Incompressibility smectics

Theoretical work on dislocations in smectic liquid crystals was first done by de Gennes Q) followed by Pershan (2). For an incompressible smectic A liquid crystal in the hnear hydrodynamic approximation, the elastic strain can be described in terms of a single variable w(x,y,z) Aat specifies local displacements of the smectic layers. The stress-strain field was derived for an isolated dislocation in an unbounded liquid crystal media and extended their results to bounded media using the concept of an "image dislocation." The solution is valid for a thick wedge (relative to a characteristic length of the dislocation) of small angle. However, the... 

Consider the isothermal incompressible smectic A liquid crystal in the absence of significant body forces due to electromagnetic or gravitational fields. The elastic strain for the liquid crystal can be descril d in terms of a single variable w (x, y, z ) that specifies the local displacement of the smectic layers. The theoretical pr iction [2,3 or 7] is that w (x, y, z ) satisfies the differential equation... 

Many structural defects compatible with the incompressible smectic layers can be observed under a microscope. Among them are cylinders, tores and hemispheres observed at the surfaces, radial hedgehogs observed in smectic drops, etc. Three of them are presented in Fig. 8.29a-c. Note that in aU defect structures of this type, the splay distortion plays the fundamental role but bend and twist are absent. Other, more special defects, namely, the walls composed of screw dislocations, are observed in the TGBA phase. 

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)... 

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... 

An extension of rubber elasticity (i.e. of the description of large, static and incompressible deformations) to nematic elastomers has been given in a large number of papers [52, 61-66]. Abrupt transitions between different orientations of the director under external mechanical stress have been predicted in a model without spatial nonuniformities in the strain field [52,63]. The effect of electric fields on rubber elasticity of nematics has been incorporated [65]. Finally the approach of rubber elasticity was also applied recently to smectic A [67] and to smectic C [68] elastomers. Comparisons with experiments on smectic elastomers do not appear to exist at this time. Recently a rather detailed review of the model of an-... 

These are the most striking features of smectic textures [19]. Smectic layers of constant thickness (incompressible, modulus B— oo) form surfaces called Dupin cyclides. We have seen some of them, which have the form of tori including disclinations, see Fig. 4.7b. Such cyclides can fill any volume of a liquid crystal by cones of different size. An example is afocal-conic pair, namely, two cones with a common base. The common base is an ellipse with apices at A and C and foci at O and O , see Fig. 8.30a. The hyperbola B-B passes through focus O. The focus of... 

As the membrane is in the fluid state of matter (i.e. smectic A in liquid crystal terminology), it cannot withstand shear in its plane. Moreover, the solubility of the double chain phospholipids is extremely low. Therefore, there is practically no exchange of material between membrane and solution. This fact, together with the small compressibility of the membrane, implies that for almost all phenomena the membrane can be considered as locally incompressible. 

Flexoelectric phenomena in smectic C phases are considerably more complicated than in smectic A phases, even in the case of preserved layer thickness. Thus under the assumption of incompressible layers, there are not less than nine independent flexoelectric contributions. This was first shown in [ 127]. If we add such deformations which do not preserve the layer spacing, there are a total of 14 flexocoefficients [128]. If we add chirality, as in smectic C, we thus have, in principle, 15 different sources of polarization to take into account. 

The general problem of tracing the deformations coupled to polarization in smectic C phases is complex, as has already been stated. We refer to [127] for the derivation in the incompressible case and to the discussion in [128] for the general case. In the following we only want to illustrate the results in the simplest terms possible. This is done in Fig. 42, where we describe the deformations with regard to the reference system, k. 

To proceed from simple to more complicated, let us first assume that the smectic layers (Figure 4.10) are incompressible, i.e., only those deformations are allowed that leave the smectic layer spacing constant. It is obvious that... 

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