Distortion splay

A phase itself, the amplitude of the density modulation is constant and twist and splay distortions are forbidden, thus the expression for the free energy density simplifies to equation (C2.2.10). 

Consider elementary distortions of a nematic. The undistorted director n = (0,0,1) is aligned along the z-axis. Fig. 8.5a. For instance, at a distance 8x from the origin of the Cartesian frame O the director has been turned through some angle in the zOx plane like in Fig. 8.5b. The relative distortion is then described by the ratio of hn, an absolute change of the x-component of the director, to distance 8x, over which the distortion occurs. In the same sketch, but in the zOy plane we see similar fan-shape or splay distortion 8/iy. Thus for the two elementary splay distortions we write ... 

Why does the free energy density acquire this particular form First, in the curvature term with modulus Ku, we must use the second derivatives because the first derivatives correspond to a pure rotation of all the layers that does not cost energy. The higher derivatives are ignored for small distortions. For the compressibility term, the first derivative (du/dz) is sufficient. Second, both the compressibility and the curvature terms must be squared due to head-to-tail symmetry and parabolic form of the density increment gdisrgo as a function of distortion (Hooke s law). However, the question arises why is only splay modulus taken into account in (8.44) and not the other two Frank moduli K22 and 33. Considering the splay and bend distortions of the SmA phase in Fig. 8.24 we can see that only the splay distortion is allowed because it leaves the interlayer distance and the... 

Fig. 8.25 Distortion of a honieotropically aligned SmA liquid crystal by a corrugated surface of solid boundary plate with the dotted line pictured an exponential decay of the distortion (a) and the wave-like splay distortions in a thin layer with the arrows indicating the direction of the induced local pressure (b)...

Consider a SmA layer of thickness d between two glasses. Fig. 8.25b. Flat surfaces stabilise the parallel arrangement of the layers while thermal fluctuations excite wave-Uke splay distortions. In thin cells these fluctuations are markedly suppressed for d Lp they are very strong, for d Lp they are quenched for any wavevector. For a fixed cell thickness, we can find a critical wavevector q for... 

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. 11.25 Quadrupolar flexoelectric polarization. Undistorted nematic phase consisted solely of molecular quadrupoles (a) and appearance of a polar axis and flexoelectric polarization due to splay distortion (b). Note that in the lower part of (b) the density of positive charges is larger than in the upper part whereas in sketch (a) these densities are equal...

The director at one of the boundaries of a hybrid cell is aligned homeotropically, at the opposite boundary homogeneously as was shown earlier in Fig. 10.11. Therefore, a hybrid cell has intrinsic bend-splay distortion and must have a projection of the macroscopic polarization along the cell normal. We can clearly see in Figs. 10.11 and 11.25 how the quadrupolar polarization emerges. The molecules may have positive (eo > 0) quadrupoles shown in Fig. 11.25 or negative ones (fio < 0) seen in Inset to Fig. 11.26b. 

In the test cells to be discussed below, the values of the helical pitch and the tunable cell thickness are close to each other (about 28 pm). Therefore, as shown in Fig. 12.17 the full pitch structure (n = 2) is the most stable n means a number of half-pitches). The elastic energy of the two states (n = 0 and n = 2) is calculated with allowance for the twist, bend and splay distortions. Solid lines in Fig. 12.18 demonstrate dependencies of the elastic energy of the two states on thickness-to-pitch ratio in the absence of an external field. In the figure, the free energy is normalized to the unit cell area and factor dlK22. It is seen that the free energy for... 

Fig. 4. Because these three elastic constants are usually of similar magnitude for small-molecule nematics, one often refers to a single elastic constant, K, for the material. For polymeric materials, on the other hand, the three elastic constants can be very different and are indeed found to be very different experimentally [7]. For instance, in order to have a splay distortion, there must be a net excess of tails over heads of molecules, defined by the molecular orientation along the splay direction. In a polymeric system in which the molecular length is large, the density of chain ends is small, so splay becomes more and more energetically expensive with increasing molecular length. This and many other issues associated with polymeric liquid crystals are reviewed by Meyer [6].

The first term expresses the energy density associated with compressmn or expansion of the layers y is the compression or expansion layer-strain and B is the layer compressibility (not to be conf used with bulk compressibility). The second term represents the energy associated with a splay distortion (the molecules under splay. 

Let us go back to the discussion of the Frederiks transition in a homogeneously oriented nematic with positive dielectric anisotropy (splay distortion). A conventional sandwich cell is used which is very convenient in this case, because the Kerr effect is not observed when the light wave vector coincides with the field direction. Let us imagine that we are measuring the temperature dependence of the anchoring energy of the nematic using the saturation field for the complete director reorientation. For 5CB we have the left part of Fig. 4.39 [226]. 

FIGURE 4.44. Calculated values of the critical field in units (7r/d)(R 22/Ax) for the periodic splay-twist and uniform splay distortions [244]. 

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... 

Thus, even in this favorable case of the allowed splay-distortion, the Prederiks transition is, in fact, unobservable (ghost-transition). Instead, we observe a texture transition accompanied by the appearance of a number of defects. 

Polarized photomicrographs of smectic A samples show so-called focal-conic fan textures (Fig. 6.29). Similar but not identical structures are also found in smectic C phases. The origin of these structures is the preference for splay distortion as opposed to the unfavourable twist and bend distortions in these smectics. 

Figure 10. Principal radii of curvature in a saddle-splay distortion.

Here, Eq is the amplitude of the incident optical field, cOo the frequency of the incident light, V the scattering volume and R the distance between the scattering volume and the detector. The scattered light consists of two modes, splay/bend (a=l) and twist/bend (a=2), as shown in Fig. 2. It can be seen from Fig. 3 (a) that the mode I fluctuations can contain only contributions from the bend and splay distortions. Mode 2 fluctuations are in the perpendicular plane... 

In undertaking light scattering experiments from SmA samples, it is important to exclude effects due to defects and to wall induced undulations which are particularly pronounced in homeotropic geometries. Consequently, much of the work has been done on planar samples, showing that only splay distortions occur [100]. Clark and Pershan [101] studied the light scattering from racemic p-butoxybenzal-p-(j8-methylbutyl)aniIine... 

In the SmC phase, and Kf are the Frank elastic constants for bend and splay distortion of the C-director. In the tilted hexatic phase, they are the sums of the director and bond-orientational elasticities. The last... 

Figure 6. The explanation for the tiger skin texture sometimes adopted by chromonic N (and P) phases The banded pattern seen with crossed polars is thought to result firom a twisted rope-like arrangement of the columns. Such a pattern could be a way of accommodating initial misalignment in the sample with a minimum of splay distortion, (a) The postulated arrangement of molecular stacks in two adjacent twisted ropelike assemblies, (b) The gross structure of the sample showing the large-scale organization of the assemblies represented in (a), (c) The banded appearance of a sample when viewed vertically between crossed polars.

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