Deformation splay

Note 2 A splay deformation is described by the non-zero derivatives n dn jdx ) and n dn ldx where the symbols are defined in Definition 5.2. 

Fig. 28. Schematic representation of a splay deformation (a) changes in the components of the director n, defining the orientational change (b) splay deformation of an oriented layer of a...

The three kinds of deformations are associated with the variation of n, i.e. Vn. For the splay deformation, the divergence of the n vectors, V n, is not zero for the twist deformation, n V x n 0, and for the bend deformation, n x V x n 0. In order to describe the meaning of the three formulae, it is supposed that in the undeformed sample, n points along the z direction. These three deformations can hence be written in the form of components as follows... 

It is noted that there is no twisted deformation in liquid crystals, thus the K22 term does not appear in Fs. If the deformation is not so great, the splay deformation is dominant. When the deformation becomes greater, the bend deformation becomes more important. 

Figure 6.2. The segregation of molecular terminals in the splay-deformed domain of rod-like liquid crystalline polymers.

K is usually greater than Kzz for thermotropic liquid crystalline polymers. Two neighboring splay deformations tend to compensate each other. This effect tends to lower Kn. 

Here nd are elastic constants. The first, is associated with a splay deformation, K2 is associated with a twist deformation and with bend (figure C2.2.11). These three elastic constants are termed the Frank elastic constants of a nematic phase. Since they control the variation of the director orientation, they influence the scattering of light by a nematic and so can be determined from light-scattering experiments. Other techniques exploit electric or magnetic field-induced transitions in well-defined geometries (Freedericksz transitions, see section (C2.2.4.1I [20, M]. 

Figure C2.2.12. A Freedericksz transition involving splay and bend. This is sometimes called a splay deformation, but only becomes piuely 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. 3. Bend and splay deformations in nematic liquid crystals...

Fig. 3.13.2. Interpretation of the origin of flexoelectricity in an assembly of quadrupoles (a) in the undeformed state the symmetry is such that there is no bulk polarization, (6) a splay deformation causes the positive charges to approach the upper plane and to be pushed away from the lower one. This dissymmetry gives rise...

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. 

In precisely the same way, a spontaneously splay-deformed structure must correspond to the equilibrium condition with finite coefficient fsTi 7 0 in tensor (8.13). The corresponding term should be added to the splay term with (divn). If the molecules have, e.g., pear shape they can pack as shown in Fig. 8.7b. In this case, the local symmetry is Coov (conical) with a polar rotation axis, which is compatible with existence of the spontaneous polarization. However, such packing is unstable, as seen in sketch (b), and the conventional nematic packing (a) is more probable. The splayed stmcture similar to that pictured in Fig. 8.7b can occur close to the interface with a solid substrate or when an external electric field reduces the overall symmetry (a flexoelectric ejfecf). 

For saddle-splay deformations, principal curvatures ci and C2 measured at the mean surface are such that Ci = — C2. The mean curvature is therefore zero. However, at the interfaces and 2 each monolayer, shown in... 

Fig. 9.5. Saddle-splay deformation. Molecules (not represented) are everywhere perpendicular to the surface. Note that the polyhedron ABCD is a regular tetrahedron...

The second possibility to avoid the singularity at the center of the cylinder is to escape from the splay deformation to the bend deformation, as shown in Figure 1.21(b) [30-32]. The liquid crystal director tilts to the z direction and is given by... 

In the case of pear-shaped molecules, the value of the induced polarization is proportional to the splay deformation V n, and its direction is along n. In the case of the banana-shaped molecules, the induced polarization is proportional to the bend deformation n xVxn. Including both cases, the induced polarization is given by... 

Figure 4.5 Schematic diagram showing the flexoelectric effect in the splay deformation.

Smectics. The smectic phases are more rigid than the nematic phase. In order to preserve the layer spacing, twist and bend deformations are permitted only by the creation of a series of edge and screw dislocations, respectively. This makes them energetically expensive. Only the splay deformation, in which the molecules splay and the layer planes bend, is of low energy. 

It is well known that a nematic liquid crystal is nonpolar as a result of the free or hindered rotation of its constituent molecules around their axes. In the absence of an external field the distribution of the dipoles in an undistorted nematic liquid crystal has a nonpolar cylindrical symmetry. This is shown schematically in Fig. 4.29(a). However, as Meyer [183] has shown, a polar axis can arise in a liquid crystal made up of polar pear-shaped molecules when it is subjected to splay deformations, or in a liquid crystal made up of banana-shaped molecules subjected to bend deformations. In this case, the polar structure corresponds to closer packing of the molecules (Fig. 4.29(b)). Thus, the external mechanical deformation of the nematic liquid crystal results in the occurrence of a charge at electrodes perpendicular to the polar axis, i.e., there is a similarity to the piezoelectric effect in solid crystals. 

The most direct method of finding the coefficient en would be to fill the space between the metallic coaxial cylinders with a liquid crystal having pear-shaped molecules, with the surfaces of the cylinders having been pretreated for homeotropic orientation and to measure the potential difference between the cylinders. In fact, because of the difference in radii of the cylinders, the nematic liquid crystal structure proves to be splay deformed, and if the molecules have even a small longitudinal dipole moment the plates of the coaxial capacitor would be charged. However, despite its apparent simplicity, this experiment is, in fact, complicated because of the screening of the potential caused by the flexoelectric effect by firee charges from the liquid crystal and the atmosphere. 

This problem can be considered in the framework of the Helfrich approach to the nematic, though we have to take into account the specific viscoelastic properties of smectics and a proper sign of the conductivity anisotropy. First of all, it makes sense to consider only the onset of a splay deformation in a homeotropic structure for smectic A, since => 00. This approach is developed in [121], where the following expression for the threshold field of an instability is derived ... 

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