Selection of Crystal Chirality: Equilibrium or Nonequilibrium?

To study the solution growth of crystals composed of chiral organic molecules, a spin-one Ising lattice gas model is proposed. The model turns out to be equivalent to the Blume-Emery-Griffiths model, which shows an equilibrium chiral symmetry breaking at low temperatures. The kinetic Monte Carlo simulation of crystal growth, however, demonstrates that Ostwald ripening is a very slow process with a characteristic time proportional to the system size: The dynamics is nonergodic. It is then argued that by incorporating grinding dynamics, homochirality is achieved in a short time, independent of the system size. Grinding limits Cluster sizes to a certain range independent of system size and at the same time keeps the supersaturation so high that population numbers of average-sized clusters grow. If numbers of clusters for two types of enantiomers differ by chance, the difference is amplified exponentially and the system rapidly approaches the homochiral state. Relaxation time to the final homochiral state is determined by the average cluster size. We conclude that the system should be driven and kept in a nonequilibrium state to achieve homochirality.