Chiral Crystal Growth under Grinding

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

Drift-Induced Step Instabilities Due to the Gap in the Diffusion Coefficient

On a Si(111) vicinal face near the structural transition temperature, the $1 \times 1$ structure and the $7 \times 7$ structure coexist in a terrace: the $1 \times 1$ structure is in the lower side of the step edge and the $7 \times 7$ structure in the upper side. The diffusion coefficient of adatoms is different in the two structures. Taking account of the gap in the diffusion coefficient at the step, we study the possibility of step wandering induced by drift of adatoms. A linear stability analysis shows that the step wandering always occurs with step-down drift if the diffusion coefficient has a gap at the step. Formation of straight grooves by the step wandering is expected from a nonlinear analysis. The stability analysis also shows that step bunching occurs irrespective of the drift direction if the diffusion in the lower side of the step is faster. The step bunching disturbs the formation of grooves. If step-step repulsion is strong, however, the step bunching is suppressed and the straight grooves appear. Monte Carlo simulation confirms these predictions.Comment: 5 pages, 6 figure

Effect of Cu impurities on wet etching of Si(110): formation of trapezoidal hillocks

We simulate the formation of experimentally observed trapezoidal hillocks on etched Si(110) surfaces, describing their generic geometrical shape and analyzing the relative stability and/or reactivity of the key surface sites. In our model, the hillocks are stabilized by Cu impurities in the etchant adsorbing on the surface and acting as pinning agents. A model of random adsorptions will not result in hillock formation since a single impurity is easily removed from the surface. Instead a whole cluster of Cu atoms is needed as a mask to stabilize a hillock. Therefore we propose and analyze mechanisms that drive correlated adsorptions and lead to stable Cu clusters.Peer reviewe

Step Bunching with Alternation of Structural Parameters

By taking account of the alternation of structural parameters, we study bunching of impermeable steps induced by drift of adatoms on a vicinal face of Si(001). With the alternation of diffusion coefficient, the step bunching occurs irrespective of the direction of the drift if the step distance is large. Like the bunching of permeable steps, the type of large terraces is determined by the drift direction. With step-down drift, step bunches grows faster than those with step-up drift. The ratio of the growth rates is larger than the ratio of the diffusion coefficients. Evaporation of adatoms, which does not cause the step bunching, decreases the difference. If only the alternation of kinetic coefficient is taken into account, the step bunching occurs with step-down drift. In an early stage, the initial fluctuation of the step distance determines the type of large terraces, but in a late stage, the type of large terraces is opposite to the case of alternating diffusion coefficient.Comment: 8pages, 16 figure

Simple Models for Chirality Conversion of Crystals and Molecules by Grinding

By grinding crystals in a solution, the chirality conversion of the crystal structure and the resultant conversion of molecular chirality have been realized recently. Reaction-type models that enable the reproduction of these phenomena are studied. In the models, chiral clusters are assumed to be formed by grinding and that these clusters can be integrated into crystals of the same chirality. An initial chirality imbalance is amplified exponentially, and the rate is approximately proportional to the grinding strength and to the molecular racemization rate in a relevant parameter range.journal articl

Nanoscale Equilibrium Crystal Shapes

The finite size and interface effects on equilibrium crystal shape (ECS) have been investigated for the case of a surface free energy density including step stiffness and inverse-square step-step interactions. Explicitly including the curvature of a crystallite leads to an extra boundary condition in the solution of the crystal shape, yielding a family of crystal shapes, governed by a shape parameter c. The total crystallite free energy, including interface energy, is minimized for c=0, yielding in all cases the traditional PT shape (z x3/2). Solutions of the crystal shape for c≠0 are presented and discussed in the context of meta-stable states due to the energy barrier for nucleation. Explicit scaled relationships for the ECS and meta-stable states in terms of the measurable step parameters and the interfacial energy are presented.Comment: 35 page

Novel continuum modeling of crystal surface evolution

We propose a novel approach to continuum modeling of the dynamics of crystal surfaces. Our model follows the evolution of an ensemble of step configurations, which are consistent with the macroscopic surface profile. Contrary to the usual approach where the continuum limit is achieved when typical surface features consist of many steps, our continuum limit is approached when the number of step configurations of the ensemble is very large. The model can handle singular surface structures such as corners and facets. It has a clear computational advantage over discrete models.Comment: 4 pages, 3 postscript figure

Steady chirality conversion by grinding crystals : Supercritical and subcritical bifurcations

The chirality conversion of crystals by grinding in an open system under circulation is studied. A generalized classical nucleation model and simple reaction-type models both show that steady perfect chirality conversion is possible if circulation is weak. By increasing the circulation rate λ, the amplification rate of a chirality imbalance becomes small and the imbalance of chiral products in the steady state decreases with λ. The chiral steady state disappears above a critical strength of circulation. The transition between the racemic and the chiral state is continuous in the system of achiral molecules and discontinuous in the system of chiral molecules.journal articl

The Classical Nucleation Model : Entire Process of Crystal Growth and Application to Chirality Conversion

With the use of the classical nucleation model, the passage from the initial stage of nucleation to the final stage of Ostwald ripening is surveyed. Direct numerical integration confirms that, under weak initial supersaturation, the drop of supersaturation occurs with a long delay but the relative drop is deep. For various initial supersaturation, the cluster size distribution approaches the single Lifshits‐Slyozov‐Wagner form. Based on an analysis of current in the cluster size space, a single variable model to describe the time evolution of supersaturation is proposed. The classical nucleation model is generalized and applied to the problem of chirality conversion with grinding crystals.14th International Summer School on Crystal Growth ( 1–7 August 2010, Dalian (China))conference pape