3 research outputs found
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
Disordered spherical bead packs are anisotropic
Investigating how tightly objects pack space is a long-standing problem, with relevance for many disciplines from discrete mathematics to the theory of glasses. Here we report on the fundamental yet so far overlooked geometric property that disordered mono-disperse spherical bead packs have significant local structural anisotropy manifest in the shape of the free space associated with each bead. Jammed disordered packings from several types of experiments and simulations reveal very similar values of the cell anisotropy, showing a linear decrease with packing fraction. Strong deviations from this trend are observed for unjammed configurations and for partially crystalline packings above 64%. These findings suggest an inherent geometrical reason why, in disordered packings, anisotropic shapes can fill space more efficiently than spheres, and have implications for packing effects in non-spherical liquid crystals, foams and structural glasses