Smectic layers, density wave

On passing from the least ordered smectic A phase down to the more ordered crystal H and K phases the layer planes tend to sharpen up. In the smectic A and C phases the layers are therefore very diffuse, and can be thought of as one-dimensional density waves relative to the director. Thus, locally these two phases are very similar in structure to the nematic phase. 

Liquid-like layers with the molecules upright on the average (fig. 1.1.5(a)) negligible in-plane and interlayer positional correlations. Thus the structure may be described as an orientationally ordered fluid on which is superimposed a onedimensional density wave. A number of polymorphic types of smectic A have been discovered (see 5.6). 

As will be seen later ( 5.3.1), experiments have confirmed that the density wave in smectic A is, in fact, very well represented by a sinusoidal function, indicating that higher terms in the Fourier expansion can be neglected. The form of the potential (5.2.2) ensures that the energy is a minimum when the molecule is in the smectic layer with its axis along z s and O are order parameters which we shall define presently. 

Fig. 6.9 Below. A schematic picture of molecular packing in the vertically oiicmled smectic layers. Above. Average density p modulated with amplitude pi and period / of the density wave...

For discussion of dynamics of lamellar smectic phases it is important to include another variable, the layer displacement u (r) [3] or, more generally, the phase of the density wave [4]. This variable is also hydrodynamic for a weak compression or dilatation of a very thick stack of smectic layers (L oo) the relaxation would require infinite time. On the other hand, the director in the smectic A phase is no longer independent variable because it must always be perpendicular to the smectic layers. Therefore, total number of hydrodynamic variables for a SmA is six. For the smectic C phase, the director acquires a degree of freedom for rotation about the normal to the layers and the number of variables again becomes seven. 

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. 

Fig. 1. Schematic representation of (a) nematic, (b) smectic and (c) cholesteric (or chiral nematic) liquid crystalline phases. In the nematic phase only orientational correlations are present with a mean alignment in the direction of the director n. In the smectic phase there are additional layer-like correlations between the molecules in planes perpendicular to the director. The planes, drawn as broken lines, are in reality due to density variations in the direction of the director. The interplane separation then corresponds to the period of these density waves. In the cholesteric phase the molecules lie in planes (defined by broken lines) twisted with respect to each other. Since the molecules in one plane exhibit nematic-like order with a mean alignment defined by the director n, the director traces out a right- or left-handed helix on translation through the cholesteric medium in a direction perpendicular to the planes. When the period of this helix is of the order of the wavelength of light, the cholesteric phase exhibits bright Bragg-like reflections. In these illustrations the space between the molecules (drawn as ellipsoids for simplicity) will be filled with the alkyl chains, etc., to give a fairly high packing...

The Landau-de Gennes theory for the nematic isotropic transition can be extended to the smectic A-nematic transition. The order parameter for this transition is rl), the amplitude of the density wave describing the formation of layers in the smectic A phase. Since the difference between a value of rlr and -Irlrl only amonnts to a shift of one half layer spacing in the location of all the layers (and therefore no change in the free energy per nnit volume), the expansion in terms of powers of rlr can only contain even powers. Hence the free energy per unit volume in the smectic A phase can be written as... 

The Maier-Saupe theory can also be extended to describe the smectic A-nematic transition in what is called McMillan s model. Two order parameters are introduced into the mean-field potential energy function, the usual orientational order parameter S and an order parameter a related to the amplitude of the density wave describing the smectic A layers,... 

The mesophases differ from each other regarding the positional order of the molecules (Fig. l). In the nematic phase there is no long range positional order at all just as in isotropic liquids. Nematics are normally uniaxial, however biaxial nematics were discovered very recently. In the smectic phases the centre of masses of the molecules are concentrated in layers forming a one-dimensional density wave. In the smectic A and C phases there is no long-range positional order within the layers. The smectic A phase is uniaxial, the director (n) is parallel with the layer normal, 1. In the C phase the director is tilted with respect to the layer normal. This phase is biaxial although the deviation from uniaxiality is usually small. There are further smectic phases in which the molecules form two-dimensional lattices within the layers (ordered smectic phases). The difference between ordered... 

So far, we have assumed that the crosslinks pin the smectic layers at a number of points but do not disturb the smectic density wave. However, a sufficient large density of crosslinks might lead to layer distortions that could destroy the quasi-long-range order of ID lamellar lattices [130, 131]. The crosslinks are randomly functionalized into the polymer backbone, and local density variations lead to quenched random disorder. This manifests itself as a mechanical random field that disturbs local layer positions and orientations. The effect of crosslinks on the smectic layer structure can be introduced via a corrugated potential that penalizes deviations of crosslinks from the local layer positions [4,132] ... 

Smectic- (orthogonal) and smectic-C (tilted) are the least-ordered smectic phases. The layer structure of these phases actually consists of a onedimensional density wave. The shape of the density wave appears to be almost purely sinusoidal as can be concluded from the absence or very weak intensity of higher-order reflections in X-ray experiments. Thus, one should... 

One can conveniently describe the layering in smectic liquid crystals by employing a density wave vector a, which following Oseen [43] and de Gennes [8] is subject to the constraint... 

The simple, popular picture of a smectic phase (Sm) consisting of elongated molecules in sharp, distinct layers is rather misleading a more realistic picture is one where the molecules are arranged to provide a single sinusoidal density wave [1] with its wave vector either parallel to the molecular director (orthogonal phases such as SmA, SmB, etc. - see Fig. 1) or at some angle to it (tilted phases such as SmC, SmI, etc.). However, it is often convenient to refer to the layered nature of the smectic phase to help explain phenomena such as conductivity anisotropy. 

Molecules within the layers move freely, with no defined packing arrangement. There is no correlation between molecules from layer to layer in this phase. The layer-to-layer ordering of the smectic phase can be approximated to a density wave, and this can be incorporated in a modified form of the order parameter. 

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