Liquid crystal director elastic deformation

When a nematic liquid crystal is in the ground state the direction of the liquid crystal director n is uniform in space. When n is deformed, there will be elastic energy. Although liquid crystal director deformations cost energy, they do occur in reality because of surface anchoring, spatial confinements, impurities, irregularities, and externally applied fields [1,23]. In this section, we consider possible director deformations, associated elastic energies, and transformation between deformations 

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In nematic phase, the liquid crystal director it is uniform in space in the ground state. In reality, the liquid crystal director it may vary spatially because of confinements or external fields. This spatial variation of the director, called the deformation of the direetor, eosts energy. When the variation occurs over a distance much larger than the moleeular size, the orientational order parameter does not change, and the deformation ean be deseribed by a continuum theory in analogue to the classic elastic theory of a solid. The elastie energy is proportional to the square of the spatial variation rate. 

Smectic liquid crystals possess partial positional orders besides the orientational order exhibited in nematic and cholesteric liquid crystals. Here we only consider the simplest case smectic-A. The elastic energy of the deformation of the liquid crystal director in smectic-A is the same as in nematic. In addition, the dilatation (compression) of the smectic layer also costs energy, which is given by [23]... 

For a disclination with the strength S, as one approaches the center of the disclination, the elastic energy diverges, as shown by Equation (1.124). In reality this will not occur. The liquid crystal will transform either into isotropic phase at the center of the disclination or a different deformation where there is no singularity. Here we only discuss the cases of cylindrical confinements (two-dimensional confinement) where it is possible to obtain analytical solutions. The mechanism of liquid crystal director escape in spherical confinement (three-dimensional confinement) is similar to that of two-dimensional. 

We consider a nematic liquid crystal confined in a cylinder with a radius of R. The anchoring condition on the surface of the cylinder is perpendicular, as shown in Figure 1.21. The liquid crystal director aUgns along the radial axis direction, as shown in Figure 1.21(a), and is described by n =f. The elastic deformation of the liquid crystal director is splay with the strength of 5= 1. The elastic energy is... 

For most liquid crystals, the twist elastic constant is smaller than the bend elastic constant Therefore it is possible to reduce the total elastie energy by escaping fi om the bend deformation to the twist deformation as shown in Figure 1.21(c). The liquid crystal director is no longer on the r-z plane but twists out of the plane and is given by... 

In the toroidal droplet, the liquid crystal director is aligned along concentric circles on planes perpendicular to a diameter, as shown in Figure 11.15(b). There is a line defect along the diameter of the droplet. There is a rotational symmetry around the defect line. The bend elastic deformation is the only one involved. Toroidal droplets exist when bend elastic constant is smaller than the splay elastic constant otherwise the droplets take the bipolar configuration. Toroidal droplets rarely exist because for most liquid crystals the bend elastic constant is usually larger than the splay elastic constant. Nevertheless, toroidal droplets have been... 

The basic difference between deformations in a liquid crystal and in a solid is that in liquid crystals there is no translational displacement of the molecules on distortion of a sample. This is due to slippage between liquid layers. A purely shear deformation of a liquid crystal conserves elastic energy. The elasticity of an isotropic liquid is related to changes in density. In liquid crystals, variations in density can also be characterized by a suitable modulus, but the elasticity which is related to the local variation in the orientation of the director is their principal characteristic. 

The constants iFi=i-3 are the Frank elastic constants and correspond to the three fundamental deformations of the director field splay, twist and bend. Then-magnitude is of the order of 10 N, which is a fairly small value, meaning that on a length scale of several hundred microns (a typical size of a bulk sample), the nematic liquid crystal can easily deform or be deformed. It should be added that the Frank elastic constants depend on the order in the liquid crystal, being approximately proportional to 5. Consequently, the Frank constants Ki are temperature dependent and increase with decreasing temperature. 

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... 

Fgi is the elastic Frank free energy which describes the slowly varying spatial distortions of the director the free energy density f i is a function of the elastic modes of deformation of a nematic liquid crystal and is given by [19,23]... 

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

The stratified structure of a smectic liquid crystal imposes certain restrictions on the types of deformation that can take place in it. A compression of the layers requires considerable energy - very much more than for a curvature elastic distortion in a nematic - and therefore only those deformations are easily possible that tend to preserve the interlayer spacing. Consider the smectic A structure in which each layer is, in effect, a two-dimensional fluid with the director n normal to its surface. Assuming the layers to be incompressible, the integral... 

Fig. 5. The three principal elastic deformations induced by electric fields in nematic liquid crystals. In (a) the molecules are aligned in a planar texture with the director n parallel to the substrate for E = 0. In (b) the initial alignment is homeotropic with the director perpendicular to the substrate. The arrow f or circle indicates that E is applied either perpendicular or parallel to the substrate, respectively.

The situation is the same as in the twist geometry considered earher, in that the director has only x and y components, the field is in the y direction, and only twist deformation is present. The only differences are (1) there is an extra term in the elastic energy per unit volume due to the intrinsic chirality of the liquid crystal, and (2) the field energy per imit volume is due to a magnetic field rather than an electric field. With these two changes, the energy per unit volume is... 

Deformed cholesteric. Assume that Eq, (8) describes le initial structure of the director, but that the cell is filled ith a cholesteric liquid crystal with equilibrium pitch = lir/q. For p = we have an ordinary Grandjean struc-ire. Voxp q elastic stresses are present which in principle [lould be revealed when the cholesteric interacts with light, n the Mauguin limit, the equations describing this interac-on are given by (11) except that 2K -K 2K ... 

The only curvature strains of the director field which must be considered correspond to the splay, bend, and twist distortions (Fig. 2.17). Other types of deformation either do not change the elastic energy (e.g., the above mentioned pure shears) or are forbidden due to the symmetry. In nematic liquid crystals the cylindrical symmetry of the structure, as well as the absence of polarity (head to tail symmetry) must be taken into account. 

The theory of the elasticity of smectic liquid crystals has its own features. Deformations related to a change in the spacing between the layers are common to all smectic phases. The deformations is, in general, not related to a change in director orientation, and here an additional modulus of elasticity B occurs. 

The splay and bend elastic moduli, Ku and 33, also play an important role in the determination of a liquid crystal layer sensitivity and resolution. The smaller the elastic moduli, the larger the director deformation amplitude at a given voltage, Vq, i.e., sensitivity increases. This effect is particularly evident for low Ku values. The decrease in Ku leads to a considerable improvement in the layer spatial resolution. On the other hand, it would be unreasonable to decrease the X33 value dramatically, since, in this case, the resolution, t max, becomes worse. Consequently, we come to the conclusion that the optimum sensitivity and resolution of a liquid crystal layer with a homeotropic alignment can be achieved at K33 < 10 dyne and the highest possible elastic moduli ratio, KzzjKu [157, 163]. 

In the oblique evaporation process, a micro columnar structure is realized on the substrate surface, due to the self shadowing effect as shown in Fig. 3.3.1. When a nematic liquid crystal contacts such a surface, elastic deformation of the liquid crystal along the surface induces an interaction energy between the surface and the nematic material. This is thought to be the driving force for alignment of the nematic director. 

The bending elasticity of fluid membranes is closely related to the director field elasticity of liquid crystals. Of the three elastic deformations in nematics, which are splay, bend, and twist, only splay remains as it does in the case of smectics. In fact, a membrane is like an isolated smectic layer and this is why membrane curvature is sometimes expressed in terms of splay and saddle splay. 

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