(655c) Modeling the Step Velocity of Non-Centrosymmetric Growth Units and Accounting for Stable/Unstable Layers

The velocities of step edges on crystal faces are anisotropic. This must be accurately captured for a mechanistic model of crystal growth to be successful, since both a spiral growth regime and a birth-and-spread two-dimensional nucleation and growth regime depend on step velocities. The key components to the step velocity are the density of kinks along the step edge (the kink density) and the net rate of growth unit incorporation into kinks (the kink rate).

Non-centrosymmetric growth units represent the majority of industrially relevant molecules, but introduce significant complexity into this problem. The pattern of different growth units along the step edge produces multiple kink sites, and so the simple centrosymmetric expressions for kink density [1] and kink rate [2] are no longer applicable. Kuvadia and Doherty [3] introduced a framework for the mechanistic modeling of crystals with non-centrosymmetric growth units: they considered hypothetical transformations to form an analogous Boltzmann distribution for the kink density and determined an expression for the kink rate for a general number of growth units. Importantly, they also introduced the concept of stable/unstable step edge layers, which are unique to non-centrosymmetric growth units.

We present several improvements and generalizations of this framework that enable a more universal application and provide the ability to better capture the effect of stable/unstable layers. First, we use surface energies around each step edge site to form the Boltzmann-distributed kink density, which enables calculations of specific site densities. This also ensures the required collapse to the centrosymmetric case under appropriate limits. Second, we consider that specific kink sites are unable to move at their native kink rate velocity indefinitely; they can become limited following collision events with other kink sites. This effect can be captured by considering characteristic timescales for possible collisions and a characteristic timescale (based on kink annihilation) for edge reorganization. Each kink is allowed to advance at its native velocity for a fraction of the reorganization timescale and is then appropriately limited. Physically, this corresponds to the exclusion of overhang growth along the step surface. To calculate the step velocity, the individual contributions from each type of kink on the surface are summed (i.e., specific kink densities and specific native kink rates, which may become limited).

We have tested this expression for the case of a 2-layer step with alternating layers of â??Aâ?? and â??Bâ?? growth units against kinetic Monte Carlo simulations, varying the edge energy of B. The predicted step velocities are in excellent agreement with simulations and can capture the trend that as the B edge energy is increased, double kinks become dominant on the step edge surface. The resulting models enable accurate morphology prediction for a wide class of molecular solutes.

References

1. Frenkel, J. On the surface motion of particles in crystals and the natural roughness of crystalline faces. J. Phys. (Moscow) 1945, 9, 392-398.
2. Voronkov, V. The movement of an elementary step by means of the formation of one-dimensional nuclei. Sov. Phys. Crystallogr. 1970, 15, 8-13.
3. Kuvadia, Z. B.; Doherty, M. F. Spiral Growth Model for Faceted Crystals of Non-Centrosymmetric Organic Molecules Grown from Solution. Cryst. Growth Des. 2011, 11, 2780-2802.