Convex Geometry for the Morphological Modeling and Characterization of Crystal Shapes

Abstract

AbstractTo characterize crystals and other particles not only with respect to their size but also to their shape there has been increasing interest over the last decade. Though there are several studies on the geometric problems for single crystals and studies using morphological population balance equations for specific cases there is a lack of a systematic way to construct models for the growth of crystal populations with arbitrary shape. This is due to the geometrical complexity. The aim of this work is to exploit mathematical theories in the area of convex geometry to overcome this deficiency. By introducing an algebra for convex bodies, any convex crystal shape can be decomposed into a set of simple shapes called structuring elements. The decomposition is a linear combination from which generic equations for the calculation of any measure, like volume, surface area or mean diameter, can be easily derived. Importantly, all the required parameters for these measure calculations can be calculated a priori to any dynamic simulation. Ultimately, this allows the generic construction of morphological population balances to model crystal growth. Additionally, the combinatorial complexity is explored with respect to the computational effort. Although the concepts are developed for faceted crystals only, the framework may apply to a much broader class of convex particles.