Splayed-twist state

It is reported that the splayed-twist state can be stabilized by the conditions shown below [18-21]. 

Fig. 4.7.9 LC molecular configuration of splayed-twist state (a) and reverse-twist state (b).

We mentioned that T- P=0 at the chevron interface, which means that even if P changes direction abruptly, Pj, (as well as any other component) is continuous across that surface (Fig. 96). As the boundary conditions at the substrates do not normally correspond to the director condition at the chevron, dPJdx is nonzero between the chevron and the substrates, but is small enough to be ignored. That this is not true when we have polar boundary conditions, is shown in Fig. 107. Let us, for instance, assume that the polarization P prefers to be directed from the boundary into the liquid crystal. Whether we have a chevron or not, we will have a splay state P x) corresponding to a splay-twist state in the director. With a chevron the splay is taking place in the upper or lower half of the cell when this is in its... 

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. 

According to Eq. (7.1) P is zero for the two cases of uniform director fields and pure twist. Hence both cases can serve as a zero state as far as flexoelectric excitations are concerned. It is important to note that a twist is not associated with a polarization (i.e. C2 is identically zero, cf. Fig. 7.2). An imstrained nematic has a centre of symmetry (centre of inversion). On the other hand, none of the elementary deformations - splay, twist or bend have a centre of symmetry. According to Curie s principle they could then be associated with the separation of charges analogous to the piezoeffect in solids. This is true for splay and bend but not for twist because of an additional symmetry in that case if we twist the adjacent directors in a nematic on either side of a reference point, there is always a two-fold symmetry axis along the director of the reference point. In fact, any axis perpendicular to the twist axis is such an axis. Due to this symmetry no vectorial property can exist perpendicular to the director. In other words, a twist does not lead to the separation of charges. This is the reason why twist states appear naturally in liquid crystals and are extremely common. It also means that an electric field cannot induce a twist just by itself in the bulk of a nematic. If anything it reduces the twist. A twist can only be induced in a situation where a field turns the director out of a direction that has previously been fixed by boundary conditions (which, for instance, happens in the pixels of an IPS display). 

The fiexoelectric coupling is not chiral, so what is the role of chirality in this case The answer is that the helically twisted state is the only one that is fiexoelectrically neutral (there is no local polarization related to twist) and therefore the only state from where a splay-bend deformation can increase continuously from zero in a symmetric fashion independent of the direction of E, while allowing for a homogeneously space-filling splay-bend. How it increases is illustrated in Fig. 7.5. 

Feiastic is the free energy density in the steady state due to the splay, twist, and bend deformations, and the helix twist, as given in Eq. (67),... 

As discussed in Sec. 2.2.2.1, the foundations of the continuum model were laid by Oseen [61] and Zocher [107] some seventy years ago, and this model was reexamined by Frank [65], who introduced the concept of curvature elasticity to describe the equilibrium free energy. This theory is used, to this day, to determine splay, twist, and bend distortions in nematic materials. The dynamic models or how the director field behaves in changing from one equilibrium state to another have taken much longer to evolve. This is primarily due to the interdependency of the director it (r, t) and v (r, /) fields, which in the case of chiral nematics is made much more complex due to the long-range, spiraling structural correlations. The most widely used dynamic theory for chiral... 

The free energy must be positive definite, otherwise the undistorted state would not correspond to the minimal energy. Because it is possible to generate pure splay, twist and bend, > 0 (i = 1,2,3). The saddle-splay deformation involves other deformations, too, so K24 is not necessarily positive. It can be shown that IK241< K22 and I - K22 - K241<... 

The three classical Freedericksz transitions will be considered in what are commonly called the splay, twist and bend geometries. Full details will be given for the splay geometry, with the corresponding results for the twist and bend geometries being stated since their analysis is analogously similar. The Section ends with a qualitative discussion for the one-constant approximation. 

Thus, in both [66] and [67], the 1,8-substituents appear to be attracting, rather than repelling one another and carbonyl substituents are twisted out of the plane of the aromatic ring and splayed back, in accord with the picture of the transition state given in [68] (Schweizer et al.. 1978). 

The three elastic constants of a liquid crystal are important physical parameters which depend on the interaction between the molecules in the liquid crystalline state. While a large number of theoretical and experimental investigations on the elastic constants are contained in the literature for thermotropic liquid crystals, very little is known about them in the case of lyotropic polymer liquid crystals such as those formed by poly-Y-benzyl-L-glutamate (PBLG) in various organic solvents. Some theoretical investigations have been carried out 3 the experimental data is limited largely to measurements of the twist elastic constant and a few recent measurements of the bend and splay constants. ... 

In the test cells to be discussed below, the values of the helical pitch and the tunable cell thickness are close to each other (about 28 pm). Therefore, as shown in Fig. 12.17 the full pitch structure (n = 2) is the most stable n means a number of half-pitches). The elastic energy of the two states (n = 0 and n = 2) is calculated with allowance for the twist, bend and splay distortions. Solid lines in Fig. 12.18 demonstrate dependencies of the elastic energy of the two states on thickness-to-pitch ratio in the absence of an external field. In the figure, the free energy is normalized to the unit cell area and factor dlK22. It is seen that the free energy for... 

The energy required to effect any deformation must be a quadratic function of the displacements. This is just the equivalent of Hooke s law as it is usually stated for solids. Since the energy must be invariant under any rotation of the reference frame, it can only be a function of scalar quantities, in this case the square of each of the terms taken separately. The contribution of the third term can be expressed as the sum of a surface term together with splay and twist terms which add to those found previously. 

FIGURE 14 Director configurations occurring during operation of a pi-cell. The fully relaxed splayed configuration H is switched into a fully ON state V, which can then be switched rapidly to V and back. On removal of the switching waveform, the cell relaxes first to form a 180° twisted structure T and eventually back to H. 

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