(141g) Population Balance Modeling of Morphology Distribution of Asymmetric Crystals
AIChE Annual Meeting
Crystallization of Pharmaceutical and Biological Molecules 2: In Honor of the Late Dr. Jeff Wei of Eli Lilly
Thursday, November 11, 2010 - 10:35am to 10:55am
The association of important properties with crystal shape, like solubility of drugs, activity of catalyst, efficiency of solar cells etc., has inspired researchers to study the dynamics of crystal shape towards its control. Shape of a faceted crystal is expressed by a set of the normal distances of crystal faces from a specific center. The vector of the distances of all faces for a given crystal is called the h-vector. Usually, the evolution of the shape of a population of crystals is described by a population balance model (PBM). For most cases, such PBM is associated with a large number of internal coordinates. Maximum symmetric form of crystals is usually assumed to cope with the computational load associated with such models. However, an efficient modeling framework for asymmetric crystals is much needed as most of the crystals bear some degree of asymmetry. In this work we demonstrate a new technique for formulation and efficient solution of PBM for asymmetric crystals.
This new technique is based on the observation that although a large number of internal coordinates may exist for a crystal, many of them have identical growth rates. Therefore the internal coordinate vector of a crystal can be classified into various sets (or families) characterized by identical growth rates. Owing to the identical growth rates, the difference between any internal coordinate in a family with a fixed reference coordinate in the same family remains time invariant. Therefore, a suitable transformation in the internal coordinate space of the crystal leads to classification of the internal coordinates of a family into only one dynamic coordinate (the reference) and a set of time invariant coordinates (the differences). Because it is needless to follow the invariant coordinates dynamically, such transformation of coordinates leads to the minimum effective dimensionality to the PBM. Such redefinition of internal coordinates leads to the important feature that the number of dynamic coordinates for an asymmetric crystal is the same as that of a symmetric crystal and therefore consideration of asymmetry does not lead to extra computation.
The number of time invariant coordinates depends on the symmetry of the crystal and for a crystal of maximum symmetry this reduces to zero. A population of asymmetric crystals can have crystals with different degrees of symmetry. Symmetry of a given crystal can be quantified using the number of symmetry elements present in the crystal. In this work we show that group theory can be used effectively to obtain the symmetry elements of a given crystal. Once the symmetry group of the crystal is obtained, the minimum number of time invariant coordinates can be obtained readily. Thus, crystals are classified into several symmetry classes and PBE for each class can be written. The number of internal coordinate of crystal may also change due to disappearance of a crystal face due to its overgrowth which leads to flux of one kind of population into another.
Simulation of a population of asymmetric crystal is carried out using the abovementioned strategy. Our result shows that the population of asymmetric crystals has natural tendency to gravitate towards more symmetric form. However, this natural movement can be controlled by manipulating the supersaturation and thereby a population of crystal can be driven towards a desired symmetry. It has also been shown that symmetry related properties like sphericity or specific surface area can be controlled by manipulating supersaturation.
Reference:  Chakraborty, J., et al., Modeling of crystal morphology distributions. Towards crystals with preferred asymmetry. Chemical Engineering Science (2010), doi:10.1016/j.ces.2010.03.026