Symmetry and Chirality in Liquid Crystals

Goodby J W 1998 Symmetry and chirality in liquid crystals Handbook of Liquid Crystals Vol 1. Fundamentals ed D Demus, J Goodby, G W Gray, H-W Spiess and V Vill (New York Wiley-VCH)... 

Finally, we conclude that with his interest in chirality in liquid crystals for many years now, an interest we have shared, Professor Gerd Heppke has demonstrated his perspicacity and good taste in scientific problems. May you have many more years of happy experiences offered by the ineluctable pleasures of chirality—and the Broken Symmetries of Life—particularly broken time reversal symmetry. 

However, there are a number of different symmetry concepts utilized in liquid crystals which often generate considerable confusion. This is because chemists and physicists use different symmetry arguments and different labelling and language in describing various systems. In the following sections, the various symmetry arguments and definitions that are more commonly used to describe the structures of chiral liquid crystals will be discussed. 

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. 

The local translational and orientational order of atoms or molecules in a sample may be destroyed by singular points, lines or walls. The discontinuities associated with the translational order are the dislocations while the defects associated with the orientational order are the disclinations. Another kind of defect, dispirations, are related to the singularities of the chiral symmetry of a medium. The dislocations were observed long after the research on them began. The dislocations in crystals have been extensively studied because of the requirement in industry for high strength materials. On the contrary, the first disclination in liquid crystals was observed as early as when the liquid crystal was discovered in 1888, but the theoretical treatment on disclinations was quite a recent endeavor. 

N. Vaupotic and M. Copic, Polarization modulation instability in liquid crystals with spontaneous chiral symmetry breaking, Phys. Rev. E. 72(3), 031701/1-4, (2005). doi 10.1103/PhysRevE.72.031701... 

Since the discovery of spontaneous break of mirror symmetry [39, 43], many new, so-called banana-form compounds have been synthesised and hundreds of papers published on that subject [44]. It became a hot topic in modem physics and chemistry of liquid crystals. In the present book there is no space for discussirm of different aspects of this fascinating phenomenon and I have decided to finish my narration here. I believe very soon the books shall appear devoted solely to this important subject related not only to liquid crystals, but to the general problems of chirality of the matter. 

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. 

The symmetry approach to ferroelectricity in liquid crystals can be realized not only for individual substances but also for multicomponent systems. For low-molar-mass ferroelectric liquid crystals, most applications use LC mixtures with two main components a nonchiral matrix providing the tilted smectic structure and a chiral dopant [7]. As for the preparation of FLCPs, mixing of a smectic C polymer with a chiral dopant also results in a ferroelectric chiral smectic system [74]. Japanese authors [75,76] have carried out systematic studies on mixing tilted smectic polymers with low-molar-mass ferroelectric liquid crystals. 

The second living system feature we select is that they are chiral. We take this to mean broken mirror symmetry. In liquid crystals, chirality is expressed as a macroscopic helix with a pitch, />(,. In cholesteric liquid crystals, 1500 A < Pq < oo, and qo, its wavevector (wavenumber qo = 2it/po), is perpendicular to n, the direction of orientational order. The existence of this intrinsic equilibrium length has several implications of which we mention two particularly insightful ones [1], [2]. 

The existence or nonexistence of mirror symmetry plays an important role in nature. The lack of mirror symmetry, called chirality, can be found in systems of all length scales, from elementary particles to macroscopic systems. Due to the collective behavior of the molecules in liquid crystals, molecular chirality has a particularly remarkable influence on the macroscopic physical properties of these systems. Probably, even the flrst observations of thermotropic liquid crystals by Planer (1861) and Reinitzer (1888) were due to the conspicuous selective reflection of the helical structure that occurs in chiral liquid crystals. Many physical properties of liquid crystals depend on chirality, e.g., certain linear and nonlinear optical properties, the occurrence of ferro-, ferri-, antiferro- and piezo-electric behavior, the electroclinic effect, and even the appearance of new phases. In addition, the majority of optical applications of liquid crystals is due to chiral structures, namely the ther-mochromic effect of cholesteric liquid crystals, the rotation of the plane of polarization in twisted nematic liquid crystal displays, and the ferroelectric and antiferroelectric switching of smectic liquid crystals. 

Chirality has become arguably the most important topic of research in liquid crystals today. The reduced symmetry in these organized phases leads to a variety of novel phase structures, properties, and applications. Molecular asymmetry imparts form chirality to liquid crystal phases, which is manifested in the formation of helical ordering of the constituent molecules of the phase. Similarly, molecular asymmetry imposes a reduction in the space symmetry, which leads to some phases having unusual nonlinear properties, such as ferroelectric-ity and pyroelectricity. 

Thus, dissymmetric molecules commonly have a simple axis of symmetry, and in asymmetric molecules this axis is absent however, both species are usually optically active. In liquid crystal systems both types of material are capable of exhibiting chiral properties. Table 1 summarizes the relationships between optical activity, molecular structure, and rotational symmetry operations [1]. 

There are a variety of ways in which the space or environmental symmetry and asymmetry can be expressed in liquid crystals, with the most commonly discussed system being that of the chiral smectic C phase. Thus, for the purposes of describing space symmetry in liquid crystals, the structure and symmetry properties of the smectic C phase will be described in the following sections [5, 6]. 

Presumed ferroelectric effects in liquid crystals were reported by Williams at RCA in Princeton, U. S. A., as early as 1963, and thus at the very beginning of the modern era of liquid crystal research [5]. By subjecting nematics to rather high dc fields, he provoked domain patterns that resembled those found in solid ferroelectrics. The ferroelectric interpretation seemed to be strengthened by subsequent observations of hysteresis loops by Kapustin and Vistin and by Williams and Heilmeier [7]. However, these patterns turned out to be related to electrohydrodynamic instabilities, which are well understood today (see, for instance, [8], Sec. 2.4.3 or [9], Sec. 2.4.2), and it is also well known that certain loops (similar to ferroelectric hysteresis) may be obtained from a nonlinear lossy material (see [10], Sec. 2.4.2). As we know today, nematics do not show ferroelectric or even polar properties. In order to find such properties we have to lower the symmetry until we come to the tilted smectics, and further lowering their symmetry by making them chiral. The prime example of such a liquid crystal phase is the smectic C. ... 

Bearing in mind the odd-even effect, I tried to synthesize new polymer types in order to express the ferroelectric phase in an achiral system. The opportunity to study chiral ferroelectric liquid crystals was ripe at that time, as I already mentioned in the opening sentence. Their discoverer Mayer [118] argued that by introducing a chiral molecule into a Sc phase, the twofold axis of the Sc phase becomes the polar axis because of the extinction of a vertical mirror symmetry (Fig. 9.20). The reduction of the symmetry by the introduction of chirality into the system is in a sense conventional, simple, and straightforward. 

As witli tlie nematic phase, a chiral version of tlie smectic C phase has been observed and is denoted SniC. In tliis phase, tlie director rotates around tlie cone generated by tlie tilt angle [9,32]. This phase is helielectric, i.e. tlie spontaneous polarization induced by dipolar ordering (transverse to tlie molecular long axis) rotates around a helix. However, if tlie helix is unwound by external forces such as surface interactions, or electric fields or by compensating tlie pitch in a mixture, so tliat it becomes infinite, tlie phase becomes ferroelectric. This is tlie basis of ferroelectric liquid crystal displays (section C2.2.4.4). If tliere is an alternation in polarization direction between layers tlie phase can be ferrielectric or antiferroelectric. A smectic A phase foniied by chiral molecules is sometimes denoted SiiiA, altliough, due to the untilted symmetry of tlie phase, it is not itself chiral. This notation is strictly incorrect because tlie asterisk should be used to indicate the chirality of tlie phase and not tliat of tlie constituent molecules. 

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