Twist-bend distortions

For each in a uniaxial phase there are two normal modes corresponding to a splay-bend distortion n q) and a twist-bend distortion ri2(q) biaxial liquid crystal phases have five normal modes for each value of q. The free energy density can be written in terms of the normal coordinates for torsional displacement in a uniaxial nematic as ... 

Defect-stabilised mesophase created when a smectic A mesophase is subjected to a twist or bend distortion. 

Note 1 The twist and bend distortions can be stabilised by an array of screw or edge dislocations. 

The three elastic constants are the Frank elastic constants, called after Frank, who introduced them already in 1958. They originate from the deformation of the director field as shown in Fig. 15.52. A continuous small deformation of an oriented material can be distinguished into three basis distortions splay, twist and bend distortions They are required to describe the resistance offered by the nematic phase to orientational distortions. As an example, values for Miesowicz viscosities and Frank elastic constants are presented in Table 15.10. It should be mentioned that those material constants are not known for many LCs and LCPs. Nevertheless, they have to be substituted in specific rheological constitutive equations in order to describe the rheological peculiarities of LCPs. Accordingly, the viscosity and the dynamic moduli will be functions of the Miesowicz viscosities and/or the Frank elastic constants. Several theories have been presented that are more or less able to explain the rheological peculiarities. Well-known are the Leslie-Ericksen theory and the Larson-Doi theory. It is far beyond the scope of this book to go into detail of these theories. The reader is referred to, e.g. Aciemo and Collyer (General References, 1996). 

Fig. 3.13.3. A hybrid aligned cell for the determination of the anisotropy of the flexoelectric coefficients. In this geometry, the director has a splay-bend distortion which gives rise to a flexoelectric polarization P. On applying an electric field E, the director is twisted by an angle (j> cc — which can be measured optically.

In other words, both twist and bend distortions are absent, leaving only the splay term in the Oseen-Frank free energy expression (3.3.7). It is seen from fig. 5.3.1, that by merely bending or corrugating the layers a splay deformation can be readily achieved without affecting the layer thickness. 

First detailed dynamic light scattering (DLS) experiments using bulk liquid crystal samples have confirmed the theoretically predicted existence of two dissipative fluctuation eigenmodes in the nematic liquid crystalline phase the first mode being a combination of splay and bend distortion and the second one a combination of twist and bend fluctuations [55,56]. Both modes are overdamped and the relaxation rate 1/r of each mode depends on the fluctuation wave vector q and viscoelastic properties of the sample [57] ... 

Fig. 7.3. The deviation of the optic axis in a cholesteric (hard-twisted chiral nematic, p <, where p is the cholesteric pitch and A is the wavelength of light) when an electric field E is applied perpendicular to the helical axis. The cholesteric geometry allows a fiexoelectric polarization to be induced in the direction of E. The plane containing the director, which is perpendicular to the page in the middle figure and is shown in the lower figure, illustrates the splay-bend distortion and the corresponding polarization that arises. (After Rudquist, inspired by Meyer and Patel.

In Fig. 8.10b, we see that the fluctuation mode i(q) is a mixture of the splay and bend distortions, and the component 2(q) is a mixture of twist and bend distortions. This may be clarified as follows the splay-bend (SB) mode on the left side of Fig. 8.10b corresponds to realignment of the molecules within the, z-plane as q evolves and there is no twist here. In contrast, on the right side of the same figure the molecules are deflected from the q z-plane of the figure therefore, the twist and bend are present but the splay is absent (TB mode). 

The splay and bend distortions are described by angle 0 while the twist distortion is related to a slight change of the period of the helical structure. The maximum values of two variables 9 and Aq are coupled to each other by equation... 

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... 

FIGURE 4 Splay, twist, and bend distortions of a nematic are shown. (Adapted from Ref. 3.)... 

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. 

In general, the free-energy density of the system consists of the terms from the three elastic torques (bend, splay, and twist) and the optional torque. The free-energy density term F associated with the bending distortion is given by... 

FIGURE 4.4. Crossed electric and magnetic fields, (a, b) Magnetic field stabilizes director orientation, the first-order Frederiks transitions in an electric field are possible, (c, d, e) Both magnetic and electric fields are destabilizing (c) twist, (d) splay-bend, and (e) twist-splay-bend distortions are possible dependent on the value of the electric and magnetic fields. 

The behavior of weakly twisted structures depends on the relative values of the elastic constants in (5.3) and (5.4). As we shall see in the next section, splay and bend distortions are often relaxed by twist. It is therefore important to know the elastic constants for different types of deformations these constants are specified by molecular structures and interactions. 

Polarized photomicrographs of smectic A samples show so-called focal-conic fan textures (Fig. 6.29). Similar but not identical structures are also found in smectic C phases. The origin of these structures is the preference for splay distortion as opposed to the unfavourable twist and bend distortions in these smectics. 

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