Director distortion

Yokoyama, H., and Van Sprang, H. A. A novel method for determining the anchoring energy function at a nematic liquid crystal-wall interface from director distortions at high fields./. Appl. Phys. 57, 4520 (1985). 

Fig. 8. a Elastic dipoles and quadrupoles are easily recognizable between crossed polarizers since the director distortions regions differ in the two configurations. 1 two quadrupoles between uncrossed polarizers. 2 the same two quadrupoles between crossed polarizers. 3 dipoles in a chain. Scale bar. 6.3 pm. b Schematic representation of a chain of dipoles... 

We now consider a twist disclination loop in a twisted nematic. The nematic is supposed to have a planar structure with the director parallel to the xy plane and an imposed twist of q per unit length about the z axis, and the disclination loop of radius R is supposed to be in the xy plane. The director distortions are planar, = cos = sin = 0. On going once round the disclination line at any point on the loop, the director orientation changes by 2tis, the sign of which may be either the same as that of q or opposite. 

The nature of the core still remains an interesting unsolved problem. We have seen in 3.1.1 that director distortions have stresses associated with them as given by (3.3.4). In the case of a single disclination the stress is a tension which can be expressed as... 

If the director distortion is not constrained as in the above, and can depend on an additional coordinate (x or y), or if it is not confined to a plane, the bulk flexotorque is not zero, and these geometries can be used to measure the flexocoefficients, as we shall describe later. 

Frost and Pershan developed an interdigitated electrode technique (Fig. 2.1), which overcomes all the problems mentioned above. The field-induced director distortion generates a phase grating whose spacing due to... 

Homeotropic cells offer another way to detect the flexoelectric response via observing the bend director distortions induced by an electric field parallel to the substrates (the Helfrich method ). Takezoe s group applied it to ClPbislOBB and found 63 20 pCm (which is the order of flexocoef-... 

Preedericksz transition in planar geometry is uniform in the plane of the layer and varies only in the z direction. However, in some exceptional cases, when the splay elastic constant Ki is much larger than the twist elastic constant K2 (e.g., in liquid crystal polymers), a spatially periodic out-of-plane director distortion becomes energetically favourable. The resulting splay-twist (ST) Freedericksz state is manifested in experiments in the form of a longitudinal stripe pattern running parallel to the initial director alignment no x. 

Besides the elastic and the electric torques the so-called flexoelectric (or flexo) torques on the director play an important role as well. Their effect on pattern-forming instabilities in nematics is the main issue of this chapter. Flexotorques originate from the fact that typically (in some loose analogy to piezoelectricity) any director distortion is accompanied by an electric flexopolarization Pa (characterized by the two ffexocoefScients ei, 63). From a microscopic point of view, finite ei and 03 naturally arise when the nematic molecules have a permanent dipole moment. But also for molecules with a quadrupolar moment, finite ei and 63 are possible (see also Chapter 1 in this book ). Flexopolarization has to be incorporated into the free energy P n) for finite E. It is not surprising that this leads to quantitative modifications of phenomena, which exist also for ci = 63 = 0. Though, for example, the Freedericksz threshold field Ep is not modified, the presence of flexoelectricity leads to considerable modifications of the Freedericksz distorted state for E > Ep- ... 

More frequently, instead of the equilibrium pattern sketched so far, one observes electroconvection (EC) patterns in nematics, which present dissipative structmes characterized by director distortions, space charges and material flow. A necessary requirement for their existence is the presence of charge carriers in the nematic. In a distorted nematic, where n is neither parallel nor perpendicular to E, the generation of a non-zero space charge, pei, by charge separation is then inevitable. The resulting Coulomb force in the flow equations (generalized Navier-Stokes equations) drives a... 

The theoretical analysis of ffexodomains in planar nematics, which is briefly reviewed in the following, exploits the balance of torques by minimizing the total free energy E n). If not otherwise stated, an assumption of strong anchoring of the director n at the confining plates is used i.e. the director at the boundaries remains parallel to no x, irrespective of director distortions in the bulk of the nematic layer. 

The solutions of the linear equations for the director distortions nz y,z,t) and ny y,z,t) are periodic in the y direction as in the DC case, but depend explicitly on time. Since the hnear equations are invariant against a time shift by half a period, T t t + n/ut, we have two classes of solutions characterized by Triz(t) = iprizit) with p = 1. Which symmetry class is realized for flexodomains depends on w. The case p = 1 defines solutions with the so-called conductive symmetry , where the time average of over one period is finite the complementary case p = — 1 corresponds to solutions with dielectric symmetry where the time average of nz t) vanishes. The time symmetry of the in-plane director component is opposite, i.e. Triy t) = —pny t). 

There has been only one investigation of flexodomains in the weakly nonlinear regime for U > Vein the DC case. Based on a clever variational ansatz for the director distortion it has been demonstrated that the director amplitudes grow continuously as (U — (forward bifurcation). As a... 

Fig. 10.17 Twisted structure with a rigid boundary condition at z = 0 and soft boundary condition at z = d. The geometry of the director distortion (a) and illustration of the extrapolation length b and linear dependence of the director angle 4)(z) (b)...

Fig. 12.23 Flexoelectric distortion in a cholesteric liquid crystal, (a) The d.c. field from the source U is applied to the cell along the x-axis. (b) The field induced director distortion for positive and negative field directed perpendicular to the plane of the figure along the x-axis it is seen how the cholesteric quasi-layers are tilted though angle from their field-OFF configuration within the x, y-plane shown in the central sketch...

Figure 2.10 shows the director distortion for which only the first term on the right hand side of each of the above equations is non-zero. These distortions have been named splay, twist, and bend thus Kj, K, and are the splay, twist, and bend distortion constants. 

The hybrid (or homeoplanar) director profile was considered for both the field in the plane of the director distortion [15] and that perpendicular to it [16]. It was shown that in the latter case the effect is of a threshold character, and the threshold is lower than that for the homogeneous orientation [15]. 

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