Effects of compression on the vibrational modes of marginally jammed solids

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.

Jamming transition and normal modes of polydispersed soft particle packing

The jamming transition of soft particles characterized by narrow size distributions has been well studied by physicists. However, polydispersed systems are more relevant to engineering, and the influence of polydispersity on jamming phenomena is still unexplored. Here, we numerically investigate jamming transitions of polydispersed soft particles in two dimensions. We find that polydispersity strongly influences contact forces, local coordination, and the jamming transition density. In contrast, the critical scaling of pressure and elastic moduli is not affected by the particle size distribution. Consistent with this observation, we find that the vibrational density of states is also insensitive to the polydispersity. Our results suggest that, regardless of particle size distributions, both mechanical and vibrational properties of soft particle packings near jamming are governed by the distance to jamming.

Tuning collective actuation of active solids by optimizing activity localization

Active solids, more specifically elastic lattices embedded with polar active units, exhibit collective actuation when the elasto-active feedback, generically present in such systems, exceeds some critical value. The dynamics then condensates on a small fraction of the vibrational modes, the selection of which obeys non trivial rules rooted in the nonlinear part of the dynamics. So far, the complexity of the selection mechanism has limited the design of specific actuation. Here, we investigate numerically how localizing activity to a fraction of modes enables the selection of non-trivial collective actuation. We perform numerical simulations of an agent-based model on triangular and disordered lattices and vary the concentration and the localization of the active agents on the lattice nodes. Both contribute to the distribution of the elastic energy across the modes. We then introduce an algorithm, which, for a given fraction of active nodes, evolves the localization of the activity in such a way that the energy distribution on a few targeted modes is maximized - or minimized. We illustrate on a specific targeted actuation, how the algorithm performs as compared to manually chosen localization of the activity. While, in the case of the ordered lattice, a well-educated guess performs better than the algorithm, and the latter outperform the manual trials in the case of the disordered lattice. Finally, the analysis of the results in the case of the ordered lattice leads us to introduce a design principle based on a measure of the susceptibility of the modes to be activated along certain activation paths.

The yielding of granular matter is marginally stable and critical

Amorphous materials undergo a transition from liquid-like to solid-like states through processes like rapid quenching or densification. Under external loads, they exhibit yielding, with minimal structural changes compared to crystals. However, these universal characteristics are rarely explored comprehensively in a single granular experiment due to the added complexity of inherent friction. The discernible differences between static configurations before and after yielding are largely unaddressed, and a comprehensive examination from both statistical physics and mechanical perspectives is lacking. To address these gaps, we conducted experiments using photoelastic disks, simultaneously tracking particles and measuring forces. Our findings reveal that the yielding transition demonstrates critical behavior from a statistical physics standpoint and marginal stability from a mechanical perspective, akin to the isotropic jamming transition. This criticality differs significantly from spinodal criticality in frictionless amorphous solids, highlighting unique characteristics of granular yielding. Furthermore, our analysis confirms the marginal stability of granular yielding by assessing the contact number and evaluating the balance between weak forces and small gaps. These factors serve as structural indicators for configurations before and after yielding. Our results not only contribute to advancing our understanding of the fundamental physics of granular materials but also bear significant implications for practical applications in various fields.

Experimental measurements of the granular density of modes via impact

The jamming transition is an important feature of granular materials, with prior work showing an excess of low-frequency modes in the granular analog to the density of states, the granular density of modes. In this work, we present an experimental method for acoustically measuring the granular density of modes using a single impact event to excite vibrational modes in an experimental, three-dimensional, granular material. We test three different granular materials, all of which are composed of spherical beads. The first two systems are monodisperse collections of either 6 mm or 8 mm diameter beads. The third system is a bidisperse mixture of the previous two bead sizes. During data collection, the particles are confined to a box; on top of this box, and resting on the granular material, is a light, rigid sheet onto which pressure can be applied to the system. To excite the material, a steel impactor ball is dropped on top of the system. The response of the granular material to the impact pulse is recorded by piezoelectric sensors buried throughout the material, and the density of modes is computed from the spectrum of the velocity autocorrelation of these sensors. Our measurements of the density of modes show more low-frequency modes at low pressure, consistent with previous experimental and numerical results, as well as several low-frequency peaks in the density of modes that shift with applied pressure. Our method represents an experimentally simple technique for investigating the granular density of modes and may increase the accessibility and number of such measurements.

An explicit granular-mechanics approach to marine sediment acoustics

Here, we theoretically and computationally study the frequency dependence of phase speed and attenuation for marine sediments from the perspective of granular mechanics. We leverage recent theoretical insights from the granular physics community as well as discrete-element method simulations, where the granular material is treated as a packing of discrete objects that interact via pairwise forces. These pairwise forces include both repulsive contact forces as well as dissipative terms, which may include losses from the fluid as well as losses from inelasticity at grain-grain contacts. We show that the structure of disordered granular packings leads to anomalous scaling laws for frequency-dependent phase speed and attenuation that do not follow from a continuum treatment. Our results demonstrate that granular packing structure, which is not explicitly considered in existing models, may play a crucial role in a complete theory of sediment acoustics. While this simple approach does not explicitly treat sound propagation or inertial effects in the interstitial fluid, it provides a starting point for future models that include these and other more complex features.

Effects of coordination and stiffness scale separation in disordered elastic networks

Many fibrous materials are modeled as elastic networks featuring a substantial separation between the stiffness scales that characterize different microscopic deformation modes of the network's constituents. This scale separation has been shown to give rise to emergent complexity in these systems' linear and nonlinear mechanical response. Here we study numerically a simple model featuring said stiffness scale separation in two-dimensions and show that its mechanical response is governed by the competition between the characteristic stiffness of collective nonphononic soft modes of the stiff subsystem, and the characteristic stiffness of the soft interactions. We present and rationalize the behavior of the shear modulus of our complex networks across the unjamming transition at which the stiff subsystem alone loses its macroscopic mechanical rigidity. We further establish a relation in the soft-interaction-dominated regime between the shear modulus, the characteristic frequency of nonphononic vibrational modes, and the mesoscopic correlation length that marks the crossover from a disorder-dominated response to local mechanical perturbations in the near field, to a linear, continuumlike response in the far field. The effects of spatial dimension on the observed scaling behavior are discussed, in addition to the interplay between stiffness scales in strain-stiffened networks, which is relevant to understanding the nonlinear mechanics of non-Brownian fibrous biomatter.

Elasticity of self-organized frustrated disordered spring networks

There have been some interesting recent advances in understanding the notion of mechanical disorder in structural glasses and the statistical mechanics of these systems' low-energy excitations. Here we contribute to these advances by studying a minimal model for structural glasses' elasticity in which the degree of mechanical disorder-as characterized by recently introduced dimensionless quantifiers-is readily tunable over a very large range. We comprehensively investigate a number of scaling laws observed for various macro, meso and microscopic elastic properties, and rationalize them using scaling arguments. Interestingly, we demonstrate that the model features the universal quartic glassy vibrational density of states as seen in many atomistic and molecular models of structural glasses formed by cooling a melt. The emergence of this universal glassy spectrum highlights the role of self-organization (toward mechanical equilibrium) in its formation, and elucidates why models featuring structural frustration alone do not feature the same universal glassy spectrum. Finally, we discuss relations to existing work in the context of strain stiffening of elastic networks and of low-energy excitations in structural glasses, in addition to future research directions.

Connecting polymer collapse and the onset of jamming

Previous studies have shown that the interiors of proteins are densely packed, reaching packing fractions that are as large as those found for static packings of individual amino-acid-shaped particles. How can the interiors of proteins take on such high packing fractions given that amino acids are connected by peptide bonds and many amino acids are hydrophobic with attractive interactions? We investigate this question by comparing the structural and mechanical properties of collapsed attractive disk-shaped bead-spring polymers to those of three reference systems: static packings of repulsive disks, of attractive disks, and of repulsive disk-shaped bead-spring polymers. We show that the attractive systems quenched to temperatures below the glass transition T≪T_{g} and static packings of both repulsive disks and bead-spring polymers possess similar interior packing fractions. Previous studies have shown that static packings of repulsive disks are isostatic at jamming onset, i.e., the number of interparticle contacts N_{c} matches the number of degrees of freedom, which strongly influences their mechanical properties. We find that repulsive polymer packings are hypostatic at jamming onset (i.e., with fewer contacts than degrees of freedom) but are effectively isostatic when including stabilizing quartic modes, which give rise to quartic scaling of the potential energy with displacements along these modes. While attractive disk and polymer packings are often considered hyperstatic with excess contacts over the isostatic number, we identify a definition for interparticle contacts for which they can also be considered as effectively isostatic. As a result, we show that the mechanical properties (e.g., scaling of the potential energy with excess contact number and low-frequency contribution to the density of vibrational modes) of weakly attractive disk and polymer packings are similar to those of isostatic repulsive disk and polymer packings. Our results demonstrate that static packings generated via attractive collapse or compression of repulsive particles possess similar structural and mechanical properties.

Jamming on convex deformable surfaces

Jamming is a fundamental transition that governs the behavior of particulate media, including sand, foams and dense suspensions. Upon compression, such media change from freely flowing to a disordered, marginally stable solid that exhibits non-Hookean elasticity. While the jamming process is well established for fixed geometries, the nature and dynamics of jamming for a diverse class of soft materials and deformable substrates, including emulsions and biological matter, remains unknown. Here we propose a new scenario, metric jamming, where rigidification occurs on a surface that has been deformed from its ground state. Unlike classical jamming processes that exhibit discrete mechanical transitions, surprisingly we find that metric jammed states possess mechanical properties continuously tunable between those of classically jammed and conventional elastic media. The compact and curved geometry significantly alters the vibrational spectra of the structures relative to jamming in flat Euclidean space, and metric jammed systems also possess new types of vibrational mode that couple particle and shape degrees of freedom. Our work provides a theoretical framework that unifies our understanding of solidification processes that take place on deformable media and lays the groundwork to exploit jamming for the control and stabilization of shape in self-assembly processes.

Microrheology near jamming

The jamming transition is a nonequilibrium critical phenomenon, which governs characteristic mechanical properties of jammed soft materials, such as pastes, emulsions, and granular matters. Both experiments and theory of jammed soft materials have revealed that the complex modulus measured by conventional macrorheology exhibits a characteristic frequency dependence. Microrheology is a new type of method to obtain the complex modulus, which transforms the microscopic motion of probes to the complex modulus through the generalized Stokes relation (GSR). Although microrheology has been applied to jammed soft materials, its theoretical understanding is limited. In particular, the validity of the GSR near the jamming transition is far from obvious since there is a diverging length scale lc, which characterizes the heterogeneous response of jammed particles. Here, we study the microrheology of jammed particles by theory and numerical simulation. First, we develop a linear response formalism to calculate the response function of the probe particle, which is transformed to the complex modulus via the GSR. Then, we apply our formalism to a numerical model of jammed particles and find that the storage and loss modulus follow characteristic scaling laws near the jamming transition. Importantly, the observed scaling law coincides with that in macrorheology, which indicates that the GSR holds even near the jamming transition. We rationalize this equivalence by asymptotic analysis of the obtained formalism and numerical analysis on the displacement field of jammed particles under a local perturbation.

Slow and fast particles in shear-driven jamming: Critical behavior

We do extensive simulations of a simple model of shear-driven jamming in two dimensions to determine and analyze the velocity distribution at different densities ϕ around the jamming density ϕ_{J} and at different low shear strain rates, γ[over ̇]. We then find that the velocity distribution is made up of two parts which are related to two different physical processes which we call the slow process and the fast process as they are dominated by the slower and the faster particles, respectively. Earlier scaling analyses have shown that the shear viscosity η, which diverges as the jamming density is approached from below, consists of two different terms, and we present strong evidence that these terms are related to the two different processes: the leading divergence is due to the fast process, whereas the correction-to-scaling term is due to the slow process. The analysis of the slow process is possible thanks to the observation that the velocity distribution for different γ[over ̇] and ϕ at and around the shear-driven jamming transition has a peak at low velocities and that the distribution has a constant shape up to and slightly above this peak. We then find that it is possible to express the contribution to the shear viscosity due to the slow process in terms of height and position of the peak in the velocity distribution and find that this contribution matches the correction-to-scaling term, determined through a standard critical scaling analysis. A further observation is that the collective particle motion is dominated by the slow process. In contrast to the usual picture in critical phenomena with a direct link between the diverging correlation length and a diverging order parameter, we find that correlations and shear viscosity decouple since they are controlled by different sets of particles and that shear-driven jamming is thus an unusual kind of critical phenomenon.

Boson-peak vibrational modes in glasses feature hybridized phononic and quasilocalized excitations

A hallmark of structural glasses and other disordered solids is the emergence of excess low-frequency vibrations on top of the Debye spectrum DDebye(ω) of phonons (ω denotes the vibrational frequency), which exist in any solid whose Hamiltonian is translationally invariant. These excess vibrations-a signature of which is a THz peak in the reduced density of states D(ω)/DDebye(ω), known as the boson peak-have resisted a complete theoretical understanding for decades. Here, we provide direct numerical evidence that vibrations near the boson peak consist of hybridizations of phonons with many quasilocalized excitations; the latter have recently been shown to generically populate the low-frequency tail of the vibrational spectra of structural glasses quenched from a melt and of disordered crystals. Our results suggest that quasilocalized excitations exist up to and in the vicinity of the boson-peak frequency and, hence, constitute the fundamental building blocks of the excess vibrational modes in glasses.

Comparison of compression versus shearing near jamming, for a simple model of athermal frictionless disks in suspension

Using a simplified model for a non-Brownian suspension, we numerically study the response of athermal, overdamped, frictionless disks in two dimensions to isotropic and uniaxial compression, as well as to pure and simple shearing, all at finite constant strain rates ε[over ̇]. We show that isotropic and uniaxial compression result in the same jamming packing fraction ϕ_{J}, while pure-shear- and simple-shear-induced jamming occurs at a slightly higher ϕ_{J}^{*}, consistent with that found previously for simple shearing. A critical scaling analysis of pure shearing gives critical exponents consistent with those previously found for both isotropic compression and simple shearing. Using orientational order parameters for contact bond directions, we compare the anisotropy of the force and contact networks at both lowest nematic order, as well as higher 2n-fold order.

Gas-liquid phase separation at zero temperature: mechanical interpretation and implications for gelation

The relationship between glasses and gels has been intensely debated for decades; however, the transition between these two phases remains elusive. To investigate a gel formation process in the zero-temperature limit and its relation to the glass phase, we conducted numerical experiments on athermal quasistatic decompression. During decompression, the system experiences a cavitation event similar to phase separation and this is a gelation process at zero temperature. A normal mode analysis revealed that the phase separation is signaled by the vanishing of the lowest eigenenergy, similar to plastic events of glasses under shear. One primary difference from the shear-induced plasticity is that the vanishing mode experiences a qualitative change in its spatial energy distribution at the phase separation point. These findings enable us to define the glass-gel phase boundary based on mechanics.

Elastogranular columns and beams

String and grains can be combined to create structures capable of bearing significant loads. In this work, we prepare columns and beams through a layer-by-layer deposition of granular matter and loops of fiber strings, and characterize their mechanical properties. The loops cause the grains to jam, and the inter-grain contact leads to a Hertzian-like constitutive response. Initially, one force chain that propagates vertically through the column bears most of the compressive load. As the magnitude of the load is increased, more force chains form in the column, which act in parallel to increase its stiffness, akin to a "super-Hertzian" regime. Applying a compressive prestress enables the structures to withstand shear, enabling the fabrication of cantilevered beams. This work provides a mechanical framework to use elastogranular jamming to create rapid, reusable infrastructure components, such as columns, beams, and arches from inexpensive, commonplace materials, such as rocks and string.

Low-frequency vibrational states in ideal glasses with random pinning

Glasses exhibit spatially localized vibrations in the low-frequency regime. These localized modes emerge below the boson peak frequency ω_{BP}, and their vibrational densities of state follow g(ω)∝ω^{4} (ω is frequency). Here, we attempt to address how the localized vibrations behave through the ideal glass transition. To do this, we employ a random pinning method, which enables us to study the thermodynamic glass transition. We find that the localized vibrations survive even in equilibrium glass states. Remarkably, the localized vibrations still maintain the properties of appearance below ω_{BP} and g(ω)∝ω^{4}. Our results provide important insight into the material properties of ideal glasses.

Vibrational density of states of jammed packing at high dimensions: Mean-field theory

Several mean-field theories predict that the Hessian matrix of amorphous solids converges the Wishart matrix in the limit of the large spatial dimensions d→∞. Motivated by these results, we calculate here the density of states of random packing of harmonic spheres by mapping the Hessian of the original system to the Wishart matrix. We compare our result with that of previous numerical simulations of harmonic spheres in several spatial dimensions d=3, 5, and 9. For small pressure p≪1 (near jamming), we find a good agreement even in d=3, and obtain better agreements in larger d, suggesting that the approximation becomes exact in the limit d→∞.

Viscoelastic Scaling Regimes for Marginally Rigid Fractal Spring Networks

A family of marginally rigid (isostatic) spring networks with fractal structure up to a controllable length was devised, and the viscoelastic spectra G^{*}(ω) calculated. Two nontrivial scaling regimes were observed, (i) G^{'}≈G^{''}∝ω^{Δ} at low frequencies, consistent with Δ=1/2, and (ii) G^{'}∝G^{''}∝ω^{Δ^{'}} for intermediate frequencies corresponding to fractal structure, consistent with a theoretical prediction Δ^{'}=(ln3-ln2)/(ln3+ln2). The crossover between these two regimes occurred at lower frequencies for larger fractals in a manner suggesting diffusivelike dispersion. Solid gels generated by introducing internal stresses exhibited similar behavior above a low-frequency cutoff, indicating the relevance of these findings to real-world applications.

Experimental observations of marginal criticality in granular materials

SignificanceAmorphous materials, such as grains, foams, colloids, and glasses, are ubiquitous in nature and our daily life. They can undergo glass transitions or jamming transitions to obtain rigidity either by fast quench or compression, but show subtle changes in the structures compared to the liquid states or liquid-like states. Recent progress on the first-principle replica theory unifies the glass transition and the jamming transition and points out the marginal phase with fractal free-energy landscape within the stable glass phase. Independently, marginal stability analysis predicts the relations between the exponents of the marginal phase. Here, we perform experiments with photoelastic disks and provide direct evidence of these theories in real-world amorphous materials.

Phonon transport properties of particulate physical gels

Particulate physical gels are sparse, low-density amorphous materials in which clusters of glasses are connected to form a heterogeneous network structure. This structure is characterized by two length scales, ξ s and ξ G: ξ s measures the length of heterogeneities in the network structure and ξ G is the size of glassy clusters. Accordingly, the vibrational states (eigenmodes) of such a material also exhibit a multiscale nature with two characteristic frequencies, [Formula: see text] and ω G, which are associated with ξ s and ξ G, respectively: (i) phonon-like vibrations in the homogeneous medium at [Formula: see text], (ii) phonon-like vibrations in the heterogeneous medium at [Formula: see text], and (iii) disordered vibrations in the glassy clusters at ω > ω G. Here, we demonstrate that the multiscale characteristics seen in the static structures and vibrational states also extend to the phonon transport properties. Phonon transport exhibits two distinct crossovers at frequencies ω* and ω G (or at wavenumbers of [Formula: see text] and [Formula: see text]). In particular, both transverse and longitudinal phonons cross over between Rayleigh scattering at [Formula: see text] and diffusive damping at [Formula: see text]. Remarkably, the Ioffe–Regel limit is located at the very low frequency of ω*. Thus, phonon transport is localized above ω*, even where phonon-like vibrational states persist. This markedly strong scattering behavior is caused by the sparse, porous structure of the gel.

Criticality and marginal stability of the shear jamming transition of frictionless soft spheres

We study numerically the critical behavior and marginal stability of the shear jamming transition for frictionless soft spheres, observed to occur over a finite range of densities, associated with isotropic jamming for densities above the minimum jamming (J-point) density. Several quantities are shown to scale near the shear jamming point in the same way as the isotropic jamming point. We compute the exponents associated with the small force distribution and the interparticle gap distribution and show that the corresponding exponents are consistent with the marginal stability condition observed for isotropic jamming and with predictions of the mean-field theory of jamming in hard spheres.

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.

Universality of stress-anisotropic and stress-isotropic jamming of frictionless spheres in three dimensions: Uniaxial versus isotropic compression

We numerically study a three-dimensional system of athermal, overdamped, frictionless spheres, using a simplified model for a non-Brownian suspension. We compute the bulk viscosity under both uniaxial and isotropic compression as a means to address the question of whether stress-anisotropic and stress-isotropic jamming are in the same critical universality class. Carrying out a critical scaling analysis of the system pressure p, shear stress σ, and macroscopic friction μ=σ/p, as functions of particle packing fraction ϕ and compression rate ε[over ̇], we find good agreement for all critical parameters comparing the isotropic and anisotropic cases. In particular, we determine that the bulk viscosity diverges as p/ε[over ̇]∼(ϕ_{J}-ϕ)^{-β}, with β=3.36±0.09, as jamming is approached from below. We further demonstrate that the average contact number per particle Z can also be written in a scaling form as a function of ϕ and ε[over ̇]. Once again, we find good agreement between the uniaxial and isotropic cases. We compare our results to prior simulations and theoretical predictions.

Average density of states of amorphous Hamiltonians: role of phonon mediated coupling of nano-clusters

Based on a description of an amorphous solid as a collection of coupled nanosize molecular clusters referred as basic blocks, we analyse the statistical properties of its Hamiltonian. The information is then used to derive the ensemble averaged density of the vibrational states (non-phonon) which turns out to be a Gaussian in the bulk of the spectrum and an Airy function in the low frequency regime. A comparison with experimental data for six glasses confirms validity of our theoretical predictions.

Structural, mechanical, and vibrational properties of particulate physical gels

Our lives are surrounded by a rich assortment of disordered materials. In particular, glasses are well known as dense, amorphous materials, whereas gels exist in low-density, disordered states. Recent progress has provided a significant step forward in understanding the material properties of glasses, such as mechanical, vibrational, and transport properties. In contrast, our understanding of particulate physical gels is still highly limited. Here, using molecular dynamics simulations, we study a simple model of particulate physical gels, the Lennard-Jones (LJ) gels, and provide a comprehensive understanding of their structural, mechanical, and vibrational properties, all of which are markedly different from those of LJ glasses. First, the LJ gels show sparse, heterogeneous structures, and the length scale ξ_s of the structures grows as the density is lowered. Second, the LJ gels are extremely soft, with both shear G and bulk K moduli being orders of magnitude smaller than those of LJ glasses. Third, many low-frequency vibrational modes are excited, which form a characteristic plateau with the onset frequency ω_* in the vibrational density of states. Structural, mechanical, and vibrational properties, characterized by ξ_s, G, K, and ω_*, respectively, show power-law scaling behaviors with the density, which establishes a close relationship between them. Throughout this work, we also reveal that LJ gels are multiscale, solid-state materials: (i) homogeneous elastic bodies at long lengths, (ii) heterogeneous elastic bodies with fractal structures at intermediate lengths, and (iii) amorphous structural bodies at short lengths.

Achieving adjustable elasticity with non-affine to affine transition

For various engineering and industrial applications it is desirable to realize mechanical systems with broadly adjustable elasticity to respond flexibly to the external environment. Here we discover a topology-correlated transition between affine and non-affine regimes in elasticity in both two- and three-dimensional packing-derived networks. Based on this transition, we numerically design and experimentally realize multifunctional systems with adjustable elasticity. Within one system, we achieve solid-like affine response, liquid-like non-affine response and a continuous tunability in between. Moreover, the system also exhibits a broadly tunable Poisson's ratio from positive to negative values, which is of practical interest for energy absorption and for fracture-resistant materials. Our study reveals a fundamental connection between elasticity and network topology, and demonstrates its practical potential for designing mechanical systems and metamaterials.

Moduli and modes in the Mikado model

We determine how low frequency vibrational modes control the elastic shear modulus of Mikado networks, a minimal mechanical model for semi-flexible fiber networks. From prior work it is known that when the fiber bending modulus is sufficiently small, (i) the shear modulus of 2D Mikado networks scales as a power law in the fiber line density, G ∼ ρα+1, and (ii) the networks also possess an anomalous abundance of soft (low-frequency) vibrational modes with a characteristic frequency ωκ ∼ ρβ/2. While it has been suggested that α and β are identical, the preponderance of evidence indicates that α is larger than theoretical predictions for β. We resolve this inconsistency by measuring the vibrational density of states in Mikado networks for the first time. Supported by these results, we then demonstrate analytically that α = β + 1. In so doing, we uncover new insights into the coupling between soft modes and shear, as well as the origin of the crossover from bending- to stretching-dominated response.

Testing mean-field theory for jamming of non-spherical particles: contact number, gap distribution, and vibrational density of states

We perform numerical simulations of the jamming transition of non-spherical particles in two dimensions. In particular, we systematically investigate how the physical quantities at the jamming transition point behave when the shapes of the particle deviate slightly from the perfect disks. For efficient numerical simulation, we first derive an analytical expression of the gap function, using the perturbation theory around the reference disks. Starting from disks, we observe the effects of the deformation of the shapes of particles by the n-th-order term of the Fourier series [Formula: see text]. We show that the several physical quantities, such as the number of contacts, gap distribution, and characteristic frequencies of the vibrational density of states, show the power-law behaviors with respect to the linear deviation from the reference disks. The power-law behaviors do not depend on n and are fully consistent with the mean-field theory of the jamming of non-spherical particles. This result suggests that the mean-field theory holds very generally for nearly spherical particles.

Long-Range Anomalous Decay of the Correlation in Jammed Packings

We numerically study the structure of the interactions occurring in three-dimensional systems of hard spheres at jamming, focusing on the large-scale behavior. Given the fundamental role in the configuration of jammed packings, we analyze the propagation through the system of the weak forces and of the variation of the coordination number with respect to the isostaticity condition, ΔZ. We show that these correlations can be successfully probed by introducing a correlation function weighted on the density-density fluctuations. The results of this analysis can be further improved by introducing a representation of the system based on the contact points between particles. In particular, we find evidence that the weak forces and the ΔZ fluctuations support the hypothesis of randomly jammed packings of spherical particles being hyperuniform by exhibiting an anomalous long-range decay. Moreover, we find that the large-scale structure of the density-density correlation exhibits a complex behavior due to the superimposition of two exponentially damped oscillating signals propagating with linearly depending frequencies.

Ioffe-Regel criterion and viscoelastic properties of amorphous solids

We show that viscoelastic effects play a crucial role in the damping of vibrational modes in harmonic amorphous solids. The relaxation of a given plane elastic wave is described by a memory function of a semi-infinite one-dimensional mass-spring chain. The initial vibrational energy spreads from the first site of the chain to infinity. In the beginning of the chain, there is a barrier, which significantly reduces the decay of vibrational energy below the Ioffe-Regel frequency. To obtain the parameters of the chain, we present a numerically stable method, based on the Chebyshev expansion of the local vibrational density of states.

Sound damping in soft particle packings: the interplay between configurational disorder and inelasticity

We numerically investigate sound damping in disordered two-dimensional soft particle packings. We simulate evolution of standing waves of particle displacements and analyze time correlation functions of particle velocities and power spectra. We control the strength of inelastic interactions between the particles in contact to show how the inelasticity affects anomalous sound characteristics of disordered systems: Increasing the strength of inelastic interactions, we find that (i) sound softening vanishes and (ii) attenuation coefficients exhibit a transition from the Rayleigh law to quadratic growth. We also report (iii) how the Ioffe-Regel limit frequencies depend on the strength of inelasticity as useful information for experiments and applications of the sound in disordered media. Our findings suggest that sound damping in soft particle packings is determined by the interplay between elastic heterogeneities and inelasticity.

Nonaffine displacements below jamming under athermal quasistatic compression

Critical properties of frictionless spherical particles below jamming are studied using extensive numerical simulations, paying particular attention to the nonaffine part of the displacements during the athermal quasistatic compression. It is shown that the squared norm of the nonaffine displacement exhibits a power-law divergence toward the jamming transition point. A possible connection between this critical exponent and that of the shear viscosity is discussed. The participation ratio of the displacements vanishes in the thermodynamic limit at the transition point, meaning that the nonaffine displacements are localized marginally with a fractal dimension. Furthermore, the distribution of the displacement is shown to have a power-law tail, the exponent of which is related to the fractal dimension.

Critical scaling of compression-driven jamming of athermal frictionless spheres in suspension

We study numerically a system of athermal, overdamped, frictionless spheres, as in a non-Brownian suspension, in two and three dimensions. Compressing the system isotropically at a fixed rate ε[over ̇], we investigate the critical behavior at the jamming transition. The finite compression rate introduces a control timescale, which allows one to probe the critical timescale associated with jamming. As was found previously for steady-state shear-driven jamming, we find for compression-driven jamming that pressure obeys a critical scaling relation as a function of packing fraction ϕ and compression rate ε[over ̇], and that the bulk viscosity p/ε[over ̇] diverges upon jamming. A scaling analysis determines the critical exponents associated with the compression-driven jamming transition. Our results suggest that stress-isotropic, compression-driven jamming may be in the same universality class as stress-anisotropic, shear-driven jamming.

Mechanical disorder of sticky-sphere glasses. I. Effect of attractive interactions

Recent literature indicates that attractive interactions between particles of a dense liquid play a secondary role in determining its bulk mechanical properties. Here we show that, in contrast with their apparent unimportance to the bulk mechanics of dense liquids, attractive interactions can have a major effect on macro- and microscopic elastic properties of glassy solids. We study several broadly applicable dimensionless measures of stability and mechanical disorder in simple computer glasses, in which the relative strength of attractive interactions-referred to as "glass stickiness"-can be readily tuned. We show that increasing glass stickiness can result in the decrease of various quantifiers of mechanical disorder, on both macro- and microscopic scales, with a pair of intriguing exceptions to this rule. Interestingly, in some cases strong attractions can lead to a reduction of the number density of soft, quasilocalized modes, by up to an order of magnitude, and to a substantial decrease in their core size, similar to the effects of thermal annealing on elasticity observed in recent works. Contrary to the behavior of canonical glass models, we provide compelling evidence indicating that the stabilization mechanism in our sticky-sphere glasses stems predominantly from the self-organized depletion of interactions featuring large, negative stiffnesses. Finally, we establish a fundamental link between macroscopic and microscopic quantifiers of mechanical disorder, which we motivate via scaling arguments. Future research directions are discussed.

Mean-Field Predictions of Scaling Prefactors Match Low-Dimensional Jammed Packings

No known analytic framework precisely explains all the phenomena observed in jamming. The replica theory for glasses and jamming is a mean-field theory which attempts to do so by working in the limit of infinite dimensions, such that correlations between neighbors are negligible. As such, results from this mean-field theory are not guaranteed to be observed in finite dimensions. However, many results in mean field for jamming have been shown to be exact or nearly exact in low dimensions. This suggests that the infinite dimensional limit is not necessary to obtain these results. In this Letter, we perform precision measurements of jamming scaling relationships between pressure, excess packing fraction, and number of excess contacts from dimensions 2-10 in order to extract the prefactors to these scalings. While these prefactors should be highly sensitive to finite dimensional corrections, we find the mean-field predictions for these prefactors to be exact in low dimensions. Thus the mean-field approximation is not necessary for deriving these prefactors. We present an exact, first-principles derivation for one, leaving the other as an open question. Our results suggest that mean-field theories of critical phenomena may compute more for d≥d_{u} than has been previously appreciated.

Inferring the particle-wise dynamics of amorphous solids from the local structure at the jamming point

Jamming is a phenomenon shared by a wide variety of systems, such as granular materials, foams, and glasses in their high density regime. This has motivated the development of a theoretical framework capable of explaining many of their static critical properties with a unified approach. However, the dynamics occurring in the vicinity of the jamming point has received little attention and the problem of finding a connection with the local structure of the configuration remains unexplored. Here we address this issue by constructing physically well defined structural variables using the information contained in the network of contacts of jammed configurations, and then showing that such variables yield a resilient statistical description of the particle-wise dynamics near this critical point. Our results are based on extensive numerical simulations of systems of spherical particles that allow us to statistically characterize the trajectories of individual particles in terms of their first two moments. We first demonstrate that, besides displaying a broad distribution of mobilities, particles may also have preferential directions of motion. Next, we associate each of these features with a structural variable computed uniquely in terms of the contact vectors at jamming, obtaining considerably high statistical correlations. The robustness of our approach is confirmed by testing two types of dynamical protocols, namely molecular dynamics and Monte Carlo, with different types of interaction. We also provide evidence that the dynamical regime we study here is dominated by anharmonic effects and therefore it cannot be described properly in terms of vibrational modes. Finally, we show that correlations decay slowly and in an interaction-independent fashion, suggesting a universal rate of information loss.

Novel elastic instability of amorphous solids in finite spatial dimensions

Recently, progress has been made in the understanding of anomalous vibrational excitations in amorphous solids. In the lowest-frequency region, the vibrational spectrum follows a non-Debye quartic law, which persists up to zero frequency without any frequency gap. This gapless vibrational density of states (vDOS) suggests that glasses are on the verge of instability. This feature of marginal stability is now highlighted as a key concept in the theories of glasses. In particular, the elasticity theory based on marginal stability predicts the gapless vDOS. However, this theory yields a quadratic law and not the quartic law. To address this inconsistency, we presented a new type of instability, which is different from the conventional one, and proposed that amorphous solids are marginally stable considering the new instability in the preceding study [M. Shimada, H. Mizuno and A. Ikeda, Soft Matter, 2020, 16, 7279]. In this study, we further extend and detail the results for these instabilities. By analyzing various examples of disorder, we demonstrate that real glasses in finite spatial dimensions can be marginally stable by the proposed novel instability.

Thermal origin of quasilocalized excitations in glasses

Key aspects of glasses are controlled by the presence of excitations in which a group of particles can rearrange. Surprisingly, recent observations indicate that their density is dramatically reduced and their size decreases as the temperature of the supercooled liquid is lowered. Some theories predict these excitations to cause a gap in the spectrum of quasilocalized modes of the Hessian that grows upon cooling, while others predict a pseudogap D_{L}(ω)∼ω^{α}. To unify these views and observations, we generate glassy configurations of controlled gap magnitude ω_{c} at temperature T=0, using so-called breathing particles, and study how such gapped states respond to thermal fluctuations. We find that (i) the gap always fills up at finite T with D_{L}(ω)≈A_{4}(T)ω^{4} and A_{4}∼exp(-E_{a}/T) at low T, (ii) E_{a} rapidly grows with ω_{c}, in reasonable agreement with a simple scaling prediction E_{a}∼ω_{c}^{4} and (iii) at larger ω_{c} excitations involve fewer particles, as we rationalize, and eventually become stringlike. We propose an interpretation of mean-field theories of the glass transition, in which the modes beyond the gap act as an excitation reservoir, from which a pseudogap distribution is populated with its magnitude rapidly decreasing at lower T. We discuss how this picture unifies the rarefaction as well as the decreasing size of excitations upon cooling, together with a stringlike relaxation occurring near the glass transition.

Acoustic localisation in a two-dimensional granular medium

We focus on localizing the source of acoustic emissions within a compressed two-dimensional granular ensemble of photoelastic disks, having as main information the arrival times of the acoustic signal to 6 sensors located in the boundaries of the system. By estimating, thanks to the photoelasticity of the grains, the wave speed at every point of the structure, we are able to compute the arrival times from every point of the system to the sensors. A comparison between the arrival time differences between every set of computed values to those from the actual measurements allows finding the source of the acoustic emissions.

Critical point of jamming transition in two-dimensional monodisperse systems

The existence of amorphous packings in two-dimensional monodisperse system is a classical unsolved problem. We get the energy minimum state by the energy minimization method of enthalpy under constant pressure conditions. Firstly, we find that there are two peaks in the experiment, which demonstrate the interesting features of the coexistence of crystals and amorphous crystals. And then, we confirm the critical point of jamming transition of the two-dimensional monodisperse is [Formula: see text]. Finally, we prove that the jamming scaling is still satisfied in two-dimensional monodispersed system: [Formula: see text] and vanishes as [Formula: see text], and the boson peak shifts to lower frequencies for less compressed systems.

Dynamic length scales in athermal, shear-driven jamming of frictionless disks in two dimensions

We carry out numerical simulations of athermally sheared, bidisperse, frictionless disks in two dimensions. From an appropriately defined velocity correlation function, we determine that there are two diverging length scales, ξ and ℓ, as the jamming transition is approached. We analyze our results using a critical scaling ansatz for the correlation function and argue that the more divergent length ℓ is a consequence of a dangerous irrelevant scaling variable and that it is ξ, which is the correlation length that determines the divergence of the system viscosity as jamming is approached from below in the liquid phase. We find that ξ∼(ϕ_{J}-ϕ)^{-ν} diverges with the critical exponent ν=1. We provide evidence that ξ measures the length scale of fluctuations in the rotation of the particle velocity field, while ℓ measures the length scale of fluctuations in the divergence of the velocity field.

Contact network changes in ordered and disordered disk packings

We investigate the mechanical response of packings of purely repulsive, frictionless disks to quasistatic deformations. The deformations include simple shear strain at constant packing fraction and at constant pressure, "polydispersity" strain (in which we change the particle size distribution) at constant packing fraction and at constant pressure, and isotropic compression. For each deformation, we show that there are two classes of changes in the interparticle contact networks: jump changes and point changes. Jump changes occur when a contact network becomes mechanically unstable, particles "rearrange", and the potential energy (when the strain is applied at constant packing fraction) or enthalpy (when the strain is applied at constant pressure) and all derivatives are discontinuous. During point changes, a single contact is either added to or removed from the contact network. For repulsive linear spring interactions, second- and higher-order derivatives of the potential energy/enthalpy are discontinuous at a point change, while for Hertzian interactions, third- and higher-order derivatives of the potential energy/enthalpy are discontinuous. We illustrate the importance of point changes by studying the transition from a hexagonal crystal to a disordered crystal induced by applying polydispersity strain. During this transition, the system only undergoes point changes, with no jump changes. We emphasize that one must understand point changes, as well as jump changes, to predict the mechanical properties of jammed packings.

Intermittent rearrangements accompanying thermal fluctuations distinguish glasses from crystals

Glasses exhibit vibrational and thermal properties that are markedly different from those of crystals. While recent works have advanced our understanding of vibrational excitations in glasses in the harmonic approximation limit, efforts in understanding finite-temperature anharmonic processes have been limited. In crystals, phonon-phonon coupling provides an extremely efficient mechanism for anharmonic decay that is also important in glasses. By using extensive molecular dynamics simulation of model atomic systems, here we first describe, both numerically and analytically, the anharmonic couplings in the crystal and the glass by focusing on the temperature dependence of the associated decay rates. Next, we show that an additional anharmonic channel of different origin emerges in the amorphous case, which induces unconventional intermittent rearrangements of particles. We have found that thermal vibrations in glasses trigger transitions among numerous different local minima of the energy landscape, which, however, are located within the same wide (meta)basin. These processes generate motions that are different from both diffusive and out-of-equilibrium aging dynamics. We suggest that (i) the observed intermittent rearrangements accompanying thermal fluctuations are crucial features distinguishing glasses from crystals and (ii) they can be considered as relics of the liquid state that survive the complete dynamic arrest taking place at the glass transition temperature.

Marginally jammed states of hard disks in a one-dimensional channel

We have studied a class of marginally jammed states in a system of hard disks confined in a narrow channel-a quasi-one-dimensional system-whose exponents are not those predicted by theories valid in the infinite dimensional limit. The exponent γ which describes the distribution of small gaps takes the value 1 rather than the infinite dimensional value 0.41269⋯. Our work shows that there exist jammed states not found within the tiling approach of Ashwin and Bowles. The most dense of these marginal states is an unusual state of matter that is asymptotically crystalline.

Emergent solidity of amorphous materials as a consequence of mechanical self-organisation

Amorphous solids have peculiar properties distinct from crystals. One of the most fundamental mysteries is the emergence of solidity in such nonequilibrium, disordered state without the protection by long-range translational order. A jammed system at zero temperature, although marginally stable, has solidity stemming from the space-spanning force network, which gives rise to the long-range stress correlation. Here, we show that such nonlocal correlation already appears at the nonequilibrium glass transition upon cooling. This is surprising since we also find that the system suffers from giant anharmonic fluctuations originated from the fractal-like potential energy landscape. We reveal that it is the percolation of the force-bearing network that allows long-range stress transmission even under such circumstance. Thus, the emergent solidity of amorphous materials is a consequence of nontrivial self-organisation of the disordered mechanical architecture. Our findings point to the significance of understanding amorphous solids and nonequilibrium glass transition from a mechanical perspective.

Jamming Below Upper Critical Dimension

Extensive numerical simulations in the past decades proved that the critical exponents of the jamming of frictionless spherical particles are the same in two and three dimensions. This implies that the upper critical dimension is d_{u}=2 or lower. In this Letter, we study the jamming transition below the upper critical dimension. We investigate a quasi-one-dimensional system: disks confined in a narrow channel. We show that the system is isostatic at the jamming transition point as in the case of standard jamming transition of the bulk systems in two and three dimensions. Nevertheless, the scaling of the excess contact number shows the linear scaling. Furthermore, the gap distribution remains finite even at the jamming transition point. These results are qualitatively different from those of the bulk systems in two and three dimensions.

Structured randomness: jamming of soft discs and pins

Simulations are used to find the zero temperature jamming threshold, φj, for soft, bidisperse disks in the presence of small fixed particles, or "pins", arranged in a lattice. The presence of pins leads, as one expects, to a decrease in φj. Structural properties of the system near the jamming threshold are calculated as a function of the pin density. While the correlation length exponent remains ν = 1/2 at low pin densities, the system is mechanically stable with more bonds, yet fewer contacts than the Maxwell criterion implies in the absence of pins. In addition, as pin density increases, novel bond orientational order and long-range spatial order appear, which are correlated with the square symmetry of the pin lattice.

Jamming with Tunable Roughness

We introduce a new model to study the effect of surface roughness on the jamming transition. By performing numerical simulations, we show that for a smooth surface, the jamming transition density and the contact number at the transition point both increase upon increasing asphericity, as for ellipsoids and spherocylinders. Conversely, for a rough surface, both quantities decrease, in quantitative agreement with the behavior of frictional particles. Furthermore, in the limit corresponding to the Coulomb friction law, the model satisfies a generalized isostaticity criterion proposed in previous studies. We introduce a counting argument that justifies this criterion and interprets it geometrically. Finally, we propose a simple theory to predict the contact number at finite friction from the knowledge of the force distribution in the infinite friction limit.