Anomalous local coordination, density fluctuations, and void statistics in disordered hyperuniform many-particle ground states

Structural properties of hyperuniform Voronoi networks

Disordered hyperuniform many-particle systems are recently discovered exotic states of matter, characterized by the complete suppression of normalized infinite-wavelength density fluctuations, as in perfect crystals, while lacking conventional long-range order, as in liquids and glasses. In this work, we begin a program to quantify the structural properties of nonhyperuniform and hyperuniform networks. In particular, large two-dimensional (2D) Voronoi networks (graphs) containing approximately 10,000 nodes are created from a variety of different point configurations, including the antihyperuniform hyperplane intersection process (HIP), nonhyperuniform Poisson process, nonhyperuniform random sequential addition (RSA) saturated packing, and both non-stealthy and stealthy hyperuniform point processes. We carry out an extensive study of the Voronoi-cell area distribution of each of the networks by determining multiple metrics that characterize the distribution, including their average areas and corresponding variances as well as higher-order cumulants (i.e., skewness γ_{1} and excess kurtosis γ_{2}). We show that the HIP distribution is far from Gaussian, as evidenced by a high skewness (γ_{1}=3.16) and large positive excess kurtosis (γ_{2}=16.2). The Poisson (with γ_{1}=1.07 and γ_{2}=1.79) and non-stealthy hyperuniform (with γ_{1}=0.257 and γ_{2}=0.0217) distributions are Gaussian-like distributions, since they exhibit a small but positive skewness and excess kurtosis. The RSA (with γ_{1}=0.450 and γ_{2}=-0.0384) and the highest stealthy hyperuniform distributions (with γ_{1}=0.0272 and γ_{2}=-0.0626) are also non-Gaussian because of their low skewness and negative excess kurtosis, which is diametrically opposite of the non-Gaussian behavior of the HIP. The fact that the cell-area distributions of large, finite-sized RSA and stealthy hyperuniform networks (e.g., with N≈10,000 nodes) are narrower, have larger peaks, and smaller tails than a Gaussian distribution implies that in the thermodynamic limit the distributions should exhibit compact support, consistent with previous theoretical considerations. Moreover, we compute the Voronoi-area correlation functions C_{00}(r) for the networks, which describe the correlations between the area of two Voronoi cells separated by a given distance r [M. A. Klatt and S. Torquato, Phys. Rev. E 90, 052120 (2014)1539-375510.1103/PhysRevE.90.052120]. We show that the correlation functions C_{00}(r) qualitatively distinguish the antihyperuniform, nonhyperuniform, and hyperuniform Voronoi networks considered here. Specifically, the antihyperuniform HIP networks possess a slowly decaying C_{00}(r) with large positive values, indicating large fluctuations of Voronoi cell areas across scales. While the nonhyperuniform Poisson and RSA network possess positive and fast decaying C_{00}(r), we find strong anticorrelations in C_{00}(r) (i.e., negative values) for the hyperuniform networks. The latter indicates that the large-scale area fluctuations are suppressed by accompanying large Voronoi cells with small cells (and vice versa) in the systems in order to achieve hyperuniformity. In summary, we have shown that cell-area distributions and pair correlation functions of Voronoi networks enable one to distinguish quantitatively antihyperuniform, standard nonhyperuniform, and hyperuniform networks from one another.

Three-dimensional construction of hyperuniform, nonhyperuniform, and antihyperuniform disordered heterogeneous materials and their transport properties via spectral density functions

Rigorous theories connecting physical properties of a heterogeneous material to its microstructure offer a promising avenue to guide the computational material design and optimization. The spectral density function χ[over ̃]_{_{V}}(k), which can be obtained experimentally from scattering data, enables accurate determination of various transport and wave propagation characteristics, including the time-dependent diffusion spreadability S(t) and effective dynamic dielectric constant ε_{e} for electromagnetic wave propagation. Moreover, χ[over ̃]_{_{V}}(k) determines rigorous upper bounds on the fluid permeability K. Given the importance of χ[over ̃]_{_{V}}(k), we present here an efficient Fourier-space based computational framework to construct three-dimensional (3D) statistically isotropic two-phase heterogeneous materials corresponding to targeted spectral density functions. In particular, we employ a variety of analytical functional forms for χ[over ̃]_{_{V}}(k) that satisfy all known necessary conditions to construct disordered stealthy hyperuniform, standard hyperuniform, nonhyperuniform, and antihyperuniform two-phase heterogeneous material systems at varying phase volume fractions. We show that by tuning the correlations in the system across length scales via the targeted functions, one can generate a rich spectrum of distinct structures within each of the above classes of materials. Importantly, we present the first realization of antihyperuniform two-phase heterogeneous materials in 3D, which are characterized by autocovariance function χ_{_{V}}(r) with a power-law tail, resulting in microstructures that contain clusters of dramatically different sizes and morphologies. We also determine the diffusion spreadability S(t) and estimate the fluid permeability K associated with all of the constructed materials directly from the corresponding spectral densities. Although it is well established that the long-time asymptotic scaling behavior of S(t) only depends on the functional form of χ[over ̃]_{_{V}}(k), with the stealthy hyperuniform and antihyperuniform media, respectively, achieving the most and least efficient transport, we find that varying the length-scale parameter within each class of χ[over ̃]_{_{V}}(k) functions can also lead to orders of magnitude variation of S(t) at intermediate and long time scales. Moreover, we find that increasing the solid volume fraction ϕ_{1} and correlation length a in the constructed media generally leads to a decrease in the dimensionless fluid permeability K/a^{2}, while the antihyperuniform media possess the largest K/a^{2} among the four classes of materials with the same ϕ_{1} and a. These results indicate the feasibility of employing parameterized spectral densities for designing composites with targeted transport properties.

Hyperuniformity classes of quasiperiodic tilings via diffusion spreadability

Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent α>0, that characterizes the power-law scaling behavior of the structure factor S(k) as a function of wave number k≡|k| in the vicinity of the origin, e.g., S(k)∼|k|^{α} in cases where S(k) varies continuously with k as k→0. In this paper, we show that the spreadability is an effective method for determining α for quasiperiodic systems where S(k) is discontinuous and consists of a dense set of Bragg peaks. It has been shown in [Phys. Rev. E 104, 054102 (2021)10.1103/PhysRevE.104.054102] that, for media with finite α, the long-time behavior of the excess spreadability S(∞)-S(t) can be fit to a power law of the form t^{-(d-α)/2}, where d is the space dimension, to accurately extract α for the continuous case. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density χ[over ̃]_{_{V}}(k) of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract α for the one-dimensional (1D) limit-periodic period-doubling chain (α=1) and the 1D quasicrystalline Fibonacci chain (α=3) to within 0.02% of the analytically known exact results. Moreover, we obtain a value of α=5.97±0.06 for the two-dimensional Penrose tiling and present plausible theoretical arguments strongly suggesting that α is exactly equal to six. We also show that, due to the self-similarity of the structures examined here, one can truncate the small-k region of the scattering information used to compute the spreadability and obtain an accurate value of α, with a small deviation from the untruncated case that decreases as the system size increases. This strongly suggests that one can obtain a good estimate of α for an infinite self-similar quasicrystal from a modestly sized finite sample. The methods described here offer a simple and general procedure to characterize accurately the large-scale translational order present in quasicrystalline and limit-periodic media in any space dimension that are self-similar. Moreover, the scattering information extracted from these two-phase media encoded in χ[over ̃]_{_{V}}(k), can be used to estimate their physical properties, such as their effective dynamic dielectric constants, effective dynamic elastic constants, and fluid permeabilities.

Computational design of anisotropic stealthy hyperuniform composites with engineered directional scattering properties

Disordered hyperuniform materials are an emerging class of exotic amorphous states of matter that endow them with singular physical properties, including large isotropic photonic band gaps, superior resistance to fracture, and nearly optimal electrical and thermal transport properties, to name but a few. Here we generalize the Fourier-space-based numerical construction procedure for designing and generating digital realizations of isotropic disordered hyperuniform two-phase heterogeneous materials (i.e., composites) developed by Chen and Torquato [Acta Mater. 142, 152 (2018)1359-645410.1016/j.actamat.2017.09.053] to anisotropic microstructures with targeted spectral densities. Our generalized construction procedure explicitly incorporates the vector-dependent spectral density function χ[over ̃]_{_{V}}(k) of arbitrary form that is realizable. We demonstrate the utility of the procedure by generating a wide spectrum of anisotropic stealthy hyperuniform microstructures with χ[over ̃]_{_{V}}(k)=0 for k∈Ω, i.e., complete suppression of scattering in an "exclusion" region Ω around the origin in Fourier space. We show how different exclusion-region shapes with various discrete symmetries, including circular-disk, elliptical-disk, square, rectangular, butterfly-shaped, and lemniscate-shaped regions of varying size, affect the resulting statistically anisotropic microstructures as a function of the phase volume fraction. The latter two cases of Ω lead to directionally hyperuniform composites, which are stealthy hyperuniform only along certain directions and are nonhyperuniform along others. We find that while the circular-disk exclusion regions give rise to isotropic hyperuniform composite microstructures, the directional hyperuniform behaviors imposed by the shape asymmetry (or anisotropy) of certain exclusion regions give rise to distinct anisotropic structures and degree of uniformity in the distribution of the phases on intermediate and large length scales along different directions. Moreover, while the anisotropic exclusion regions impose strong constraints on the global symmetry of the resulting media, they can still possess structures at a local level that are nearly isotropic. Both the isotropic and anisotropic hyperuniform microstructures associated with the elliptical-disk, square, and rectangular Ω possess phase-inversion symmetry over certain range of volume fractions and a percolation threshold ϕ_{c}≈0.5. On the other hand, the directionally hyperuniform microstructures associated with the butterfly-shaped and lemniscate-shaped Ω do not possess phase-inversion symmetry and percolate along certain directions at much lower volume fractions. We also apply our general procedure to construct stealthy nonhyperuniform systems. Our construction algorithm enables one to control the statistical anisotropy of composite microstructures via the shape, size, and symmetries of Ω, which is crucial to engineering directional optical, transport, and mechanical properties of two-phase composite media.

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.

Structural characterization of many-particle systems on approach to hyperuniform states

The study of hyperuniform states of matter is an emerging multidisciplinary field, impinging on topics in the physical sciences, mathematics, and biology. The focus of this work is the exploration of quantitative descriptors that herald when a many-particle system in d-dimensional Euclidean space R^{d} approaches a hyperuniform state as a function of the relevant control parameter. We establish quantitative criteria to ascertain the extent of hyperuniform and nonhyperuniform distance-scaling regimes as well as the crossover point between them in terms of the "volume" coefficient A and "surface-area" coefficient B associated with the local number variance σ^{2}(R) for a spherical window of radius R. The larger the ratio B/A, the larger the hyperuniform scaling regime, which becomes of infinite extent in the limit B/A→∞. To complement the known direct-space representation of the coefficient B in terms of the total correlation function h(r), we derive its corresponding Fourier representation in terms of the structure factor S(k), which is especially useful when scattering information is available experimentally or theoretically. We also demonstrate that the free-volume theory of the pressure of equilibrium packings of identical hard spheres that approach a strictly jammed state either along the stable crystal or metastable disordered branch dictates that such end states be exactly hyperuniform. Using the ratio B/A, as well as other diagnostic measures of hyperuniformity, including the hyperuniformity index H and the direct-correlation function length scale ξ_{c}, we study three different exactly solvable models as a function of the relevant control parameter, either density or temperature, with end states that are perfectly hyperuniform. Specifically, we analyze equilibrium systems of hard rods and "sticky" hard-sphere systems in arbitrary space dimension d as a function of density. We also examine low-temperature excited states of many-particle systems interacting with "stealthy" long-ranged pair interactions as the temperature tends to zero, where the ground states are disordered, hyperuniform, and infinitely degenerate. We demonstrate that our various diagnostic hyperuniformity measures are positively correlated with one another. The same diagnostic measures can be used to detect the degree to which imperfections in nearly hyperuniform systems cause deviations from perfect hyperuniformity. Moreover, the capacity to identify hyperuniform scaling regimes should be particularly useful in analyzing experimentally or computationally generated samples that are necessarily of finite size.

Characterizing the hyperuniformity of ordered and disordered two-phase media

The hyperuniformity concept provides a unified means to classify all perfect crystals, perfect quasicrystals, and exotic amorphous states of matter according to their capacity to suppress large-scale density fluctuations. While the classification of hyperuniform point configurations has received considerable attention, much less is known about the classification of hyperuniform two-phase heterogeneous media, which include composites, porous media, foams, cellular solids, colloidal suspensions, and polymer blends. The purpose of this article is to begin such a program for certain two-dimensional models of hyperuniform two-phase media by ascertaining their local volume-fraction variances σ_{_{V}}^{2}(R) and the associated hyperuniformity order metrics B[over ¯]_{V}. This is a highly challenging task because the geometries and topologies of the phases are generally much richer and more complex than point-configuration arrangements, and one must ascertain a broadly applicable length scale to make key quantities dimensionless. Therefore, we purposely restrict ourselves to a certain class of two-dimensional periodic cellular networks as well as periodic and disordered or irregular packings of circular disks, some of which maximize their effective transport and elastic properties. Among the cellular networks considered, the honeycomb networks have minimal values of the hyperuniformity order metrics B[over ¯]_{V} across all volume fractions. On the other hand, for all packings of circular disks examined, the triangular-lattice packings have the smallest values of B[over ¯]_{V} for the possible range of volume fractions. Among all structures studied here, the triangular-lattice packing of circular disks have the minimal values of the order metric for almost all volume fractions. Our study provides a theoretical foundation for the establishment of hyperuniformity order metrics for general two-phase media and a basis to discover new hyperuniform two-phase systems with desirable bulk physical properties by inverse design procedures.

Stone-Wales defects preserve hyperuniformity in amorphous two-dimensional networks

Disordered hyperuniformity (DHU) is a recently discovered novel state of many-body systems that possesses vanishing normalized infinite-wavelength density fluctuations similar to a perfect crystal and an amorphous structure like a liquid or glass. Here, we discover a hyperuniformity-preserving topological transformation in two-dimensional (2D) network structures that involves continuous introduction of Stone-Wales (SW) defects. Specifically, the static structure factor [Formula: see text] of the resulting defected networks possesses the scaling [Formula: see text] for small wave number k, where [Formula: see text] monotonically decreases as the SW defect concentration p increases, reaches [Formula: see text] at [Formula: see text], and remains almost flat beyond this p. Our findings have important implications for amorphous 2D materials since the SW defects are well known to capture the salient feature of disorder in these materials. Verified by recently synthesized single-layer amorphous graphene, our network models reveal unique electronic transport mechanisms and mechanical behaviors associated with distinct classes of disorder in 2D materials.

Realizable hyperuniform and nonhyperuniform particle configurations with targeted spectral functions via effective pair interactions

The capacity to identify realizable many-body configurations associated with targeted functional forms for the pair correlation function g_{2}(r) or its corresponding structure factor S(k) is of great fundamental and practical importance. While there are obvious necessary conditions that a prescribed structure factor at number density ρ must satisfy to be configurationally realizable, sufficient conditions are generally not known due to the infinite degeneracy of configurations with different higher-order correlation functions. A major aim of this paper is to expand our theoretical knowledge of the class of pair correlation functions or structure factors that are realizable by classical disordered ensembles of particle configurations, including exotic "hyperuniform" varieties. We first introduce a theoretical formalism that provides a means to draw classical particle configurations from canonical ensembles with certain pairwise-additive potentials that could correspond to targeted analytical functional forms for the structure factor. This formulation enables us to devise an improved algorithm to construct systematically canonical-ensemble particle configurations with such targeted pair statistics, whenever realizable. As a proof of concept, we test the algorithm by targeting several different structure factors across dimensions that are known to be realizable and one hyperuniform target that is known to be nontrivially unrealizable. Our algorithm succeeds for all realizable targets and appropriately fails for the unrealizable target, demonstrating the accuracy and power of the method to numerically investigate the realizability problem. Subsequently, we also target several families of structure-factor functions that meet the known necessary realizability conditions but are not known to be realizable by disordered hyperuniform point configurations, including d-dimensional Gaussian structure factors, d-dimensional generalizations of the two-dimensional one-component plasma (OCP), and the d-dimensional Fourier duals of the previous OCP cases. Moreover, we also explore unusual nonhyperuniform targets, including "hyposurficial" and "antihyperuniform" examples. In all of these instances, the targeted structure factors are achieved with high accuracy, suggesting that they are indeed realizable by equilibrium configurations with pairwise interactions at positive temperatures. Remarkably, we also show that the structure factor of nonequilibrium perfect glass, specified by two-, three-, and four-body interactions, can also be realized by equilibrium pair interactions at positive temperatures. Our findings lead us to the conjecture that any realizable structure factor corresponding to either a translationally invariant equilibrium or nonequilibrium system can be attained by an equilibrium ensemble involving only effective pair interactions. Our investigation not only broadens our knowledge of analytical functional forms for g_{2}(r) and S(k) associated with disordered point configurations across dimensions but also deepens our understanding of many-body physics. Moreover, our work can be applied to the design of materials with desirable physical properties that can be tuned by their pair statistics.

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.

Methodology to construct large realizations of perfectly hyperuniform disordered packings

Disordered hyperuniform packings (or dispersions) are unusual amorphous two-phase materials that are endowed with exotic physical properties. Such hyperuniform systems are characterized by an anomalous suppression of volume-fraction fluctuations at infinitely long-wavelengths, compared to ordinary disordered materials. While there has been growing interest in such singular states of amorphous matter, a major obstacle has been an inability to produce large samples that are perfectly hyperuniform due to practical limitations of conventional numerical and experimental methods. To overcome these limitations, we introduce a general theoretical methodology to construct perfectly hyperuniform packings in d-dimensional Euclidean space R^{d}. Specifically, beginning with an initial general tessellation of space by disjoint cells that meets a "bounded-cell" condition, hard particles of general shape are placed inside each cell such that the local-cell particle packing fractions are identical to the global packing fraction. We prove that the constructed packings with a polydispersity in size are perfectly hyperuniform in the infinite-sample-size limit, regardless of particle shapes, positions, and numbers per cell. We use this theoretical formulation to devise an efficient and tunable algorithm to generate extremely large realizations of such packings. We employ two distinct initial tessellations: Voronoi as well as sphere tessellations. Beginning with Voronoi tessellations, we show that our algorithm can remarkably convert extremely large nonhyperuniform packings into hyperuniform ones in R^{2} and R^{3}. Implementing our theoretical methodology on sphere tessellations, we establish the hyperuniformity of the classical Hashin-Shtrikman multiscale coated-spheres structures, which are known to be two-phase media microstructures that possess optimal effective transport and elastic properties. A consequence of our work is a rigorous demonstration that packings that have identical tessellations can either be nonhyperuniform or hyperuniform by simply tuning local characteristics. It is noteworthy that our computationally designed hyperuniform two-phase systems can easily be fabricated via state-of-the-art methods, such as 2D photolithographic and 3D printing technologies. In addition, the tunability of our methodology offers a route for the discovery of novel disordered hyperuniform two-phase materials.

Hyperuniformity order metric of Barlow packings

The concept of hyperuniformity has been a useful tool in the study of density fluctuations at large length scales in systems ranging across the natural and mathematical sciences. One can rank a large class of hyperuniform systems by their ability to suppress long-range density fluctuations through the use of a hyperuniformity order metric Λ[over ¯]. We apply this order metric to the Barlow packings, which are the infinitely degenerate densest packings of identical rigid spheres that are distinguished by their stacking geometries and include the commonly known fcc lattice and hcp crystal. The "stealthy stacking" theorem implies that these packings are all stealthy hyperuniform, a strong type of hyperuniformity, which involves the suppression of scattering up to a wave vector K. We describe the geometry of three classes of Barlow packings, two disordered classes and small-period packings. In addition, we compute a lower bound on K for all Barlow packings. We compute Λ[over ¯] for the aforementioned three classes of Barlow packings and find that, to a very good approximation, it is linear in the fraction of fcc-like clusters, taking values between those of least-ordered hcp and most-ordered fcc. This implies that the value of Λ[over ¯] of all Barlow packings is primarily controlled by the local cluster geometry. These results highlight the special nature of anisotropic stacking disorder, which provides impetus for future research on the development of anisotropic order metrics and hyperuniformity properties.

Hyperuniformity with no fine tuning in sheared sedimenting suspensions

Particle suspensions, present in many natural and industrial settings, typically contain aggregates or other microstructures that can complicate macroscopic flow behaviors and damage processing equipment. Recent work found that applying uniform periodic shear near a critical transition can reduce fluctuations in the particle concentration across all length scales, leading to a hyperuniform state. However, this strategy for homogenization requires fine tuning of the strain amplitude. Here we show that in a model of sedimenting particles under periodic shear, there is a well-defined regime at low sedimentation speed where hyperuniform scaling automatically occurs. Our simulations and theoretical arguments show that the homogenization extends up to a finite length scale that diverges as the sedimentation speed approaches zero.

Suspensions appear in a wide range of industrial settings, and dispersing particles in a uniform manner throughout a fluid remains challenging for applications. Wang et al. obtain hyperuniform mixtures without fine tuning by harnessing self-organized criticality due to slow sedimentation and shear.

Classical many-particle systems with unique disordered ground states

Classical ground states (global energy-minimizing configurations) of many-particle systems are typically unique crystalline structures, implying zero enumeration entropy of distinct patterns (aside from trivial symmetry operations). By contrast, the few previously known disordered classical ground states of many-particle systems are all high-entropy (highly degenerate) states. Here we show computationally that our recently proposed "perfect-glass" many-particle model [Sci. Rep. 6, 36963 (2016)10.1038/srep36963] possesses disordered classical ground states with a zero entropy: a highly counterintuitive situation . For all of the system sizes, parameters, and space dimensions that we have numerically investigated, the disordered ground states are unique such that they can always be superposed onto each other or their mirror image. At low energies, the density of states obtained from simulations matches those calculated from the harmonic approximation near a single ground state, further confirming ground-state uniqueness. Our discovery provides singular examples in which entropy and disorder are at odds with one another. The zero-entropy ground states provide a unique perspective on the celebrated Kauzmann-entropy crisis in which the extrapolated entropy of a supercooled liquid drops below that of the crystal. We expect that our disordered unique patterns to be of value in fields beyond glass physics, including applications in cryptography as pseudorandom functions with tunable computational complexity.

Microstructure and mechanical properties of hyperuniform heterogeneous materials

A hyperuniform random heterogeneous material is one in which the local volume fraction fluctuations in an observation window decay faster than the reciprocal window volume as the window size increases. Recent studies show that this class of materials are endowed with superior physical properties such as large isotropic photonic band gaps and optimal transport properties. Here we employ a stochastic optimization procedure to systematically generate realizations of hyperuniform heterogeneous materials with controllable short-range order, which is partially quantified using the two-point correlation function S_{2}(r) associated with the phase of interest. Specifically, our procedure generalizes the widely used Yeong-Torquato reconstruction procedure by including an additional constraint for hyperuniformity, i.e., the volume integral of the autocovariance function χ(r)=S_{2}(r)-ϕ^{2} over the whole space is zero. In addition, we only require the reconstructed S_{2} to match the target function up to a certain cutoff distance γ, in order to give the system sufficient degrees of freedom to satisfy the hyperuniform condition. By systematically increasing the γ value for a given S_{2}, one can produce a spectrum of hyperuniform heterogeneous materials with varying degrees of partial short-range order compatible with the specified S_{2}. The mechanical performance including both elastic and brittle fracture behaviors of the generated hyperuniform materials is analyzed using a volume-compensated lattice-particle method. For the purpose of comparison, the corresponding nonhyperuniform materials with the same short-range order (i.e., with S_{2} constrained up to the same γ value) are also constructed and their mechanical performance is analyzed. Here we consider two specific S_{2} including the positive exponential decay function and the correlation function associated with an equilibrium hard-sphere system. For the constructed systems associated with these two specific functions, we find that although the hyperuniform materials are softer than their nonhyperuniform counterparts, the former generally possess a significantly higher brittle fracture strength than the latter. This superior mechanical behavior is attributed to the lower degree of stress concentration in the material resulting from the hyperuniform microstructure, which is crucial to crack initiation and propagation.

Structural Characterization and Statistical-Mechanical Model of Epidermal Patterns

In proliferating epithelia of mammalian skin, cells of irregular polygon-like shapes pack into complex, nearly flat two-dimensional structures that are pliable to deformations. In this work, we employ various sensitive correlation functions to quantitatively characterize structural features of evolving packings of epithelial cells across length scales in mouse skin. We find that the pair statistics in direct space (correlation function) and Fourier space (structure factor) of the cell centroids in the early stages of embryonic development show structural directional dependence (statistical anisotropy), which is a reflection of the fact that cells are stretched, which promotes uniaxial growth along the epithelial plane. In the late stages, the patterns tend toward statistically isotropic states, as cells attain global polarization and epidermal growth shifts to produce the skin's outer stratified layers. We construct a minimalist four-component statistical-mechanical model involving effective isotropic pair interactions consisting of hard-core repulsion and extra short-range soft-core repulsion beyond the hard core, whose length scale is roughly the same as the hard core. The model parameters are optimized to match the sample pair statistics in both direct and Fourier spaces. By doing this, the parameters are biologically constrained. In contrast with many vertex-based models, our statistical-mechanical model does not explicitly incorporate information about the cell shapes and interfacial energy between cells; nonetheless, our model predicts essentially the same polygonal shape distribution and size disparity of cells found in experiments, as measured by Voronoi statistics. Moreover, our simulated equilibrium liquid-like configurations are able to match other nontrivial unconstrained statistics, which is a testament to the power and novelty of the model. The array of structural descriptors that we deploy enable us to distinguish between normal, mechanically deformed, and pathological skin tissues. Our statistical-mechanical model enables one to generate tissue microstructure at will for further analysis. We also discuss ways in which our model might be extended to better understand morphogenesis (in particular the emergence of planar cell polarity), wound healing, and disease-progression processes in skin, and how it could be applied to the design of synthetic tissues.

Enhanced hyperuniformity from random reorganization

Diffusion relaxes density fluctuations toward a uniform random state whose variance in regions of volume [Formula: see text] scales as [Formula: see text] Systems whose fluctuations decay faster, [Formula: see text] with [Formula: see text], are called hyperuniform. The larger [Formula: see text], the more uniform, with systems like crystals achieving the maximum value: [Formula: see text] Although finite temperature equilibrium dynamics will not yield hyperuniform states, driven, nonequilibrium dynamics may. Such is the case, for example, in a simple model where overlapping particles are each given a small random displacement. Above a critical particle density [Formula: see text], the system evolves forever, never finding a configuration where no particles overlap. Below [Formula: see text], however, it eventually finds such a state, and stops evolving. This "absorbing state" is hyperuniform up to a length scale [Formula: see text], which diverges at [Formula: see text] An important question is whether hyperuniformity survives noise and thermal fluctuations. We find that hyperuniformity of the absorbing state is not only robust against noise, diffusion, or activity, but that such perturbations reduce fluctuations toward their limiting behavior, [Formula: see text], a uniformity similar to random close packing and early universe fluctuations, but with arbitrary controllable density.

Noise, Diffusion, and Hyperuniformity

We consider driven many-particle models which have a phase transition between an active and an absorbing phase. Like previously studied models, we have particle conservation, but here we introduce an additional symmetry-when two particles interact, we give them stochastic kicks which conserve the center of mass. We find that the density fluctuations in the active phase decay in the fastest manner possible for a disordered isotropic system, and we present arguments that the large scale fluctuations are determined by a competition between a noise term which generates fluctuations, and a deterministic term which reduces them. Our results may be relevant to shear experiments and may further the understanding of hyperuniformity which occurs at the critical point.

The Perfect Glass Paradigm: Disordered Hyperuniform Glasses Down to Absolute Zero

Rapid cooling of liquids below a certain temperature range can result in a transition to glassy states. The traditional understanding of glasses includes their thermodynamic metastability with respect to crystals. However, here we present specific examples of interactions that eliminate the possibilities of crystalline and quasicrystalline phases, while creating mechanically stable amorphous glasses down to absolute zero temperature. We show that this can be accomplished by introducing a new ideal state of matter called a "perfect glass". A perfect glass represents a soft-interaction analog of the maximally random jammed (MRJ) packings of hard particles. These latter states can be regarded as the epitome of a glass since they are out of equilibrium, maximally disordered, hyperuniform, mechanically rigid with infinite bulk and shear moduli, and can never crystallize due to configuration-space trapping. Our model perfect glass utilizes two-, three-, and four-body soft interactions while simultaneously retaining the salient attributes of the MRJ state. These models constitute a theoretical proof of concept for perfect glasses and broaden our fundamental understanding of glass physics. A novel feature of equilibrium systems of identical particles interacting with the perfect-glass potential at positive temperature is that they have a non-relativistic speed of sound that is infinite.

Disordered hyperuniform heterogeneous materials

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.

Hyperuniformity and its generalizations

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."

Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations

Systems of particles interacting with "stealthy" pair potentials have been shown to possess infinitely degenerate disordered hyperuniform classical ground states with novel physical properties. Previous attempts to sample the infinitely degenerate ground states used energy minimization techniques, introducing algorithmic dependence that is artificial in nature. Recently, an ensemble theory of stealthy hyperuniform ground states was formulated to predict the structure and thermodynamics that was shown to be in excellent agreement with corresponding computer simulation results in the canonical ensemble (in the zero-temperature limit). In this paper, we provide details and justifications of the simulation procedure, which involves performing molecular dynamics simulations at sufficiently low temperatures and minimizing the energy of the snapshots for both the high-density disordered regime, where the theory applies, as well as lower densities. We also use numerical simulations to extend our study to the lower-density regime. We report results for the pair correlation functions, structure factors, and Voronoi cell statistics. In the high-density regime, we verify the theoretical ansatz that stealthy disordered ground states behave like "pseudo" disordered equilibrium hard-sphere systems in Fourier space. The pair statistics obey certain exact integral conditions with very high accuracy. These results show that as the density decreases from the high-density limit, the disordered ground states in the canonical ensemble are characterized by an increasing degree of short-range order and eventually the system undergoes a phase transition to crystalline ground states. In the crystalline regime (low densities), there exist aperiodic structures that are part of the ground-state manifold but yet are not entropically favored. We also provide numerical evidence suggesting that different forms of stealthy pair potentials produce the same ground-state ensemble in the zero-temperature limit. Our techniques may be applied to sample the zero-temperature limit of the canonical ensemble of other potentials with highly degenerate ground states.

Hyperuniformity of critical absorbing states

The properties of the absorbing states of nonequilibrium models belonging to the conserved directed percolation universality class are studied. We find that, at the critical point, the absorbing states are hyperuniform, exhibiting anomalously small density fluctuations. The exponent characterizing the fluctuations is measured numerically, a scaling relation to other known exponents is suggested, and a new correlation length relating to this ordering is proposed. These results may have relevance to photonic band-gap materials.

Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale packing problem

Optimal spatial sampling of light rigorously requires that identical photoreceptors be arranged in perfectly regular arrays in two dimensions. Examples of such perfect arrays in nature include the compound eyes of insects and the nearly crystalline photoreceptor patterns of some fish and reptiles. Birds are highly visual animals with five different cone photoreceptor subtypes, yet their photoreceptor patterns are not perfectly regular. By analyzing the chicken cone photoreceptor system consisting of five different cell types using a variety of sensitive microstructural descriptors, we find that the disordered photoreceptor patterns are "hyperuniform" (exhibiting vanishing infinite-wavelength density fluctuations), a property that had heretofore been identified in a unique subset of physical systems, but had never been observed in any living organism. Remarkably, the patterns of both the total population and the individual cell types are simultaneously hyperuniform. We term such patterns "multihyperuniform" because multiple distinct subsets of the overall point pattern are themselves hyperuniform. We have devised a unique multiscale cell packing model in two dimensions that suggests that photoreceptor types interact with both short- and long-ranged repulsive forces and that the resultant competition between the types gives rise to the aforementioned singular spatial features characterizing the system, including multihyperuniformity. These findings suggest that a disordered hyperuniform pattern may represent the most uniform sampling arrangement attainable in the avian system, given intrinsic packing constraints within the photoreceptor epithelium. In addition, they show how fundamental physical constraints can change the course of a biological optimization process. Our results suggest that multihyperuniform disordered structures have implications for the design of materials with novel physical properties and therefore may represent a fruitful area for future research.

Improved reconstructions of random media using dilation and erosion processes

By using the most sensitive two-point correlation functions introduced to date, we reconstruct the microstructures of two-phase random media with heretofore unattained accuracy. Such media arise in a host of contexts, including porous and composite media, ecological structures, biological media, and astrophysical structures. The aforementioned correlation functions are special cases of the so-called canonical n-point correlation function H(n) and generalize the ones that have been recently employed to advance our ability to reconstruct complex microstructures [Y. Jiao, F. H. Stillinger, and S. Torquato, Proc. Natl. Acad. Sci. 106, 17634 (2009)]. The use of these generalized correlation functions is tantamount to dilating or eroding a reference phase of the target medium and incorporating the additional topological information of the modified media, thereby providing more accurate reconstructions of percolating, filamentary, and other topologically complex microstructures. We apply our methods to a multiply connected "donut" medium and a dilute distribution of "cracks" (a set of essentially zero measure), demonstrating improved accuracy in both cases with implications for higher-dimensional and biconnected two-phase systems. The high information content of the generalized two-point correlation functions suggests that it would be profitable to explore their use to characterize the structural and physical properties of not only random media, but also molecular systems, including structural glasses.

Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. II. Anisotropy in particle shape

We extend the results from the first part of this series of two papers by examining hyperuniformity in heterogeneous media composed of impenetrable anisotropic inclusions. Specifically, we consider maximally random jammed (MRJ) packings of hard ellipses and superdisks and show that these systems both possess vanishing infinite-wavelength local-volume-fraction fluctuations and quasi-long-range pair correlations scaling as r(-(d+1)) in d Euclidean dimensions. Our results suggest a strong generalization of a conjecture by Torquato and Stillinger [Phys. Rev. E 68, 041113 (2003)], namely, that all strictly jammed saturated packings of hard particles, including those with size and shape distributions, are hyperuniform with signature quasi-long-range correlations. We show that our arguments concerning the constrained distribution of the void space in MRJ packings directly extend to hard-ellipse and superdisk packings, thereby providing a direct structural explanation for the appearance of hyperuniformity and quasi-long-range correlations in these systems. Additionally, we examine general heterogeneous media with anisotropic inclusions and show unexpectedly that one can decorate a periodic point pattern to obtain a hard-particle system that is not hyperuniform with respect to local-volume-fraction fluctuations. This apparent discrepancy can also be rationalized by appealing to the irregular distribution of the void space arising from the anisotropic shapes of the particles. Our work suggests the intriguing possibility that the MRJ states of hard particles share certain universal features independent of the local properties of the packings, including the packing fraction and average contact number per particle.

Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. I. Polydisperse spheres

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.