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Klatt MA, Ziff RM, Torquato S. Critical pore radius and transport properties of disordered hard- and overlapping-sphere models. Phys Rev E 2021; 104:014127. [PMID: 34412300 DOI: 10.1103/physreve.104.014127] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/20/2021] [Accepted: 06/29/2021] [Indexed: 02/01/2023]
Abstract
Transport properties of porous media are intimately linked to their pore-space microstructures. We quantify geometrical and topological descriptors of the pore space of certain disordered and ordered distributions of spheres, including pore-size functions and the critical pore radius δ_{c}. We focus on models of porous media derived from maximally random jammed sphere packings, overlapping spheres, equilibrium hard spheres, quantizer sphere packings, and crystalline sphere packings. For precise estimates of the percolation thresholds, we use a strict relation of the void percolation around sphere configurations to weighted bond percolation on the corresponding Voronoi networks. We use the Newman-Ziff algorithm to determine the percolation threshold using universal properties of the cluster size distribution. The critical pore radius δ_{c} is often used as the key characteristic length scale that determines the fluid permeability k. A recent study [Torquato, Adv. Wat. Resour. 140, 103565 (2020)10.1016/j.advwatres.2020.103565] suggested for porous media with a well-connected pore space an alternative estimate of k based on the second moment of the pore size 〈δ^{2}〉, which is easier to determine than δ_{c}. Here, we compare δ_{c} to the second moment of the pore size 〈δ^{2}〉, and indeed confirm that, for all porosities and all models considered, δ_{c}^{2} is to a good approximation proportional to 〈δ^{2}〉. However, unlike 〈δ^{2}〉, the permeability estimate based on δ_{c}^{2} does not predict the correct ranking of k for our models. Thus, we confirm 〈δ^{2}〉 to be a promising candidate for convenient and reliable estimates of the fluid permeability for porous media with a well-connected pore space. Moreover, we compare the fluid permeability of our models with varying degrees of order, as measured by the τ order metric. We find that (effectively) hyperuniform models tend to have lower values of k than their nonhyperuniform counterparts. Our findings could facilitate the design of porous media with desirable transport properties via targeted pore statistics.
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Affiliation(s)
- Michael A Klatt
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA.,Institut für Theoretische Physik, FAU Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany
| | - Robert M Ziff
- Center for the Study of Complex Systems and Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA
| | - Salvatore Torquato
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA.,Department of Chemistry, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA
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2
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Klatt MA, Steinhardt PJ, Torquato S. Gap Sensitivity Reveals Universal Behaviors in Optimized Photonic Crystal and Disordered Networks. Phys Rev Lett 2021; 127:037401. [PMID: 34328757 DOI: 10.1103/physrevlett.127.037401] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2020] [Accepted: 05/27/2021] [Indexed: 06/13/2023]
Abstract
Through an extensive series of high-precision numerical computations of the optimal complete photonic band gap (PBG) as a function of dielectric contrast α for a variety of crystal and disordered heterostructures, we reveal striking universal behaviors of the gap sensitivity S(α)≡dΔ(α)/dα, the first derivative of the optimal gap-to-midgap ratio Δ(α). In particular, for all our crystal networks, S(α) takes a universal form that is well approximated by the analytic formula for a 1D quarter-wave stack, S_{QWS}(α). Even more surprisingly, the values of S(α) for our disordered networks converge to S_{QWS}(α) for sufficiently large α. A deeper understanding of the simplicity of this universal behavior may provide fundamental insights about PBG formation and guidance in the design of novel photonic heterostructures.
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Affiliation(s)
- Michael A Klatt
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
- Institut für Theoretische Physik, University of Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany
| | - Paul J Steinhardt
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Salvatore Torquato
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
- Department of Chemistry, Princeton Institute for the Science and Technology of Materials,and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA
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3
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Hong S, Klatt MA, Schröder-Turk G, François N, Saadatfar M. Dynamical arrest of topological defects in 2D hyperuniform disk packings. EPJ Web Conf 2021. [DOI: 10.1051/epjconf/202124915002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We investigate collective motions of points in 2D systems, orchestrated by Lloyd algorithm. The algorithm iteratively updates a system by minimising the total quantizer energy of the Voronoi landscape of the system. As a result of a tradeoff between energy minimisation and geometric frustration, we find that optimised systems exhibit a defective landscape along the process, where strands of 5- and 7-coordinated dislocations are embedded in the hexatic phase. In particular, dipole defects, each of which is the simplest possible pair of a pentagon and a heptagon, come into the picture of dynamical arrest, as the system freezes down to a disordered hyperuniform state. Moreover, we explore the packing fractions of 2D disk packings associated to the obtained hyperuniform systems by considering the maximum inscribed disks in their Voronoi cells.
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Hain TM, Klatt MA, Schröder-Turk GE. Low-temperature statistical mechanics of the Quantizer problem: Fast quenching and equilibrium cooling of the three-dimensional Voronoi liquid. J Chem Phys 2020; 153:234505. [DOI: 10.1063/5.0029301] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023] Open
Affiliation(s)
- Tobias M. Hain
- Institut für Mathematik, Universität Potsdam, Karl-Liebknecht-Str. 24-25, D-14476 Potsdam OT Golm, Germany
- Physical Chemistry, Center for Chemistry and Chemical Engineering, Lund University, Lund 22100, Sweden
- Murdoch University, College of Science, Health, Engineering and Education, Mathematics and Statistics, 90 South St., Murdoch WA 6150, Australia
| | - Michael A. Klatt
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Gerd E. Schröder-Turk
- Physical Chemistry, Center for Chemistry and Chemical Engineering, Lund University, Lund 22100, Sweden
- Murdoch University, College of Science, Health, Engineering and Education, Mathematics and Statistics, 90 South St., Murdoch WA 6150, Australia
- Department of Food Science, University of Copenhagen, Rolighedsvej 26, 1958 Frederiksberg C, Denmark
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5
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Klatt MA, Kim J, Torquato S. Cloaking the underlying long-range order of randomly perturbed lattices. Phys Rev E 2020; 101:032118. [PMID: 32289999 DOI: 10.1103/physreve.101.032118] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2019] [Accepted: 01/25/2020] [Indexed: 06/11/2023]
Abstract
Random, uncorrelated displacements of particles on a lattice preserve the hyperuniformity of the original lattice, that is, normalized density fluctuations vanish in the limit of infinite wavelengths. In addition to a diffuse contribution, the scattering intensity from the the resulting point pattern typically inherits the Bragg peaks (long-range order) of the original lattice. Here we demonstrate how these Bragg peaks can be hidden in the effective diffraction pattern of independent and identically distributed perturbations. All Bragg peaks vanish if and only if the sum of all probability densities of the positions of the shifted lattice points is a constant at all positions. The underlying long-range order is then "cloaked" in the sense that it cannot be reconstructed from the pair correlation function alone. On the one hand, density fluctuations increase monotonically with the strength of perturbations a, as measured by the hyperuniformity order metric Λ[over ¯]. On the other hand, the disappearance and reemergence of long-range order, depending on whether the system is cloaked as the perturbation strength increases, is manifestly captured by the τ order metric. Therefore, while the perturbation strength a may seem to be a natural choice for an order metric of perturbed lattices, the τ order metric is a superior choice. It is noteworthy that cloaked perturbed lattices allow one to easily simulate very large samples (with at least 10^{6} particles) of disordered hyperuniform point patterns without Bragg peaks.
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Affiliation(s)
- Michael A Klatt
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Jaeuk Kim
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Salvatore Torquato
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
- Department of Chemistry, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA
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Klatt MA, Lovrić J, Chen D, Kapfer SC, Schaller FM, Schönhöfer PWA, Gardiner BS, Smith AS, Schröder-Turk GE, Torquato S. Universal hidden order in amorphous cellular geometries. Nat Commun 2019; 10:811. [PMID: 30778054 PMCID: PMC6379405 DOI: 10.1038/s41467-019-08360-5] [Citation(s) in RCA: 27] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2018] [Accepted: 01/03/2019] [Indexed: 12/04/2022] Open
Abstract
Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. Here we investigate the Quantizer problem, defined as the optimisation of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that tile space are preferred. We employ Lloyd’s centroidal Voronoi diagram algorithm to solve this problem and find that it converges to disordered states associated with deep local minima. These states are universal in the sense that their structure factors are characterised by a complete independence of a wide class of initial conditions they evolved from. They moreover exhibit an anomalous suppression of long-wavelength density fluctuations and quickly become effectively hyperuniform. Our findings warrant the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties. Disordered hyperuniformity implies a hidden order on length scales that can be found in various amorphous materials. Klatt et al. analyse the evolution of random point patterns using Llyod’s algorithm and show that they converge to an effectively hyperuniform state regardless of the initial conditions.
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Affiliation(s)
- Michael A Klatt
- Institute of Stochastics, Karlsruhe Institute of Technology (KIT), Englerstr. 2, 76131, Karlsruhe, Germany.,Department of Physics, Princeton University, Princeton, NJ, 08544, USA
| | - Jakov Lovrić
- Division of Physical Chemistry, Ruđer Bošković Institute, Bijenička 54, 10 000 Zagreb, Croatia.,School of Engineering and Information Technology, Murdoch University, 90 South St, Murdoch, WA, 6150, Australia.,PULS Group, Department of Physics and Interdisciplinary Center for Nanostructured Films, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 3, 91058 Erlangen, Germany
| | - Duyu Chen
- Department of Chemistry, Princeton University, Princeton, NJ, 08544, USA
| | - Sebastian C Kapfer
- Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, 91058, Erlangen, Germany
| | - Fabian M Schaller
- Institute of Stochastics, Karlsruhe Institute of Technology (KIT), Englerstr. 2, 76131, Karlsruhe, Germany.,Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, 91058, Erlangen, Germany
| | - Philipp W A Schönhöfer
- School of Engineering and Information Technology, Murdoch University, 90 South St, Murdoch, WA, 6150, Australia.,Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, 91058, Erlangen, Germany
| | - Bruce S Gardiner
- School of Engineering and Information Technology, Murdoch University, 90 South St, Murdoch, WA, 6150, Australia.,School of Computer Science and Software Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia
| | - Ana-Sunčana Smith
- Division of Physical Chemistry, Ruđer Bošković Institute, Bijenička 54, 10 000 Zagreb, Croatia.,PULS Group, Department of Physics and Interdisciplinary Center for Nanostructured Films, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 3, 91058 Erlangen, Germany
| | - Gerd E Schröder-Turk
- School of Engineering and Information Technology, Murdoch University, 90 South St, Murdoch, WA, 6150, Australia.,Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, 91058, Erlangen, Germany.,Department of Applied Mathematics, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, ACT, 0200, Australia
| | - Salvatore Torquato
- Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544, USA.
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Ebner B, Henze N, Klatt MA, Mecke K. Goodness-of-fit tests for complete spatial randomness based on Minkowski functionals of binary images. Electron J Stat 2018. [DOI: 10.1214/18-ejs1467] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Klatt MA, Torquato S. Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions. Phys Rev E 2018; 97:012118. [PMID: 29448326 DOI: 10.1103/physreve.97.012118] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/15/2017] [Indexed: 06/08/2023]
Abstract
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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Affiliation(s)
- Michael A Klatt
- Institute of Stochastics, Department of Mathematics, Karlsruhe Institute of Technology, Englerstraße 2, 76131 Karlsruhe, Germany
| | - Salvatore Torquato
- Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA
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9
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Klatt MA, Schröder-Turk GE, Mecke K. Mean-intercept anisotropy analysis of porous media. I. Analytic formulae for anisotropic Boolean models. Med Phys 2017; 44:3650-3662. [DOI: 10.1002/mp.12281] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2016] [Revised: 03/05/2017] [Accepted: 04/03/2017] [Indexed: 11/09/2022] Open
Affiliation(s)
- Michael A. Klatt
- Institute of Stochastics; Karlsruhe Institute of Technology (KIT); Englerstraße 2 76131 Karlsruhe Germany
- Institut für Theoretische Physik; Universität Erlangen-Nürnberg; Staudtstr. 7 91058 Erlangen Germany
| | - Gerd E. Schröder-Turk
- School of Engineering & IT; Murdoch University; 90 South Street Murdoch WA 6150 Australia
| | - Klaus Mecke
- Institut für Theoretische Physik; Universität Erlangen-Nürnberg; Staudtstr. 7 91058 Erlangen Germany
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10
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Klatt MA, Schröder-Turk GE, Mecke K. Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative. Med Phys 2017; 44:3663-3675. [DOI: 10.1002/mp.12280] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2016] [Revised: 03/05/2017] [Accepted: 04/03/2017] [Indexed: 11/08/2022] Open
Affiliation(s)
- Michael A. Klatt
- Institute of Stochastics; Karlsruhe Institute of Technology (KIT); Englerstraße 2 76131 Karlsruhe Germany
- Institut für Theoretische Physik; Universität Erlangen-Nürnberg; Staudtstr. 7 91058 Erlangen Germany
| | - Gerd E. Schröder-Turk
- School of Engineering & IT; Murdoch University; 90 South Street Murdoch WA 6150 Australia
| | - Klaus Mecke
- Institut für Theoretische Physik; Universität Erlangen-Nürnberg; Staudtstr. 7 91058 Erlangen Germany
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11
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Wilts BD, Apeleo Zubiri B, Klatt MA, Butz B, Fischer MG, Kelly ST, Spiecker E, Steiner U, Schröder-Turk GE. Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development. Sci Adv 2017; 3:e1603119. [PMID: 28508050 PMCID: PMC5406134 DOI: 10.1126/sciadv.1603119] [Citation(s) in RCA: 44] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/2016] [Accepted: 02/23/2017] [Indexed: 05/05/2023]
Abstract
The formation of the biophotonic gyroid material in butterfly wing scales is an exceptional feat of evolutionary engineering of functional nanostructures. It is hypothesized that this nanostructure forms by chitin polymerization inside a convoluted membrane of corresponding shape in the endoplasmic reticulum. However, this dynamic formation process, including whether membrane folding and chitin expression are simultaneous or sequential processes, cannot yet be elucidated by in vivo imaging. We report an unusual hierarchical ultrastructure in the butterfly Thecla opisena that, as a solid material, allows high-resolution three-dimensional microscopy. Rather than the conventional polycrystalline space-filling arrangement, a gyroid occurs in isolated facetted crystallites with a pronounced size gradient. When interpreted as a sequence of time-frozen snapshots of the morphogenesis, this arrangement provides insight into the formation mechanisms of the nanoporous gyroid material as well as of the intracellular organelle membrane that acts as the template.
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Affiliation(s)
- Bodo D. Wilts
- Adolphe Merkle Institute, University of Fribourg, Chemin des Verdiers 4, CH-1700 Fribourg, Switzerland
| | - Benjamin Apeleo Zubiri
- Institute of Micro- and Nanostructure Research and Center for Nanoanalysis and Electron Microscopy, Department of Materials Science and Engineering, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 6, D-91058 Erlangen, Germany
| | - Michael A. Klatt
- Institute of Stochastics, Karlsruhe Institute of Technology, Englerstraße 2, D-76131 Karlsruhe, Germany
| | - Benjamin Butz
- Institute of Micro- and Nanostructure Research and Center for Nanoanalysis and Electron Microscopy, Department of Materials Science and Engineering, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 6, D-91058 Erlangen, Germany
| | - Michael G. Fischer
- Adolphe Merkle Institute, University of Fribourg, Chemin des Verdiers 4, CH-1700 Fribourg, Switzerland
| | - Stephen T. Kelly
- Carl Zeiss X-Ray Microscopy, 4385 Hopyard Road, Pleasanton, CA 94588, USA
| | - Erdmann Spiecker
- Institute of Micro- and Nanostructure Research and Center for Nanoanalysis and Electron Microscopy, Department of Materials Science and Engineering, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 6, D-91058 Erlangen, Germany
| | - Ullrich Steiner
- Adolphe Merkle Institute, University of Fribourg, Chemin des Verdiers 4, CH-1700 Fribourg, Switzerland
| | - Gerd E. Schröder-Turk
- School of Engineering and Information Technology, Mathematics and Statistics, Murdoch University, Murdoch, 90 South Street, Western Australia 6150, Australia
- Department of Applied Mathematics, Research School of Physics and Engineering, 60 Mills Road, Australian National University, Canberra, Australian Capital Territory 2601, Australia
- Institute of Theoretical Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7B, D-91058 Erlangen, Germany
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12
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Klatt MA, Torquato S. Characterization of maximally random jammed sphere packings. II. Correlation functions and density fluctuations. Phys Rev E 2016; 94:022152. [PMID: 27627291 DOI: 10.1103/physreve.94.022152] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2016] [Indexed: 06/06/2023]
Abstract
In the first paper of this series, we introduced Voronoi correlation functions to characterize the structure of maximally random jammed (MRJ) sphere packings across length scales. In the present paper, we determine a variety of different correlation functions that arise in rigorous expressions for the effective physical properties of MRJ sphere packings and compare them to the corresponding statistical descriptors for overlapping spheres and equilibrium hard-sphere systems. Such structural descriptors arise in rigorous bounds and formulas for effective transport properties, diffusion and reactions constants, elastic moduli, and electromagnetic characteristics. First, we calculate the two-point, surface-void, and surface-surface correlation functions, for which we derive explicit analytical formulas for finite hard-sphere packings. We show analytically how the contact Dirac delta function contribution to the pair correlation function g_{2}(r) for MRJ packings translates into distinct functional behaviors of these two-point correlation functions that do not arise in the other two models examined here. Then we show how the spectral density distinguishes the MRJ packings from the other disordered systems in that the spectral density vanishes in the limit of infinite wavelengths; i.e., these packings are hyperuniform, which means that density fluctuations on large length scales are anomalously suppressed. Moreover, for all model systems, we study and compute exclusion probabilities and pore size distributions, as well as local density fluctuations. We conjecture that for general disordered hard-sphere packings, a central limit theorem holds for the number of points within an spherical observation window. Our analysis links problems of interest in material science, chemistry, physics, and mathematics. In the third paper of this series, we will evaluate bounds and estimates of a host of different physical properties of the MRJ sphere packings that are based on the structural characteristics analyzed in this paper.
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Affiliation(s)
- Michael A Klatt
- Karlsruhe Institute of Technology (KIT), Institute of Stochastics, Englerstraße 2, 76131 Karlsruhe, Germany
| | - Salvatore Torquato
- Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA
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13
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Scholz C, Wirner F, Klatt MA, Hirneise D, Schröder-Turk GE, Mecke K, Bechinger C. Direct relations between morphology and transport in Boolean models. Phys Rev E Stat Nonlin Soft Matter Phys 2015; 92:043023. [PMID: 26565348 DOI: 10.1103/physreve.92.043023] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2015] [Indexed: 06/05/2023]
Abstract
We study the relation of permeability and morphology for porous structures composed of randomly placed overlapping circular or elliptical grains, so-called Boolean models. Microfluidic experiments and lattice Boltzmann simulations allow us to evaluate a power-law relation between the Euler characteristic of the conducting phase and its permeability. Moreover, this relation is so far only directly applicable to structures composed of overlapping grains where the grain density is known a priori. We develop a generalization to arbitrary structures modeled by Boolean models and characterized by Minkowski functionals. This generalization works well for the permeability of the void phase in systems with overlapping grains, but systematic deviations are found if the grain phase is transporting the fluid. In the latter case our analysis reveals a significant dependence on the spatial discretization of the porous structure, in particular the occurrence of single isolated pixels. To link the results to percolation theory we performed Monte Carlo simulations of the Euler characteristic of the open cluster, which reveals different regimes of applicability for our permeability-morphology relations close to and far away from the percolation threshold.
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Affiliation(s)
- Christian Scholz
- 2. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
- Institut für Multiskalensimulation, Nägelsbachstraße 49b, Friedrich-Alexander Universität Erlangen-Nürnberg, 91052 Erlangen, Germany
| | - Frank Wirner
- 2. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Michael A Klatt
- Institut für Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstraße 7B, 91058 Erlangen, Germany
| | - Daniel Hirneise
- 2. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Gerd E Schröder-Turk
- Institut für Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstraße 7B, 91058 Erlangen, Germany
- Murdoch University, School of Engineering & IT, Maths & Stats, 90 South Str., Murdoch WA 6150, Australia
| | - Klaus Mecke
- Institut für Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstraße 7B, 91058 Erlangen, Germany
| | - Clemens Bechinger
- 2. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
- Max-Planck-Institut für Intelligente Systeme, Heisenbergstraße 3, 70569 Stuttgart, Germany
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14
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Klatt MA, Torquato S. Characterization of maximally random jammed sphere packings: Voronoi correlation functions. Phys Rev E Stat Nonlin Soft Matter Phys 2014; 90:052120. [PMID: 25493753 DOI: 10.1103/physreve.90.052120] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/10/2014] [Indexed: 06/04/2023]
Abstract
We characterize the structure of maximally random jammed (MRJ) sphere packings by computing the Minkowski functionals (volume, surface area, and integrated mean curvature) of their associated Voronoi cells. The probability distribution functions of these functionals of Voronoi cells in MRJ sphere packings are qualitatively similar to those of an equilibrium hard-sphere liquid and partly even to the uncorrelated Poisson point process, implying that such local statistics are relatively structurally insensitive. This is not surprising because the Minkowski functionals of a single Voronoi cell incorporate only local information and are insensitive to global structural information. To improve upon this, we introduce descriptors that incorporate nonlocal information via the correlation functions of the Minkowski functionals of two cells at a given distance as well as certain cell-cell probability density functions. We evaluate these higher-order functions for our MRJ packings as well as equilibrium hard spheres and the Poisson point process. It is shown that these Minkowski correlation and density functions contain visibly more information than the corresponding standard pair-correlation functions. We find strong anticorrelations in the Voronoi volumes for the hyperuniform MRJ packings, consistent with previous findings for other pair correlations [A. Donev et al., Phys. Rev. Lett. 95, 090604 (2005)PRLTAO0031-900710.1103/PhysRevLett.95.090604], indicating that large-scale volume fluctuations are suppressed by accompanying large Voronoi cells with small cells, and vice versa. In contrast to the aforementioned local Voronoi statistics, the correlation functions of the Voronoi cells qualitatively distinguish the structure of MRJ sphere packings (prototypical glasses) from that of not only the Poisson point process but also the correlated equilibrium hard-sphere liquids. Moreover, while we did not find any perfect icosahedra (the locally densest possible structure in which a central sphere contacts 12 neighbors) in the MRJ packings, a preliminary Voronoi topology analysis indicates the presence of strongly distorted icosahedra.
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Affiliation(s)
- Michael A Klatt
- Department of Chemistry, Department of Physics, Princeton University, Princeton, New Jersey 08544, USA and Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Institut für Theoretische Physik, Staudtstraße 7, 91058 Erlangen, Germany
| | - Salvatore Torquato
- Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA
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Schuetrumpf B, Klatt MA, Iida K, Maruhn JA, Mecke K, Reinhard PG. Time-Dependent Hartree-Fock Approach to Nuclear Pasta at Finite Temperature. ACTA ACUST UNITED AC 2013. [DOI: 10.1088/1742-6596/426/1/012009] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Schröder-Turk GE, Mickel W, Kapfer SC, Klatt MA, Schaller FM, Hoffmann MJF, Kleppmann N, Armstrong P, Inayat A, Hug D, Reichelsdorfer M, Peukert W, Schwieger W, Mecke K. Minkowski tensor shape analysis of cellular, granular and porous structures. Adv Mater 2011; 23:2535-2553. [PMID: 21681830 DOI: 10.1002/adma.201100562] [Citation(s) in RCA: 52] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
Predicting physical properties of materials with spatially complex structures is one of the most challenging problems in material science. One key to a better understanding of such materials is the geometric characterization of their spatial structure. Minkowski tensors are tensorial shape indices that allow quantitative characterization of the anisotropy of complex materials and are particularly well suited for developing structure-property relationships for tensor-valued or orientation-dependent physical properties. They are fundamental shape indices, in some sense being the simplest generalization of the concepts of volume, surface and integral curvatures to tensor-valued quantities. Minkowski tensors are based on a solid mathematical foundation provided by integral and stochastic geometry, and are endowed with strong robustness and completeness theorems. The versatile definition of Minkowski tensors applies widely to different types of morphologies, including ordered and disordered structures. Fast linear-time algorithms are available for their computation. This article provides a practical overview of the different uses of Minkowski tensors to extract quantitative physically-relevant spatial structure information from experimental and simulated data, both in 2D and 3D. Applications are presented that quantify (a) alignment of co-polymer films by an electric field imaged by surface force microscopy; (b) local cell anisotropy of spherical bead pack models for granular matter and of closed-cell liquid foam models; (c) surface orientation in open-cell solid foams studied by X-ray tomography; and (d) defect densities and locations in molecular dynamics simulations of crystalline copper.
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Affiliation(s)
- G E Schröder-Turk
- Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany.
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