Symmetry reflection

The simplest argument is that of Agmon based on Fig. 5-15. Because of the reflection symmetry at point T, the triangles ATR and BTP are similar, so we can write AT/AR = TB/PB, or, if we express the ordinate in free energies. 

Reflective symmetry boundary conditions are specified at the sides of the sample as... 

Figure 6.7 The 3D EXSY-EXSY spectrum of heptamethylbenzenonium sulfate in H.2SO4 representing o i = o)j reflection symmetry. Cross-peaks lying on identical planes (coi, a s) are connected by lines. (Reprinted from J. Mag. Reson. 84, C. Griesinger, et al, 14, copyright (1989), with permission from Academic Press, Inc.)...

The first four facets are rotationally equivalent to each other as are the final four. The two sets are related by reflectional symmetry to each other. When a chiral adsorbate, for example, S-lysine, is used, the reflectional symmetry is no longer valid and only rotationally equivalent facets should be formed. This was demonstrated elegantly by Zhao with STM [53], The driving force for facet formation is proposed to be a three-point interaction involving the carboxylate group, the a-amino group, and the amino-terminated side chain. The simultaneous optimization of adsorbate-adsorbate and adsorbate-substrate interactions determines the stereochemistry of the facet. 

The reflection symmetry of the Lagrangian has apparently been broken by choosing the ground state stable vacuum. This is achieved by defining the field... 

The first term in this free energy is the standard isotropic curvature rigidity. The second term is a chiral term with coefficient Xhp, which can exist only in chiral membranes and is prohibited by reflection symmetry in nonchiral membranes. This term favors curvature in a direction 45° from m. Thus, it gives an intrinsic bending force in any membrane with both chirality and tilt order. 

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... 

Polar structures may have rotation symmetry and reflection symmetry. However, there can be no rotation or reflection normal to the principal rotation axis. Thus, the presence of the mirror plane normal to the C2 axis precludes any properties in the SmC requiring polar symmetry the SmC phase is nonpolar. 

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. 

This idea is elegant for its simplicity and also for its usefulness. While often in phenomenological theories of materials, control of parameters with molecular structure would provide useful properties, but the parameters are not related in any obvious way to controllable molecular structural features. Meyer s idea, however, is just the opposite. Chemists have the ability to control enantiomeric purity and thus can easily create an LC phase lacking reflection symmetry. In the case of the SmC, the macroscopic polar symmetry of this fluid phase can lead to a macroscopic electric dipole, and such a dipole was indeed detected by Meyer and his collaborators in a SmC material, as reported in 1975.2... 

Spontaneous reflection symmetry breaking in achiral LCs is also well known, driven by specific boundary conditions. A very simple example of this type of chiral domain formation is illustrated in Figure 8.11. Suppose we start with two uniaxial solid substrates, which provide strong azimuthal anchoring ... 

The starting system is achiral (plates at 90° with isotropic fluid between), but leads to the formation of a chiral TN structure when the fluid becomes nematic. In this case, enantiomeric domains must be formed with equal likelihood and this is precisely what happens. The size of these domains is determined by the geometry and physics of the system, but they are macroscopic. Though the output polarization is identical for a pair of heterochiral domains, domain walls between them can be easily observed by polarized light microscopy. This system represents a type of spontaneous reflection symmetry breaking, leading to formation of a conglomerate of chiral domains. 

Many other interesting examples of spontaneous reflection symmetry breaking in macroscopic domains, driven by boundary conditions, have been described in LC systems. For example, it is well known that in polymer disperse LCs, where the LC sample is confined in small spherical droplets, chiral director structures are often observed, driven by minimization of surface and bulk elastic free energies.24 We have reported chiral domain structures, and indeed chiral electro-optic behavior, in cylindrical nematic domains surrounded by isotropic liquid (the molecules were achiral).25... 

In all of these cases the symmetry breaking is derived from the manipulation of surface forces. Prior to 1997, spontaneous reflection symmetry breaking had never been reported in any thermodynamic bulk fluid phase. In the LC field, this empirical fact led to the generally believed assumptions that any LC phase composed of achiral or racemic compounds must be achiral, and... 

It is interesting to point out here that with all of the theoretical speculation in the literature about polar order (both ferroelectric and antiferroelectric) in bilayer chevron smectics, and about reflection symmetry breaking by formation of a helical structure in a smectic with anticlinic layer interfaces, the first actual LC structure proven to exhibit spontaneous reflection symmetry breaking, the SmCP structure, was never, to our knowledge, suggested prior to its discovery. 

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. 

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