Population Balance Modeling of Growth and Secondary Nucleation by Attrition and Ripening

Abstract

Secondary nucleation is ubiquitous in nature and of fundamental importance for both batch and continuous crystallization processes. Attrition is the mechanism through which fragments are formed after the collision of a crystal with a stirrer. Those fine fragments, if small enough, are considered secondary nuclei. In this work, starting from the mechanistic description of attrition by Gahn and Mersmann (Crystallization Technology HandbookCRC Press2001), two population balance equation models to simulate secondary nucleation processes have been derived. The first simulates attrition as a breakage term, and growth rate is the result of size-dependent solubility. The second model considers attrition as a boundary condition at zero crystal size, where the expression for secondary nucleation rate already takes into account the effect of supersaturation, while the growth rate is size-independent. The two models are proven equivalent in the growth regime, thus where secondary nucleation and growth are the dominant phenomena. At extremely low values of supersaturation, thanks to size-dependent solubility, the first model yields to further development of the crystal population, e.g., ripening and aging. The main result is that secondary nucleation by attrition can be described as a birth/death term or, alternatively, as a source term according to the final application of the model. Since the two approaches have very different computational intensities, one can choose the right model based on the objective of the simulation study. The evolution of the crystal population for high values of supersaturation will be the same in both cases.

Overview of Secondary Nucleation: From Fundamentals to Application

“…118 nucleation rates are given by an empirical relationship that depends on the number of suspended crystals (volume or surface), the level of supersaturation (either directly or through growth rate), and the degree of mixing (usually through average energy dissipation). 48 The rate of secondary nucleation increases with the increase of these three quantities. 113 Secondary nucleation is an autocatalytic process that significantly affects the crystallization outcome, such as the CSD, polymorphism, and chirality of the crystal products.…”

“…These fragments then grow or dissolve depending on the degree of supersaturation in the system. 48 Bosetti and Mazzotti 49 conducted attrition experiments in ethanol at different stirring speeds, and the results showed that by increasing energy input, nuclei appeared faster and in greater quantities. Synowiec et al 50 studied the attrition of potassium alum in the presence of an antisolvent (ethanol) and established a model that takes into account the fine particles formed due to the turbulent nature of the flow field.…”

Secondary Nucleation by Interparticle Energies. III. Nucleation Rate Model

“…As far as r is concerned, there are two limit behaviors, whereby ϵ(0) ≃ 0 and ϵ( r → ∞) ≃ 1, and two threshold values, namely, the just suspended and the uniformly suspended cases. It is worth noting in passing that for very intense stirring, i.e., for r → ∞, secondary nucleation will be controlled by the attrition mechanism, 48 whereby the role of SNIPE will be negligible.…”

Solid-State Deracemization via Temperature Cycles in Continuous Operation: Model-Based Process Design

“…The latter mechanism must clearly be prevented, namely, by keeping the supersaturation level below a (secondary nucleation-) specific threshold, which is even lower than the previous one. Such physical complexity is accounted for in the model by allowing for the formation of new nuclei of the desired enantiomer via secondary nucleation by attrition, 41 as described in Table 2 , and by constraining the simulations in such a way that the supersaturation with respect to the undesired enantiomer is kept below a given threshold, so as to avoid both its secondary and its primary nucleation. To model the mechanisms that occur in this process in a simple manner, both breakage and the formation of new nuclei of the desired enantiomer by secondary nucleation can be described as secondary nucleation via attrition by a simple breakage model, 41 as Table 2 describes.…”