Layer normal

Chiral Smectic. In much the same way as a chiral compound forms the chiral nematic phase instead of the nematic phase, a compound with a chiral center forms a chiral smectic C phase rather than a smectic C phase. In a chiral smectic CHquid crystal, the angle the director is tilted away from the normal to the layers is constant, but the direction of the tilt rotates around the layer normal in going from one layer to the next. This is shown in Figure 10. The distance over which the director rotates completely around the layer normal is called the pitch, and can be as small as 250 nm and as large as desired. If the molecule contains a permanent dipole moment transverse to the long molecular axis, then the chiral smectic phase is ferroelectric. Therefore a device utilizing this phase can be intrinsically bistable, paving the way for important appHcations. 

For smectic phases the defining characteristic is their layer structure with its one dimensional translational order parallel to the layer normal. At the single molecule level this order is completely defined by the singlet translational distribution function, p(z), which gives the probability of finding a molecule with its centre of mass at a distance, z, from the centre of one of the layers irrespective of its orientation [19]. Just as we have seen for the orientational order it is more convenient to characterise the translational order in terms of translational order parameters t which are the averages of the Chebychev polynomials, T (cos 2nzld)-, for example... 

Fig. 9a,b. The intermoleailar scattering patterns calculated for a the smectic A b the smectic B phases of the mesogen GB(4.4, 20.0, 1, 1) with the scattering vector parallel to the layer normal. The scaled scattering vectors Q and Q range from -8n to 8n... 

The compounds crystallise in noncentrosymmetric space groups namely PI, P2i, C2, and P2i2i2i (but with priority of P2i) due to the chirality of the molecules. Most of the compounds have a tilted layer structure in the crystalline state. The tilt angle of the long molecular axes with respect to the layer normal in the crystal phase of the compounds is also presented in Table 18. Some compounds show larger tilt angles in the crystalline state than in the smectic phase. In the following only the crystal structures of some selected chiral liquid crystals will be discussed. 

Tilt angle of the long molecular axes with respect to the layer normal in the crystal structures. These compounds show a bent structure. 

Smectic A and C phases are characterized by a translational order in one dimension and a liquid-like positional order in two others. In the smectic A phase the molecules are oriented on average in the direction perpendicular to the layers, whereas in the smectic C phase the director is tilted with respect to the layer normal. A simple model of the smectic A phase has been proposed by McMillan [8] and Kobayashi [9] by extending the Maier-Saupe approach for the case of one-dimensional density modulation. The corresponding mean field, single particle potential can be expanded in a Fourier series retaining only the leading term ... 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

The IR dichroism measurements allowed a fairly precise determination of the preferential molecular conformations both in the smectic Ai and X phases (see Sect. 2.3). In the smectic Ai phase the biphenyl moiety is parallel on average to the layer normal, while the hydrocarbon and perfluorinated fragments are tilted at angles 18 and 32°, respectively. The phase transition to the smectic X phase is accompanied by a dramatic change in the main molecular conformation - now all the fragments are strongly tilted with respect to the layer normal (especially the biphenyl core which tilts at an angle of around 56°) (Fig. 12). 

In the standard setup W (y) is the profile of the primary beam in horizontal direction. In order to solve the smearing integral, the orientation distribution of the layer normals, g (), is approximated by a Poisson kernel121 and W (y) is approximated by a shape function with the integral breadth 2ymax of the primary beam perpendicular to the plane of incidence. In the simplified result... 

Figure 8.4 Illustration showing layer normal (z), director (n), and other parts of the SmC structure. Twofold rotation axis of symmetry of SmC phase for singular point in center of layer is also illustrated. There is also mirror plane of symmetry parallel to plane of page, leading to C2h designation for the symmetry of phase. This phase is nonpolar and achiral.

It is now instructive to ask why the achiral calamitic SmC a (or SmC) is not antiferroelectric. Cladis and Brand propose a possible ferroelectric state of such a phase in which the tails on both sides of the core tilt in the same direction, with the cores along the layer normal. Empirically this type of conformational ferroelectric minimum on the free-energy hypersurface does not exist in known calamitic LCs. Another type of ferroelectric structure deriving from the SmCA is indicated in Figure 8.13. Suppose the calamitic molecules in the phase were able to bend in the middle to a collective free-energy minimum structure with C2v symmetry. In this ferroelectric state the polar axis is in the plane of the page. 

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