Zusammenfassung
In the chemical industries in general and in the specialty and fine chemical industries, in particular, the crystal shape and its distribution are of considerable interest because it can remarkably influence the product downstream processing and usability. Also bioavailability and dissolution behavior of active pharmaceutical ingredient is governed by the nature of prevailing crystal faces. Consequently it is important to predict, optimize and control shape distributions for industrial applications.
The shape of a convex crystal can be completely defined by a set of surfaces (planes) along with their normal distances from the crystal center (Zhang et al., 2006). One can then obtain boundaries in the shape space, which stand for the appearance/ disappearance of distinct morphological properties of a crystal (Borchert et al., 2008). An evolutionary crystal shape reaching such a boundary looses a face (or faces of similar kind) on the crystal surface in the sense that it grows out of the crystal. Therefore its dynamics is dimensionally smaller by one than it was before because only the real faces on the crystal surface determine the evolution in shape space. This imparts the system a hybrid character with varying degrees of freedom. This makes computations of single crystal shape evolution trajectories more complex than for purely continuous processes.
Based on the systematic procedure of Borchert et al. (2008), we identify morphological subspaces and define corresponding crystal number densities. In order to describe the dynamics of crystal population in these subspaces the identification of multidimensional population balance equations for each of the subspaces is required. Clearly, since a crystal can evolve from one shape to another, meaning transitions of number densities from one sub-space to another and vice versa, the dynamics of these number densities is coupled. This aspect requires the specification of clear boundary conditions and exchange terms, which will be the significant part of discussion. There are two different cases of how the crystals can be exchanged between morphological domains one by continuous influx into a lower or equally dimensional domain and second by discontinuous (instantaneous) jump from a lower to a higher dimensional domain. Due to this peculiarity one obtains a varying set of partial di
fferential equations that describes the evolution of crystal populations.
Clearly the dynamics is governed by the supersaturation which is the result of the joint crystal growth. The dependency of boundary conditions and exchange terms on supersaturation is an interesting aspect. Since supersaturation determines the growth behavior of the individual faces one can manipulate the crystal shape towards a desired direction in shape space by changing supersaturation. Hence a thorough analysis of the joint state space, which consists of the shape space and the supersaturation coordinate, is necessary to keep track of possible discrete morphological changes, which will be a key part of presentation. We also present an interesting model case study that illustrates the proposed methodology.
References
Zhang, Y., Sizemore J. P. & Doherty M. F. (2006). Shape Evolution of 3-Dimensional Faceted Crystals. AIChE Journal, 52, 1906-1915.
Borchert, C. B., Nere, N. K., Ramkrishna, D., Voigt, A., and Sundmacher, K. On the Prediction of Crystal Shape Distributions in a Steady State Continuous Crystallizer. Chemical Engineering Science, under revision, 2008.
Borchert ,C. B., Nere, N. K., Ramkrishna, D., Voigt, A., and Sundmacher, K. Evolution of Crystal Shape Distributions and Morphology Classification. ISIC17, Maastricht, 2008.
