Modelling the stochastic behaviour of primary nucleation

Abstract

We study the stochastic nature of primary nucleation and how it manifests itself in a crystallisation process at different scales and under different operating conditions. Such characteristics of nucleation are evident in many experiments where detection times of crystals are not identical, despite identical experimental conditions, but instead are distributed around an average value. While abundant experimental evidence has been reported in the literature, a clear theoretical understanding and an appropriate modelling of this feature is still missing. In this contribution, we present two models describing a batch cooling crystallisation, where the interplay between stochastic nucleation and deterministic crystal growth is described differently in each. The nucleation and growth rates of the two models are estimated by a comprehensive set of measurements of paracetamol crystallisation from aqueous solution in a 1 mL vessel [Kadam et al., Chemical Engineering Science, 2012, 72, 10-19]. Both models are applied to the cooling crystallisation process above under different operating conditions, i.e. different volumes, initial concentrations, cooling rates. The advantages and disadvantages of the two approaches are illustrated and discussed, with particular reference to their use across scales of nucleation rate measured in very small crystallisers.

Continuous Generation of Millimeter-Sized Glycine Crystals in Non-Seeded Millifluidic Slug Flow

“…The total solution volume for slug flow experiments (5 mL) was the same as Batch 2's total volume (5 mL), but the volume of each slug (e.g., 14-28 µL) was much smaller than the batch experiment. A smaller solution volume not only reduced the secondary nucleation probability as found by other studies [52][53][54]56], but also had a more evident effect on the total CSD. The reason is that for a small total number of crystals, a small change in primary nuclei number still means a large change in portion, thus a large change in CSD per slug and total, as in this case.…”

“…In this study, nucleation occurred directly in slugs, which had a much smaller individual volume than a continuous stream. As primary nucleation is a stochastic process that depends on both the supersaturation and solution volume [56], such a process is intrinsically more difficult. Figure 2 shows that primary nucleation can be induced, with a few crystals formed in each slug, at a relative supersaturation (C/C*, where C is the solution concentration and C* is the equilibrium saturation concentration) of about three.…”

Validation of Models Predicting Nucleation Rates from Induction Times and Metastable Zone Widths

“…There are several approaches for the estimation of nucleation kinetics from MSZW or induction times, including the stochastic probability distribution model approach [14][15][16][17][18][19][20][21][22][23] and the empirical power law model approach [11,[24][25][26][27][28][29][30][31][32][33]. Each approach builds on different assumptions.…”

Determination of kinetics in batch cooling crystallization processes—A sequential parameter estimation approach

“…Therefore, two colliding spherical particles are assumed to lead to an agglomerate with spherical shape and volume equivalent to the sum of volumes of the colliding particles, instead of considering the real configuration of the newly formed crystal. Additionally, the deterministic classical approach has been adopted in this work to model primary nucleation, while stochastic modeling of this phenomenon has been proposed to explain the experimental evidence …”