Smectic-A Liquid Crystals

In an ideal single-domain SmA sample, in which the layers are parallel and equidistantly separated, the director axis components and riy are related to the layer displacement u(x,y,z), in the limit of small distortion, by the following relationships  

In the equilibrium case only u(xy,z) and its spatial derivatives are needed to describe elastic distortion in SmA. For example, a small director axis reorientation may be represented by 

The energy associated with this distortion, which corresponds to a compression of the layer, is given by 

This process is analogous to the compressibility of an isotropic liquid crystal. Typically, the compressibility B is on the order of 10 -10 erg cm . Assuming further that (i) there is no long-range transitional order in the plane, (ii) z and -z are equivalent (no ferroelectricity), and (iii) the deformation is small so that the molecules at any 

The first term on the right-hand side is the unperturbed free energy. The second term is the energy associated with layer compression. The third term is the splay distortion energy 2, which is identical in form to that in nematics. Similarly, 

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Synthesis of the first mesoionic nematic and smectic A liquid crystals derived from sydnones has been described and their self-organization into liquid crystal phases has been studied by optical, calorimetric, and powder X-ray diffraction methods <2005CC1552>. 

Figure 4.28 Schematic representation of (a) nematic (b) chiral nematic and (c) smectic A liquid crystals.

Similar considerations of symmetry apply in other systems, for example nematic liquid crystals and aligned short fibre composites have symmetry D h, smectic A liquid crystals D , while in copolymers and certain fibre composites examples of hexagonal symmetry may be found and translational symmetry may also be present, which is not found in petrology. 

In the framework of irreversible thermodynamics (compare, for example, [31, 32]) the macroscopic variables of a system can be divided into those due to conservation laws (here mass density p, momentum density g = pv with the velocity field v and energy density e) and those reflecting a spontaneously broken continuous symmetry (here the layer displacement u characterizes the broken translational symmetry parallel to the layer normal). For a smectic A liquid crystal the director h of the underlying nematic order is assumed to be parallel to the layer normal p. So far, only in the vicinity of a nematic-smectic A phase transition has a finite angle between h and p been shown to be of physical interest [33],... 

Smectic A liquid crystals are known to be rather sensitive to dilatations of the layers. As shown in [34, 35], a relative dilatation of less than 10-4 parallel to the layer normal suffices to cause an undulation instability of the smectic layers. Above this very small, but finite, critical dilatation the liquid crystal develops undulations of the layers to reduce the strain locally. Later on Oswald and Ben-Abraham considered dilated smectic A under shear [36], When a shear flow is applied (with a parallel orientation of the layers), the onset for undulations is unchanged only if the wave vector of the undulations points in the vorticity direction (a similar situation was later considered in [37]). Whenever this wave vector has a component in the flow direction, the onset of the undulation instability is increased by a portion proportional to the applied shear rate. 

In a smectic A liquid crystal one can easily define two directions the normal to the layers p and an average over the molecular axes, the director, h. In the standard formulation of smectic A hydrodynamics these two directions are parallel by construction. Only in the vicinity of phase transitions (either the nematic-smectic A or smectic A-smectic C ) has it been shown that director fluctuations are of physical interest [33, 44, 45], Nevertheless h and p differ significantly in their interaction with an applied shear flow. 

Ferroelectric liquid crystalline monomers (Fig. 21, XXXVII) bearing two terminal vinyl groups were polymerized directly from their smectic A" liquid crystal phase using a Grubbs-type initiator. 

The most simple molecular topology of such systems reported so far is a tetrahedral supermolecule obtained by reacting tetrakis(dimethylsiloxy)-silane with alkenyloxy-cyanobiphenyls (Fig. 22), as discussed previously. Such tetramers exhibit smectic A liquid crystal phases [179]. For such end-on materials, microsegregation at the molecular level favors the formation of the smectic A phases in preference to the nematic phase exhibited by the mesogenic monomers themselves. The use of different polyhedral silox-ane systems (Fig. 24) or the Ceo polyhedron as the template for multi- and polypedal hexakis(methano)fullerenes (Fig. 70) substituted with a large number of terminally attached mesogenic groups confirm the same tendency to the formation of smectic A phases (vide supra). 

Figure 13.21 Apparent moduli G (closed symbols) and G" (open symbols) as functions of frequency at a strain amplitude of 10% for (a) a PS-PI lamellar block copolymer and (b) a smectic-A liquid crystal, 8CB. In both (a) and (b), the curves labeled MD are macroscopically disordered, or unaligned, states...

The bend and especially the compressive moduli of lamellar block copolymers are therefore typically lower, or at least no higher, than those of small-molecule smectic-A liquid crystals, while the viscosities of the former are usually much larger than the latter. By analogy with nematics, for layered materials one can define characteristic Ericksen numbers as Erj = r]yh /K and Er = where is a characteristic viscosity and... 

There has also been a study of the flow properties of a version of the Gay-Berne fluid that can form smectic A liquid crystals [36]. It becomes flow unstable close to the nematic-smectic A (N-S ) transition point. This is in agreement with the theory by Brochard and Jahnig [37]. They predicted that the twist viscosity would diverge at this transition. Therefore the correlation function P (r) P"(0))g. i2 must also diverge. This means that the equality... 

Figure 1.27. Typical dislocations (a, b) and disclinations (c-f) in smectic A liquid crystals. (From Kurik Lavrentovich, 1988.)...

Shashidhar R and Ratna B R 1989 Phase-transitions and critical phenomena in polar smectic-A liquid-crystals— plenary lecture Liq. Cryst. 5 421-42... 

Garland C W, Nounesis G and Stine J J 1989 XY behavior for the heat-capacity at nematic-smectic-A liquid-crystal transitions Phys.Rev. A 39 4919-22... 

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... 

Structural forces due to long-range positional order are quite easily observed in the smectic A liquid crystals. SFA measurements have been performed on lamellar lyotropic smectics [42,43] and in thermotropic smectics [44-46]. These works extend to a nanometer scale the early studies on elasticity, viscoelastic response and layers instability of smectic A, observed in macroscopic wedge-shaped piezoelectric cells [47,48]. 

J.B. Fournier, G. Durand, Focal conic faceting in smectic-A liquid crystals. J. Phys. II Fr 1, 845-870 (1991)... 

A. Honglawan, D.A. Beller, M. Cavallaro, R.D. Kamien, K.J. Stebe, S. Yang, Pillar-assisted epitaxial assembly of toric focal conic domains of smectic-A liquid crystals. Adv. Mater. 23, 5519-5523 (2011)... 

D.J. Gardiner, S.M. Morris, H.J. Coles, High-efficiency multistable switchable glazing using smectic A liquid crystals. Sol. Energy Mat. Sol. C. 93, 301-306 (2009)... 

S. Shojaei-Zadeh, S.L. Anna, Role of surface anchoring and geometric confinement on focal conic textures in smectic-A liquid crystals. Langmuir 22, 9986-9993 (2006)... 

T. Ohzono, Y. Takenaka, J. Fukuda, Focal conics in a smectic-A liquid crystal in microwrinkle grooves. Soft Matter 8, 2438-2444 (2012)... 

It has also been used to study the second sound resonance in smectic A liquid crystals and measure the compression modulus. For measuring the flexocoefficients (ei — 63) and (ei + 63), hybrid-aligned nematic cells have been used extensively. AC techniques avoid problems associated with ionic impurities, but require elaborate numerical fitting of the data. Some observations on the pnblished measurements of flexo-coefHcients are made in Section 2.4, which ends with a few concluding remarks. 

The intensity of the scattering from the smectic A liquid crystal can then be written as... 

The equations presented above are derived based on the least ordered smectic liquid crystalline polymers, i.e., the smectic A liquid crystals. The more ordered smectic systems may generate X-ray scattering patterns similar to those of crystal structures. 

Note that, for infinitely thick parallelepiped (A oo), there is no diffraction, only directly transmitted beam is left and the integral becomes 8-function. Generally, the larger parallelepiped dimensions the narrower is the central peak. We shall come back to this point when discussing the diffraction on thin layers of a smectic A liquid crystal. 

However, there is a shift of the entire parallelepiped diffraction spectrum by q on the wavevector scale the curve for a parallelepiped without density modulation is centered at <7 = 0 whereas the curve for the modulated structure is centered at q = qo. Such a shifted angular spectrum of diffraction intensity is very similar to that observed on the freely suspended films of smectic A liquid crystals. It allows the determination of both the smectic layer period and the film thickness. 

In Section 5.7.2 we discussed a general problem of stability of one, two- and three-dimensional phases. Here, we shall analyze stability of the smectic A liquid crystal, which is three-dimensional structure with one-dimensional periodicity. The question of stability is tightly related to the elastic properties of the smectic A phase. Consider a stack of smectic layers (each of thickness Z) with their normal along the z-direction. The size of the sample along z is L, along x and y it is L, the volume is V = Lj L. Fluctuations of layer displacement u(r) = u(z, r i) along z and in bofli directions perpendicular to z can be expanded in the Fourier series with wavevec-tors q and q (normal modes) ... 

Prost, J., Pershan, P.S. Flexoelectricity in nematic and smectic-A liquid crystals. J. AppL Phys. 47, 2298-2313 (1976)... 

Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals... 

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