The Construction of the Dupin Cyclides in a Smectic A Polygonal Texture

We report experimental observations with optical microscopy of the usual so-called polygonal texture. We have made simulations of the Dupin cyclides in one small region of the sample. 1. Introduction Many experiments on smectic A phases of liquid crystals report on the existence of macroscopic defective textures since the pioneer work of Friedel and Grandjean of 1910 [1]. Such a liquid crystal phase put in evidence an interconnection between physics and mathematics as nicely illustrated in [2]. Because of the nature of the singular lines in a smectic phase, two confocal lines (ellipse and hyperbola or two confocal parabolas) both located in perpendicular plans, smectic layers take the shape of Dupin cyclides in order to keep their equidistance. Notice that other defective lines, the double helices, exist also in the smectic A phase of a liquid crystal. They were first reported by Williams [3] and the smectic layers at their vicinity take the form of ruled helicoids [4]. Let us focus in this paper on the FCD textures that we describe here. The cover of [5] shows a nice polygonal domain, notion first introduced by Friedel [6]. This texture is highly regular and presents different properties. Figure 1 shows the same kind of texture. We can distinguish two different networks of polygons while focusing on the bottom of the substrate (Figure 1(a)) or on the top (Figure 1(b)) for a 75？μm sample thickness. Figure 1: Observation with crossed polarizers optical microscope of the polygonal texture with two different focusing: (a) on the bottom of the substrate and (b) on the top, 75？ μm sample thickness. Let us focus on a part of the ellipses network, which appear inside the white frame (see Figure 1(a)). Figure 2(a) shows a zoom of one part of the experimental network of the sample and Figure 2(b) shows the simulation of the only several ellipses: the biggest one and the other six, which are in contact with the biggest one. One of the ellipses being too small, we will not take this ellipse into account in what follows. For this simulation, we first assume that the biggest ellipse is in the plane of the top of the substrate. The six other ellipses are drawing after the realization of a small rotation of different angles in the plane of the bottom of the substrate. Figure 2: Investigated part of the sample observed under optical crossed polarizers: (a) zoom of a part of Figure 1 and (b) simulation of seven ellipses. After measuring different parameters like semimajor and semiminor axis for the seven ellipses and the distances between apices from the experimental