Pattern formation and chiral phase separation by compression: Racemic monolayer viewed as Bragg-Williams binary cholesterics

Abstract

In nature, it is so abundant that the two molecular enantiomers of left- and right-handed chirality usually combine to form a racemic dimer. The means of separation of the dimers into distinct phases of optical activity, the chiral phase separation ~CPS!, are of critical importance in physics, chemistry, biology, and geology @1#, and have been realized through experiments going back to Pasteur in 1848 in isolation of enantiomers of sodium tartrate by crystallization @2#, and to Jungfleish in 1882 in the localization of crystallization of individual enantiomers on suitably disposed seeds @3#. However, the major challenge in the study is, on the theoretical side, to understand the mechanism of chiral symmetry breaking ~CSB! in CPS. The study of CPS in the threedimensional ~3D! state is especially difficult because both left- and right-handed chiral molecules have been classically regarded as exact mirror images and, therefore, are energetically equivalent in bulk due to the molecular rotation. It is natural to think that this difficulty may be reduced in a monolayer of racemic amphiphiles at the air/water interface, because the molecular rotation of the amphiphiles is limited due to the hydrophobic interaction between the amphiphiles and the water. In fact, a remarkable progress of the study in a 2D system was performed by Lundquist in the 1960s @4#. Recognizing the importance of steric factors in the living system, such as in the architecture of the lipoprotein membranes of cells, Lundquist found that at a temperature below 20 °C the racemic monolayer of the methyl ester of 2-methylhexacosanoic acid shows a pressure plateau in each pressure-area isotherm. This was interpreted as a 2D phase transition region, from the expanded phase to the condensed 2D crystalline phase. At that time, however, there were no methods available for the direct crystallographic study for the crystalline phase. In 1993, a significant advance was made by Eckhardt et al. @5#. By transferring such a monolayer of surface pressure from a water surface onto mica, and using atomic-force microscopy, they detected that the phase transition at the region of small molecular areas is a CPS transition, specifically a transition from rectangular lattices of a molecular packing structure ~racemic phase! into a chiral phase in which the enantiomers form two antipodal oblique lattices distinguished by some stripes ~see Figs. 1‐4 in @5#!. In physics, the CSB and stripe formation have been extensively studied in hexatic liquid-crystal ~LC! films @6#,