Our consideration will be limited to spatially uniform fields E the case when E depends on coordinates is discussed in Chapter 5. We also discuss semiphenomenological approaches for the determination of nematic flexoelectric moduli ei and 63. Different types of electrooptical phenomena, where flexoelectric distortion plays a dominant role will be considered. Some of them are promising for potential applications. 

External field distortions in SmC and chiral SmC phases have been investigated [38], but the large number of elastic terms in the free-energy, and the coupling between the permanent polarization and electric fields for chiral phases considerably complicates the description. In the chiral smectic C phase a simple helix unwinding Fr6ede-ricksz transition can be detected for the c director. This is similar to the chiral nematic-nematic transition described by Eq. (83), and the result is identical for the SmC phase. Indeed it appears that at least in interactions with magnetic fields in the plane of the layers, SmC and SmC phases behave as two dimensional nematics [39]. 

Distortions due to Direct Interaction of a Field with the Director... 

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