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Skolnick M, Torquato S. Understanding degeneracy of two-point correlation functions via Debye random media. Phys Rev E 2021; 104:045306. [PMID: 34781573 DOI: 10.1103/physreve.104.045306] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2021] [Accepted: 09/27/2021] [Indexed: 11/07/2022]
Abstract
It is well known that the degeneracy of two-phase microstructures with the same volume fraction and two-point correlation function S_{2}(r) is generally infinite. To elucidate the degeneracy problem explicitly, we examine Debye random media, which are entirely defined by a purely exponentially decaying two-point correlation function S_{2}(r). In this work, we consider three different classes of Debye random media. First, we generate the "most probable" class using the Yeong-Torquato construction algorithm [Yeong and Torquato, Phys. Rev. E 57, 495 (1998)1063-651X10.1103/PhysRevE.57.495]. A second class of Debye random media is obtained by demonstrating that the corresponding two-point correlation functions are effectively realized in the first three space dimensions by certain models of overlapping, polydisperse spheres. A third class is obtained by using the Yeong-Torquato algorithm to construct Debye random media that are constrained to have an unusual prescribed pore-size probability density function. We structurally discriminate these three classes of Debye random media from one another by ascertaining their other statistical descriptors, including the pore-size, surface correlation, chord-length probability density, and lineal-path functions. We also compare and contrast the percolation thresholds as well as the diffusion and fluid transport properties of these degenerate Debye random media. We find that these three classes of Debye random media are generally distinguished by the aforementioned descriptors, and their microstructures are also visually distinct from one another. Our work further confirms the well-known fact that scattering information is insufficient to determine the effective physical properties of two-phase media. Additionally, our findings demonstrate the importance of the other two-point descriptors considered here in the design of materials with a spectrum of physical properties.
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Affiliation(s)
- Murray Skolnick
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
| | - 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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Barbosa R, Escobar B, Rodríguez A, Andaverde J. A study on the conduction efficiency of solid materials that evolves from a particulate system to an overlapping discs agglomerate. POWDER TECHNOL 2021. [DOI: 10.1016/j.powtec.2021.06.041] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Malmir H, Sahimi M, Rahimi Tabar MR. Statistical characterization of microstructure of packings of polydisperse hard cubes. Phys Rev E 2017; 95:052902. [PMID: 28618643 DOI: 10.1103/physreve.95.052902] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2017] [Indexed: 06/07/2023]
Abstract
Polydisperse packings of cubic particles arise in several important problems. Examples include zeolite microcubes that represent catalytic materials, fluidization of such microcubes in catalytic reactors, fabrication of new classes of porous materials with precise control of their morphology, and several others. We present the results of detailed and extensive simulation and microstructural characterization of packings of nonoverlapping polydisperse cubic particles. The packings are generated via a modified random sequential-addition algorithm. Two probability density functions (PDFs) for the particle-size distribution, the Schulz and log-normal PDFs, are used. The packings are analyzed, and their random close-packing density is computed as a function of the parameters of the two PDFs. The maximum packing fraction for the highest degree of polydispersivity is estimated to be about 0.81, much higher than 0.57 for the monodisperse packings. In addition, a variety of microstructural descriptors have been calculated and analyzed. In particular, we show that (i) an approximate analytical expression for the structure factor of Percus-Yevick fluids of polydisperse hard spheres with the Schulz PDF also predicts all the qualitative features of the structure factor of the packings that we study; (ii) as the packings become more polydisperse, their behavior resembles increasingly that of an ideal system-"ideal gas"-with little or no correlations; and (iii) the mean survival time and mean relaxation time of a diffusing species in the packings increase with increasing degrees of polydispersivity.
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Affiliation(s)
- Hessam Malmir
- Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, California 90089-1211, USA
| | - Muhammad Sahimi
- Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, California 90089-1211, USA
| | - M Reza Rahimi Tabar
- Department of Physics, Sharif University of Technology, Tehran 11365-9161, Iran
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Torquato S. Disordered hyperuniform heterogeneous materials. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:414012. [PMID: 27545746 DOI: 10.1088/0953-8984/28/41/414012] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Disordered hyperuniform many-body systems are distinguishable states of matter that lie between a crystal and liquid: they are like perfect crystals in the way they suppress large-scale density fluctuations and yet are like liquids or glasses in that they are statistically isotropic with no Bragg peaks. These systems play a vital role in a number of fundamental and applied problems: glass formation, jamming, rigidity, photonic and electronic band structure, localization of waves and excitations, self-organization, fluid dynamics, quantum systems, and pure mathematics. Much of what we know theoretically about disordered hyperuniform states of matter involves many-particle systems. In this paper, we derive new rigorous criteria that disordered hyperuniform two-phase heterogeneous materials must obey and explore their consequences. Two-phase heterogeneous media are ubiquitous; examples include composites and porous media, biological media, foams, polymer blends, granular media, cellular solids, and colloids. We begin by obtaining some results that apply to hyperuniform two-phase media in which one phase is a sphere packing in d-dimensional Euclidean space [Formula: see text]. Among other results, we rigorously establish the requirements for packings of spheres of different sizes to be 'multihyperuniform'. We then consider hyperuniformity for general two-phase media in [Formula: see text]. Here we apply realizability conditions for an autocovariance function and its associated spectral density of a two-phase medium, and then incorporate hyperuniformity as a constraint in order to derive new conditions. We show that some functional forms can immediately be eliminated from consideration and identify other forms that are allowable. Specific examples and counterexamples are described. Contact is made with well-known microstructural models (e.g. overlapping spheres and checkerboards) as well as irregular phase-separation and Turing-type patterns. We also ascertain a family of autocovariance functions (or spectral densities) that are realizable by disordered hyperuniform two-phase media in any space dimension, and present select explicit constructions of realizations. These studies provide insight into the nature of disordered hyperuniformity in the context of heterogeneous materials and have implications for the design of such novel amorphous materials.
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Affiliation(s)
- Salvatore Torquato
- Department of Chemistry, Princeton University, Princeton, NJ 08544, USA. Department of Physics, Princeton University, Princeton, NJ 08544, USA. Princeton Institute for the Science and Technology of Materials, Princeton, NJ 08544, USA. Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
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Abstract
Disordered many-particle hyperuniform systems are exotic amorphous states of matter that lie between crystal and liquid: They are like perfect crystals in the way they suppress large-scale density fluctuations and yet are like liquids or glasses in that they are statistically isotropic with no Bragg peaks. These exotic states of matter play a vital role in a number of problems across the physical, mathematical as well as biological sciences and, because they are endowed with novel physical properties, have technological importance. Given the fundamental as well as practical importance of disordered hyperuniform systems elucidated thus far, it is natural to explore the generalizations of the hyperuniformity notion and its consequences. In this paper, we substantially broaden the hyperuniformity concept along four different directions. This includes generalizations to treat fluctuations in the interfacial area (one of the Minkowski functionals) in heterogeneous media and surface-area driven evolving microstructures, random scalar fields, divergence-free random vector fields, and statistically anisotropic many-particle systems and two-phase media. In all cases, the relevant mathematical underpinnings are formulated and illustrative calculations are provided. Interfacial-area fluctuations play a major role in characterizing the microstructure of two-phase systems (e.g., fluid-saturated porous media), physical properties that intimately depend on the geometry of the interface, and evolving two-phase microstructures that depend on interfacial energies (e.g., spinodal decomposition). In the instances of random vector fields and statistically anisotropic structures, we show that the standard definition of hyperuniformity must be generalized such that it accounts for the dependence of the relevant spectral functions on the direction in which the origin in Fourier space is approached (nonanalyticities at the origin). Using this analysis, we place some well-known energy spectra from the theory of isotropic turbulence in the context of this generalization of hyperuniformity. Among other results, we show that there exist many-particle ground-state configurations in which directional hyperuniformity imparts exotic anisotropic physical properties (e.g., elastic, optical, and acoustic characteristics) to these states of matter. Such tunability could have technological relevance for manipulating light and sound waves in ways heretofore not thought possible. We show that disordered many-particle systems that respond to external fields (e.g., magnetic and electric fields) are a natural class of materials to look for directional hyperuniformity. The generalizations of hyperuniformity introduced here provide theoreticians and experimentalists new avenues to understand a very broad range of phenomena across a variety of fields through the hyperuniformity "lens."
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Affiliation(s)
- Salvatore Torquato
- Department of Chemistry, Department of Physics, Princeton Center for Theoretical Science, Program of Applied and Computational Mathematics, Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA
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Derossi A, De Pilli T, Severini C. Use of Lineal-Path Distribution Function as Statistical Descriptor of the Crumb Structure of Bread. FOOD BIOPHYS 2013. [DOI: 10.1007/s11483-013-9289-0] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Zachary CE, Jiao Y, Torquato S. Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. I. Polydisperse spheres. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:051308. [PMID: 21728526 DOI: 10.1103/physreve.83.051308] [Citation(s) in RCA: 41] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2011] [Revised: 03/14/2011] [Indexed: 05/31/2023]
Abstract
Hyperuniform many-particle distributions possess a local number variance that grows more slowly than the volume of an observation window, implying that the local density is effectively homogeneous beyond a few characteristic length scales. Previous work on maximally random strictly jammed sphere packings in three dimensions has shown that these systems are hyperuniform and possess unusual quasi-long-range pair correlations decaying as r(-4), resulting in anomalous logarithmic growth in the number variance. However, recent work on maximally random jammed sphere packings with a size distribution has suggested that such quasi-long-range correlations and hyperuniformity are not universal among jammed hard-particle systems. In this paper, we show that such systems are indeed hyperuniform with signature quasi-long-range correlations by characterizing the more general local-volume-fraction fluctuations. We argue that the regularity of the void space induced by the constraints of saturation and strict jamming overcomes the local inhomogeneity of the disk centers to induce hyperuniformity in the medium with a linear small-wave-number nonanalytic behavior in the spectral density, resulting in quasi-long-range spatial correlations scaling with r(-(d+1)) in d Euclidean space dimensions. A numerical and analytical analysis of the pore-size distribution for a binary maximally random jammed system in addition to a local characterization of the n-particle loops governing the void space surrounding the inclusions is presented in support of our argument. This paper is the first part of a series of two papers considering the relationships among hyperuniformity, jamming, and regularity of the void space in hard-particle packings.
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Affiliation(s)
- Chase E Zachary
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA.
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Quintanilla J. Microstructure and properties of random heterogeneous materials: A review of theoretical results. POLYM ENG SCI 2004. [DOI: 10.1002/pen.11446] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
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Lu B, Torquato S. Chord‐length and free‐path distribution functions for many‐body systems. J Chem Phys 1993. [DOI: 10.1063/1.464812] [Citation(s) in RCA: 58] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Torquato S, Lu B. Chord-length distribution function for two-phase random media. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:2950-2953. [PMID: 9960331 DOI: 10.1103/physreve.47.2950] [Citation(s) in RCA: 159] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Lu B, Torquato S. Lineal-path function for random heterogeneous materials. II. Effect of polydispersivity. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 45:7292-7301. [PMID: 9906804 DOI: 10.1103/physreva.45.7292] [Citation(s) in RCA: 29] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Lu B, Torquato S. Nearest-surface distribution functions for polydispersed particle systems. PHYSICAL REVIEW A 1992; 45:5530-5544. [PMID: 9907651 DOI: 10.1103/physreva.45.5530] [Citation(s) in RCA: 201] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Lu B, Torquato S. Lineal-path function for random heterogeneous materials. PHYSICAL REVIEW A 1992; 45:922-929. [PMID: 9907058 DOI: 10.1103/physreva.45.922] [Citation(s) in RCA: 182] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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