PhD thesis – Fabian Schaller

Abstract

In particulate systems with short-range interactions, such as granular matter or simple fluids, the local structure has crucial influence on the macroscopic physical properties. This thesis advances our understanding of granular matter by a comprehensive study of Voronoi-based local structure metrics applied to amorphous ellipsoid configurations. In particular, a methodology for a local, density-resolved analysis of structural properties is developed. These methods are then applied to address the question of when the global packing fraction alone is a sufficient descriptor of the structure, and situations for which this is not the case.

Packings of monodisperse spherical particles are a common simple model for granular matter and packing problems. This work focuses on packings of ellipsoidal particles, a system which offers the possibility to study the influence of particle shape on packing properties, in particular particle anisotropy.

A large scale experimental study of jammed packings of oblate mm-sized ellipsoids with various aspect ratios is performed. Packings are prepared with different preparation protocols to achieve different global packing fractions and imaged by X-ray tomography. Additional datasets of packings are created by Discrete Element Method simulations of frictional and frictionless particles with and without gravity. Furthermore, packings of Ottawa sand samples are analyzed, in an attempt to investigate the relevance of the ellipsoid model system for real world granulates.

The structure of the packings is analyzed by Set Voronoi diagrams, an extension of the conventional Voronoi diagram to aspherical particles. We find some surprising structural properties, specifically related to the local packing fraction, defined as particle volume divided by its Voronoi cell volume. A universality is found in the probability density function to find a particle with a local packing fraction in a given packing. The width of the density function is independent of the aspect ratio. For spheres, Aste et al. [EPL 79:24003, 2007] proposed an analytic model for the distribution of Voronoi cell volumina. Their model strongly depends on the locally densest configuration, a quantity that was, prior to this work, not known for ellipsoids. We numerically investigate the locally densest structures and analyze their occurrence as local building blocks of granular packings. Knowledge of the densest structures allows to rescale the Voronoi volume distributions onto the single-parameter family of $k$-Gamma distributions. Remaining deviations are explained by an excessive formation of distorted icosahedral clusters.

A robust tool to characterize spatial structures is provided by Minkowski tensors, which generalize the concepts of interface and moment tensors. Here, we investigate the shape properties of the Voronoi cells by anisotropy indices derived from these tensors. These local anisotropy indices point towards a significant difference in the local structure of random packings of spheres and ellipsoids. While the average cell shape of all cells with a given local packing fraction is very similar in dense and loose jammed sphere packings, the structure of dense and loose ellipsoid packings differs substantially such that this does not hold true.

The average number of contacts of a particle with its neighbors is an important predictor of the mechanical properties of a packing as forces in granular matter are transmitted through contacts. This conceptually straightforward parameter is, however, difficult to analyze, since contact detection is hindered by experimental noise and is often connected to a numerical cut-off. It is less reliable than the continuously defined analysis by Minkowski tensors. In our jammed packings of ellipsoids, we find, that the average number of contacts is a monotonously increasing function of the global packing fraction for all aspect ratios. This dependence can be explained by a local analysis where each particle is described by its local packing fraction and the average number of neighbors.

Our results clearly demonstrate the need for a local analysis when ellipsoid packings are analyzed. Local analyses of this type reveal differences in the structure, which are not captured by global averages. This points to an important structural difference to the sphere pack case where the global packing fraction seems to suffice to rationalize most observed properties, at least to a good approximation. While our study specifically addressed granular matter models of hard particles subject to gravity, these finding are likely to also rationalize observations of other soft matter particulate systems, including colloidal particles or other micron- or nanometer-sized particles.