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A cluster growth model is proposed to study chiral symmetry breaking in stirred crystallization from an achiral element. Achiral monomers are assumed to coagulate to form chiral clusters from dimers to hexamers. Due to the stirring, the hexamers break into dimers. The coagulation of two dimers into a tetramer also occurs. The fixed point analysis of coupled rate equations of cluster densities and their numerical integrations show that the chiral symmetry becomes broken if dimers are critical and able to dissociate back to achiral monomers.The chirality selection is understood by showing that, in a quasi-steady approximation, the rate equations reduce to those of dimers with nonlinear autocatalysis and decomposition. KEYWORDS: homochirality, crystal growth, enantiomer, rate equation, dynamical chiral symmetry breakingIt has been known for a long time that organic molecules in life are homochiral with a completely broken chiral symmetry, and their origin has intrigued many scientists. 1 Frank showed theoretically that an autocatalytic reaction with cross inhibition between two enatiomers leads to the selection of one type of enantiomer in an open system. 2 We have recently proven that a system with nonlinear autocatalysis and back reaction as is described by the following rate equationsṙcan lead to complete homochirality in a closed system. 3-5 Here, a denotes the concentration of an achiral substrate, r and s are those of chiral enantiomers, and the conservation relation a + r + s =const. is assumed to hold. The model is motivated by the chemical reaction system discovered by Soai et al.,6,7 which is the first example of chemical reaction to show the amplification of enantiomeric excess in a closed system. Chiral symmetry breaking, however, is found not only in biological and chemical systems, but also in other systems such as in crystallization. 8 Even achiral inorganic molecules can gather together to form chiral crystals. Examples are a quartz of SiO 2 crystal, or a crystal of sodium chlorate (Na ClO 3 ). When the sodium chlorate crystal grows from a solution in a cell, a racemic mixture of D and L crystals emerges. On the other hand, when the crystal is *

To study the establishment of homochirality observed in the crystal growth experiment of chiral molecules from a solution under grinding, we extend the lattice gas model of crystal growth as follows. A lattice site can be occupied by a chiral molecule in R or S form, or can be empty. Molecules form homoclusters by nearest neighbor bonds. They change their chirality if they are isolated monomers in the solution. Grinding is incorporated by cutting and shafling the system randomly. It is shown that Ostwald ripening without grinding is extremely slow to select chirality, if possible. Grinding alone also cannot achieve chirality selection. For the accomplishment of homochirality, we need an enhanced chirality change on crystalline surface. With this "autocatalytic effect" and the recycling of monomers due to rinding, an exponential increase of crystal enantiomeric excess to homochiral state is realized.Comment: 10 pages, 5 figure

The deformation of vesicles with conducting membranes in external electric fields has been studied in the framework of the perturbation theory. A simple model for dynamics of deformation is proposed, and the results of numerical calculations for typical combinations of the vesicle size and the electric field are presented. When the conductivity inside a spherical vesicle is larger than that of the exterior medium, the deformations, both static and dynamic, are prolate and, otherwise, they are oblate.