80 research outputs found
How bees and foams respond to curved confinement:Level set boundary representations in the Surface Evolver
We investigate the equilibrium properties of a single area-minimising bubble trapped between two narrowly-separated parallel curved plates. We begin with the simple case of a bubble trapped between concentric spherical plates. We develop a model that shows that the surface tension energy of the bubble is lower when confined between spherical plates as compared to a bubble trapped between flat plates. We confirm our findings by comparing against Surface Evolver simulations. Next, we derive a simple model for a bubble between arbitrarily curved parallel plates. The energy is found to be higher when the local Gaussian curvature of the plates is negative and lower when the curvature is positive. To check the validity of the model we consider a bubble trapped between concentric tori. In the toroidal case we find that the sensitivity of the bubble's energy to the local curvature acts as a geometric potential capable of driving bubbles from regions with negative to positive curvature
Curvature driven motion of a bubble in a toroidal Hele-Shaw cell
We investigate the equilibrium properties of a single area-minimizing bubble trapped between two narrowly separated parallel curved plates. We begin with the case of a bubble trapped between concentric spherical plates. We develop a model which shows that the surface energy of the bubble is lower when confined between spherical plates than between flat plates. We confirm our findings by comparing against Surface Evolver simulations. We then derive a simple model for a bubble between arbitrarily curved parallel plates. The energy is found to be higher when the local Gaussian curvature of the plates is negative and lower when the curvature is positive. To check the validity of the model, we consider a bubble trapped between concentric tori. In the toroidal case, we find that the sensitivity of the bubble’s energy to the local curvature acts as a geometric potential capable of driving bubbles from regions with negative to positive curvature
Second order analysis of geometric functionals of Boolean models
This paper presents asymptotic covariance formulae and central limit theorems
for geometric functionals, including volume, surface area, and all Minkowski
functionals and translation invariant Minkowski tensors as prominent examples,
of stationary Boolean models. Special focus is put on the anisotropic case. In
the (anisotropic) example of aligned rectangles, we provide explicit analytic
formulae and compare them with simulation results. We discuss which information
about the grain distribution second moments add to the mean values.Comment: Chapter of the forthcoming book "Tensor Valuations and their
Applications in Stochastic Geometry and Imaging" in Lecture Notes in
Mathematics edited by Markus Kiderlen and Eva B. Vedel Jensen. (The second
version mainly resolves minor LaTeX problems.
Minkowski Tensors of Anisotropic Spatial Structure
This article describes the theoretical foundation of and explicit algorithms
for a novel approach to morphology and anisotropy analysis of complex spatial
structure using tensor-valued Minkowski functionals, the so-called Minkowski
tensors. Minkowski tensors are generalisations of the well-known scalar
Minkowski functionals and are explicitly sensitive to anisotropic aspects of
morphology, relevant for example for elastic moduli or permeability of
microstructured materials. Here we derive explicit linear-time algorithms to
compute these tensorial measures for three-dimensional shapes. These apply to
representations of any object that can be represented by a triangulation of its
bounding surface; their application is illustrated for the polyhedral Voronoi
cellular complexes of jammed sphere configurations, and for triangulations of a
biopolymer fibre network obtained by confocal microscopy. The article further
bridges the substantial notational and conceptual gap between the different but
equivalent approaches to scalar or tensorial Minkowski functionals in
mathematics and in physics, hence making the mathematical measure theoretic
method more readily accessible for future application in the physical sciences
Local Anisotropy of Fluids using Minkowski Tensors
Statistics of the free volume available to individual particles have
previously been studied for simple and complex fluids, granular matter,
amorphous solids, and structural glasses. Minkowski tensors provide a set of
shape measures that are based on strong mathematical theorems and easily
computed for polygonal and polyhedral bodies such as free volume cells (Voronoi
cells). They characterize the local structure beyond the two-point correlation
function and are suitable to define indices of
local anisotropy. Here, we analyze the statistics of Minkowski tensors for
configurations of simple liquid models, including the ideal gas (Poisson point
process), the hard disks and hard spheres ensemble, and the Lennard-Jones
fluid. We show that Minkowski tensors provide a robust characterization of
local anisotropy, which ranges from for vapor
phases to for ordered solids. We find that for fluids,
local anisotropy decreases monotonously with increasing free volume and
randomness of particle positions. Furthermore, the local anisotropy indices
are sensitive to structural transitions in these simple
fluids, as has been previously shown in granular systems for the transition
from loose to jammed bead packs
Tomographic analysis of jammed ellipsoid packings
Disordered packings of ellipsoidal particles are an important model for disordered granular matter. Here we report a way to determine the average contact number of ellipsoid packings from tomographic analysis. Tomographic images of jammed ellipsoid packings prepared by vertical shaking of loose configurations are recorded and the positions and orientations of the ellipsoids are reconstructed. The average contact number can be extracted from a contact number scaling (CNS) function. The size of the particles, that may vary due to production inaccuracies, can also be determined by this method
Minimal surfaces in nuclear pasta with the Time-Dependent Hartree-Fock approach
In continuation to the studies of the whole variety of pasta shapes in [1], we present here calculations performed with the Hartree-Fock and time-dependent Hartree- Fock method concerning the mid-density range of pasta shapes: The slab-like, connected rod-like (p-surface) and the gyroidal shapes. On the one hand we present simulations of the dynamic formation of these shapes at fi- nite temperature. On the other hand we calculate the binding energies of these shapes for varying simulation box lengths and mean densities. All of these shapes are found to be at least metastable. The slab shape has a slightly lower energy because of the lack of curvature, but among these three configurations the gyroidal shape is metastable for the widest range in mean density
Bicontinuous minimal surface nanostructures for polymer blend solar cells
This paper presents the first examination of the potential for bicontinuous structures such as the gyroid structure to produce high efficiency solar cells based on conjugated polymers. The solar cell characteristics are predicted by a simulation model that shows how the morphology influences device performance through integration of all the processes occurring in organic photocells in a specified morphology. In bicontinuous phases, the surface de. ning the interface between the electron and hole transporting phases divides the volume into two disjoint subvolumes. Exciton loss is reduced because the interface at which charge separation occurs permeates the device so excitons have only a short distance to reach the interface. As each of the component phases is connected, charges will be able to reach the electrodes more easily. In simulations of the current-voltage characteristics of organic cells with gyroid, disordered blend and vertical rod (rods normal to the electrodes) morphologies, we find that gyroids have a lower than anticipated performance advantage over disordered blends, and that vertical rods are superior. These results are explored thoroughly, with geminate recombination, i.e. recombination of charges originating from the same exciton, identified as the primary source of loss. Thus, if an appropriate materials choice could reduce geminate recombination, gyroids show great promise for future research and applications
Minkowski tensors and local structure metrics: Amorphous and crystalline sphere packings
Robust and sensitive tools to characterise local structure are essential for investigations of granular or particulate matter. Often local structure metrics derived from the bond network are used for this purpose, in particular Steinhardt's bond-orientational order parameters ql . Here we discuss an alternative method, based on the robust characterisation of the shape of the particles' Voronoi cells, by Minkowski tensors and derived anisotropy measures. We have successfully applied these metrics to quantify structural changes and the onset of crystallisation in random sphere packs. Here we specifically discuss the expectation values of these metrics for simple crystalline unimodal packings of spheres, consisting of single spheres on the points of a Bravais lattice. These data provide an important reference for the discussion of anisotropy values of disordered structures that are typically of relevance in granular systems. This analysis demonstrates that, at least for sufficiently high packing fractions above φ > 0.61, crystalline sphere packs exist whose Voronoi cells are more anisotropic with respect to a volumetric moment tensor than the average value of Voronoi cell anisotropy in random sphere packs
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