421 research outputs found
Cell shape analysis of random tessellations based on Minkowski tensors
To which degree are shape indices of individual cells of a tessellation
characteristic for the stochastic process that generates them? Within the
context of stochastic geometry and the physics of disordered materials, this
corresponds to the question of relationships between different stochastic
models. In the context of image analysis of synthetic and biological materials,
this question is central to the problem of inferring information about
formation processes from spatial measurements of resulting random structures.
We address this question by a theory-based simulation study of shape indices
derived from Minkowski tensors for a variety of tessellation models. We focus
on the relationship between two indices: an isoperimetric ratio of the
empirical averages of cell volume and area and the cell elongation quantified
by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for
these quantities, as well as for distributions thereof and for correlations of
cell shape and volume, are presented for Voronoi mosaics of the Poisson point
process, determinantal and permanental point processes, and Gibbs hard-core and
random sequential absorption processes as well as for Laguerre tessellations of
polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data
are complemented by mechanically stable crystalline sphere and disordered
ellipsoid packings and area-minimising foam models. We find that shape indices
of individual cells are not sufficient to unambiguously identify the generating
process even amongst this limited set of processes. However, we identify
significant differences of the shape indices between many of these tessellation
models. Given a realization of a tessellation, these shape indices can narrow
the choice of possible generating processes, providing a powerful tool which
can be further strengthened by density-resolved volume-shape correlations.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 Jense
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
Pore-scale Modelling of Gravity-driven Drainage in Disordered Porous Media
Multiphase flow through a porous medium involves complex interactions between
gravity, wettability and capillarity during drainage process. In contrast to
these factors, the effect of pore distribution on liquid retention is less
understood. In particular, the quantitative correlation between the fluid
displacement and level of disorder has not yet been established. In this work,
we employ direct numerical simulation by solving the Navier-Stokes equations
and using volume of fluid method to track the liquid-liquid interface during
drainage in disordered porous media. The disorder of pore configuration is
characterized by an improved index to capture small microstructural
perturbation, which is pivotal for fluid displacement in porous media. Then, we
focus on the residual volume and morphological characteristics of saturated
zones after drainage and compare the effect of disorder under different
wettability (i.e., the contact angle) and gravity (characterized by a modified
Bond number) conditions. Pore-scale simulations reveal that the
highly-disordered porous medium is favourable to improve liquid retention and
provide various morphologies of entrapped saturated zones. Furthermore, the
disorder index has a positive correlation to the characteristic curve index (n)
in van Genuchten equation, controlling the shape of the retention
characteristic curves. It is expected that the findings will benefit to a broad
range of industrial applications involving drainage processes in porous media,
e.g., drying, carbon sequestration, and underground water remediation.Comment: 22 pages, 8 figure
Local geometry of random geodesics on negatively curved surfaces
It is shown that the tessellation of a compact, negatively curved surface
induced by a typical long geodesic segment, when properly scaled, looks locally
like a Poisson line process. This implies that the global statistics of the
tessellation -- for instance, the fraction of triangles -- approach those of
the limiting Poisson line process.Comment: This version extends the results of the previous version to surfaces
with possibly variable negative curvatur
Stochastic model for the 3D microstructure of pristine and cyclically aged cathodes in Li-ion batteries
It is well-known that the microstructure of electrodes in lithium-ion
batteries strongly affects their performance. Vice versa, the microstructure
can exhibit strong changes during the usage of the battery due to aging
effects. For a better understanding of these effects, mathematical analysis and
modeling has turned out to be of great help. In particular, stochastic 3D
microstructure models have proven to be a powerful and very flexible tool to
generate various kinds of particle-based structures. Recently, such models have
been proposed for the microstructure of anodes in lithium-ion energy and power
cells. In the present paper, we describe a stochastic modeling approach for the
3D microstructure of cathodes in a lithium-ion energy cell, which differs
significantly from the one observed in anodes. The model for the cathode data
enhances the ideas of the anode models, which have been developed so far. It is
calibrated using 3D tomographic image data from pristine as well as two aged
cathodes. A validation based on morphological image characteristics shows that
the model is able to realistically describe both, the microstructure of
pristine and aged cathodes. Thus, we conclude that the model is suitable to
generate virtual, but realistic microstructures of lithium-ion cathodes
Avian Cone Photoreceptors Tile the Retina as Five Independent, Self-Organizing Mosaics
The avian retina possesses one of the most sophisticated cone photoreceptor systems among vertebrates. Birds have five types of cones including four single cones, which support tetrachromatic color vision and a double cone, which is thought to mediate achromatic motion perception. Despite this richness, very little is known about the spatial organization of avian cones and its adaptive significance. Here we show that the five cone types of the chicken independently tile the retina as highly ordered mosaics with a characteristic spacing between cones of the same type. Measures of topological order indicate that double cones are more highly ordered than single cones, possibly reflecting their posited role in motion detection. Although cones show spacing interactions that are cell type-specific, all cone types use the same density-dependent yardstick to measure intercone distance. We propose a simple developmental model that can account for these observations. We also show that a single parameter, the global regularity index, defines the regularity of all five cone mosaics. Lastly, we demonstrate similar cone distributions in three additional avian species, suggesting that these patterning principles are universal among birds. Since regular photoreceptor spacing is critical for uniform sampling of visual space, the cone mosaics of the avian retina represent an elegant example of the emergence of adaptive global patterning secondary to simple local interactions between individual photoreceptors. Our results indicate that the evolutionary pressures that gave rise to the avian retina's various adaptations for enhanced color discrimination also acted to fine-tune its spatial sampling of color and luminance
- …