A Stochastic Model for Nucleation Kinetics Determination in Droplet-Based Microfluidic Systems

Abstract

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.

Influence of History of Solution in Crystal Nucleation of Fenoxycarb: Kinetics and Mechanisms

“…Accordingly, at each set of conditions a distribution of induction times is obtained similar to studies reported in the literature. [14][15][16] To smoothe the data a lognormal function was fitted to each distribution, giving a very reasonable fit. Lognormal distributions are seen in many data sets from different scientific disciplines, and are indicative of random variation resulting from multiple independent factors which have a multiplicative effect on the measured quantity.…”

Control of self-assembly in micro- and nano-scale systems

“…For crystallization in a nanoliter droplet implemented in a microfluidic platform [18,19,31], the small scale of the system inhibits implementing conventional methods of probing the solute concentration. While visual observation of the dynamics in self-assembled systems could be accessible through advanced microscopes (such as the fluorescent imaging technique used to track the real-time movement and clustering of Janus particles [32]), such information has to be translated into a variable that can represent the assembly status for control.…”

“…The crystallization of crystals in small volumes requires the use of stochastic models (e.g., [31]). The nucleation and growth processes for the microfluidic platform in Fig.…”

“…8 can be modeled by a Master equation (3). When only the number of crystals is considered (i.e., nucleation events), the Master equation can be solved analytically as a function of time [31]. However, when the state is extended to include the length of all crystals (i.e., nucleation and growth), the number of Master equations becomes infeasible to construct/solve directly and a KMC strategy must be used to generate approximate statistical results from the model.…”

“…The rate expressions and parameter values used in all model simulations are given in [19,31] for lysozyme. For all simulations, the droplet initially has a total volume of 5 L, a protein loading (C protein ) of 18 g/L, a salt concentration (C salt ) of 0.36 M, and a solubility curve defined by C protein-eq = 1.57C 2.94 salt g/L, where C protein-eq denotes the equilibrium (saturated) protein concentration.…”