Cyclides

Note 2 The smectic layers within a focal-conic domain adopt the arrangement of Dupin cyclides, since as in these figures there appear concentric circles resulting from the intersection of ellipses and hyperbolae. They also have the distinctive property of preserving an equal distance between them. 

Fig. 24. Dupin cyclide and perfect focal-conic domain construction, (a) Vertical section showing layers of the structure thick lines indicate the ellipse, hyperbola, Dupin cyclide, and central domain, (b) Focal-conic domain showing structural layers with a representation of the arrangement of the molecules within one of them.

The Oseen theory embraces smectic mesophases, but is not really required for this case. The interpretation of the equilibrium structures assumed by smectic substances under a particular system of external influences may be carried out by essentially geometric arguments alone. The structures are conditioned by the existence of layers of uniform thickness, which may be freely curved, but in ways which do not require a breach of the layering in regions of greater extension than lines. These conditions automatically require the layers to be Dupin cyclides and the singular lines to be focal conics. Nothing, essentially, has been added... 

Parabolic focal conics are a special case of generic focal conic defects, which are composed of layers curved to form toroidal surfaces called Dupin cyclides (see Fig. 10-31). Each such structure contains a pair of disclination lines—one an ellipse and the other... 

Figure 10.31 Smectic A layers forming Dupin cyclides. In (a), the disclination lines are an ellipse and a hyperbola the ellipse passes through the focus of the hyperbola and the hyperbola passes through the focus of the ellipse. In the degenerate case (b), the ellipse becomes a circle and the hyperbola becomes a line. The cone is an isolated confocal domain. (From Lavrentovich, reprinted with permission from Sov. Phys. JETP 64 984, Copyright 1986, American Institute of Physics.)...

For a Dupin cyclide comprised a circle base and a straight line core, shown in Figure 1.28(b),... 

If the base is an ellipse and the core is a hyperbolic curve, shown in Figure 1.28(c), the Dupin cyclide has the energy... 

Figure 1.28. The Dupin cyclides (a) a point and sphere (b) a circle and a straight line (c) an ellipse and a hyperbohc curve.

Under the polarized optical microscope, the liquid crystal films show colorful patterns, i.e., the optical textures. Each liquid crystal phase shows its typical texture which provides the means to identify the phase of the liquid crystals. The typical textures are the Schlieren, threadlike, homeotropic, homogeneous, marble, finger-print, focal-conic, Dupin cyclide, fan-shape, sanded, mosaic, and so on. They are the integrations of many typical defects mentioned above. Demus Richter (1978) were the first to review... 

The formation of focal-conic textures is characteristic of smectic phases (Gray and Goodby, 1984). These textures are the result of smectic layers arranged in Dupin cyclides (Friedel, 1922). The common defects included in these arrangements have the form of ellipses and hyperbolae in certain confocal relationship. Dependent on the direction of observation relative to the defects, the dark lines shown in Figure 4.20 may be observed in focal-conic domains. 

Fig. 5.4.1. (a) Smectic layers in concentric cylinders to form a myelin sheath with a singular line L along the axis (b) the cylinders are closed to form tori there are two singular lines, a circle and a straight line (c) the general case when the smectic layers form Dupin cyclides the circle becomes an ellipse and the straight... 

Based on the fact that the layers are bent into Dupin cyclides, one can derive expressions for the principal curvatures (Tj and thus work out... 

Fig. 5.4.3. (a) Geometry of a pair of focal conics. (After Friedel. ). b) A section of a) in the plane of the hyperbola, showing parts of Dupin cyclides. (After... 

Curvature of the layers in the smectic state is compatible with the formation of cylindrical, tore and Dupin cyclide defects. The most common defect is the Dupin cyclide and therefore is discussed in more detail. When the smectic A phase nucleates from either the Hquid or the nematic phase, it can do so with the formation of curved structures called baton-nets (see Figure 6), i.e., layers of molecules add to a... 

These are the most striking features of smectic textures [19]. Smectic layers of constant thickness (incompressible, modulus B— oo) form surfaces called Dupin cyclides. We have seen some of them, which have the form of tori including disclinations, see Fig. 4.7b. Such cyclides can fill any volume of a liquid crystal by cones of different size. An example is afocal-conic pair, namely, two cones with a common base. The common base is an ellipse with apices at A and C and foci at O and O , see Fig. 8.30a. The hyperbola B-B passes through focus O. The focus of... 

Fig. 8.30 Focal-conic defect structure in SmA A pair of cones with a common elliptical base and a hyperbola connecting cone apices (a) cross-section of the upper cone by plane ABC with gaps between lines (Dupin cyclides) indicating the smectic layers (b) filling the space of the sample by cones of different size (c)...

Figure 9.5. A vertical cross-section through a set of Dupin cyclides.

Combining the POM observations and structural features resulted in the supposition that the droplet texture was polygonal and that the smectic layers were arranged as Duppin cyclides with the cone ellipses oriented on the plane of the droplet wall and the peak to its center. 

The SMLC cholesterics most frequently present polygonal textures with domains of a negative Gaussian curvature. In these domains the focal conditions are not exaetly satisfied [91], and unlike the situation in smectics, the cholesteric layers might deviate from the exact geometry of Dupin cyclides... 

Smectic A phases in which the layers are not uniformly parallel to the glass slides confining the sample (i.e. not in a planar orientation) are characterized by fan-like textures (Fig. 5.10b), made up of focal conics (Fig. 5.12). A focal conic is an intersection in the plane of a geometric object called a Dupin cyclide (Fig. 5.13), which results from lamellae forming a concentric roll (like a Swiss roll) being bent into an object based on an elliptical torus of non-uniform cross-section. The straight line that would define the rotation axis of the torus is distorted into a hyperbola in the Dupin cyclide. 

Figure 5.13 Sketch of a Dupin cyclide formed by rolled-up layers in the SmA phase. Here E represents an ellipse and H a hyperbola...

Figure 29. Cyclides are surfaces the lines of curvature of which are circles, (a) An envelope Z of a family of spheres S, centred in A describing L, and contact circles y. (b) Nested tori, the normals of which pass through the circle L and the axis L, one sheet being suppressed to avoid intersection of the surfaces.

Smectic A phases often contain pairs of focal conics (see Fig. 7 a and b) that are well contrasted in the liquid, even in the absence of polarizers. A more common situation is the presence of arcs of conics associated in pairs (Fig. 29 d). The parallel Dupin cyclides and the associated revolution cones form three mutually orthogonal systems of surfaces [7], and therefore the lines of curvature of any surface in one system are its intersections with the surfaces of the other two systems. The toroidal shape of Dupin s... 

Figure 30. Dupin s cyclides and their deformations, (a) A toroidal Dupin cyclide and its circular lines of curvature y and /, the planes of which intersect along the axes A (joining O and Q) and A, respectively. Two tangential planes along two symmetrical circles T intersect along A. (b) A parabolic Dupin s cyclide, with its two focal parabolae P and P, its circular curvature lines y and /, and its axes A and A, (c) Three parallel, but separated, parabolic Dupin s cyclides. The lines represent sections by planes x=0,y = 0 and = 0 at the top of the figure, two conical points are present, as in (b), but the spindle-shaped sheet has been removed (Rosenblatt et al.

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