Dupin cyclides textures

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. 

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... 

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. 

We have already considered the polygonal networks of cholesteric liquids, which assemble saddle-shaped layers close to Dupin s cyclides, and the focal segments of which differ from those of focal conics (see Sec. 7.4.3). Friedel and Grandjean [21] described in smectic A phases a texture much closer to Dupin s cyclides, with genuine focal conics. As for cholesterics, this texture is obtained when a thin film of the isotropic phase of the mesogenic compound separates the mesophase from the two glasses. 

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