Director deformations, smectics

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]... 

Figure 40. The easy director deformation in a smectic A is pure splay which preserves the layer spacing everywhere.

The director deformations described by that do not lead to layer compressions, in the continuum range where the wavelengths A of the deformation are much larger than the molecular dimensions (A 10 nm) can be induced by stress K 27t/pf <10T N/m. This is usually smaller than of the layer compression modulus B l(f N/m , For this reason, deformations that do not lead to layer compression (such as splay in SmA) are usually called soft deformations, whereas those that require layer compression (such as bend and twist in SmA) are the so-called hard deformations. In SmC there will be six soft and three hard deformations, so it is basically impossible to take into account all elastic terms while keeping the transparent physics. (In the chiral smectic C materials, additional three terms are needed, as shown by de Gennes. ) Fortunately, however, the larger number of soft deformations enable for the material to avoid the hard deformations, which makes it possible to understand most of the elastic effects, even in SmC materials. 

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... 

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... 

Theoretical investigations by Brand [ 135] and Brand and Pleiner [136] predicted that a monodomain liquid-crystalline elastomer exhibiting a cholesteric or a chiral smectic C phase should display piezoelectric properties due to a modification of the pitch of the helix under strain. So, a piezoelectric voltage should be observed across the sample when a mechanical field is applied parallel to the helicoidal axis. In this description, the crosslinking density is supposed to be weak enough to allow the motion of the director, and deformations of the sample (compression, elongation, etc.) are assumed to be much smaller than those that should lead to a suppression of the helix. The possibility of a piezoelectric effect do not only concern cholesteric and chiral smectic C phases, but was also theoretically outlined for more exotic chiral layered systems such as chiral smectic A mesophases [137]. 

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-... 

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. 

Smectic elastomers, due to their layered structure, exhibit distinct anisotropic mechanical properties and mechanical deformation processes that are parallel or perpendicular to the normal orientation of the smectic layer. Such elastomers are important due to their optical and ferroelectric properties. Networks with a macroscopic uniformly ordered direction and a conical distribution of the smectic layer normal with respect to the normal smetic direction are mechanically deformed by uniaxial and shear deformations. Under uniaxial deformations two processes were observed [53] parallel to the direction of the mechanical field directly couples to the smectic tilt angle and perpendicular to the director while a reorientation process takes place. This process is reversible for shear deformation perpendicular and irreversible by applying the shear force parallel to the smetic direction. This is illustrated in Fig. 2.14. 

The term piezoelectric was borrowed from the physics of solids by analogy to the piezoelectric effect in crystals without center of symmetry. As a rule, the piezoelectric polarization manifests itself as a charge on the surfaces of a crystal due to a translational deformation, e.g. compression or extension. Piezo-effects are also characteristic of polar liquid crystalline phases, e.g., of the chiral smectic C phase. The polarization, we are interested now, is caused by the mechanical curvature (or flexion) of the director field, and, following De Gennes, we call it flexoelectric. 

The smectic A is an untilted phase in which the mass density wave is parallel to the director. The cost in free energy of buckling the layers into saddle-shaped deformations is low, with the result that it is relatively easy to construct devices that show bistability between a scattering focal conic director configuration in which the layers are buckled and a clear homeotropic configuration in which the director is perpendicular to the cell walls and the layers parallel to the walls. Transitions between these two textures have been exploited in laser-written projection displays and in both thermo-optic and electrooptic matrix displays. The various mechanisms employed are summarized in Fig. 12. 

Figure 1.12 Schematic diagram showing the deformation of the liquid crystal director and the smectic layer in the smectic-A liquid crystal.

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. 

In smectic A liquid crystals the only allowed deformation is specific undulation of the smectic layers, such that interlayer distance is kept constant and the director remains normal to the layer. According to [31] this deformation imposes the following limitation to the director field ... 

The same authors [116] theoretically studied deformations of the director of the Sc phase for the case of a field lying in the plane formed by the director and the normal to the smectic layers. Here the angle between the field and the normal to the layers varied within the Umits 0 < a < tt. It turned out that for tt/2 < a < tt rotations of the director around the normal to the layers should be observed, with the director deviating out of the plane containing the field. Depending on a, these field-induced transitions can have the character of first- or second-order transitions. 

Fig. 16 Smectic-C elastomer with uniform director orientation but conical layer distribution subjected to a second uniaxial mechanical deformation under an angle of 0 90° with respect to the first deformation axis (0 is the Sc tilt angle) [80, 116]...

In Sect. 3.2 it was shown that uniaxial stretching or compression of Sc elastomers does not produce macroscopically oriented samples. Due to the Sc symmetry, uniaxial deformation only induces a uniform orientation of the director but leaves the smectic layer normals conically distributed around the stress axis. Usually such elastomers remain opaque. For the preparation of Sc LSCEs a more complex orientation strategy is necessary which can be realized by deploying two successive orientation processes. 

The first effect, piezoelectricity, is in principle well known for many types of polar materials [119]. The asymmetry of the switching observed for photo-crosslinked elastomers (see Fig. 15) indicates stabilization of one particular switched state. This means that the sample must have a permanent dipole. If the sample is deformed, the director, and thereby the direction of the polar axis, changes. This effect is detected as an electric signal. Indeed a piezoelectric signal can be measured if the sample is deformed in the direction of the polar axis. As a consequence of the stabilization of the polar state, the sample shows a piezoelectric effect not only in the smectic-C phase, but also in the smectic-A phase [36]. The signal vanishes when the sample becomes isotropic (see Fig. 18). 

Electric field is also expected as an effective external field to drive finite and fast deformation in LCEs, because, as is well known for low molecular mass LCs (LMM-LCs), an electric field is capable of inducing fast rotation of the director toward the field direction [6]. This electrically driven director rotation results in a large and fast change in optical birefringence that is called the electro-optical (EO) effect. The EO effect is a key principle of LC displays. Electrically induced deformation of LCEs is also attractive when they are used for soft actuators a fast actuation is expected, and electric field is an easily controlled external variable. However, in general, it is difficult for LCEs in the neat state to exhibit finite deformation in response to the modest electric fields accessible in laboratories. Some chiral smectic elastomers in the neat state show finite deformation stemming from electroclinic effects [7,8], but that is beyond the scope of this article we focus on deformation by director rotation. 

The described fluorinated compound shows interesting elastic properties [159]. The sample remains fully transparent when stretched either parallel or perpendicular to the director (Sect. 5.1). These results have been correlated with high-resolution X-ray scattering [135]. An increase in the FWHM of the smectic peak found during stretching corresponds to a decrease of the average domain size from the original 180 nm down to about 45 nm at the threshold to plastic deformation. At this level, at... 

Figure 10.2. (a)-(c) Basic deformation modes of a nematic director field (a) splay deformation (divn + 0) (b) twist deformation (n curln 7 0) and (c) bend deformation (n X curln + 0). (d)-(f) The same deformations of the director field in a smectic phase. Only the splay deformation of the director field (d) is compatible with the constant layer spacing. A twist deformation (e) [bend deformation (f)j is only possible if screw dislocations [edge dislocations] appear. 

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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