Bend directors

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-... 

Mode 1 splay/bend director confined to the x-z plane and... 

Mode 2 twist/bend director confined to the y-z ( 2- o) plane. 

Figure 37. An electric field E applied perpendicular to the helix axis of a cholesteric will turn the director an angle ) and thereby the optic axis by the same amount. The director tilt is coupled to the periodic splay—bend director pattern shown below, which is generated in all cnts perpendicular to the new optic axis. In this inverse flexoelectric effect, splay and bend will cooperate if and have the same sign. The relation between E and 0 is shown for a positive helical wave vector k (right-handed helix) and a positive average flexoelectric coefficient e= (Cs+ b)-When the sign ofE is reversed, the optic axis tilts in the opposite direction (0—>-0).

Illxistration of deformation of layered structures, (a) Layer splay (director bend) (b) twist (c) layer bend (director splay). It can be seen that only the layer bend leaves the layer spacing constant. 

Figure C2.2.11. (a) Splay, (b) twist and (c) bend defonnations in a nematic liquid crystal. The director is indicated by a dot, when nonnal to the page. The corresponding Frank elastic constants are indicated (equation(C2.2.9)).

If we compare with figure C2.2.I I, we can see that this defonnation involves bend and splay of the director field. This field-induced transition in director orientation is called a Freedericksz transition [9, 106, 1071. We can also define Freedericksz transitions when the director and field are both parallel to the surface, but mutually orthogonal or when the director is nonnal to the surface and the field is parallel to it. It turns out there is a threshold voltage for attaining orientation in the middle of the liquid crystal cell, i.e. a deviation of the angle of the director [9, 107]. For all tliree possible geometries, the threshold voltage takes the fonn [9, 107]... 

Figure C2.2.12. A Freedericksz transition involving splay and bend. This is sometimes called a splay defonnation, but only becomes purely splay in the limit of infinitesimal displacements of the director from its initial position [106]. The other two Freedericksz geometries ( bend and twist ) are described in the text.

Fig. 29. Schematic representation of a bend deformation (a) changes in the components of the director, n defining the orientation change (b) bend deformation of an oriented layer of a nematic liquid crystal.

Electric polarization resulting from a splay or bend deformation of the director of a nematic liquid crystal. 

Nonlinearity (or splay) of the material, where bending occurs perpendicular to the director... 

Bend of the material, where the distortion is parallel to the director and mesogen axis... 

For a nematic LC, the preferred orientation is one in which the director is parallel everywhere. Other orientations have a free-energy distribution that depends on the elastic constants, K /. The orientational elastic constants K, K22 and K33 determine respectively splay, twist and bend deformations. Values of elastic constants in LCs are around 10 N so that free-energy difference between different orientations is of the order of 5 x 10 J m the same order of magnitude as surface energy. A thin layer of LC sandwiched between two aligned surfaces therefore adopts an orientation determined by the surfaces. This fact forms the basis of most electrooptical effects in LCs. Display devices based on LCs are discussed in Chapter 7. 

Fig. 17a-c. Elastic constants for a splay b twist c bend deformations of a nematic phase. The full lines represent the director... 

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). 

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... 

Table 3.13 Transition temperatures (°C) and some values for the dielectric anisotropy (Ae), the ratio of the dielectric anisotropy and dielectric constant measured parallel to the director fAe/ej. and the ratio of the bend (kjj) and splay (kjj) elastic constants for the nitriles 36, 41, 53, 39,49 and 50... 

In thermotropic (solvent-free) smectic-A phases, two types of distortion are permitted, namely, splaying of the director (which corresponds to bending of the layers) and layer compression. Note The material itself is assumed to remain incompressible only the layers compress.) For weak distortions, the free energy cost of these is given by (de Gennes and Frost 1993)... 

Here u is the position of a layer plane and z is the position coordinate locally parallel to the director n, where n is parallel to the average molecular axis, which is assumed to remain normal to the layer plane, du/dz = e is the compressional (or dilational) strain. Thus, layer bending and layer compression are characterized by a splay (or layer-bend) modulus K and a compression modulus B. Other kinds of distortion present in nematics, such as bend or twisting of the director n, are not compatible with layers that remain nearly parallel, and hence are forbidden. Equation (10-36) is not invariant to rotations of frame, and its validity is limited to weak distortions a rotationally invariant expression has been given by -Grinstein and Pelcovits (1981).---------------------------------------------------------... 

Because of the difficulty with which polymeric nematic monodomains are prepared, there are few measurements of Leslie viscosities and Frank constants for LCPs reported in the literature. The most complete data sets are for PBG solutions, reported by Lee and Meyer (1990), who dissolved the polymer in a mixed solvent of 18% dioxane and 82% dichloromethane with a few percent added dimethylformamide. Some of these data, measured by light scattering and by the response of the nematic director to an applied magnetic field, are shown in Figs. 11-19 and 11-20 and in Table 11-1. While the twist constant has a value of around K2 0.6 x 10 dyn, which is believed to be roughly independent of concentration and molecular weight, the splay and bend constants ATj and K3 are sensitive to concentration and molecular weight. 

The liquid crystals can be deformed by applying external fields. Even a small electric or magnetic field, shear force, surface anchoring, etc., is able to make significant distortion or deformation to liquid crystals. Thus, n is actually a function of position r. According to the symmetry of liquid crystals there exist three kinds of deformations in liquid crystals splay, twist and bend deformations, shown in Figure 1.17. The short bars in the figure represent the projections of the local directors. 

The strain increases the energy of the solid as a stress is applied. The distortion of the director in liquid crystals causes an additional energy in a similar way. The energy is proportional to the square of the deformations and the correspondent coefficients are defined as the splay elastic constant, K, twisted elastic constant K22 and bend elastic constant Kx, i.e., the respective energies are the half of... 

The deformations in the smectic A phase liquid crystals are the bending of the smectic layer (accordingly to the splay of the directors) and the dilation or compression of the layers. The energy is thus... 

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