(279a) Non-Equilibrium Kink Density Model for Organic Molecular Crystals Containing Two Growth Units

In layered regimes, crystals grow via lateral progression of steps flowing across a planar surface at a constant step velocity. Under supersaturated conditions, growth units (e.g., molecule, dimer) mainly integrate into specific sites called kink sites along a step or edge. This is attributed to the fact that kink sites regenerate themselves and attachment occurs with no surface energy penalty. Hence, the step velocity depends on (1) the rate of attachment at kink sites and (2) density of kink sites. Kink density is often approximated by its equilibrium value given by the Boltzmann distribution. However, it is known that kink density depends on supersaturation and quickly deviates from the equilibrium approximation.¹ Several models have been proposed in the literature for non-equilibrium kink density calculations.^2,3 However, these approaches have been built for the Kossel crystal and do not readily extend to any realistic molecular crystal systems. Another modeling endeavor is the Simplified Steady-State Framework (SSSF) proposed by Padwal and Doherty¹, which provides non-equilibrium kink density estimation by accounting for the surface kinetics of the most-likely surface events. Such a framework differs from other approaches in that it is simple enough to be extended to more complex molecules.

In this work, SSSF is extended to the case of asymmetric organic molecular crystals with two growth units i.e., Z=2, Z'=1 or Z=2, Z'=2. Such a study is the first of its kind attempted for asymmetric molecules. The two growth units, A and B, are identical in the solution phase and integrate into the crystal lattice in different orientations. An AB crystal system comprises of faces with different growth unit patterns and constitute edges with different row configurations. Different edges consist of different types of kinks differing in densities and rates of detachment. Different kinks are formed and destroyed via different surface processes along the edge. Based on the SSSF, steady-state master equations are constructed for the most stable kink types accounting for the more-likely events that form or destroy these kinks. This is implemented using a rate model for the physical reaction of growth unit attachment/detachment and a stochastic approach for modeling densities of various types of sites along the edge. The steady-state master equations constructed are then solved numerically to yield non-equilibrium kink densities. Step velocity is then estimated as a function of non-equilibrium densities followed by crystal growth rate estimates to yield morphology predictions.⁴ Inputs required are growth environment conditions such as temperature, supersaturation, solvent information; and an appropriate atom-atom forcefield. Such an analysis is applied to real organic molecular crystal systems to demonstrate its applicability to morphology prediction and control. This will serve as a blueprint to solve the problem of non-equilibrium kink density prediction in systems with more than two growth units, thereby marking the first step to accurate growth models for complex organic molecules with any number of growth units.

References:

1. Padwal, N. A.; Doherty, M. F. A Simple Accurate Non-equilibrium Step Velocity Model for Crystal Growth of Symmetric Organic Molecules. Crystal Growth and Design (under review).
2. Cuppen, H. M.; Meekes, H.; van Veenendaal, E.; van Enckevort, W. J. P.; Bennema, P.; Reedijk, M. F.; Arsic, J.; Vlieg, E. Kink Density and Propagation Velocity of the [010] Step on the Kossel (100) Surface. Surface Science 2002, 506 (3), 183–195. https://doi.org/10.1016/S0039-6028(02)01427-9.
3. Joswiak, M. N.; Peters, B.; Doherty, M. F. Nonequilibrium Kink Density from One-Dimensional Nucleation for Step Velocity Predictions. Crystal Growth & Design 2018, 18 (2), 723–727. https://doi.org/10.1021/acs.cgd.7b01092.
4. Li, J.; Tilbury, C. J.; Kim, S. H.; Doherty, M. F. A Design Aid for Crystal Growth Engineering. Progress in Materials Science 2016, 82, 1–38. https://doi.org/10.1016/j.pmatsci.2016.03.003.