Splay smectic

Here Fq is tire free energy of the isotropic phase. As usual, tire z direction is nonnal to tire layers. Thus, two elastic constants, B (compression) and (splay), are necessary to describe tire elasticity of a smectic phase [20,19, 86]. 

The free energy density terms introduced so far are all used in the description of the smectic phases made by rod-like molecules, the electrostatic term (6) being characteristic for the ferroelectric liquid crystals made of chiral rod-like molecules. To describe phases made by bent-core molecules one has to add symmetry allowed terms which include the divergence of the polar director (polarization splay) and coupling of the polar director to the nematic director and the smectic layer normal ... 

The first term in (7) describes the coupling between the polarization splay and tilt of the molecules with respect to the smectic layer normal. This coupling is responsible for the chiral symmetry breakdown in phases where bent-core molecules are tilted with respect to the smectic layer normal [32, 36]. The second term in (7) stabilizes a finite polarization splay. The third term with positive parameter Knp describes the preferred orientation of the molecular tips in the direction perpendicular to the tilt plane (the plane defined by the nematic director and the smectic layer normal). However, if Knp is negative, this term prefers the molecular tips to lie in the tilt plane. The last term in (7) stabilizes some general orientation (a) of the polar director (see Fig. 7) which leads to a general tilt (SmCo) structure. 

Let us first consider the case where the preferred orientation of the polar director is perpendicular to the tilt plane (K > 0). The spatial variation of the layer normal and the nematic and polar directors is shown in Fig. 12. We see that regions of favorable splay (called blocks or layer fragments in Sect. 2) are intersected by regions of unfavorable splay (defects, walls). In the region of favorable splay the smectic layer is flat. In the defects regions the tilt angle decreases to reduce energy... 

Fig. 12 Layer and director structure in 2D phases which occur due to the preference of the system to polarization splay. Upper line, side view on the layer lower line, top view on the layer. Red arrows show the polar director. Blue nails show the projection of the nematic director to the smectic plane. There is no blue nail in the centre of the wall, meaning that the cone angle is reduced to zero...

Bent-core liquid crystals are especially interesting materials for basic research as in these systems the polar and tilt order are decoupled and polarization splay seems to be an inherent property of the system. Both effects lead to a variety of structures with unusual properties, e.g., formation of the 2D density modulated phases built of the smectic layers fragments. We have presented the current knowledge... 

Coleman DA, Jones CD, Nakata M, Clark NA, Walba DM, Weissflog W, Fodor-Csorba K, Watanabe J, Novotna V, Hamplova V (2008) Polarization splay as the origin of modulation in the B1 and B7 smectic phases of bent-core molecules. Phys Rev E 77 021703... 

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 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... 

Fig. 17. 2-dimensional cut of the undulation local order schematic picture. Only a part of the smectic layer undulates, i.e. the hackhones and the spacers. The layers undulations induce mounds and wells regions where the spacers adopt a splay-like configuration... 

The smectic A free energy reduces to the Landau-Ginsburg superconductor free energy provided that = 0 for splay and kt = kb for twist and bend. The full extent of the relationships between liquid crystals and superconductors in the de Gennes analogy are summarized in Table 1 [23],... 

Lyotropic liquid crystals are principally systems that are made up of amphiphiles and suitable solvents or liquids. In essence an amphiphilic molecule has a dichotomous structure which has two halves that have vastly different physical properties, in particular their ability to mix with various liquids. For example, a dichotomous material may be made up of a fluorinated part and a hydrocarbon part. In a fluorinated solvent environment the fluorinated part of the material will mix with the solvent whereas the hydrocarbon part will be rejected. This leads to microphase separation of the two systems, i.e., the hydrocarbon parts of the amphiphile stick together and the fluorinated parts and the fluorinated liquid stick together. The reverse is the case when mixing with a hydrocarbon solvent. When such systems have no bend or splay curvature, i.e., they have zero curvature, lamellar sheets can be formed. In the case of hydrocarbon/fluorocarbon systems, a mesophase is formed where there are sheets of fluorocarbon species separated from other such sheets by sheets of hydrocarbon. This phase is called the La phase. In the La phase the molecules are orientationally ordered but positionally disordered, and as a consequence the amphiphiles are arranged perpendicular to the lamellae. The La phase of lyotropics is therefore equivalent to the smectic A phase of thermotropic liquid crystals. 

I. 1. Smalyukh, N.A. Clark, Organization of the polarization splay modulated smectic liquid crystal phase by topographic confinement. Proc. Nat. Acad. Sci. USA 107, 21311-21315 (2010)... 

For a single, two-dimensional liquid crystal hpid bilayer, or a membrane, in a smectic A-like state, the director field is represented by the membrane normal n. Fiexoelectricity is then defined as a curvature-induced area membrane polarization, or, conversely, as an electric field-induced membrane curvature. Lipids and proteins are oriented parallel to each other along the local membrane normal in the flat state. A curvature of the membrane surface leads, indeed, to a splay type deformation of the molecular local director, with a splay vector S = (si - - S2)n, while a bend deformation along the membrane normal is not allowed because there is no third dimension. Then, obviously, the only polarization component points along the membrane normal. 

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. 

Fig. 19. Elastic constants ku and (i.e. splay and bend) as a function of concentration for a polysiloxane smectic copolymer of the type shown in Fig. 3(b) dissolved in a low molar mass cyanobiphenyl liquid crystal host.

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. 

In the smectic A phase the director is always perpendicular to the plane of the smectic layers. Thus, only the splay distortion leaves the interlayer distance unchanged [7], and only the elastic modulus K i is finite while K22 and Kzz diverge when approaching the smectic A phase from the nematic phase. On the other hand, the compressibility of the layered structure and the corresponding elastic modulus B is taken into account when discussing the elastic properties of smectic phases. The free energy density for the smectic A phase, subjected to the action of an external electric field, is... 

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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