Smectic A fluctuations

Figure 5.21 Smectic A fluctuations, termed cybotactic clusters , in the nematic phase...

Figure 5. Susceptibility % and coherence lengths of smectic A fluctuations in the nematic phase of compound 40.8 (see Table 1) (after [37]).

The normal cybotactic phase, which is similar to the preceding, but where a is close to 90° (Fig. 2.7(b)). This corresponds to smectic A fluctuations. 

The smectic A phase is a liquid in two dimensions, i.e. in tire layer planes, but behaves elastically as a solid in the remaining direction. However, tme long-range order in tliis one-dimensional solid is suppressed by logaritlimic growth of tliennal layer fluctuations, an effect known as tire Landau-Peierls instability [H, 12 and 13]... 

Wlren a nematic phase is cooled towards a smectic A phase, fluctuations of smectic order build up. These fluctuations were called cybotactic clusters in tire early literature. Regardless of tire physical picture of such fluctuations. 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

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. 

C - nematic mesophase sequence, and in the case of n=12 the tilt angle of the smectic C phases decreased with temperature, resulting in the formation of an additional smectic A phase [73]. The phase transitions of these polymers are given in Table 13. WAXD studies of poly-(XX-n), n=2-7 confirmed that these polymers had a nematic phase. Some additional structural features in the X-ray pattern were interpreted as smectic C fluctuations. Poly-(XX-8), however, showed a smectic C mesophase similar to those of poly-(XX-n), n=9-12 [74]. 

This approximate expression, using the Maier-Saupe theory for S2 and 54 and taking R(p) 1, agrees reasonably well with measurements of X for a variety of liquid crystals (see Fig. 10-10), as long as there is no transition to a smectic phase near the temperature range considered. When a smectic-A phase is nearby, as is the case for 8CB, then smecticlike fluctuations of the nematic state can significantly reduce A. For 8CB, for example, A drops to around 0.3-0.4 when T — 34°C (Kneppe et al. 1981 Mather et al. 1995), which is around 0.7°C above the transition to the smectic-A phase. 

In a chiral smectic (Sc ) phase, the tilt angle is the same within a layer, but the tilt direction processes and traces a helical path through a stack of layers (Figure 43). It has been demonstrated that when such a helix is completely unwound, as in a surface stabilized ferroelectric liquid crystal cell, then changing the tilt of the molecules fi om +0 to —0 by alternating the direction of an applied field results in a substantial electro-optic effect, which has the features of veiy fast switching (%1 - lOps), high contrast and bistability [87]. The smectic A phase of chiral molecules may also exhibit an electro-optic effect, this arises due to molecular tilt fluctuations which transition is approached, which are combined with a... 

The ratio reaches up to one million times. This explains why the liquid crystal is in fact very turbid while ordinary liquid is transparent. The light scattering varies for different phases of the liquid crystals. For example, owing to the suppression of molecules into layers the light scattering of the smectic A phase is less than the nematic liquid crystal. For the smectic C phase, the fluctuation of the projection of tilted molecules onto smectic layers (the c-vector) causes stronger scattering than that in the smectic A phase. 

Als-Nielsen J, Litster J D, Birgeneau R J, Kaplan M and Safinya C R 1980 Lower marginal dimensionality. X-ray scattering from the smectic-A phase of liquid crystals Ordering in Strongly Fluctuating Condensed Matter Systems ed T Riste (New York Plenum)... 

The scan carried out just in the liquid phase of the tetradecyloxy homologue showed a diffuse peak with a peak position that was comparable to the layer peak position measured well within the temperature range of the smectic phase. Thus, this study showed that there were strong smectic-like fluctuations occurring in the liquid state before the TGBA is formed. 

To construct a continuum theory of S, we have to take into account firstly, the orientational fluctuations of the director about the layer normal (z axis), and secondly, as in smectic A, the distortions of the layers themselves. Expressions for the former were given by Saupe, but the complete theory including the latter contributions and the coupling between the two was derived by the Orsay group. We chose a cartesian... 

We can imagine a cholesteric as a smck of nematic quasi-layers of molecular thickness a with the director slightly turned by ( ) from one layer to the next one. In fact it is Oseen model [18]. Such a structure is, to some extent, similar to lamellar phase. Indeed, the quasi-nematic layers behave like smectic layers in formation of defects, in flow experiments, etc. Then, according to the Landau-Peierls theorem, the fluctuations of molecular positions in the direction of the helical axis blur the one-dimensional, long-range, positional (smectic A phase like) helical order but in reality the corresponding scale for this effect is astronomic. 

The result obtained has very interesting consequences (i) to have well aligned SmA samples, very flat glasses without corrugation are needed (ii) even small dust particles or other inhomogeneities create characteristic defects in the form of semispheres (see Fig. 8.29b below) and weU seen under an optical microscope (iii) layers are often broken (not bent) by external factors in particular, strong molecular chirality may result in the formatimi of defect phases like twist-grain-boimdary phase (iv) the thermal fluctuations of director in smectic A phase are weak and the smectic samples are not as opaque as nematic samples. In fact there is a critical cell thickness for short-wave fluctuations. 

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) ... 

According to Fig. 8.23 in Section 8.5.1, a fluctuation component of the layer displacement u = u along the x-direction (within the smectic layer) is described by u x) = ucosq x. Since the director angle 9/ Sm/Sx = q xw. the free energy density in Eq. (8.49) can be rewritten in terms of the 9y-angle fluctuations ... 

Fig. 11.12 Light scattering by the smectic A phase. Huctuating elastic modes in the z and x directions in a planar cell with director no II z (a) and typical geometry of scattering on fluctuating smectic layers (b)...

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