Convex Geometry for the Morphological Modeling and Characterization of Crystal Shapes

Abstract

To 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.

Modeling crystal growth: Polyhedra with faces parallel to the planes x1 ± x2 ± x3 = 0

“…In [4] authors study families of polyhedra homothetic to summands of one fixed polyhedron. In this paper we replaced the restriction of "homothetic to summands of one fixed polyhedron" with "faces parallel to fixed set of planes", specifically, we study the cone of G-polyhedra "with faces parallel to the planes x 1 ± x 2 ± x 3 = 0".…”

“…In this paper we replaced the restriction of "homothetic to summands of one fixed polyhedron" with "faces parallel to fixed set of planes", specifically, we study the cone of G-polyhedra "with faces parallel to the planes x 1 ± x 2 ± x 3 = 0". In [4] the authors do not modify Minkowski addition. Here we modify it and this G-addition is more adequate to the physical nature of crystal growth.…”

“…In a similar way also A th = tA h for negative t if and only if the set A h is a singleton. The vector h A represents the polyhedron A in a manner of [4].…”

Simulations of crystal aggregation and growth: Towards correct crystal area

“…This work follows the crystal representations used in prior publications of the Briesen group and therein developed MATLAB code. In the “geometrically complex” method, each primary particle is described by a set of unit face normals a i , as well as their distances h i from the coordinate system origin (H‐Representation).…”

“…This method, as well as the involved crystallization phenomena and challenges related to shape control have been reviewed in the literature . One of the main challenges lies in the complexity of the crystal shape and the changes it undergoes during crystallization . Crystals may change the set of visible faces through growth and dissolution, as well as undergo breakage or form aggregates of several crystals.…”