Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles

Abstract

A set of lower bounds on the continuum percolation threshold ${\eta }_c$ of overlapping convex hyperparticles of general nonspherical (anisotropic) shape with a specified orientational probability distribution in $d$-dimensional Euclidean space have been derived [S. Torquato, J. Chem. Phys. {\bf 136}, 054106 (2012)]. The simplest of these lower bounds is given by ${\eta }_c\ge v/v_{ex}$, where $v_{ex}$ is the $d$-dimensional exclusion volume of a hyperparticle and $v$ is its $d$-dimensional volume. In order to study the effect of dimensionality on the threshold ${\eta }_c$ of overlapping nonspherical convex hyperparticles with random orientations here, we obtain a scaling relation for ${\eta }_c$ that is based on this lower bound and a conjecture that hyperspheres provide the highest threshold among all convex hyperparticle shapes for any $d$. This scaling relation exploits the principle that low-dimensional continuum percolation behavior encodes high-dimensional information. We derive a formula for the exclusion volume $v_{ex}$ of a hyperparticle in terms of its $d$-dimensional volume $v$, surface area $s$ and {\it radius of mean curvature} $\bar{R}$ (or, equivalently, {\it mean width}). These basic geometrical properties are computed for a wide variety of nonspherical hyperparticle shapes with random orientations across all dimensions, including, among other shapes, various polygons for $d=2$, Platonic solids, spherocylinders, parallepipeds and zero-volume plates for $d=3$ and their appropriate generalizations for $d\ge 4$. We then compute the lower bound and scaling relation for ${\eta }_c$ for this comprehensive set of continuum percolation models across dimensions. We show that the scaling relation provides accurate {\it upper-bound} estimates of the threshold ${\eta }_c$ across dimensions and becomes increasingly accurate as the space $d$ increases.

37 pages, 5 figures, 9 tables, to appear in Phys. Rev. E

References in corpus (9)

• Dense Packings of the Platonic and Archimedean Solids
• Continuum Percolation Thresholds in Two Dimensions
• Dense Packings of Polyhedra: Platonic and Archimedean Solids
• Optimal Packings of Superballs
• The continuum percolation threshold for interpenetrating squares and cubes
• Effect of Dimensionality on the Continuum Percolation of Overlapping Hyperspheres and Hypercubes: II. Simulation Results and Analyses
• Improved reconstructions of random media using dilation and erosion processes
• Organizing Principles for Dense Packings of Nonspherical Hard Particles: Not All Shapes Are Created Equal
• High-dimensional generalizations of the kagome and diamond crystals and the decorrelation principle for periodic sphere packings

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• Anisotropy in finite continuum percolation: Threshold estimation by Minkowski functionals
• Connectedness percolation of hard convex polygonal rods and platelets
• Modeling Multi-Cellular Dynamics Regulated by ECM-Mediated Mechanical Communication via Active Particles with Polarized Effective Attraction
• Exclusion Volumes of Convex Bodies in High Space Dimensions: Applications to Virial Coefficients and Continuum Percolation
• Geometric measures of uniaxial solids of revolution in ${\mathbb{R}}^4$ and their relation to the second virial coefficient
• Excluded volume geometry and packing fraction in binary convex hyperparticle mixtures