(146e) Secondary Nucleation Rates: Comparison of Mechanisms and Models

For decades, the production of large-scale food products, e.g., sugar and table salt, has relied upon continuous crystallization. Continuous crystallization is a critical process for realizing the continuous manufacturing of active pharmaceutical ingredients (API), thus currently attracting increasing interest from the pharmaceutical industries and regulatory agencies [1]. In continuous crystallization, secondary nucleation is of particular importance, because it largely dominates the formation of new crystals. Despite progress in our understanding of secondary nucleation, there is still a lack of precise physical understanding and thus a clear need for reliable mathematical models. The secondary nucleation mechanisms are usually classified in three families: attrition, nuclei breeding, and secondary nucleation by interparticle energies (see Figure 1)[2-4].

The first family attributes the secondary nucleation to the micro-breakage (i.e., attrition) of seed crystals due to the mechanical collision with physical objects like a stirrer blade. Another characteristic of this family is that supersaturation plays a marginal role in the secondary nucleation process. This family includes all the mechanisms resulting from a mechanical collision, i.e., initial breeding [5], fragmentation, and attrition [6-8]. Initial breeding refers to the removal of already existing crystal fines from the crystal surface due to the action of fluid shear. By contrast, both attrition and fragmentation result from the collisions among the seed crystals and/or their collision with the walls and the impellers of the reactor. These collisions lead to the formation of nuclei as small fragments broken from the seed crystals, thus having the identical crystal structure[6].

The second family is usually named nuclei/surface breeding [9], or contact nucleation [10] and nucleation is supposed to be triggered by the effect of the crystal surface on the nucleation thermodynamics and/or mechanical perturbations originated from several sources: for instance, stirring blades and the hydrodynamic shear of the system[11].

The third family is an extension of the classical nucleation theory (CNT) in which molecular clusters are present in the bulk solution because of local short-lived density fluctuations. This group of mechanisms results as an extension of the CNT, whereby these clusters are more concentrated near the seed crystal surfaces because of interparticle interactions. The surfaces of the seed crystal enhance the nucleation of these clusters by decreasing the activation energies of the nucleation process. From this perspective, secondary nucleation can be attributed to the van der Walls interaction energies as demonstrated in the embryo coagulation secondary nucleation mechanism (ECSN) [12] or to a more comprehensive set of interparticle energies as in the secondary nucleation by interparticle energy mechanism (SNIPE) [13].

The goal of this work is to share a unifying perspective to the various secondary nucleation mechanisms and corresponding secondary nucleation rate expressions by means of mathematical modeling. To achieve this goal, we present a systematic and quantitative comparison of the various secondary nucleation rates and further characterize the influence of the fluid dynamics on the secondary nucleation. Moreover, we offer guidelines for necessary experimental investigation to validate some of the mechanisms.

Methods and results

In this work, we examine the nucleation mechanisms by attrition [7] and by interparticle energies [13-14] in detail and further derived mechanistic secondary nucleation rates. Taking SNIPE as an example, we investigate the underlying chemical physics of the secondary nucleation process by advancing the thermodynamic understanding of nucleation near the crystal surface with a through consideration of the interparticle interactions involved. By identifying critical parameters and incorporating them into the framework of the ECSN model, we have successfully derived a more general secondary nucleation rate expression.

Furthermore, we have performed a comparison of the nucleation rates from a modeling and an experimental point of view. We have analyzed the functional dependencies of the secondary nucleation rates, in particular the effect of supersaturation, and of the operating conditions (e.g., energy input). Moreover, we have focused on the properties of the seed population that can vary the nucleation rate significantly, which are usually the specific area for nucleation by surface mechanisms and the specific volume or the number for attrition-like mechanisms. Finally, we have analyzed fluid dynamic aspects of the crystallizer through computation fluid dynamic simulations, and we have derived a correlation among fluid dynamic quantities (e.g., shear rates, energy dissipation rates), mechanisms of secondary nucleation, and the secondary nucleation rates.

This problem is complex because secondary nucleation can be the outcome of multiple phenomena, whose interplay results in a lumped secondary nucleation rate. Empirically derived secondary nucleation rates are usually expressed as [15]

J_S = E_t(θ)F₁(c)F₂(f)

where E_t and F₁ are generic functions, θ represents the operating conditions (e.g., stirring rate, reactor geometry, etc. ); F₁ accounts for the effect of the concentration, e.g., through the supersaturation or growth rate; F₂is a functional of the particle size distribution f [8], e.g., the second or third moment of the population.

Our thorough analysis of these dependencies allows for a more careful planning of the experimental validation of the models proposed. For example, by changing the supersaturation, one should be able to access and decouple different mechanisms of secondary nucleation. For instance, at low supersaturation, attrition is the decisive mechanism, whereas, at high supersaturation, other secondary nucleation mechanisms take the lead. Likewise, by changing the geometry of the reactor and the intensity of stirring, one could describe the mechanism and the rate of secondary nucleation in a different way.

We successfully identify the key operating parameters to achieve desired degree of secondary nucleation of a certain mechanism by decomposing the available empirical correlations of the secondary nucleation into principal elements and further controlling them.

Conclusion

Summarizing, in this work, we have achieved a general understanding of the theories available in the literature, and we have further expanded and completed the analysis to other surface-based mechanisms (SNIPE). Furthermore, we have compared the functional forms of secondary nucleation rates both from a modeling point of view, as well as from experiments, where with the aid of computational fluid dynamics, we were able to test experimental conditions, where one can isolate and observe the different mechanisms.

References

1. Brown, C. J.; Mcglone, T.; Yerdelen, S.; Srirambhatla, V.; Mabbott, F.; Gurung, R.; Briuglia, M. L.; Ahmed, B.; Polyzois, H.; Mcginty, J.; et al. Mol. Syst. Des. Eng. 2018, 3 (3), 518–549.
2. Agrawal, S. G.; Paterson, A. H. J. Eng. Commun. 2015, 202 (5), 698–706.
3. Zhang, D.; Wang, X.; Ulrich, J.; Tang, W.; Xu, S.; Li, Z.; Rohani, S.; Gong, J. Growth Des. 2019, 19, 3070–3084.
4. Xu, S.; Wang, Y.; Hou, Z.; Chuai, X. Eng. Chem. Res. 2020, 59 (41), 18335–18356.
5. Steendam, R. R. E.; Frawley, P. J. Growth Des. 2019, 19 (6), 3453–3460.
6. Mersmann, A. Crystallization technology handbook. CRC press, 2001.
7. van der Heijden, A. E. D. M.; van der Eerden, J. P.; van Rosmalen, G. M. Eng. Sci. 1994, 49 (18), 3103–3113.
8. Bosetti, L.; Mazzotti, M. Growth Des. 2019, 20 (1), 307–319.
9. Anwar, J.; Khan, S.; Lindfors, L. Chemie - Int. Ed. 2015, 54 (49), 14681–14684.
10. Cui, Y.; Myerson, A. S. Growth Des. 2014, 14 (10), 5152–5157.
11. Yousuf, M.; Frawley, P. J. Process Res. Dev. 2019, 23 (9), 2009–2019.
12. Qian, R. Y.; Botsaris, G. D. Eng. Sci. 1997, 52 (20), 3429–3440.
13. Bosetti, L.; Ahn, B.; Mazzotti, M. (in preparation) 2021.
14. Israelachvili, J. N. Intermolecular and Surface Forces, 3rd ed.; Elsevier Inc., 2011.
15. Botsaris, G. D. Secondary Nucleation - A Review. In Industrial Crystallization; 1976; pp 3–22.

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 2-73959-18.