Local number fluctuations in ordered and disordered phases of water across temperatures: Higher-order moments and degrees of tetrahedrality.

The isothermal compressibility (i.e., related to the asymptotic number variance) of equilibrium liquid water as a function of temperature is minimal under near-ambient conditions. This anomalous non-monotonic temperature dependence is due to a balance between thermal fluctuations and the formation of tetrahedral hydrogen-bond networks. Since tetrahedrality is a many-body property, it will also influence the higher-order moments of density fluctuations, including the skewness and kurtosis. To gain a more complete picture, we examine these higher-order moments that encapsulate many-body correlations using a recently developed, advanced platform for local density fluctuations. We study an extensive set of simulated phases of water across a range of temperatures (80-1600 K) with various degrees of tetrahedrality, including ice phases, equilibrium liquid water, supercritical water, and disordered nonequilibrium quenches. We find clear signatures of tetrahedrality in the higher-order moments, including the skewness and excess kurtosis, which scale for all cases with the degree of tetrahedrality. More importantly, this scaling behavior leads to non-monotonic temperature dependencies in the higher-order moments for both equilibrium and non-equilibrium phases. Specifically, under near-ambient conditions, the higher-order moments vanish most rapidly for large length scales, and the distribution quickly converges to a Gaussian in our metric. However, under non-ambient conditions, higher-order moments vanish more slowly and hence become more relevant, especially for improving information-theoretic approximations of hydrophobic solubility. The temperature non-monotonicity that we observe in the full distribution across length scales could shed light on water's nested anomalies, i.e., reveal new links between structural, dynamic, and thermodynamic anomalies.

Wave propagation and band tails of two-dimensional disordered systems in the thermodynamic limit.

Understanding the nature and formation of band gaps associated with the propagation of electromagnetic, electronic, or elastic waves in disordered materials as a function of system size presents fundamental and technological challenges. In particular, a basic question is whether band gaps in disordered systems exist in the thermodynamic limit. To explore this issue, we use a two-stage ensemble approach to study the formation of complete photonic band gaps (PBGs) for a sequence of increasingly large systems spanning a broad range of two-dimensional photonic network solids with varying degrees of local and global order, including hyperuniform and nonhyperuniform types. We discover that the gap in the density of states exhibits exponential tails and the apparent PBGs rapidly close as the system size increases for nearly all disordered networks considered. The only exceptions are sufficiently stealthy hyperuniform cases for which the band gaps remain open and the band tails exhibit a desirable power-law scaling reminiscent of the PBG behavior of photonic crystals in the thermodynamic limit.

Critical pore radius and transport properties of disordered hard- and overlapping-sphere models.

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.

Gap Sensitivity Reveals Universal Behaviors in Optimized Photonic Crystal and Disordered Networks.

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.

Low-temperature statistical mechanics of the Quantizer problem: Fast quenching and equilibrium cooling of the three-dimensional Voronoi liquid.

The quantizer problem is a tessellation optimization problem where point configurations are identified such that the Voronoi cells minimize the second moment of the volume distribution. While the ground state (optimal state) in 3D is almost certainly the body-centered cubic lattice, disordered and effectively hyperuniform states with energies very close to the ground state exist that result as stable states in an evolution through the geometric Lloyd's algorithm [M. A. Klatt et al. Nat. Commun. 10, 811 (2019)]. When considered as a statistical mechanics problem at finite temperature, the same system has been termed the "Voronoi liquid" by Ruscher, Baschnagel, and Farago [Europhys. Lett. 112, 66003 (2015)]. Here, we investigate the cooling behavior of the Voronoi liquid with a particular view to the stability of the effectively hyperuniform disordered state. As a confirmation of the results by Ruscher et al., we observe, by both molecular dynamics and Monte Carlo simulations, that upon slow quasi-static equilibrium cooling, the Voronoi liquid crystallizes from a disordered configuration into the body-centered cubic configuration. By contrast, upon sufficiently fast non-equilibrium cooling (and not just in the limit of a maximally fast quench), the Voronoi liquid adopts similar states as the effectively hyperuniform inherent structures identified by Klatt et al. and prevents the ordering transition into a body-centered cubic ordered structure. This result is in line with the geometric intuition that the geometric Lloyd's algorithm corresponds to a type of fast quench.

Cloaking the underlying long-range order of randomly perturbed lattices.

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.

Phoamtonic designs yield sizeable 3D photonic band gaps.

We show that it is possible to construct foam-based heterostructures with complete photonic band gaps. Three-dimensional foams are promising candidates for the self-organization of large photonic networks with combinations of physical characteristics that may be useful for applications. The largest band gap found is based on 3D Weaire-Phelan foam, a structure that was originally introduced as a solution to the Kelvin problem of finding the 3D tessellation composed of equal-volume cells that has the least surface area. The photonic band gap has a maximal size of 16.9% (at a volume fraction of 21.6% for a dielectric contrast [Formula: see text]) and a high degree of isotropy, properties that are advantageous in designing photonic waveguides and circuits. We also present results for 2 other foam-based heterostructures based on Kelvin and C15 foams that have somewhat smaller but still significant band gaps.

Universal hidden order in amorphous cellular geometries.

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.

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

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.

Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development.

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.

Mean-intercept anisotropy analysis of porous media. I. Analytic formulae for anisotropic Boolean models.

PURPOSE: Structure-property relations, which relate the shape of the microstructure to physical properties such as transport or mechanical properties, need sensitive measures of structure. What are suitable fabric tensors that quantify the shape of anisotropic heterogeneous materials? The mean intercept length is among the most commonly used characteristics of anisotropy in porous media, for example, of trabecular bone in medical physics. METHODS: We analyze the orientation-biased Boolean model, a versatile stochastic model that represents microstructures as overlapping grains with an orientation bias towards a preferred direction. This model is an extension of the isotropic Boolean model, which has been shown to truthfully reproduce multi-functional properties of isotropic porous media. We explain the close relationship between the concept of intersections with test lines to the elaborate mathematical theory of queues, and how explicit results from the latter can be directly applied to characterize microstructures. RESULTS: In this series of two papers, we provide analytic formulas for the anisotropic Boolean model and demonstrate often overlooked conceptual shortcomings of this approach. Queuing theory is used to derive simple and illustrative formulas for the mean intercept length. It separates into an intensity-dependent and an orientation-dependent factor. The global average of the mean intercept length can be expressed by local characteristics of a single grain alone. CONCLUSIONS: We thus identify which shape information about the random process the mean intercept length contains. The connection between global and local quantities helps to interpret observations and provides insights into the possibilities and limitations of the analysis. In the second paper of this series, we discuss, based on the findings in this paper, short-comings of the mean intercept analysis for (bone-)microstructure characterization. We will suggest alternative and better defined sensitive anisotropy measures from integral geometry.

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

PURPOSE: Structure-property relations, which relate the shape of the microstructure to physical properties such as transport or mechanical properties, need sensitive measures of structure. What are suitable fabric tensors to quantify the shape of anisotropic heterogeneous materials? The mean intercept length is among the most commonly used characteristics of anisotropy in porous media, e.g., of trabecular bone in medical physics. Yet, in this series of two papers we demonstrate that it has conceptual shortcomings that limit the validity of its results. METHODS: We test the validity of general assumptions regarding the properties of the mean-intercept length tensor using analytical formulas for the mean-intercept lengths in anisotropic Boolean models (derived in part I of this series), augmented by numerical simulations. We discuss in detail the functional form of the mean intercept length as a function of the test line orientations. RESULTS: As the most prominent result, we find that, at least for the example of overlapping grains modeling porous media, the polar plot of the mean intercept length is in general not an ellipse and hence not represented by a second-rank tensor. This is in stark contrast to the common understanding that for a large collection of grains the mean intercept length figure averages to an ellipse. The standard mean intercept length tensor defined by a least-square fit of an ellipse is based on a model mismatch, which causes an intrinsic lack of accuracy. CONCLUSIONS: Our analysis reveals several shortcomings of the mean intercept length tensor analysis that pose conceptual problems and limitations on the information content of this commonly used analysis method. We suggest the Minkowski tensors from integral geometry as alternative sensitive measures of anisotropy. The Minkowski tensors allow for a robust, comprehensive, and systematic approach to quantify various aspects of structural anisotropy. We show the Minkowski tensors to be more sensitive, in the sense, that they can quantify the remnant anisotropy of structures not captured by the mean intercept length analysis. If applied to porous tissue and microstructures, this improved structure characterization can yield new insights into the relationships between geometry and material properties.

Characterization of maximally random jammed sphere packings. II. Correlation functions and density fluctuations.

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.

Direct relations between morphology and transport in Boolean models.

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.

Characterization of maximally random jammed sphere packings: Voronoi correlation functions.

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.