Symmetry, liquid crystals

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. 

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

The maintenance of a connection to experiment is essential in that reliability is only measurable against experimental results. However, in practice, the computational cost of the most reliable conventional quantum chemical methods has tended to preclude their application to the large, low-symmetry molecules which form liquid crystals. There have however, been several recent steps forward in this area and here we will review some of these newest developments in predictive computer simulation of intramolecular properties of liquid crystals. In the next section we begin with a brief overview of important molecular properties which are the focus of much current computational effort and highlight some specific examples of cases where the molecular electronic origin of macroscopic properties is well established. 

An essential requirement for device applications is that the orientation of the molecules at the cell boundaries be controllable. At present there are many techniques used to control liquid crystal alignment which involve either chemical or mechanical means. However the relative importance of these two is uncertain and the molecular origin of liquid crystal anchoring remains unclear. Phenomenological models invoke a surface anchoring energy which depends on the so-called surface director , fij. In the case where there exists cylindrical symmetry about a preferred direction, hp the potential is usually expressed in the form of Rapini and Popoular [48]... 

We start with some elementary information about anisotropic intermolec-ular interactions in liquid crystals and molecular factors that influence the smectic behaviour. The various types of molecular models and commonly accepted concepts reproducing the smectic behaviour are evaluated. Then we discuss in more detail the breaking of head-to-tail inversion symmetry in smectic layers formed by polar and (or) sterically asymmetric molecules and formation of particular phases with one and two dimensional periodicity. We then proceed with the description of the structure and phase behaviour of terminally fluorinated and polyphilic mesogens and specific polar properties of the achiral chevron structures. Finally, different possibilities for bridging the gap between smectic and columnar phases are considered. 

However, the main research result from those years was the discovery of the room-temperature single-electron phenomenon. In the 1990s, STM experiments on liquid crystal had shown a very weak staircase (Nejoh 1991) only one year later, the clear observations of the coulomb blockade and the coulomb staircase were demonstrated on gold nanoparticles (Shonenberger et al. 1992a) and the role of system symmetry on the appearance of these two phenomena was outlined (Shonenberger et al. 1992b). 

The interference pattern depends both on the symmetry of the liquid crystal mesophase and on the arrangement of the molecules between the glass cover slips. Three examples are given in Fig. 8. 

Several kinds of intermediate states exist between the state of highest order in a crystal having translational symmetry in three dimensions and the disordered distribution of particles in a liquid. Liquid crystals are closest to the liquid state. They behave macroscopically like liquids, their molecules are in constant motion, but to a certain degree there exists a crystal-like order. 

Obviously, the model is crude and does not take into account many of the factors operating in a real molecular stack. Lack of symmetry with respect to the polar axis and the fact that dipoles may not necessarily be situated in one plane represent additional complications. The angle a could also be field dependent which is ignored in the model. The model also requires that interactions between molecules in adjacent stacks be very weak in order for fields of 10 to 20KV/cm to overcome barriers for field induced reorientation. The cores are then presumably composed of a more or less ordered assembly of stacks with a structure similar to smectic liquid crystals. 

The earliest approach to explain tubule formation was developed by de Gen-nes.168 He pointed out that, in a bilayer membrane of chiral molecules in the Lp/ phase, symmetry allows the material to have a net electric dipole moment in the bilayer plane, like a chiral smectic-C liquid crystal.169 In other words, the material is ferroelectric, with a spontaneous electrostatic polarization P per unit area in the bilayer plane, perpendicular to the axis of molecular tilt. (Note that this argument depends on the chirality of the molecules, but it does not depend on the chiral elastic properties of the membrane. For that reason, we discuss it in this section, rather than with the chiral elastic models in the following sections.)... 

The second issue concerns the anisotropy of the membrane. The models presented in this section all assume that the membrane has the symmetry of a chiral smectic-C liquid crystal, so that the only anisotropy in the membrane plane comes from the direction of the molecular tilt. With this assumption, the membrane has a twofold rotational symmetry about an axis in the membrane plane, perpendicular to the tilt direction. It is possible that a membrane... 

It is also possible that a membrane might have an even lower symmetry than a chiral smectic-C liquid crystal in particular, it might lose the twofold rotational symmetry. This would occur if the molecular tilt defines one orientation in the membrane plane and the direction of one-dimensional chains defines another orientation. In that case, the free energy would take a form similar to Eq. (5) but with additional elastic constants favoring curvature. The argument for tubule formation presented above would still apply, but it would become more mathematically complex because of the extra elastic constants. As an approximation, we can suppose that there is one principal direction of elastic anisotropy, with some slight perturbations about the ideal twofold symmetry. In that approximation, we can use the results presented above, with 4) representing the orientation of the principal elastic anisotropy. 

So far we have considered the formation of tubules in systems of fixed molecular chirality. It is also possible that tubules might form out of membranes that undergo a chiral symmetry-breaking transition, in which they spontaneously break reflection symmetry and select a handedness, even if they are composed of achiral molecules. This symmetry breaking has been seen in bent-core liquid crystals which spontaneously form a liquid conglomerate composed of macroscopic chiral domains of either handedness.194 This topic is extensively discussed in Walba s chapter elsewhere in this volume. Some indications of this effect have also been seen in experiments on self-assembled aggregates.195,196... 

Spontaneous Reflection Symmetry Breaking in Liquid Crystals... 

Molecular chirality, however, proved an extremely powerful tool in the quest for polar LCs. In 1974 Robert Meyer presented to participants of the 5th International Liquid Crystal Conference his now famous observation that a SmC phase composed of an enantiomerically enriched compound (a chiral SmC, denoted SmC ) could possess no reflection symmetry.1 This would leave only the C2 symmetry axis for a SmC a subgroup of C. The SmC phase is therefore necessarily polar, with the polar axis along the twofold rotation axis. 

The author was supported by the Ferroelectric Liquid Crystal Materials Research Center (National Science Foundation MRSEC award No. DMR-9809555) during the writing of this chapter. The author thanks Professors Tom Lubensky, Leo Radzihovsky, and Joseph Gal for helpful discussions around the issue of terminology for reflection symmetry breaking, and especially Professor Noel Clark for his help on this and many other banana-phase issues. The author also thanks Dr. Renfan Shao for the photomicrographs shown in Figures 8.32 and 8.33. 

This volume of Topics in Stereochemistry could not be complete without hearing about ferroelectric liquid crystals, where chirality is the essential element behind the wide interest in this mesogenic state. In Chapter 8, Walba, a pioneering contributor to this area, provides a historical overview of the earlier key developments in this field and leads us to the discovery of the unique banana phases. This discussion is followed by a view of the most recent results, which involve, among others, the directed design of chiral ferroelectric banana phases, which display spontaneous polar symmetry breaking in a smectic liquid crystal. 

Ferroelectric liquid crystals where a continuous symmetry group is broken at Tc and the doubly degenerate relaxational soft mode of the high-temperature phase splits below Tc into an amphtudon -type soft mode and a symmetry restoring Goldstone (i.e., phason ) mode [e.g., p-decyloxybenzylidene p -amino-2-methylbutylcinnamate (DOBAMBC)]. 

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