Diagnosing hyperuniformity in two-dimensional, disordered, jammed packings of soft spheres

Universal Hyperuniform Organization in Looped Leaf Vein Networks

The leaf vein network is a hierarchical vascular system that transports water and nutrients to the leaf cells. The thick primary veins form a branched network, while the secondary veins can develop closed loops forming a well-defined cellular structure. Through extensive analysis of a variety of distinct leaf species, we discover that the apparently disordered cellular structures of the secondary vein networks exhibit a universal hyperuniform organization and possess a hidden order on large scales. Disorder hyperuniform systems lack conventional long-range order, yet they completely suppress normalized infinite-wavelength density fluctuations like crystals. Specifically, we find that the distributions of the geometric centers associated with the vein network loops possess a vanishing static structure factor in the limit that the wave number k goes to 0, i.e., S(k)∼k^{α}, where α≈0.64±0.021, providing an example of class III hyperuniformity in biology. This hyperuniform organization leads to superior efficiency of diffusive transport, as evidenced by the much faster convergence of the time-dependent spreadability S(t) to its longtime asymptotic limit, compared to that of other uncorrelated or correlated disordered but nonhyperuniform organizations. Our results also have implications for the discovery and design of novel disordered network materials with optimal transport properties.

Vibrational properties of disordered stealthy hyperuniform 1D atomic chains

Disorder hyperuniformity is a recently discovered exotic state of many-body systems that possess a hidden order in between that of a perfect crystal and a completely disordered system. Recently, this novel disordered state has been observed in a number of quantum materials including amorphous 2D graphene and silica, which are endowed with unexpected electronic transport properties. Here, we numerically investigate 1D atomic chain models, including perfect crystalline, disordered stealthy hyperuniform (SHU) as well as randomly perturbed atom packing configurations to obtain a quantitative understanding of how the unique SHU disorder affects the vibrational properties of these low-dimensional materials. We find that the disordered SHU chains possess lower cohesive energies compared to the randomly perturbed chains, implying their potential reliability in experiments. Our inverse partition ratio (IPR) calculations indicate that the SHU chains can support fully delocalized states just like perfect crystalline chains over a wide range of frequencies, i.e.ω∈(0,100)cm-1, suggesting superior phonon transport behaviors within these frequencies, which was traditionally considered impossible in disordered systems. Interestingly, we observe the emergence of a group of highly localized states associated withω∼200cm-1, which is characterized by a significant peak in the IPR and a peak in phonon density of states at the corresponding frequency, and is potentially useful for decoupling electron and phonon degrees of freedom. These unique properties of disordered SHU chains have implications in the design and engineering of novel quantum materials for thermal and phononic applications.

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.

Observation of magnetic structural universality and jamming transition with NMR

Nuclear magnetic resonance (NMR) has been instrumental in deciphering the structure of proteins. Here we show that transverse NMR relaxation, through its time-dependent relaxation rate, is distinctly sensitive to the structure of complex materials or biological tissues at the mesoscopic scale, from micrometers to tens of micrometers. Based on the ideas of universality, we show analytically and numerically that the time-dependent transverse relaxation rate approaches its long-time limit in a power-law fashion, with the dynamical exponent reflecting the universality class of mesoscopic magnetic structure. The spectral line shape acquires the corresponding non-analytic power law singularity at zero frequency. We experimentally detect the change in the dynamical exponent as a result of the transition into maximally random jammed state characterized by hyperuniform correlations. The relation between relaxational dynamics and magnetic structure opens the way for noninvasive characterization of porous media, complex materials and biological tissues.

Exploring generic principles of compartmentalization in a developmental in vitro model

Self-organization of cells into higher-order structures is key for multicellular organisms, for example via repetitive replication of template-like founder cells or syncytial energids. Yet, very similar spatial arrangements of cell-like compartments ('protocells') are also seen in a minimal model system of Xenopus egg extracts in the absence of template structures and chromatin, with dynamic microtubule assemblies driving the self-organization process. Quantifying geometrical features over time, we show here that protocell patterns are highly organized with a spatial arrangement and coarsening dynamics similar to that of two-dimensional foams but without the long-range ordering expected for hexagonal patterns. These features remain invariant when enforcing smaller protocells by adding taxol, i.e. patterns are dominated by a single, microtubule-derived length scale. Comparing our data to generic models, we conclude that protocell patterns emerge by simultaneous formation of randomly assembling protocells that grow at a uniform rate towards a frustrated arrangement before fusion of adjacent protocells eventually drives coarsening. The similarity of protocell patterns to arrays of energids and cells in developing organisms, but also to epithelial monolayers, suggests generic mechanical cues to drive self-organized space compartmentalization.

Role of initial conditions in one-dimensional diffusive systems: Compressibility, hyperuniformity, and long-term memory

We analyze the long-lasting effects of initial conditions on dynamical fluctuations in one-dimensional diffusive systems. We consider the mean-squared displacement of tracers in homogeneous systems with single-file diffusion, and current fluctuations for noninteracting diffusive particles. In each case we show analytically that the long-term memory of initial conditions is mediated by a single static quantity: a generalized compressibility that quantifies the density fluctuations of the initial state. We thereby identify a universality class of hyperuniform initial states whose dynamical variances coincide with the quenched cases studied previously, alongside a continuous family of other classes among which equilibrated (or annealed) initial conditions are but one member. We verify our predictions through extensive Monte Carlo simulations.

Hyperuniform Active Chiral Fluids with Tunable Internal Structure

Large density fluctuations observed in active systems and hyperuniformity are two seemingly incompatible phenomena. However, the formation of hyperuniform states has been recently predicted in nonequilibrium fluids formed by chiral particles performing circular motion with the same handedness. Here we report evidence of hyperuniformity realized in a chiral active fluid comprised of pear-shaped Quincke rollers of arbitrary handedness. We show that hyperuniformity and large density fluctuations, triggered by dynamic clustering, coexist in this system at different length scales. The system loses its hyperuniformity as the curvature of particles' motion increases, transforming them into localized spinners. Our results experimentally demonstrate a novel hyperuniform active fluid and provide new insights into an interplay between chirality, activity, and hyperuniformity.

Geometrical properties of mechanically annealed systems near the jamming transition

Geometrical properties of two-dimensional mixtures near the jamming transition point are numerically investigated using harmonic particles under mechanical training. The configurations generated by the quasi-static compression and oscillatory shear deformations exhibit anomalous suppression of the density fluctuations, known as hyperuniformity, below and above the jamming transition. For the jammed system trained by compression above the transition point, the hyperuniformity exponent increases. For the system below the transition point under oscillatory shear, the hyperuniformity exponent also increases until the shear amplitude reaches the threshold value. The threshold value matches with the transition point from the point-reversible phase where the particles experience no collision to the loop-reversible phase where the particles' displacements are non-affine during a shear cycle before coming back to an original position. The results demonstrated in this paper are explained in terms of neither of universal criticality of the jamming transition nor the nonequilibrium phase transitions.

Dynamic evolution of hyperuniformity in a driven dissipative colloidal system

Hyperuniformity is evolving to become a unifying concept that can help classify and characterize equilibrium and nonequilibrium states of matter. Therefore, understanding the extent of hyperuniformity in dissipative systems is critical. Here, we study the dynamic evolution of hyperuniformity in a driven dissipative colloidal system. We experimentally show and numerically verify that the hyperuniformity of a colloidal crystal is robust against various lattice imperfections and environmental perturbations. This robustness even manifests during crystal disassembly as the system switches between strong (class I), logarithmic (class II), weak (class III), and non-hyperuniform states. To aid analyses, we developed a comprehensive computational toolbox, enabling real-time characterization of hyperuniformity in real- and reciprocal-spaces together with the evolution of several order metric features, and measurements showing the effect of external perturbations on the spatiotemporal distribution of the particles. Our findings provide a new framework to understand the basic principles that drive a dissipative system to a hyperuniform state.

Quantifying the long-range structure of foams and other cellular patterns with hyperuniformity disorder length spectroscopy

We investigate the local and long-range structure of several space-filling cellular patterns: bubbles in a quasi-two-dimensional foam, and Voronoi constructions made around points that are uncorrelated (Poisson patterns), low discrepancy (Halton patterns), and displaced from a lattice by Gaussian noise (Einstein patterns). We study local structure with distributions of quantities including cell areas and side numbers. The former is the widest for the bubbles making foams the most locally disordered, while the latter show no major differences between the cellular patterns. To study long-range structure, we begin by representing the cellular systems as patterns of points, both unweighted and weighted by cell area. For this, foams are represented by their bubble centroids and the Voronoi constructions are represented by the centroids as well as the points from which they are created. Long-range structure is then quantified in two ways: by the spectral density, and by a real-space analog where the variance of density fluctuations for a set of measuring windows of diameter D is made more intuitive by conversion to the distance h(D) from the window boundary where these fluctuations effectively occur. The unweighted bubble centroids have h(D) that collapses for the different ages of the foam with random Poissonian fluctuations at long distances. The area-weighted bubble centroids and area-weighted Voronoi points all have constant h(D)=h_{e} for large D; the bubble centroids have the smallest value h_{e}=0.084sqrt[〈a〉], meaning they are the most uniform. Area-weighted Voronoi centroids exhibit collapse of h(D) to the same constant h_{e}=0.084sqrt[〈a〉] as for the bubble centroids. A similar analysis is performed on the edges of the cells and the spectra of h(D) for the foam edges show h(D)∼D^{1-ε} where ε=0.30±0.15.

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.

Light scattering from colloidal aggregates on a hierarchy of length scales

Disordered dielectrics with structural correlations on length scales comparable to visible light wavelengths exhibit interesting optical properties. Such materials exist in nature, leading to beautiful structural non-iridescent color, and they are also increasingly used as building blocks for optical materials and coatings. In this article, we explore the angular resolved single-scattering properties of micron-sized, disordered colloidal assemblies. The aggregates act as structurally colored supraparticles or as building blocks for macroscopic photonic glasses. We obtain first experimental data for the differential scattering and transport cross-section. Based on existing macroscopic models, we develop a theoretical framework to describe the scattering from densely packed colloidal assemblies on a hierarchy of length scales.

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.

Non-Gaussian tail in the force distribution: a hallmark of correlated disorder in the host media of elastic objects

Inferring the nature of disorder in the media where elastic objects are nucleated is of crucial importance for many applications but remains a challenging basic-science problem. Here we propose a method to discern whether weak-point or strong-correlated disorder dominates based on characterizing the distribution of the interaction forces between objects mapped in large fields-of-view. We illustrate our proposal with the case-study system of vortex structures nucleated in type-II superconductors with different pinning landscapes. Interaction force distributions are computed from individual vortex positions imaged in thousands-vortices fields-of-view in a two-orders-of-magnitude-wide vortex-density range. Vortex structures nucleated in point-disordered media present Gaussian distributions of the interaction force components. In contrast, if the media have dilute and randomly-distributed correlated disorder, these distributions present non-Gaussian algebraically-decaying tails for large force magnitudes. We propose that detecting this deviation from the Gaussian behavior is a fingerprint of strong disorder, in our case originated from a dilute distribution of correlated pinning centers.

Impact of particle size and multiple scattering on the propagation of waves in stealthy-hyperuniform media

Propagation of waves in materials that exhibit stealthy-hyperuniform long-range correlations is investigated. By using a modal decomposition of the field that takes multiple scattering into account at all orders, we study the impact of the concentration of particles on the transparency of such materials at low frequency. An upper frequency limit for transparency is defined that include both the particle size and the degree of stealthiness. We show that the independent scattering approximation is not relevant to calculate elastic mean free paths when wavelength becomes comparable to the size of particles. We find that transparency is very robust with regard to the degree of heterogeneity of the host random medium and the polydispersity of particles. Finally, it is shown that resonances can be used as the frequency filter.

Disordered hyperuniformity in two-dimensional amorphous silica

Disordered hyperuniformity (DHU) is a recently proposed new state of matter, which has been observed in a variety of classical and quantum many-body systems. DHU systems are characterized by vanishing infinite-wavelength normalized density fluctuations and are endowed with unique novel physical properties. Here, we report the discovery of disordered hyperuniformity in atomic-scale two-dimensional materials, i.e., amorphous silica composed of a single layer of atoms, based on spectral-density analysis of high-resolution transmission electron microscopy images. Moreover, we show via large-scale density functional theory calculations that DHU leads to almost complete closure of the electronic bandgap compared to the crystalline counterpart, making the material effectively a metal. This is in contrast to the conventional wisdom that disorder generally diminishes electronic transport and is due to the unique electron wave localization induced by the topological defects in the DHU state.

Hyperuniform states generated by a critical friction field

Hyperuniform states are an efficient way to fill up space for disordered systems. In these states the particle distribution is disordered at the short scale but becomes increasingly uniform when looked at large scales. Hyperuniformity appears in several systems, in static or quasistatic regimes, as well as close to transitions to absorbing states. Here, we show that a vibrated granular layer, at the critical point of the liquid-to-solid transition, displays dynamic hyperuniformity. Prior to the transition, patches of the solid phase form, with length scales and mean lifetimes that diverge critically at the transition point. When reducing the wave number, density fluctuations encounter increasingly more patches that block their propagation, resulting in a static structure factor that tends to zero for small wave numbers at the critical point, which is a signature of hyperuniformity. A simple model demonstrates that this coupling of a density field to a highly fluctuating scalar friction field gives rise to dynamic hyperuniform states. Finally, we show that the structure factor detects better the emergence of hyperuniformity, compared to the particle number variance.

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.

Foam as a self-assembling amorphous photonic band gap material

We show that slightly polydisperse disordered 2D foams can be used as a self-assembled template for isotropic photonic band gap (PBG) materials for transverse electric (TE) polarization. Calculations based on in-house experimental and simulated foam structures demonstrate that, at sufficient refractive index contrast, a dry foam organization with threefold nodes and long slender Plateau borders is especially advantageous to open a large PBG. A transition from dry to wet foam structure rapidly closes the PBG mainly by formation of bigger fourfold nodes, filling the PBG with defect modes. By tuning the foam area fraction, we find an optimal quantity of dielectric material, which maximizes the PBG in experimental systems. The obtained results have a potential to be extended to 3D foams to produce a next generation of self-assembled disordered PBG materials, enabling fabrication of cheap and scalable photonic devices.

Spherical structure factor and classification of hyperuniformity on the sphere

Understanding how particles are arranged on the surface of a sphere is not only central to numerous physical, biological, soft matter, and materials systems but also finds applications in computational problems, approximation theory, and analysis of geophysical and meteorological measurements. Objects that lie on a sphere experience constraints that are not present in Euclidean (flat) space and that influence both how the particles can be arranged as well as their statistical properties. These constraints, coupled with the curved geometry, require a careful extension of quantities used for the analysis of particle distributions in Euclidean space to distributions confined to the surface of a sphere. Here, we introduce a framework designed to analyze and classify structural order and disorder in particle distributions constrained to the sphere. The classification is based on the concept of hyperuniformity, which was first introduced 15 years ago and since then studied extensively in Euclidean space, yet has only very recently been considered also for spherical surfaces. We employ a generalization of the structure factor on the sphere, related to the power spectrum of the corresponding multipole expansion of particle density distribution. The spherical structure factor is then shown to couple with cap number variance, a measure of density variations at different scales, allowing us to analytically derive different forms of the variance pertaining to different types of distributions. Based on these forms, we construct a classification of hyperuniformity for scale-free particle distributions on the sphere and show how it can be extended to include other distribution types as well. We demonstrate that hyperuniformity on the sphere can be defined either through a vanishing spherical structure factor at low multipole numbers or through a scaling of the cap number variance-in both cases extending the Euclidean definition, while at the same time pointing out crucial differences. Our work thus provides a comprehensive tool for detecting global, long-range order on spheres and for the analysis of spherical computational meshes, biological and synthetic spherical assemblies, and ordering phase transitions in spherically distributed particles.

Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings

This work considers the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long-wavelength fluctuations in a broad class of one-dimensional substitution tilings. A simple argument is presented which predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and numerical computations confirm that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra and limit-periodic tilings. Quasiperiodic or singular continuous cases can be constructed with α arbitrarily close to any given value between -1 and 3. Limit-periodic tilings can be constructed with α between -1 and 1 or with Fourier intensities that approach zero faster than any power law.

Local structure in deeply supercooled liquids exhibits growing lengthscales and dynamical correlations

Glasses are among the most widely used of everyday materials, yet the process by which a liquid's viscosity increases by 14 decades to become a glass remains unclear, as often contradictory theories provide equally good descriptions of the available data. Knowledge of emergent lengthscales and higher-order structure could help resolve this, but this requires time-resolved measurements of dense particle coordinates-previously only obtained over a limited time interval. Here we present an experimental study of a model colloidal system over a dynamic window significantly larger than previous measurements, revealing structural ordering more strongly linked to dynamics than previously found. Furthermore we find that immobile regions and domains of local structure grow concurrently with density, and that these regions have low configurational entropy. We thus show that local structure plays an important role at deep supercooling, consistent with a thermodynamic interpretation of the glass transition rather than a principally dynamic description.

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.

Rational design of stealthy hyperuniform two-phase media with tunable order

Disordered stealthy hyperuniform materials are exotic amorphous states of matter that have attracted recent attention because of their novel structural characteristics (hidden order at large length scales) and physical properties, including desirable photonic and transport properties. It is therefore useful to devise algorithms that enable one to design a wide class of such amorphous configurations at will. In this paper, we present several algorithms enabling the systematic identification and generation of discrete (digitized) stealthy hyperuniform patterns with a tunable degree of order, paving the way towards the rational design of disordered materials endowed with novel thermodynamic and physical properties. To quantify the degree of order or disorder of the stealthy systems, we utilize the discrete version of the τ order metric, which accounts for the underlying spatial correlations that exist across all relevant length scales in a given digitized two-phase (or, equivalently, a two-spin state) system of interest. Our results impinge on a myriad of fields, ranging from physics, materials science and engineering, visual perception, and information theory to modern data science.

Binary mixtures of charged colloids: a potential route to synthesize disordered hyperuniform materials

A new route to fabricate large samples of 2D disordered hyperuniform materials via self-assembly of mixtures of charged colloids.

Optimizing Hyperuniformity in Self-Assembled Bidisperse Emulsions

We study long range density fluctuations (hyperuniformity) in two-dimensional jammed packings of bidisperse droplets. Taking advantage of microfluidics, we systematically span a large range of size and concentration ratios of the two droplet populations. We identify various defects increasing long range density fluctuations mainly due to organization of local particle environment. By choosing an appropriate bidispersity, we fabricate materials with a high level of hyperuniformity. Interesting transparency properties of these optimized materials are established based on numerical simulations.

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.

Characterizing pixel and point patterns with a hyperuniformity disorder length

We introduce the concept of a "hyperuniformity disorder length" h that controls the variance of volume fraction fluctuations for randomly placed windows of fixed size. In particular, fluctuations are determined by the average number of particles within a distance h from the boundary of the window. We first compute special expectations and bounds in d dimensions, and then illustrate the range of behavior of h versus window size L by analyzing several different types of simulated two-dimensional pixel patterns-where particle positions are stored as a binary digital image in which pixels have value zero if empty and one if they contain a particle. The first are random binomial patterns, where pixels are randomly flipped from zero to one with probability equal to area fraction. These have long-ranged density fluctuations, and simulations confirm the exact result h=L/2. Next we consider vacancy patterns, where a fraction f of particles on a lattice are randomly removed. These also display long-range density fluctuations, but with h=(L/2)(f/d) for small f, and h=L/2 for f→1. And finally, for a hyperuniform system with no long-range density fluctuations, we consider "Einstein patterns," where each particle is independently displaced from a lattice site by a Gaussian-distributed amount. For these, at large L,h approaches a constant equal to about half the root-mean-square displacement in each dimension. Then we turn to gray-scale pixel patterns that represent simulated arrangements of polydisperse particles, where the volume of a particle is encoded in the value of its central pixel. And we discuss the continuum limit of point patterns, where pixel size vanishes. In general, we thus propose to quantify particle configurations not just by the scaling of the density fluctuation spectrum but rather by the real-space spectrum of h(L) versus L. We call this approach "hyperuniformity disorder length spectroscopy".

Hyperuniformity disorder length spectroscopy for extended particles

The concept of a hyperuniformity disorder length h was recently introduced for analyzing volume fraction fluctuations for a set of measuring windows [Chieco et al., Phys. Rev. E 96, 032909 (2017).PLEEE81539-375510.1103/PhysRevE.96.032909]. This length permits a direct connection to the nature of disorder in the spatial configuration of the particles and provides a way to diagnose the degree of hyperuniformity in terms of the scaling of h and its value in comparison with established bounds. Here, this approach is generalized for extended particles, which are larger than the image resolution and can lie partially inside and partially outside the measuring windows. The starting point is an expression for the relative volume fraction variance in terms of four distinct volumes: that of the particle, the measuring window, the mean-squared overlap between particle and region, and the region over which particles have nonzero overlap with the measuring window. After establishing limiting behaviors for the relative variance, computational methods are developed for both continuum and pixelated particles. Exact results are presented for particles of special shape and for measuring windows of special shape, for which the equations are tractable. Comparison is made for other particle shapes, using simulated Poisson patterns. And the effects of polydispersity and image errors are discussed. For small measuring windows, both particle shape and spatial arrangement affect the form of the variance. For large regions, the variance scaling depends only on arrangement but particle shape sets the numerical proportionality. The combined understanding permit the measured variance to be translated to the spectrum of hyperuniformity lengths versus region size, as the quantifier of spatial arrangement. This program is demonstrated for a system of nonoverlapping particles at a series of increasing packing fractions as well as for an Einstein pattern of particles with several different extended shapes.

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.

Large-scale structure of randomly jammed spheres

We numerically analyze the density field of three-dimensional randomly jammed packings of monodisperse soft frictionless spherical particles, paying special attention to fluctuations occurring at large length scales. We study in detail the two-point static structure factor at low wave vectors in Fourier space. We also analyze the nature of the density field in real space by studying the large-distance behavior of the two-point pair correlation function, of density fluctuations in subsystems of increasing sizes, and of the direct correlation function. We show that such real space analysis can be greatly improved by introducing a coarse-grained density field to disentangle genuine large-scale correlations from purely local effects. Our results confirm that both Fourier and real space signatures of vanishing density fluctuations at large scale are absent, indicating that randomly jammed packings are not hyperuniform. In addition, we establish that the pair correlation function displays a surprisingly complex structure at large distances, which is however not compatible with the long-range negative correlation of hyperuniform systems but fully compatible with an analytic form for the structure factor. This implies that the direct correlation function is short ranged, as we also demonstrate directly. Our results reveal that density fluctuations in jammed packings do not follow the behavior expected for random hyperuniform materials, but display instead a more complex behavior.

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.

Controlled Synthesis and Fluorescence Tracking of Highly Uniform Poly(N-isopropylacrylamide) Microgels

Stimuli-sensitive poly(N-isopropylacrylamide) (PNIPAM) microgels have various prospective practical applications and uses in fundamental research. In this work, we use single particle tracking of fluorescently labeled PNIPAM microgels as a showcase for tuning microgel size by a rapid non-stirred precipitation polymerization procedure. This approach is well suited for prototyping new reaction compositions and conditions or for applications that do not require large amounts of product. Microgel synthesis, particle size and structure determination by dynamic and static light scattering are detailed in the protocol. It is shown that the addition of functional comonomers can have a large influence on the particle nucleation and structure. Single particle tracking by wide-field fluorescence microscopy allows for an investigation of the diffusion of labeled tracer microgels in a concentrated matrix of non-labeled microgels, a system not easily investigated by other methods such as dynamic light scattering.

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."

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.

Critical slowing down and hyperuniformity on approach to jamming

Hyperuniformity characterizes a state of matter that is poised at a critical point at which density or volume-fraction fluctuations are anomalously suppressed at infinite wavelengths. Recently, much attention has been given to the link between strict jamming (mechanical rigidity) and (effective or exact) hyperuniformity in frictionless hard-particle packings. However, in doing so, one must necessarily study very large packings in order to access the long-ranged behavior and to ensure that the packings are truly jammed. We modify the rigorous linear programming method of Donev et al. [J. Comput. Phys. 197, 139 (2004)JCTPAH0021-999110.1016/j.jcp.2003.11.022] in order to test for jamming in putatively collectively and strictly jammed packings of hard disks in two dimensions. We show that this rigorous jamming test is superior to standard ways to ascertain jamming, including the so-called "pressure-leak" test. We find that various standard packing protocols struggle to reliably create packings that are jammed for even modest system sizes of N≈10^{3} bidisperse disks in two dimensions; importantly, these packings have a high reduced pressure that persists over extended amounts of time, meaning that they appear to be jammed by conventional tests, though rigorous jamming tests reveal that they are not. We present evidence that suggests that deviations from hyperuniformity in putative maximally random jammed (MRJ) packings can in part be explained by a shortcoming of the numerical protocols to generate exactly jammed configurations as a result of a type of "critical slowing down" as the packing's collective rearrangements in configuration space become locally confined by high-dimensional "bottlenecks" from which escape is a rare event. Additionally, various protocols are able to produce packings exhibiting hyperuniformity to different extents, but this is because certain protocols are better able to approach exactly jammed configurations. Nonetheless, while one should not generally expect exact hyperuniformity for disordered packings with rattlers, we find that when jamming is ensured, our packings are very nearly hyperuniform, and deviations from hyperuniformity correlate with an inability to ensure jamming, suggesting that strict jamming and hyperuniformity are indeed linked. This raises the possibility that the ideal MRJ packings have no rattlers. Our work provides the impetus for the development of packing algorithms that produce large disordered strictly jammed packings that are rattler free, which is an outstanding, challenging task.

Phyllotaxis: a framework for foam topological evolution

Phyllotaxis describes the arrangement of florets, scales or leaves in composite flowers or plants (daisy, aster, sunflower, pinecone, pineapple). As a structure, it is a geometrical foam, the most homogeneous and densest covering of a large disk by Voronoi cells (the florets), constructed by a simple algorithm: Points placed regularly on a generative spiral constitute a spiral lattice, and phyllotaxis is the tiling by the Voronoi cells of the spiral lattice. Locally, neighboring cells are organized as three whorls or parastichies, labelled with successive Fibonacci numbers. The structure is encoded as the sequence of the shapes (number of sides) of the successive Voronoi cells on the generative spiral. We show that sequence and organization are independent of the position of the initial point on the generative spiral, that is invariant under disappearance (T2 of the first Voronoi cell or, conversely, under creation of a first cell, that is under growth. This independence shows how a foam is able to respond to a shear stress, notably through grain boundaries that are layers of square cells slightly truncated into heptagons, pentagons and hexagons, meeting at four-corner vertices, critical points of T1 elementary topological transformations.

Confined disordered strictly jammed binary sphere packings

Disordered jammed packings under confinement have received considerably less attention than their bulk counterparts and yet arise in a variety of practical situations. In this work, we study binary sphere packings that are confined between two parallel hard planes and generalize the Torquato-Jiao (TJ) sequential linear programming algorithm [Phys. Rev. E 82, 061302 (2010)] to obtain putative maximally random jammed (MRJ) packings that are exactly isostatic with high fidelity over a large range of plane separation distances H, small to large sphere radius ratio α, and small sphere relative concentration x. We find that packing characteristics can be substantially different from their bulk analogs, which is due to what we term "confinement frustration." Rattlers in confined packings are generally more prevalent than those in their bulk counterparts. We observe that packing fraction, rattler fraction, and degree of disorder of MRJ packings generally increase with H, though exceptions exist. Discontinuities in the packing characteristics as H varies in the vicinity of certain values of H are due to associated discontinuous transitions between different jammed states. When the plane separation distance is on the order of two large-sphere diameters or less, the packings exhibit salient two-dimensional features; when the plane separation distance exceeds about 30 large-sphere diameters, the packings approach three-dimensional bulk packings. As the size contrast increases (as α decreases), the rattler fraction dramatically increases due to what we call "size-disparity" frustration. We find that at intermediate α and when x is about 0.5 (50-50 mixture), the disorder of packings is maximized, as measured by an order metric ψ that is based on the number density fluctuations in the direction perpendicular to the hard walls. We also apply the local volume-fraction variance σ(τ)(2)(R) to characterize confined packings and find that these packings possess essentially the same level of hyperuniformity as their bulk counterparts. Our findings are generally relevant to confined packings that arise in biology (e.g., structural color in birds and insects) and may have implications for the creation of high-density powders and improved battery designs.

A Geometric-Structure Theory for Maximally Random Jammed Packings

Maximally random jammed (MRJ) particle packings can be viewed as prototypical glasses in that they are maximally disordered while simultaneously being mechanically rigid. The prediction of the MRJ packing density ϕMRJ, among other packing properties of frictionless particles, still poses many theoretical challenges, even for congruent spheres or disks. Using the geometric-structure approach, we derive for the first time a highly accurate formula for MRJ densities for a very wide class of two-dimensional frictionless packings, namely, binary convex superdisks, with shapes that continuously interpolate between circles and squares. By incorporating specific attributes of MRJ states and a novel organizing principle, our formula yields predictions of ϕMRJ that are in excellent agreement with corresponding computer-simulation estimates in almost the entire α-x plane with semi-axis ratio α and small-particle relative number concentration x. Importantly, in the monodisperse circle limit, the predicted ϕMRJ = 0.834 agrees very well with the very recently numerically discovered MRJ density of 0.827, which distinguishes it from high-density "random-close packing" polycrystalline states and hence provides a stringent test on the theory. Similarly, for non-circular monodisperse superdisks, we predict MRJ states with densities that are appreciably smaller than is conventionally thought to be achievable by standard packing protocols.

Search for hyperuniformity in mechanically stable packings of frictionless disks above jamming

We numerically simulate mechanically stable packings of soft-core, frictionless, bidisperse disks in two dimensions, above the jamming packing fraction ϕ(J). For configurations with a fixed isotropic global stress tensor, we investigate the fluctuations of the local packing fraction ϕ(r) to test whether such configurations display the hyperuniformity that has been claimed to exist exactly at ϕ(J). For our configurations, generated by a rapid quench protocol, we find that hyperuniformity persists only out to a finite length scale and that this length scale appears to remain finite as the system stress decreases towards zero, i.e., towards the jamming transition. Our result suggests that the presence of hyperuniformity at jamming may be sensitive to the specific protocol used to construct the jammed configurations.

Nanoscale engineering of two-dimensional disordered hyperuniform block-copolymer assemblies

Disordered hyperuniform (DH) media have been recognized as a new state of disordered matter that broadens our vision of material engineering. Here, long-range correlated disordered two-dimensional patterns are fabricated by self-assembling of spherical diblock-copolymer (BCP) micelles. Control of the self-assembling parameters leads to the formation of DH patterns of micelles that can host nanoscale material inclusions, therefore providing an effective strategy for fabricating multimaterial DH structures at molecular scale. Centroidal patterns are accurately determined by virtue of BCP micelles loaded with metal nanoparticles. Our analysis reveals the signature of nearly ideal DH BCP assemblies in the local density fluctuation and a dominant linear scaling in the local number fluctuation.

Emergent Hyperuniformity in Periodically Driven Emulsions

We report the self-organization of microfluidic emulsions into anomalously homogeneous structures. Upon periodic driving confined emulsions undergo a first-order transition from a reversible to an irreversible dynamics. We evidence that this dynamical transition is accompanied by structural changes at all scales yielding macroscopic yet finite hyperuniform structures. Numerical simulations are performed to single out the very ingredients responsible for the suppression of density fluctuations. We show that, as opposed to equilibrium systems, the long-range nature of the hydrodynamic interactions are not required for the formation of hyperuniform patterns, thereby suggesting a robust relation between reversibility and hyperuniformity which should hold in a broad class of periodically driven materials.