Set Voronoi diagrams of 3D assemblies of aspherical particles

Abstract

We describe the development of a new software tool, called "Pomelo", for the calculation of Set Voronoi diagrams. Voronoi diagrams are a spatial partition of the space around the particles into separate Voronoi cells, e.g. applicable to granular materials. A generalization of the conventional Voronoi diagram for points or monodisperse spheres is the Set Voronoi diagram, also known as navigational map or tessellation by zone of influence. In this construction, a Set Voronoi cell contains the volume that is closer to the surface of one particle than to the surface of any other particle. This is required for aspherical or polydisperse systems. Pomelo is designed to be easy to use and as generic as possible. It directly supports common particle shapes and offers a generic mode, which allows to deal with any type of particles that can be described mathematically. Pomelo can create output in different standard formats, which allows direct visualization and further processing. Finally, we describe three applications of the Set Voronoi code in granular and soft matter physics, namely the problem of packings of ellipsoidal particles with varying degrees of particle-particle friction, mechanical stable packings of tetrahedra and a model for liquid crystal systems of particles with shapes reminiscent of pears. The analysis of geometries and structures on a mi-cro scale level is an important aspect of granular and soft matter physics to attain knowledge about many interesting properties of particle packings, including contact numbers , anisotropy, local volume fraction, etc. [1-3]. A well-established concept is the so called Voronoi Diagram. Here, the system is investigated by dividing the space into separate cells in respect to the positions of the center of the particles. A cell assigned to a certain particle is defined as the space (or region of space) that contains all the volume closer to the center of this specific particle than to any other one (see figure 1 left). This partition of space, however , only yields precise results for monodisperse spheres as the construction fails otherwise due to morphological properties of the objects. For nonspherical or polydisperse particles the classical Voronoi diagram is of limited use-fullnes, as shown in figure 1 (center) for a system of bidis-perse spheres. A generalized version of the Voronoi Diagram , the Set Voronoi Diagram [4], also known as nav-igational map [5] or tessellation by zone of influence [6], has to be applied. In this case the cells contain all space around the particle which is closer to the particle's surface than to the surface of any other particle. Figure 1 (right) shows the Set Voronoi Diagram of a mixture of differently shaped particles. SW and PS have contributed equally to the work described in this project Here we introduce a software tool called Pomelo, which calculates Set Voronoi Diagrams based on the algorithm described in [4]. Pomelo is particularly versatile due to its ability to generically handle any arbitrary shape which can be described mat...

Minkowski tensors of anisotropic spatial structure

“…In contrast to measures based on nearest-neighbour bonds, the Minkowski analysis naturally applies to these aspherical and possibly polydisperse particles, provided the Voronoi diagram is suitably defined. For example, with all tools in place for imaging experimental ellipsoid packings [112] and determining their Voronoi diagrams [113], the Minkowski tensor analysis may shed light on the more complicated, possibly less universal, mechanisms involved in jamming of ellipsoidal particles.…”

Characterization of microscopic disorder in reconstructed porous materials and assessment of mass transport-relevant structural descriptors

“…While this simple procedure is only valid for monosized disks and spheres, it has been extended to also be applicable to polydisperse disks and spheres as well as to irregularly shaped objects. [150][151][152][153] Voronoi cells can be evaluated with respect to the number of vertices, their edge length distributions, edge and face number distributions, area (2D) and volume (3D) distributions, coordination number, etc., and the properties thereof have been intensively studied. 101,[154][155][156][157][158][159][160][161][162][163] For crystalline structures the Wigner-Seitz cells are special cases of the Voronoi cells, 146 which consist of regular polyhedra.…”

Local Origin of Global Contact Numbers in Frictional Ellipsoid Packings

“…Finally, the Voronoi cells of the particles are computed with the algorithm described in [34]. Figure 1(d) displays the Voronoi tessellation of a small subset of particles.…”