Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative

In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here, we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical approximation formulas. These property formulas include interfacial bounds as well as universal scaling laws for the mean survival time and the fluid permeability. We also estimate the principal relaxation time associated with Brownian motion among perfectly absorbing "traps." For the propagation of electromagnetic waves in the long-wavelength limit, we show that a dispersion of dielectric MRJ spheres within a matrix of another dielectric material forms, to a very good approximation, a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio. We compare the effective properties of the MRJ sphere packings to those of overlapping spheres, equilibrium hard-sphere packings, and lattices of hard spheres. Moreover, we generalize results to micro-and macroscopically anisotropic packings of spheroids with tensorial effective properties. The analytic bounds predict the qualitative trend in the physical properties associated with these structures, which provides guidance to more time-consuming simulations and experiments. They especially provide impetus for experiments to design materials with unique bulk properties resulting from hyperuniformity, including structural-color and color-sensing applications.

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“…The first integral in the last equation is equal to the trace of a Minkowski tensor [87,88]: Note that the integrals in Eqs. (B5) and (B6)…”

Section: Moments Of Pore Sizesmentioning

confidence: 99%

Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

Klatt

Torquato

2018

Phys. Rev. E

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“…This remarkable statement has been generalized to the MTs [87,88]. Moreover, the MFs and MTs have intuitive geometric interpretations, including volume, surface area, and moments of the distributions of mass or normal vectors [34,71].…”

Section: Minkowski Tensorsmentioning

confidence: 93%

“…The MTs extend the notion of volume, surface area, and curvature to tensorial morphometric measures [34,35,68]. Thus, the MTs allow for a sensitive and comprehensive anisotropy analysis with respect to different geometrical properties, like surface area, circumference, or curvature [34,35,[69][70][71]. Because they are robust against noise and their computation time grows linearly with the system size, they are efficient and comprehensive shape descriptors for data analysis of random spatial structures in experiments [34].…”

Section: Shape Analysis Of Anisotropic Random Fieldsmentioning

confidence: 99%

“…Here we describe a two-dimensional model, where we parametrize the direction k i ∈ S 1 by the angle θ i between k i and the x-axis. We choose for the probability density of θ (similar to the anisotropic distributions in [34,45,71,97]):…”

Section: Conclusion and Outlook To A Robust Detection Of Non-gaussian...mentioning

confidence: 99%

“…For three dimensions, an explicit irreducible representation and rotational invariants of Minkowski tensors are presented in [101][102][103]. Here, the two-dimensional case of irreducible Minkowski tensors (IMTs) for the interfacial tensors W 0,s 1 for a convex body K is discussed; see also [11,71]. Note that the IMTs are equivalent to the definition of the harmonic intrinsic volumes defined in [104].…”

Section: Appendix B Irreducible Representation Of Minkowski Tensorsmentioning

confidence: 99%

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Characterization of Anisotropic Gaussian Random Fields by Minkowski Tensors

Klatt,

Hörmann,

Mecke

2021

Preprint

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Gaussian random fields are among the most important models of amorphous spatial structures and appear across length scales in a variety of physical, biological, and geological applications, from composite materials to geospatial data. Anisotropy in such systems can sensitively and comprehensively be characterized by the so-called Minkowski tensors from integral geometry.Here, we analytically calculate the expected Minkowski tensors of arbitrary rank for the level sets of Gaussian random fields. The explicit expressions for interfacial Minkowski tensors are confirmed in detailed simulations. We demonstrate how the Minkowski tensors detect and characterize the anisotropy of the level sets, and we clarify which shape information is contained in the Minkowski tensors of different rank.Using an irreducible representation of the Minkowski tensors in the Euclidean plane, we show that higher-rank tensors indeed contain additional anisotropy information compared to a rank two tensor. Surprisingly, we can nevertheless predict this information from the second-rank tensor if we assume that the random field is Gaussian. This relation between tensors of different rank is independent of the details of the model. It is, therefore, useful for a null hypothesis test that detects non-Gaussianities in anisotropic random fields.

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Characterizing digital microstructures by the Minkowski‐based quadratic normal tensor

Ernesti

Schneider

Winter

et al. 2022

Math Methods in App Sciences

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For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration and enable distinguishing microstructures with distinct effective material behavior. For this purpose, we propose using Minkowski tensors, in general, and the quadratic normal tensor, in particular, and introduce a computational algorithm applicable to voxel-based microstructure representations. Rooted in the mathematical field of integral geometry, Minkowski tensors associate a tensor to rather general geometric shapes, which make them suitable for a wide range of microstructured material classes. Furthermore, they satisfy additivity and continuity properties, which makes them suitable and robust for large-scale applications. We present a modular algorithm for computing the quadratic normal tensor of digital microstructures. We demonstrate multigrid convergence for selected numerical examples and apply our approach to a variety of microstructures. Strikingly, the presented algorithm remains unaffected by inaccurate computation of the interface area. The quadratic normal tensor may be used for engineering purposes, such as mean field homogenization or as target value for generating synthetic microstructures.

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Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative

Cited by 18 publications

References 69 publications

Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

Characterization of Anisotropic Gaussian Random Fields by Minkowski Tensors

Characterizing digital microstructures by the Minkowski‐based quadratic normal tensor

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