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.

Figure 1

(a) Evaporation-based microwell crystallization platform with droplets adhering to a glass slip, and (b) crystal formed within an evaporating hanging drop as observed from above. In (a), dark red and blue inks are used to visualize the evaporation channel and the droplet, respectively.

Figure 2

(a) Time evolution of probabilities P_n(t) for n = 0, 1, 2, …, 7 for κ = 0.1, and (b) the corresponding P_n(t) vs. n for different times. The lines in (b) are drawn to guide the eye.

Figure 3

(a) Measured and model (t_mean) mean induction times for paracetamol in water for six experimental conditions with kinetic parameters A = 3.52×10⁵ g⁻¹hr⁻¹ and B = 14.3, (b) corresponding nucleation rate expression (19), and (c) comparison of model and experimental mean induction times for five additional experimental conditions, showing 90% prediction intervals. [The nucleation kinetics were the same regardless of whether the time to grow to a visible size was taken into consideration (using growth kinetics obtained from Finnie et al., G = 0.0183(C(t)/C_sat−1)² m/hr), supporting the assumption of negligible growth time.]

Figure 4

(a) Model cumulative distribution function and (b) probability distribution function (11) for Experimental Condition #2 for paracetamol in water.

Figure 5

Lysozyme-NaCl-water system: (a) experimental and model cumulative distributions of induction times at two experimental conditions, with model (19) using parameters ln(A, g⁻¹hr⁻¹) = 12.5 and B = 9.7 fit to induction times for Experimental Condition #1, (b) probability distribution functions with model mean induction time (o) and experimental mean induction time (×), and (c) corresponding nucleation rate expression (19). The 95% confidence intervals for ln(A, g⁻¹hr⁻¹) and B are [11.9,13.1] and [7.4,12.0] using t-statistics and [11.3,13.7] and [6.8,12.7] using F-statistics (see Supporting Information for calculation and discussion of the confidence intervals). The time for a nucleus to grow large enough to be visible in these experiments is very short compared to the induction time.