Spontaneous Break of Mirror Symmetry

This phenomenon has been discovered in the liquid crystal phases consisting of so-called banana (or bent-core) shape molecules [17, 27]. A mechanical model in Fig. 4.39a illustrates the idea. Each of the two dumb-bells has symmetry Do h with infinite number of mirror planes containing the longitudinal rotation axis and one mirror plane perpendicular to that axis. Imagine now that one of the dumb-bells is lying on the table and we try to put another one on the top of the first one parallel to 

I or Iso Kh X T(3) Ordinary liquid phase with full rotational and translational symmetry 

I (chiral) K X T(3) Liquid consisted of chiral molecules showing rotation of linearly polarized light 

N Dooh X T(3) Uniaxial nematic phase possessing long range orientational order and no translational order 

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The non-equivalence of enantiomers through the spontaneous breaking of mirror-symmetry in nature is amplified by asymmetric autocatalytic reaction [34], e.g. Frank s spontaneous asymmetric synthesis [35, 36] (Fig. 7-8). Alberts and Wyn-berg have reported in enantioselective autoinduction that chiral lithium alkoxide products may be involved in the reaction to increase the enantioselectivity (Eq. (7.9)) [37]. The product % ee however does not exceed the level of catalyst % ee. In asymmetric hydrocyanation catalyzed by cyclic dipeptides, the (Si-cyanohydrin product complexes with the cyclic peptide to increase the enantioselectivity in the (S)-cyanohydrin product, the reaction going up to 95.8% ee (Eq. (7.10)) [38]. In the presence of achiral amine, (/ )-l-phenylpropan-l-ol catalyzed carbonyl-addition reaction of diethylzinc has been reported to show lower % ee than that of the catalyst employed [39]. 

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 discussed in Section 4.11, achiral molecules of the bent or banana shape may form locally chiral phases in the form of the left- and right-handed domains. This is a result of spontaneous break of the mirror symmetry [43]. 

Rg. 11.2 Single crystals of morphologically enantiomeric quartz. Note the mirror symmetry of the facets on the respective crystals. Unlike amino acids, the component silicon dioxide molecules have no chirality. The spontaneous resolution of quartz into crystals of opposite morphological handedness is an example of local symmetry breaking in the environment. 

Crystallization and reactivity in two-dimensional (2D) and 3D crystals provide a simple route for mirror-symmetry breaking. Of particular importance are the processes of the self assembly of non-chiral molecules or a racemate that undergo fast racemization prior to crystallization, into a single crystal or small number of enantiomorphous crystals of the same handedness. Such spontaneous asymmetric transformation processes are particularly efficient in systems where the nucleation of the crystals is a slow event in comparison to the sequential step of crystal growth (Havinga, 1954 Penzien and Schmidt, 1969 Kirstein et al, 2000 Ribo et al 2001 Lauceri et al, 2002 De Feyter et al, 2001). The chiral crystals of quartz, which are composed from non-chiral Si02 molecules is an exemplary system that displays such phenomenon. 

Besides the effect of strong chiral amplification in the Soai reaction there is the experimental observation of spontaneous generation of ee from entirely achiral starting conditions. Fig. 4 shows that the kinetic model can give rise to mirror-symmetry breaking. 

Spontaneous Mirror Symmetry Breaking by Formation of Clusters in Bulk-Solutions... 

Amplification of Chirality in Systems Undergoing Spontaneous Mirror Symmetry Breaking... 

The polymerization of mixtures of different racemic cr-amino acids may result in a spontaneous mirror-symmetry breaking process, as proposed for the solid-state polymerization of gwasi-racemates. Thus, in the polymerization of mixtures of different racemic Q -amino adds, the heterochiral chains contain different sequences, resulting in the stochastic mirror-symmetry breaking. ... 

Goldanskii, V. L Kuzmin, V. V. Z. Spontaneous Mirror Symmetry Breaking in Nature and the Origin of Life. Phys. Chem. (Leipzig) 1988,269,216-274. 

In the higher temperature Sm A phase of a chiral compound the molecular symmetry axes are normal to the smectic layers and the local point symmetry is 2/m. This does not allow for spontaneous polarization to occur. In the Sm C phase, on the other hand, the tilt of the long molecular axes with respect to the layer normal breaks the mirror plane symmetry and the spontaneous polarization is induced. Due to chirality of the molecules, the symmetry axis turns its direction on going from one smectic layer to another. This implies that the polarization shows a helicoidal order with a pitch distinctly larger than the layer thickness. The helix can be easily unwound by an external electric field normal to the... 

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