PHD THESES

Quasicrystals are remarkable ordered structures without periodic translational symmetry. They can possess any discrete rotational symmetry including those that must not be present in periodic crystals. An important characteristic of quasicrystals is the existence of additional degrees of freedom whose excitations lead to rearrangements of particles. Such rearrangements are called phasonic flips. Several material properties like the elasticity of quasicrystals are affected by phasonic excitations. By now, quasicrystals have been synthesized in experiments and simulations and have even been found in nature. The aim of this thesis is to contribute to the understanding of the amazing order and properties of quasicrystals. For this purpose we employ computational simulations of two-dimensional colloidal model systems where mesoscopic particles are suspended in a liquid. By implementing appropriate external or internal interactions quasicrystalline order of the colloids can be induced.
Significant new insights into the structural and dynamical complexity of quasicrystals are gained. Our results are essentially different from what is known from periodic crystals. In particular, investigations of the phase behavior of quasicrystals reveal a surprisingly rich phase diagram. Even in the solid, positional order is short-ranged due to excited phasonic degrees of freedom, and the transition to liquid is of first order. Furthermore, we illustrate how the growth of quasicrystals is affected by thermodynamic parameters and phasonic flips.
Especially the growth of nearly defect-free quasicrystals is presented. In addition, we focus on the dynamics of quasicrystals. The stability of quasicrystals against phasonic perturbations is investigated. Particles are identified which easily perform phasonic flips, while other ones are rather stable. Phasonic drifts lead to complex trajectories of the particles. Even in intrinsic quasicrystals, which form under internal interactions alone, correlated phasonic flips are found and analyzed.
Our work provides significant progress in theory and simulations of quasicrystals and our results obtained from colloidal model systems are also relevant for other fields in physics, chemistry and material science. We expect that our work motivates further theoretical and experimental research on quasicrystals and might also advance the design of novel applications based on quasicrystals.

Particulate and granular materials are ubiquitous in nature, industry and science. In such systems the spatial structure plays a central, often dominant, role in determining physical properties. In this thesis, I develop X-ray tomography methods and quantitative structure measures to examine various experimental particulate systems, such as dry and wet monodisperse sphere packings, bidisperse sphere packings, packings of frictional emulsion droplets or tetrahedral packings.

The properties of particulate systems depend strongly on the local environment of each particle, as interactions between particles are local in most cases, e.g. repulsive contact forces or cohesive liquid bridges. Therefore geometric approaches to characterize the local environment are needed. This local environment is characterized by the Set-Voronoi tessellation. While a Voronoi cell of a particle is the volume that is closer to the center of this particle than to any other particle, the Set-Voronoi cell is the volume that is closer to the boundary of the particle than to any other particle surface. Set-Voronoi tessellations can be used on arbitrary particle shapes and configurations. The Set-Voronoi tessellation allows for a local description by geometric measures, mainly the local packing fraction and different shape measures based on Minkowski functionals and Minkowski tensors. Independent of the Set-Voronoi tessellations the contact number is measured. As contacts are a key mechanism of transmitting forces through the system, the contact number is an important measure for the mechanical stability. In this work four different physical systems are investigated using structure measures:

Tribo-charging in bidisperse sphere packings Tribo-charging describes the genera- tion of electrical charge on particles by collisions. It can lead to either repulsive or attractive forces within a packing. Packings of bidisperse spheres made of polytetrafluorethylene are analyzed in order to determine the influence of tribocharging on segregation, packing fraction and contact numbers. By controlling the humidity while shaking the beads the tribocharging can be controlled. For such systems, we here show that the contact numbers are charge dependent: With increasing charge density the same-type contact numbers decrease while the opposite-type contact numbers increase.

Tetrahedral packings When compared to sphere packings, tetrahedral particles show an increased complexity due to the fact that different contact types (face-to-face, edge- to-face, edge-to-edge, vertex-to-face contacts) impose a different number of mechanical constraints. History dependence is defined as the fact that apparently identical granular samples will differ depending on their history of preparation. The effect of history de- pendence is visible in the investigated packings of plastic, injection-moulded tetrahedral particles. We perform a local analysis of the contact distribution by grouping the par- 6ticles together according to their individual local packing fractions, as obtained by the Set-Voronoi tessellation. We then show that for sufficiently tapped packings the number of face-to-face contacts becomes a universal function of the global packing fraction, while the edge-to-face and point contacts vary with the applied packing protocol.

“Skinny” emulsions Frictional emulsions are a new, interesting type of soft and de- formable particulate system. We present a first systematic analysis of the structural features of such systems using X-ray tomography on polyethylene glycol drops. While in normal granular systems the particles are assumed to be ideally hard, the droplets in emulsions are deformable. Systems with different drop sizes are investigated with respect to the pair correlation function and packing fraction distributions. The local structural properties of these system are quite interesting as some aspects are similar to packings of hard, frictional particles, like the local packing fraction distributions and the constant global packing fraction with emulsion height (Janssen effect). Other properties are quite different from hard frictional particles, for example the flat pair correlation function. When compared to other emulsion systems it becomes obvious that friction and adhesion have a major impact on the local structure of the packing.

Liquid-stabilized sphere packings The mechanical properties of granular systems change significantly when small amounts of liquid are present in the packing due to the formation of capillary bridges. The structural differences between dry and wet sphere packings are examined using a model system of monodisperse polyoxymethylene beads and bromodecane as a wetting liquid. Our analysis demonstrates that no visible struc- tural differences are found with respect to the contact numbers and packing anisotropy. Additionally the bridge number, the average amount of bridges per particle, is reported to be higher by a value of 2 than the contact number, independent of packing fraction, preparation method and liquid content.

All systems investigated in this thesis have in common that the structural properties play a governing role for the physical properties. Thus gaining insight into the internal structure by using X-ray tomography will help to get a better understanding for granular and particulate systems. The importance of Set-Voronoi tessellations as a description of the local environment and their general applicability is demonstrated in the investigated systems. Our investigations focus on granular and particulate systems. Other disciplines, for example in soft matter physics, are likely to benefit from the methods and results, which are discussed in this thesis as X-ray tomography and Set-Voronoi cells are easily applicable to those systems.

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.

In this thesis we use the bead-spring microswimmer design as a model system to study mechanical microswimming. The basic form of such a swimmer was in- troduced as the ‘three-sphere swimmer’ in Najafi and Golestanian [2004] and has found wide use in theoretical, numerical and experimental research. In our work, we have modified and extended the model in various ways, which, as explained in this thesis, allow us to gain insight into many general principles of microswim- ming, for instance the interplay between fluid drag force and swimmer elasticity in determining the efficiency of motion. The work presented here consists of both analytical solution of the equations of motion in the different investigated cases and corresponding numerical study.

Cells connect to other cells with the help of highly specific membrane- anchored adhesion proteins. This adhesion process is typically regulated by the complex biochemical network of the cell. However, on the molecu- lar level, thermal fluctuations dominate every motion of the proteins, the cell membrane and the surrounding water. These thermal fluctuations and their influence on the adhesion process between two adjacent cell mem- branes are the core of this thesis..

Triangulations, which can intuitively be described as a tessellation of space into simplicial building blocks, are structures that arise in various different branches of physics: They can be used for describing complicated and curved objects in a discretized way, e.g., in foams, gels or porous media, or for discretizing curved boundaries for fluid simulations or dissipative systems. Interpreting triangulations as (maximal planar) graphs makes it possible to use them in graph theory or statistical physics, e.g., as small-world networks, as networks of spins or in biological physics as actin networks. Since one can find an analogue of the Einstein-Hilbert action on triangulations, they can even be used for formulating theories of quantum gravity. Triangulations have also important applications in mathematics, especially in discrete topology.

Despite their wide occurrence in different branches of physics and mathematics, there are still some fundamental open questions about triangulations in general. It is a prior unknown how many triangulations there are for a given set of points or a given manifold, or even whether there are exponentially many triangulations or more, a question that relates to a well-defined behavior of certain quantum geometry models. Another major unknown question is whether elementary steps transforming triangulations into each other, which are used in computer simulations, are ergodic. Using triangulations as model for spacetime, it is not clear whether there is a meaningful continuum limit that can be identified with the usual and well-tested theory of general relativity.

Within this thesis some of these fundamental questions about triangulations are answered by the use of Markov chain Monte Carlo simulations, which are a probabilistic method for calculating statistical expectation values, or more generally a tool for calculating high-dimensional integrals. Additionally, some details about the Wang-Landau algorithm, which is the primary used numerical method in this thesis, will be examined in detail.