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

Hyperuniform disordered solids with crystal-like stability

Hyperuniform disordered solids, characterised by unusually suppressed density fluctuations at low wavenumbers (q), are of great interest due to their potentially distinct properties as a unique glass state. From the jamming perspective, there is ongoing debate about the relationship between hyperuniformity and the jamming transition, as well as whether hyperuniformity persists above the jamming point. Here, we successfully generate over-jammed disordered solids exhibiting the strongest class of hyperuniformity, characterised by a power-law density spectrum (qα with α = 4). By decompressing both hyperuniform and conventional over-jammed packings to their respective marginally jammed states, we identify protocol-independent exponents: α ≈ 0.25 for density hyperuniformity and α ≈ 2 for contact-number hyperuniformity, both associated with the jamming transition. Although both marginally jammed and conventional over-jammed packings exhibit marginal stability, we demonstrate that hyperuniform over-jammed packings possess exceptional stability across vibrational, kinetic, thermodynamic, and mechanical properties-similar to crystals. These findings suggest that hyperuniform over-jammed packings offer crucial insights into the ideal disordered solid state and stand out as promising candidates for disordered metamaterials, uniquely combining hyperuniformity with ultrastability.

Hyperuniform jammed sphere packings have anomalous material properties

A spatial distribution is hyperuniform if it has local density fluctuations that vanish in the limit of long length scales. Hyperuniformity is a well known property of both crystals and quasicrystals. Of recent interest, however, is disordered hyperuniformity: the presence of hyperuniform scaling without long-range configurational order. Jammed granular packings have been proposed as an example of disordered hyperuniformity, but recent numerical investigation has revealed that many jammed systems instead exhibit a complex set of distinct behaviors at long, emergent length scales. We use the Voronoi tessellation as a tool to define a set of rescaling transformations that can impose hyperuniformity on an arbitrary weighted point process, and show that these transformations can be used in simulations to iteratively generate hyperuniform, mechanically stable packings of athermal soft spheres. These hyperuniform jammed packings display atypical mechanical properties, particularly in the low-frequency phononic excitations, which exhibit an isolated band of highly collective modes and a band gap around zero frequency.

Universal non-Debye low-frequency vibrations in sheared amorphous solids

We study energy minimised configurations of amorphous solids with a simple shear degree of freedom. We show that the low-frequency regime of the vibrational density of states of structural glass formers is crucially sensitive to the macroscopic stress of the sampled configurations. In both two and three dimensions, shear-stabilised configurations display a D(ω_min) ∼ ω5min regime, as opposed to the ω4min regime observed under unstrained conditions. In order to isolate the source of these deviations from crystalline behaviour, we also study configurations of two dimensional, strained amorphous solids close to a plastic event. We show that the minimum eigenvalue distribution at a strain 'γ' near the plastic event occurring at 'γ_P' assumes a universal form that displays a collapse when scaled by , and with the number of particles as N^-0.22. Notably, at low frequencies, this scaled distribution displays a robust D(ω_min) ∼ ω6min power-law regime, which survives in the large N limit. Finally, we probe the properties of these configurations through a characterisation of the second and third eigenvalues of the Hessian matrix near a plastic event.

Relaxation times, rheology, and finite size effects for non-Brownian disks in two dimensions

We carry out overdamped simulations in a simple model of jamming-a collection of bidisperse soft core frictionless disks in two dimensions-with the aim to explore the finite size dependence of different quantities, both the relaxation time obtained from the relaxation of the energy and the pressure equivalent of the shear viscosity. The motivation for the paper is the observation [Nishikawa et al., J. Stat. Phys. 182, 37 (2021)0022-471510.1007/s10955-021-02710-8] that there are finite size effects in the relaxation time, τ, that give problems in the determination of the critical divergence, and the claim that this is due to a finite size dependence, τ∼lnN, which makes τ an ill-defined quantity. Beside analyses to determine the relaxation time for the whole system we determine particle relaxation times which allow us to determine both histograms of particle relaxation times and the average particle relaxation times-two quantities that are very useful for the analyses. The starting configurations for the relaxation simulations are of two different kinds-completely random or taken from steady shearing simulations-and we find that the difference between these two cases are bigger than previously noted and that the observed problems in the determination of the critical divergence obtained when starting from random configurations are not present when instead starting the relaxations from shearing configurations. We also argue that the the effect that causes the lnN dependence is not as problematic as asserted. When it comes to the finite size dependence of the pressure equivalent of the shear viscosity we find that our data don't give support for the claimed strong finite size dependence, but also that the finite size dependence is at odds with what one would normally expect for a system with a diverging correlation length, and that this calls for an alternative understanding of the phenomenon of shear-driven jamming.

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.

Non-hyperuniform metastable states around a disordered hyperuniform state of densely packed spheres: stochastic density functional theory at strong coupling

The disordered and hyperuniform structures of densely packed spheres near and at jamming are characterized by vanishing of long-wavelength density fluctuations, or equivalently by long-range power-law decay of the direct correlation function (DCF). We focus on previous simulation results that exhibit the degradation of hyperuniformity in jammed structures while maintaining the long-range nature of the DCF to a certain length scale. Here we demonstrate that the field-theoretic formulation of stochastic density functional theory is relevant to explore the degradation mechanism. The strong-coupling expansion method of stochastic density functional theory is developed to obtain the metastable chemical potential considering the intermittent fluctuations in dense packings. The metastable chemical potential yields the analytical form of the metastable DCF that has a short-range cutoff inside the sphere while retaining the long-range power-law behavior. It is confirmed that the metastable DCF provides the zero-wavevector limit of the structure factor in quantitative agreement with the previous simulation results of degraded hyperuniformity. We can also predict the emergence of soft modes localized at the particle scale by plugging this metastable DCF into the linearized Dean-Kawasaki equation, a stochastic density functional equation.

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.

Hidden Order Beyond Hyperuniformity in Critical Absorbing States

Disordered hyperuniformity is a description of hidden correlations in point distributions revealed by an anomalous suppression in fluctuations of local density at various coarse-graining length scales. In the absorbing phase of models exhibiting an active-absorbing state transition, this suppression extends up to a hyperuniform length scale that diverges at the critical point. Here, we demonstrate the existence of additional many-body correlations beyond hyperuniformity. These correlations are hidden in the higher moments of the probability distribution of the local density and extend up to a longer length scale with a faster divergence than the hyperuniform length on approaching the critical point. Our results suggest that a hidden order beyond hyperuniformity may generically be present in complex disordered systems.

Hyperuniformity and density fluctuations at a rigidity transition in a model of biological tissues

The suppression of density fluctuations at different length scales is the hallmark of hyperuniformity. Here, we explore the presence of this hidden order in a manybody interacting model of biological tissue, known to exhibit a transition, or sharp crossover, from a solid to a fluid like phase. We show that the density fluctuations in the rigid phase are only suppressed up to a finite lengthscale. This length scale monotonically increases and grows rapidly as we approach the fluid phase reminiscent to divergent behavior at a critical point, such that the system is effectively hyperuniform in the fluid phase. Furthermore, complementary behavior of the structure factor across the critical point also indicates that hyperuniformity found in the fluid phase is stealthy.

Absorbing-State Transitions in Granular Materials Close to Jamming

We consider a model for driven particulate matter in which absorbing states can be reached both by particle isolation and by particle caging. The model predicts a nonequilibrium phase diagram in which analogs of hydrodynamic and elastic reversibility emerge at low and high volume fractions respectively, partially separated by a diffusive, nonabsorbing region. We thus find a single phase boundary that spans the onset of chaos in sheared suspensions to the onset of yielding in jammed packings. This boundary has the properties of a nonequilibrium second order phase transition, leading us to write a Manna-like mean field description that captures the model predictions. Dependent on contact details, jamming marks either a direct transition between the two absorbing states, or occurs within the diffusive region.

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.

The physics of jamming for granular materials: a review

Granular materials consist of macroscopic grains, interacting via contact forces, and unaffected by thermal fluctuations. They are one of a class systems that undergo jamming, i.e. a transition between fluid-like and disordered solid-like states. Roughly twenty years ago, proposals by Cates et al for the shear response of colloidal systems and by Liu and Nagel, for a universal jamming diagram in a parameter space of packing fraction, ϕ, shear stress, τ, and temperature, T raised key questions. Contemporaneously, experiments by Howell et al and numerical simulations by Radjai et al and by Luding et al helped provide a starting point to explore key insights into jamming for dry, cohesionless, granular materials. A recent experimental observation by Bi et al is that frictional granular materials have a a re-entrant region in their jamming diagram. In a range of ϕ, applying shear strain, γ, from an initially force/stress free state leads to fragile (in the sense of Cates et al), then anisotropic shear jammed states. Shear jamming at fixed ϕ is presumably conjugate to Reynolds dilatancy, involving dilation under shear against deformable boundaries. Numerical studies by Radjai and Roux showed that Reynolds dilatancy does not occur for frictionless systems. Recent numerical studies by several groups show that shear jamming occurs for finite, but not infinite, systems of frictionless grains. Shear jamming does not lead to known ordering in position space, but Sarkar et al showed that ordering occurs in a space of force tiles. Experimental studies seeking to understand random loose and random close packings (rlp and rcp) and dating back to Bernal have probed granular packings and their response to shear and intruder motion. These studies suggest that rlp's are anisotropic and shear-jammed-like, whereas rcp's are likely isotropically jammed states. Jammed states are inherently static, but the jamming diagram may provide a context for understanding rheology, i.e. dynamic shear in a variety of systems that include granular materials and suspensions.

Two Diverging Length Scales in the Structure of Jammed Packings

At densities higher than the jamming transition for athermal, frictionless repulsive spheres we find two distinct length scales, both of which diverge as a power law as the transition is approached. The first, ξ_{Z}, is associated with the two-point correlation function for the number of contacts on two particles as a function of the particle separation. The second, ξ_{f}, is associated with contact-number fluctuations in subsystems of different sizes. On scales below ξ_{f}, the fluctuations are highly suppressed, similar to the phenomenon of hyperuniformity usually associated with density fluctuations. The exponents for the divergence of ξ_{Z} and ξ_{f} are different and appear to be different in two and three dimensions.

Local order and crystallization of dense polydisperse hard spheres

Computer simulations give precious insight into the microscopic behavior of supercooled liquids and glasses, but their typical time scales are orders of magnitude shorter than the experimentally relevant ones. We recently closed this gap for a class of models of size polydisperse fluids, which we successfully equilibrate beyond laboratory time scales by means of the swap Monte Carlo algorithm. In this contribution, we study the interplay between compositional and geometric local orders in a model of polydisperse hard spheres equilibrated with this algorithm. Local compositional order has a weak state dependence, while local geometric order associated to icosahedral arrangements grows more markedly but only at very high density. We quantify the correlation lengths and the degree of sphericity associated to icosahedral structures and compare these results to those for the Wahnström Lennard-Jones mixture. Finally, we analyze the structure of very dense samples that partially crystallized following a pattern incompatible with conventional fractionation scenarios. The crystal structure has the symmetry of aluminum diboride and involves a subset of small and large particles with size ratio approximately equal to 0.5.

Response of jammed packings to thermal fluctuations

We focus on the response of mechanically stable (MS) packings of frictionless, bidisperse disks to thermal fluctuations, with the aim of quantifying how nonlinearities affect system properties at finite temperature. In contrast, numerous prior studies characterized the structural and mechanical properties of MS packings of frictionless spherical particles at zero temperature. Packings of disks with purely repulsive contact interactions possess two main types of nonlinearities, one from the form of the interaction potential (e.g., either linear or Hertzian spring interactions) and one from the breaking (or forming) of interparticle contacts. To identify the temperature regime at which the contact-breaking nonlinearities begin to contribute, we first calculated the minimum temperatures T_{cb} required to break a single contact in the MS packing for both single- and multiple-eigenmode perturbations of the T=0 MS packing. We find that the temperature required to break a single contact for equal velocity-amplitude perturbations involving all eigenmodes approaches the minimum value obtained for a perturbation in the direction connecting disk pairs with the smallest overlap. We then studied deviations in the constant volume specific heat C[over ¯]_{V} and deviations of the average disk positions Δr from their T=0 values in the temperature regime T_{C[over ¯]_{V}}<T<T_{r}, where T_{r} is the temperature beyond which the system samples the basin of a new MS packing. We find that the deviation in the specific heat per particle ΔC[over ¯]_{V}^{0}/C[over ¯]_{V}^{0} relative to the zero-temperature value C[over ¯]_{V}^{0} can grow rapidly above T_{cb}; however, the deviation ΔC[over ¯]_{V}^{0}/C[over ¯]_{V}^{0} decreases as N^{-1} with increasing system size. To characterize the relative strength of contact-breaking versus form nonlinearities, we measured the ratio of the average position deviations Δr^{ss}/Δr^{ds} for single- and double-sided linear and nonlinear spring interactions. We find that Δr^{ss}/Δr^{ds}>100 for linear spring interactions is independent of system size. This result emphasizes that contact-breaking nonlinearities are dominant over form nonlinearities in the low-temperature range T_{cb}<T<T_{r} for model jammed systems.

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.

Anomalous stress fluctuations in athermal two-dimensional amorphous solids

We numerically study the local stress distribution within athermal, isotropically stressed, mechanically stable, packings of bidisperse frictionless disks above the jamming transition in two dimensions. Considering the Fourier transform of the local stress, we find evidence for algebraically increasing fluctuations in both isotropic and anisotropic components of the stress tensor at small wave numbers, contrary to recent theoretical predictions. Such increasing fluctuations imply a lack of self-averaging of the stress on large length scales. The crossover to these increasing fluctuations defines a length scale ℓ_{0}, however, it appears that ℓ_{0} does not vary much with packing fraction ϕ, nor does ℓ_{0} seem to be diverging as ϕ approaches the jamming ϕ_{J}. We also find similar large length scale fluctuations of stress in the inherent states of a quenched Lennard-Jones liquid, leading us to speculate that such fluctuations may be a general property of amorphous solids in two dimensions.

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.

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.

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.