Modeling the Impact of Flow Modulation on Surface Structure during the Growth of Potassium Dihydrogen Phosphate Single Crystals from Solution

We study the influence of liquid-phase flow on a potassium dihydrogen phosphate (KDP) crystal growing from solution. In particular, we focus on the effect of modulated flow on the structure of the crystalline surface both from microscopic and from macroscopic viewpoints. Our microscopic model, based on a phase-field representation of steps, correctly predicts the appearance of step bunching due to (destabilizing) solution flow in the direction of step motion; suppression of step bunches by modulating the direction of flow above the crystal surface is analyzed with this model. At the same time, our macroscopic model uncovers complications associated with flow modulation. Resultant spatiotemporal changes in the supersaturation field along the crystal surface may lead to time-dependent rates of step generation at active step sources, causing variations in the crystal slope (i.e., weak step bunching), as well as promoting periodic reversal of the step flow direction along portions of the surface. Revisiting the microscopic model, with boundary conditions modified to account for these macroscopic observations, reveals more clearly the possible impact of these complications on crystalline quality. Our microscopic and macroscopic observations suggest the need for nontrivial multi-scale analyses for the investigation of modulation or even more complex time-dependent flows as possible tools for effectively controlling flow-driven step bunching during crystal growth from solution.