Hyperbola, focal conic domains

Note 3 A focal-conic domain built around an ellipse and an hyperbola is the most common type of defect in thermotropic smectic A phases. The hyperbola passes through a focus of the ellipse and the ellipse passes through the focus of the hyperbola (see Fig. 24). Note 4 In a particular limiting case of an ellipse-hyperbola focal-conic domain, the ellipse becomes a straight line passing through the center of a circle. 

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. 

Note 6 At any point inside a focal-conic domain, the director is oriented along the straight line drawn through the point and the two defect lines (ellipse and hyperbola or two parabolae or circle and straight line). See for example BD, BC and BO in Fig. 24. 

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.

Texture composed of focal-conic domains of the ellipse-hyperbola type with visible ellipses, or parts of ellipses, located at the boundary surfaces. 

Note 3 Neighbouring domains form a family with a common apex where the hyperbolae of these domains join each other. This common point is located at the surface that is opposite to the surface containing the ellipses (see Fig. 26). Each family is bounded by a polygon formed by hyperbolic and elliptical axes these are parts of focal-conic domains that provide a smooth variation of smectic layers between the domains of different families. These domains are the tetrahedra in Fig. 26. 

Texture formed partly by focal-conic domains with their hyperbolae lying in the plane of observation. 

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. 

For purposes of simplicity, the three-dimensional focal-conic domain can be described where the ellipse is approximated to a circle and the hyperbola is equivalent to a straight... 

Friedel did however understand the layered nature of smectics, firstly through the stepped edges possessed by smectic droplets with a free surface, and secondly through his detailed studies of the optical microscopic textures of thin films of smectic phases. He understood the optical discontinuities, i.e., the defects, of the smectic focal-conic texture and saw the relationship of the black lines delineating ellipses of different eccentricities and their associated hyperbolae in terms of focal-conic domains which may be divided into a series of parallel, curved surfaces known as Dupin cy-clides. He also understood that the optically extinct homeotropic textures of smectics of the type he studied gave positive uniaxial interference figures consistent with systems of layers lying flat to the surface. His microscopic studies demonstrated the immense value of the optical microscope as a precise scientific instrument in studies of all types of liquid crystal phases. 

Fig. 2.21. Scheme of focal-conic domains. Straight lines such as AB joining points on the ellipse to points on the hyperbola mark the direction of n in the case of smectics, and the direction of the helical axis in the case of cholesterics. (From... 

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