The measured induction times in droplet-based microfluidic systems are stochastic and are not described by the deterministic population balances or moment equations commonly used to model the crystallization of amino acids, proteins, and active pharmaceutical ingredients. A stochastic model in the form of a Master equation is formulated for crystal nucleation in droplet-based microfluidic systems for any form of nucleation rate expression under conditions of time-varying supersaturation. An analytical solution is provided to describe the (1) time evolution of the probability of crystal nucleation, (2) the average number of crystals that will form at time t for a large number of droplets, (3) the induction time distribution, and (4) the mean, most likely, and median induction times. These expressions are used to develop methods for determining the nucleation kinetics. Nucleation kinetics are determined from induction times measured for paracetamol and lysozyme at high supersaturation in an evaporation-based high-throughput crystallization platform, which give low prediction errors when the nucleation kinetics were used to predict induction times for other experimental conditions. The proposed stochastic model is relevant to homogeneous and heterogeneous crystal nucleation in a wide range of droplet-based and microfluidic crystallization platforms.

This study reports the selective growth of γ-glycine crystals via concentrating microdroplets of aqueous glycine solutions through slow evaporation of water using an evaporation-based crystallization platform. In prior studies, γ-glycine crystals could only be obtained from non-neutral pH solutions, by applying electromagnetic fields, or in the presence of impurities that suppress the formation of the kinetically favored R-glycine polymorph. Here in our work, pure γ-glycine crystals form below a certain rate of evaporation (i.e. below a certain rate of supersaturation). Below this rate the crystallizing solution stays close to equilibrium throughout the evaporating process, allowing the system to sample the lowest free energy state during the formation of nuclei. These results point to the interplay of kinetic and thermodynamic effects on selective crystallization of different polymorphs. Polymorphic analysis was performed by examining all samples as randomized polycrystalline particles. The resulting multiframe diffraction patterns were combined to generate a single powder X-ray diffraction (PXRD) spectrum of each sample. In comparison to traditional powder diffraction methods, the quantitative polymorphic analysis procedure reported here eliminates the need to mechanically grind crystalline material, thereby avoiding the potential for undesired polymorphic transformations prior to data collection.

The metastable zone in solution crystallization is typically defined as a region of the phase diagram in which no appreciable nucleation occurs. Existing theoretical explanations attribute the appearance of this zone to the low probability of nucleation brought forth by the path-dependency of the nucleation rate. In this work, for the first time we present experimental data for several compounds that contradict this description. We show that the widely adopted theoretical approach which considers a time-dependent nucleation rate does not capture the observed stochastic nature of nucleation in these experiments. Instead, the experimental results are successfully explained through a probability analysis based solely on the energy barrier to nucleation. In this context, for a system that is slowly supersaturated, we develop the idea of an “induction supersaturation” as a lower boundary of metastability that does not depend on the path of the experiment. This work critically examines the limitations of the existing stochastic methods that describe nucleation under variable supersaturation and calls for a fundamental shift in the traditional view of the processes responsible for the manifestation of the metastable zone.