Double twist cylinder

Fig. 16a-c. The model proposed for blue phase II showing a,b the director arrangement within a double twist cylinder c the packing of these cylinders along three orthogonal axes... 

Fig. 4. Illustrating a cubic lattice formed by double-twist cylinders as a possible model of a...

Experimental evidence was reported for the existence of various additional phases a pre-cholesteric order in the form of a network of double-twisted cylinders, analogous to the thermotropic blue phases [27], a hexatic phase that replaces the hexagonal columnar in very long DNA fragments [31], and a structure with orthorhombic symmetry appearing in the transition to crystalline order [27]. 

Fig. 4 Model of molecule arrangements within a double-twisted cylinder...

Fig. 5 Structures of Blue Phases I and II. The rods in (a) and (c) represent double-twist cylinder. The black lines in (b) and (d) represent disclination lines...

A screw dislocation is similar in structure to that of a double twist cylinder which is the basis for the formation of Blue Phases [24], The equivalent of the... 

Fig. 31. Structure of a double twist cylinder (the molecules are shown as rods)...

Fig. 32a-f. Various models for the filamentary texture of TGBA phases. The molecules are shown as short lines, or nails when they are tilted out of the page. In each case the filaments are shown looking down the axis of the double twist cylinder (after Gilli and Kamaye [43])... 

Fig. 4.33 Double-twist cylinder (a) and the structure of the body-centered cubic phase BPl (b) and simple cubic phase BPII (c), both consisted of double-twist cylinders (adapted from [22])...

From the experimental results discussed in the above section, it is known that BPI and BPII have cubic structures. There are two theories that have successfully explained the existence of the blue phases and predict their symmetry and physical properties. One is known as the defect theory, in which the blue phases consist of packed double-twist cylinders and there... 

A defect theory for blue phases was introduced by Meiboom, Sethna, Anderson, et al. [22-27]. In this theory, the liquid crystal is assumed to form double-twist cylinders where the liquid crystal molecules twist about any radius of the cylinder, as shown in Figure 13.6. The cylinder cannot, however, cover the whole 3-D space without topological defects. Instead of a single cylinder, the blue phases consist of packed double-twist cylinders. There are defects in the regions not occupied by the cylinders. 

Figure 13.6 Schematic diagram of the double twist cylinder.

Figure 13.9 (a) The liquid crystal director configuration of the disclination formed between the doubletwist cylinders, (b) The structure of the disclinations in the simple cubic packing of the double-twist cylinders, (c) The structure of the disclinations in the body-centered cubic packing of the double-twist cylinders. 

In 3-D space, the defects formed between the double-twist cylinders are line defects and are called disclinations. The organization of the disclinations in the 3-D space has the same symmetry as the structure of the packed the double-twist cylinders. The disclinations in the simple cubic packing and body-centered cubic packing are shown in Figure 13.9(b) and (c), respectively. 

Figure 13.10 The total free energy of one unit cell as a function of radius of the double-twist cylinder.

This is the typical optical rotatory power of cholesteric liquid crystals with pitch shorter than the hght wavelength. In the blue phases, the double-twist cylinders orient along many different directions, and therefore the optical rotatory power is smaller than that of cholesteric phase. The thickness of blue phases displays are typically a few microns, and over this distance the hehcal structure in the blue phases does not change much the polarization state of light. 

Calculate the free energy per unit length of the double-twist cylinder as a function of qR from 0 to 71, where q is the chirality of the liquid crystal and R is the radius of the doubletwist cylinder. Use the following elastic constants K22, K33 = 2K22, and K24 = 0.5K22-... 

Figure 14.1 Blue phase LC structure at the microscopic level (a) double-twist alignment of LC molecules, (b) double-twist cylinder, (c) lattice cubic formed by double-twist cylinders, and (d) disclination lines.

As the temperature increases, up to three types of blue phases BPI, BPII, and BPIII may exist [14]. BPIII is believed to possess amorphous stmcture. BPI (Figure 14.2(a)) and BPII (Figure 14.2(c)) are composed of double-twist cylinders arranged in cubic lattices. Inside each cylinder, the LC director rotates spatially about any radius of the cylinder. These double-twist cylinders are then fitted into a three-dimensional stmcture. However, they cannot fill the full space without defects. Therefore, blue phase is a coexistence of double-twist cylinders and disclinations. Defects occur at the points where the cylinders are in contact (Figures. 14.2(b) and 14.2(d)). BPI is known to have body-center cubic stmcture and BPII simple cubic stmcture. 

In this structure, the director twists in moving perpendicular ftom the z-axis along any radius. Thus at the centre the director points along the z-axis, but has rotated by 45° at all pomts in the xy plane located a distance equal to one-eighth the pitch away from the origin. If the stmcture does not vary in the z-direction, then this defines a double twist cylinder. 

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