Cholesterics symmetry

When the mesogenic compounds are chiral (or when chiral molecules are added as dopants) chiral mesophases can be produced, characterized by helical ordering of the constituent molecules in the mesophase. The chiral nematic phase is also called cholesteric, taken from its first observation in a cholesteryl derivative more than one century ago. These chiral structures have reduced symmetry, which can lead to a variety of interesting physical properties such as thermocromism, ferroelectricity, and so on. 

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]. 

A vast array of covalent molecules have been synthesised over the years in the search for LCs that show the useful cholesteric and ferroelectric smectic C phases, often on a trial and error basis ignoring the interactions between the molecules. The idea that one could think of the interactions between the molecules as a kind of molecular recognition came from the careful analysis of the conformations of molecules in the layers [77,78]. The arguments are based on the symmetry limitations of the angle formed by the alkyl chain and the phenyl benzoate moiety in the molecules that were the subject of this study. A molecular recognition site within the phase was used as the basis for these speculations , which have actually proved rather successful. The actual interactions between molecules are usually weak, but the formation of layers of aromatic and aliphatic units in these mesophases gives rise to their unique properties. 

In this liquid crystal phase, the molecules have non-symmetrical carbon atoms and thus lose mirror symmetry. Otherwise optically active molecules are doped into host nematogenic molecules to induce the chiral liquid crystals. The liquid crystals consisting of such molecules show a helical structure. The most important chiral liquid crystal is the cholesteric liquid crystals. As discussed in Section 1.2, the cholesteric liquid crystal was the first discovered liquid crystal and is an important member of the liquid crystal family. In some of the literature, it is denoted as the N phase, the chiral nematic liquid crystal. As a convention, the asterisk is used in the nomenclature of liquid crystals to mean the chiral phase. Cholesteric liquid crystals have beautiful and interesting optical properties, e.g., the selective reflection of circularly polarized light, significant optical rotation, circular dichroism, etc. 

Some cholesteric materials show the blue phase as the temperature increases from that of the cholesteric phase and before it reaches that of the isotropic phase. The blue phase is a cubic phase. There have been three blue phases found so far BP I, BP II and BP III phases. It is now understood that the BP I phase is a body-centered cubic, the BP II phase is a primitive cubic and the BP III phase is a fog phase with no structural symmetry. Generally the temperature range of the blue phase is quite narrow, less than 1 degree... 

The singularities in the liquid crystals cause the deformation of the director field of liquid crystals and thus affect the symmetry of liquid crystals. This idea provides an approach to analyze the characteristics of the defects. The order vectors (or scalars, or tensors) of various liquid crystals are not the same. The director n is the order vector of the nematic liquid crystals, but the order for the cholesteric liquid crystals is a symmetric matrix, i.e., a tensor. Because the order vector space is thus a topological one, any configuration of the director field of liquid crystals is thus represented by a point in the order vector space. The order vector space (designated by M) is associated with the symmetry of liquid crystals. The topologically equivalent defects in liquid crystals constitutes the homotopy class. The complete set of homotopy classes constitutes a homotopy group, denoted Hr(M). r is the dimension of the sub-space surrounding a defect, which is related to the dimension of the defect (point, line or wall) d, and the dimension of the liquid crystal sample d by... 

When the nematic phase is composed of optically active materials (either a single component or a multicomponent mixture made up of chiral compounds or chiral compounds mixed with achiral materials), the phase itself becomes chiral and has reduced environmental space symmetry. The structure of the chiral nematic (or cholesteric) modification is one where the local molecular ordering is identical to that of the nematic phase, but in the direction normal to the director the molecules pack to form a helical macrostructure, see Fig. 5. As in the nematic phase the molecules have no long-range positional order, and no layering exists. The pitch of the helix can vary from about 0.1 x 10 m to almost infinity, and is dependent on optical purity and the degree of molecular... 

An example of this type of thermomechanical coupling appears to have been observed by Lehmann in cholesteric liquid crystals very soon after their discovery. He found that droplets of the material when heated from below seemed to be rotating violently, but from optical studies he concluded that it was not the drops themselves but the structure that was rotating. Fig. 4.4.1 shows a few of the many sketches that he made depicting his observations. Leslie s equations offer a simple explanation of the phenomenon because of the absence of mirror symmetry, an applied field, which is a polar vector, can result in a torque, which is an axial vector. 

Many synthetic polymers form cholesteric phases, and even solids showing certain of the fundamental symmetries of cholesteric liquids. The purpose of this paper is to review the main examples of biological polymers assembling into cholesteric liquids or into more or less solid analogues. We will present them according to the main chemical classes of polymers to which they belong. We will also indicate the main forces involved in creating the cholesteric twist. 

A sufficient amount of oriented chiral molecules can be obtained in an induced cholesteric liquid crystal phase if the induced helical structure has been untwisted by an electric field. In the following description tensors are needed for the sake of simplicity (At least there are three tensors required the transition moment tensor (absorption tensor ,y), the rotational strength tensor (circular dichroism tensor A ,y), and the order tensor g,y33 (i,j= 1,2,3). If the molecules do not possess any symmetry, the principal axes of all of these tensors are differently oriented with respect to the molecular frame (the coordinate system in which only the three diagonal elements of a tensor are different from zero).) The only tensorial property, needed here explicitly, is the existence of three coordinates (components) of a tensor with respect to three specially chosen mutually perpendicular axes. This means that three information instead of one information about a molecule are needed instead of one CD value, namely Ae, three CD values, namely As, (i=l, 2, 3), have to be introduced. Ac is then one-third of a sum of the three so-called tensor coordinates of the CD tensor ... 

It should be noted that cholesteric liquid crystals (chiral nematics) having point group symmetry Dqo are also periodic with flie pitch considerably exceeding a molecular size. The preferable direction of the local molecular orientatiOTi, i.e. the director oriented along the Coo axis, rotates additionally through subsequent infinitesimal angles in the direction perpendicular to that axis. Hence a helical structure forms with a screw axis and continuous translation group. 

As we shall see further on, the terms linear in dnjdx, allow us to discuss not only conventional nematics with Dooh symmetry but also some biased nematic phases. For example, we can discuss the phases with a spontaneous twist (cholesterics with broken mirror symmetry) or a spontaneous splay (uniaxial polar nematics with broken head-to-tail symmetry, n -n). For a standard nematic only quadratic terms will remain. 

Like in the nematic phase, the textures of SmC reveal blurred Schlieren patterns with linear singularities of strength s = 1. The singularities of 5 = 1/2 are not observed due to the reduced symmetry (C2h) of the SmC phase. Chiral smectics C are periodic structures and the helical pitch can be measured under a microscope either from the Grandjean lines or as a distance between the lines indicating periodicity, like in Fig. 8.22 for the cholesteric phase. On the other hand, like in... 

The Goldstone mode in an achiral SmC tries to restore the symmetry of the smectic A phase Cooh —> Dooh by free rotation of the director along the conical surface with the smectic layer normal as a rotation axis. Thus, like chiral molecules convert a nematic into a cholesteric, they convert an achiral SmC into chiral SmC without any phase transition. In addition, mixing left (L)- and right (R)-handed additives results in a partial or complete compensation of the helical pitch both in cholesterics and chiral smectic C. For example, the L- and R- isomers of the same molecule taken in the equal amounts would give us a racemic mixture, that is achiral SmC without helicity and polarity. 

The limitations on multiplexing any rms-responding monostable liquid crystal effect have been mentioned in Section II.A. Active matrix addressing, described in Sectin IV.A, is one way of overcoming these limitations. Another is to consider alternative liquid crystal effects that are bistable, or at least non-rms responding. With such effects, the maximum number of rows that can be multiplexed is usually determined by the ratio of the frame time (the time period during which the whole picture must be refreshed or updated) to the line time (the time required to address one row of pixels). This is quite demanding of the line time a frame time of 40 msec (only 25-Hz frame rate) would require a line time of 40 /xsec for 1000 lines. Bistable behavior is associated with smectic and cholesteric phases, both of which in completely different ways have translational symmetries added to nematiclike orientational order. In this section, the ferroelectric tilted smectic devices are reviewed, while (untilted) smectic A and cholesteric devices are described in Section IV.C. 

Cholesteric liquid crystals consist of chiral molecules and therefore do not have reflection symmetry. The symmetry group of cholesteric hquid crystals is >2 [1,3]- A cholesteric liquid crystal is invariant for the two-fold (180°) rotation around n, which rules out the possibility of spontaneous polarization perpendicular to n. It is also invariant for the two-fold rotation around an axis that is perpendicular to the n - (the hehcal axis) plane, which mles out the possibility of spontaneous polarization parallel to n. Therefore there is no ferroelectricity in the cholesteric phase. 

As pointed out by Meyer [14], the reflection symmetry of smectic-C liquid crystals can be removed if the constiment molecules are chiral, and thus it becomes possible to have spontaneous polarization. This phase is called the chiral smectic-C or smectic-C, and its stmcture is shown in Figure 4.7. Within a layer, the structure is the same as in smectic-C. The liquid crystal director n is, however, no longer oriented unidirectionally in space but twists from layer to layer as in the cholesteric phase [15]. The symmetry group is C2. The two-fold rotational symmetry axis is perpendicular to both the layer normal a and the director n. Now it is possible to have spontaneous polarization along the two-fold rotational symmetry axis. 

---