Effects of compression on the vibrational modes of marginally jammed solids

Microscopic approach to the nonlinear elasticity of compressed emulsions

Using confocal microscopy, we measure the packing geometry and interdroplet forces as a function of the osmotic pressure in a 3D emulsion system. We assume a harmonic interaction potential over a wide range of volume fractions and attribute the observed nonlinear elastic response of the pressure with density to the first corrections to the scaling laws of the microstructure away from the critical point. The bulk modulus depends on the excess contacts created under compression, which leads to the correction exponent α=1.5. Microscopically, the nonlinearities manifest themselves as a narrowing of the distribution of the pressure per particle as a function of the global pressure.

Jamming in systems with quenched disorder

We numerically study the effect of adding quenched disorder in the form of randomly placed pinning sites on jamming transitions in a disk packing that jams at a well-defined point J in the clean limit. Quenched disorder decreases the jamming density and introduces a depinning threshold. The onset of a finite threshold coincides with point J at the lowest pinning densities, but for higher pinning densities there is always a finite depinning threshold even well below jamming. We find that proximity to point J strongly affects the transport curves and noise fluctuations, and we observe a change from plastic behavior below jamming, where the system is highly heterogeneous, to elastic depinning above jamming. Many of the general features we find are related to other systems containing quenched disorder, including the peak effect observed in vortex systems.

Uniform shock waves in disordered granular matter

The confining pressure P is perhaps the most important parameter controlling the properties of granular matter. Strongly compressed granular media are, in many respects, simple solids in which elastic perturbations travel as ordinary phonons. However, the speed of sound in granular aggregates continuously decreases as the confining pressure decreases, completely vanishing at the jamming-unjamming transition. This anomalous behavior suggests that the transport of energy at low pressures should not be dominated by phonons. In this work we use simulations and theory to show how the response of granular systems becomes increasingly nonlinear as pressure decreases. In the low-pressure regime the elastic energy is found to be mainly transported through nonlinear waves and shocks. We numerically characterize the propagation speed, shape, and stability of these shocks and model the dependence of the shock speed on pressure and impact intensity by a simple analytical approach.

Influence of network topology on sound propagation in granular materials

Granular media, whose features range from the particle scale to the force-chain scale and the bulk scale, are usually modeled as either particulate or continuum materials. In contrast with each of these approaches, network representations are natural for the simultaneous examination of microscopic, mesoscopic, and macroscopic features. In this paper, we treat granular materials as spatially embedded networks in which the nodes (particles) are connected by weighted edges obtained from contact forces. We test a variety of network measures to determine their utility in helping to describe sound propagation in granular networks and find that network diagnostics can be used to probe particle-, curve-, domain-, and system-scale structures in granular media. In particular, diagnostics of mesoscale network structure are reproducible across experiments, are correlated with sound propagation in this medium, and can be used to identify potentially interesting size scales. We also demonstrate that the sensitivity of network diagnostics depends on the phase of sound propagation. In the injection phase, the signal propagates systemically, as indicated by correlations with the network diagnostic of global efficiency. In the scattering phase, however, the signal is better predicted by mesoscale community structure, suggesting that the acoustic signal scatters over local geographic neighborhoods. Collectively, our results demonstrate how the force network of a granular system is imprinted on transmitted waves.

Marginal stability constrains force and pair distributions at random close packing

The requirement that packings of frictionless hard spheres, arguably the simplest structural glass, cannot be compressed by rearranging their network of contacts is shown to yield a new constraint on their microscopic structure. This constraint takes the form a bound between the distribution of contact forces P(f) and the pair distribution function g(r): if P(f)∼f(θ) and g(r)∼(r-σ(0))(-γ), where σ(0) is the particle diameter, one finds that γ ≥ 1/(2 + θ). This bound plays a role similar to those found in some glassy materials with long-range interactions, such as the Coulomb gap in Anderson insulators or the distribution of local fields in mean-field spin glasses. There are grounds to believe that this bound is saturated, yielding a mechanism to explain the avalanches of rearrangements with power-law statistics that govern plastic flow in packings.

Critical scaling of a jammed system after a quench of temperature

Critical behavior of soft repulsive particles after quench of temperature near the jamming transition is numerically investigated. It is found that the plateau of the mean-square displacement of tracer particles and the pressure satisfy critical scaling laws. The critical density for the jamming transition depends on the protocol to prepare the system, while the values of the critical exponents which are consistent with the prediction of a phenomenology are independent of the protocol.

Finite-size scaling at the jamming transition

We present an analysis of finite-size effects in jammed packings of N soft, frictionless spheres at zero temperature. There is a 1/N correction to the discrete jump in the contact number at the transition so that jammed packings exist only above isostaticity. As a result, the canonical power-law scalings of the contact number and elastic moduli break down at low pressure. These quantities exhibit scaling collapse with a nontrivial scaling function, demonstrating that the jamming transition can be considered a phase transition. Scaling is achieved as a function of N in both two and three dimensions, indicating an upper critical dimension of 2.

Constraints and vibrations in static packings of ellipsoidal particles

We numerically investigate the mechanical properties of static packings of frictionless ellipsoidal particles in two and three dimensions over a range of aspect ratio and compression Δφ. While amorphous packings of spherical particles at jamming onset (Δφ=0) are isostatic and possess the minimum contact number z_{iso} required for them to be collectively jammed, amorphous packings of ellipsoidal particles generally possess fewer contacts than expected for collective jamming (z<z_{iso}) from naive counting arguments, which assume that all contacts give rise to linearly independent constraints on interparticle separations. To understand this behavior, we decompose the dynamical matrix M=H-S for static packings of ellipsoidal particles into two important components: the stiffness H and stress S matrices. We find that the stiffness matrix possesses 2N(z_{iso}-z) eigenmodes e[over ̂]_{0} with zero eigenvalues even at finite compression, where N is the number of particles. In addition, these modes e[over ̂]_{0} are nearly eigenvectors of the dynamical matrix with eigenvalues that scale as Δφ, and thus finite compression stabilizes packings of ellipsoidal particles. At jamming onset, the harmonic response of static packings of ellipsoidal particles vanishes, and the total potential energy scales as δ^{4} for perturbations by amplitude δ along these "quartic" modes, e[over ̂]_{0}. These findings illustrate the significant differences between static packings of spherical and ellipsoidal particles.

A unified framework for non-brownian suspension flows and soft amorphous solids

While the rheology of non-brownian suspensions in the dilute regime is well understood, their behavior in the dense limit remains mystifying. As the packing fraction of particles increases, particle motion becomes more collective, leading to a growing length scale and scaling properties in the rheology as the material approaches the jamming transition. There is no accepted microscopic description of this phenomenon. However, in recent years it has been understood that the elasticity of simple amorphous solids is governed by a critical point, the unjamming transition where the pressure vanishes, and where elastic properties display scaling and a diverging length scale. The correspondence between these two transitions is at present unclear. Here we show that for a simple model of dense flow, which we argue captures the essential physics near the jamming threshold, a formal analogy can be made between the rheology of the flow and the elasticity of simple networks. This analogy leads to a new conceptual framework to relate microscopic structure to rheology. It enables us to define and compute numerically normal modes and a density of states. We find striking similarities between the density of states in flow, and that of amorphous solids near unjamming: both display a plateau above some frequency scale ω(∗) ∼ |z(c) - z|, where z is the coordination of the network of particle in contact, z(c) = 2D where D is the spatial dimension. However, a spectacular difference appears: the density of states in flow displays a single mode at another frequency scale ω(min) ≪ ω(∗) governing the divergence of the viscosity.

Understanding the low-frequency quasilocalized modes in disordered colloidal systems

In disordered colloidal systems, we experimentally measure the normal modes with the covariance matrix method and clarify the origin of low-frequency quasilocalization at the single-particle level. We observe important features from both jamming and glass simulations: There is a plateau in the density of states [D(ω)] which is suppressed upon compression, as predicted by jamming; within the same systems, we also find that the low-frequency quasilocalization originates from the large vibrations of defective structures coupled with transverse excitations, consistent with a recent glass simulation. The coexistence of these features demonstrates an experimental link between jamming and glass. Extensive simulations further show that such a structural origin of quasilocalization is universally valid for various temperatures and volume fractions.

Nonaffine measures of particle displacements in sheared colloidal glasses

The nonaffine motion of particles is central to the relaxation and flow of glasses. It is usually assumed in plasticity theories that nonaffine rearrangements are localized and uncorrelated. Here we present evidence that this assumption may not hold. We investigate and compare systematically different measures of nonaffinity in a sheared colloidal glass by tracking the motion of the individual particles directly with confocal microscopy. We show that besides differences in the appearance and degree of localization of nonaffine displacements, the nature of their fluctuations is very similar. At intermediate times, all spatial correlation functions display robust power-law behavior, clearly demonstrating long-range correlations and critical behavior of the driven glass, in contrast to the assumptions of plasticity theories. We show that on long-time scales, correlations become finite and plasticity theories may apply.

Thermoluminescence at a heating rate threshold in stressed fused silica

The emissive properties of proton implanted fused silica surfaces have been studied by laser beam annealing. When submitted to a high thermal step from a focused CO2 laser, an intense near infra-red transient incandescence (TI) peak rises from stressed silica. The TI presents the characteristics of a thermoluminescent (TL) emission that occurs above a thermal rate threshold. We show that TI rises at the stress relaxation.

Microscopic theory of the jamming transition of harmonic spheres

We develop a microscopic theory to analyze the phase behavior and compute correlation functions of dense assemblies of soft repulsive particles both at finite temperature, as in colloidal materials, and at vanishing temperature, a situation relevant for granular materials and emulsions. We use a mean-field statistical mechanical approach which combines elements of liquid state theory to replica calculations to obtain quantitative predictions for the location of phase boundaries, macroscopic thermodynamic properties, and microstructure of the system. We focus, in particular, on the derivation of scaling properties emerging in the vicinity of the jamming transition occurring at large density and zero temperature. The new predictions we obtain for pair correlation functions near contact are tested using computer simulations. Our work also clarifies the conceptual nature of the jamming transition and its relation to the phenomenon of the glass transition observed in atomic liquids.

Dynamic heterogeneities, boson peak, and activation volume in glass-forming liquids

There are various arguments and models connecting the characteristic length associated with the boson peak vibrations ξ to the length scale of dynamical heterogeneity L(het). ξ is usually defined as the ratio of the transverse sound velocity to the boson peak frequency. Here we present pressure, temperature, and molecular weight dependencies of ξ, estimated using light scattering, in a few molecular and polymeric glass formers. These dependencies are compared with respective dependencies of the activation volume ΔV(#) in the same materials. Good agreement is found for the pressure and molecular weight dependencies of ξ and ΔV(#) measured at the glass transition temperature T(g). These results provide more evidence for a possible relationship between the sensitivity of structural relaxation to density (activation volume) and the heterogeneity volume. However, contrary to the expectations for L(het), ξ does not decrease with temperature above T(g) in most of the studied materials. The temperature dependence of ξ is compared to that of L(het) in glycerol and orthoterphenyl (OTP) estimated from literature data. The analysis shows a clear difference in the behavior of ξ(T) and ΔV(#)(T) at temperatures above T(g), although ΔV(#)(T)(1/3) and L(het)(T) have similar temperature dependence. Possible reasons for the observed difference are discussed.

Critical scaling near jamming transition for frictional granular particles

The critical rheology of sheared frictional granular materials near jamming transition is numerically investigated. It is confirmed that there exists a true critical density which characterizes the onset of the yield stress and two fictitious critical densities which characterize the scaling laws of rheological properties. We find the existence of a hysteresis loop between two of the critical densities for each friction coefficient. It is noteworthy that the critical scaling law for frictionless jamming transition seems to be still valid even for frictional jamming despite using fictitious critical density values.

Microscopic mean-field theory of the jamming transition

Dense particle packings acquire rigidity through a nonequilibrium jamming transition commonly observed in materials from emulsions to sandpiles. We describe athermal packings and their observed geometric phase transitions by using equilibrium statistical mechanics and develop a fully microscopic, mean-field theory of the jamming transition for soft repulsive spherical particles. We derive analytically some of the scaling laws and exponents characterizing the transition and obtain new predictions for microscopic correlation functions of jammed states that are amenable to experimental verifications and whose accuracy we confirm by using computer simulations.

New jamming scenario: from marginal jamming to deep jamming

We study the properties of jammed packings of frictionless spheres over a wide range of volume fractions. There exists a crossover volume fraction which separates deeply jammed solids from marginally jammed solids. In deeply jammed solids, all the scalings presented in marginally jammed solids are replaced with remarkably different ones with potential independent exponents. Correspondingly, there are structural changes in the pair distribution function associated with the crossover. The normal modes of vibration of deeply jammed solids also exhibit some anomalies, e.g., strengthened quasilocalization and the absence of Debye-like density of states at low frequencies. Deeply jammed systems may thus be cataloged to a new class of amorphous solids.

Finite-size scaling at the jamming transition: corrections to scaling and the correlation-length critical exponent

We carry out a finite-size scaling analysis of the jamming transition in frictionless bidisperse soft core disks in two dimensions. We consider two different jamming protocols: (i) quench from random initial positions and (ii) quasistatic shearing. By considering the fraction of jammed states as a function of packing fraction for systems with different numbers of particles, we determine the spatial correlation length critical exponent ν ≈ 1 and show that corrections to scaling are crucial for analyzing the data. We show that earlier numerical results yielding ν < 1 are due to the improper neglect of these corrections.

Fluctuations, jamming, and yielding for a driven probe particle in disordered disk assemblies

Using numerical simulations we examine the velocity fluctuations and velocity-force curve characteristics of a probe particle driven with constant force through a two-dimensional disordered assembly of disks which has a well-defined jamming point J at a density of ϕJ=0.843. As ϕ increases toward ϕJ, the average velocity of the probe particle decreases and the velocity fluctuations show an increasingly intermittent or avalanchelike behavior. When ϕ is within a few percent of the jamming density, the velocity distributions are exponential, while when ϕ is less than 1% away from jamming, the velocity distributions have a power-law character with exponents in agreement with recent experiments. The velocity power spectra exhibit a crossover from a Lorentzian form to a 1/f shape near jamming. We extract a correlation length exponent ν which is in good agreement with recent shear simulations. For ϕ>ϕJ, there is a critical threshold force F(c) that must be applied for the probe particle to move through the sample which increases with increasing ϕ. The velocity-force curves are linear below jamming, while at jamming they have a power-law form. The onset of the probe motion above ϕJ occurs via a local yielding of the particles around the probe particle which we term a local shear banding effect.

Experiments on the random packing of tetrahedral dice

Tetrahedra may be the ultimate frustrating, disordered glass forming units. Our experiments on tetrahedral dice indicate the densest (volume fraction phi=0.76+/-.02, compared with phi(sphere)=0.64), most disordered, experimental, random packing of any set of congruent convex objects to date. Analysis of MRI scans yield translational and orientational correlation functions which decay as soon as particles do not touch, much more rapidly than the approximately 6 diameters for sphere correlations to decay. Although there are only 6.3+/-.5 touching neighbors on average, face-face and edge-face contacts provide enough additional constraints, 12+/-1.6 total, to roughly bring the structure to the isostatic limit for frictionless particles. Randomly jammed tetrahedra form a dense rigid highly uncorrelated material.

Spectral responses in granular compaction

The slow compaction of a tapped granular packing is reminiscent of the low-temperature dynamics glasses. Here, I study the dynamics of granular compaction by means of a volumetric spectroscopy. While the specific packing volume v displays glassy aging and memory effects at low tapping amplitudes Gamma, the dynamic volumetric susceptibility chi(v)= partial differentialv/ partial differentialGamma displays minimal glassy effects, and its frequency spectrum gives no indication of a rapidly growing time scale. These features are contrast sharply with that found in the dielectric and magnetic susceptibilities of structural and spin glasses. Instead, chi(v) appears to exhibit the behavior of a dynamic configurational specific heat, such as that obtained from computer simulations of spin-glass models. This suggests that the glassy dynamics of granular compaction is controlled by statistically rare processes that diverge from the typical dynamics of the system. From modifications of the dynamical spectrum by finite system size, I suggest that these glassy processes derive from large-scale collective particle rearrangements.

Experimental study of the jamming transition at zero temperature

We experimentally investigate jamming in a quasi-two-dimensional granular system of automatically swelling particles and show that a maximum in the height of the first peak of the pair correlation function is a structural signature of the jamming transition at zero temperature. The same signature is also found in the second peak of the pair correlation function, but not in the third peak, reflecting the underlying singularity of jamming transition. We also study the development of clusters in this system. A static length scale extracted from the cluster structure reaches the size of the system when the system approaches the jamming point. Finally, we show that in a highly inhomogeneous system, friction causes the system to jam in series of steps. In this case, jamming may be obtained through successive buckling of force chains.

Soft modes and elasticity of nearly isostatic lattices: randomness and dissipation

The square lattice with nearest neighbor central-force springs is isostatic and does not support shear. Using the coherent potential approximation (CPA), we study how the random addition, with probability P=(z-4)/4 (z=average number of contacts), of next-nearest-neighbor (NNN) springs restores rigidity and affects phonon structure. The CPA effective NNN spring constant kappa{m}(omega), equivalent to the complex shear modulus G(omega), obeys the scaling relation, kappa{m}(omega)=kappa{m}h(omega/omega{*}), at small P, where kappa{m}=kappa{m}{'}(0) approximately P{2} and omega{*} approximately P, implying nonaffine elastic response at small P and the breakdown of plane-wave states beyond the Ioffe-Regel limit at omega approximately omega{*}. We identify a divergent length l{*} approximately P{-1}, and we relate these results to jamming.

Heat transport in model jammed solids

We calculate numerically the normal modes of vibrations in three-dimensional jammed packings of soft spheres as a function of the packing fraction and obtain the energy diffusivity, a spectral measure of transport that controls sound propagation and thermal conductivity. The crossover frequency between weak and strong phonon scattering is controlled by the coordination and shifts to zero as the system is decompressed toward the critical packing fraction at which rigidity is lost. We present a scaling analysis that relates the packing fraction dependence of the crossover frequency to the anomalous scaling of the shear modulus with compression. Below the crossover, the diffusivity displays a power-law divergence with inverse frequency consistent with Rayleigh law, which suggests that the vibrational modes are primarily transverse waves, weakly scattered by disorder. Above it, a large number of modes appear whose diffusivity plateaus at a nearly constant value before dropping to zero above the localization frequency. The thermal conductivity of a marginally jammed solid just above the rigidity threshold is calculated and related to the one measured experimentally at room temperature for most glasses.

Jamming of soft particles: geometry, mechanics, scaling and isostaticity

Amorphous materials as diverse as foams, emulsions, colloidal suspensions and granular media can jam into a rigid, disordered state where they withstand finite shear stresses before yielding. Here we review the current understanding of the transition to jamming and the nature of the jammed state for disordered packings of particles that act through repulsive contact interactions and are at zero temperature and zero shear stress. We first discuss the breakdown of affine assumptions that underlies the rich mechanics near jamming. We then extensively discuss jamming of frictionless soft spheres. At the jamming point, these systems are marginally stable (isostatic) in the sense of constraint counting, and many geometric and mechanical properties scale with distance to this jamming point. Finally, we discuss current explorations of jamming of frictional and non-spherical (ellipsoidal) particles. Both friction and asphericity tune the contact number at jamming away from the isostatic limit, but in opposite directions. This allows one to disentangle the distance to jamming and the distance to isostaticity. The picture that emerges is that most quantities are governed by the contact number and scale with the distance to isostaticity, while the contact number itself scales with the distance to jamming.

Jammed frictionless disks: Connecting local and global response

By calculating the linear response of packings of soft frictionless disks to quasistatic external perturbations, we investigate the critical scaling behavior of their elastic properties and nonaffine deformations as a function of the distance to jamming. Averaged over an ensemble of similar packings, these systems are well described by elasticity, while in single packings we determine a diverging length scale l* up to which the response of the system is dominated by the local packing disorder. This length scale, which we observe directly, diverges as 1/Deltaz , where Deltaz is the difference between contact number and its isostatic value, and appears to scale identically to the length scale which had been introduced earlier in the interpretation of the spectrum of vibrational modes. It governs the crossover from isostatic behavior at the small scale to continuum behavior at the large scale; indeed we identify this length scale with the coarse graining length needed to obtain a smooth stress field. We characterize the nonaffine displacements of the particles using the displacement angle distribution, a local measure for the amount of relative sliding, and analyze the connection between local relative displacements and the elastic moduli.

Elasticity and response in nearly isostatic periodic lattices

The square and kagome lattices with nearest-neighbor springs of spring constant k are isostatic with a number of zero-frequency modes that scale with their perimeter. We analytically study the approach to this isostatic limit as the spring constant k' for next-nearest-neighbor bonds vanishes. We identify a characteristic frequency omega* approximately square root of k' and length l* approximately square root of k/k' for both lattices. The shear modulus C(44) = k' of the square lattice vanishes with k', but that for the kagome lattice does not.

Modeling of acoustic penetration into sandy sediments: physical and geometrical aspects

Two different approaches to the problem of acoustic penetration into sandy marine sediments are considered: application of the Buckingham constitutive model for sediment with a plane surface and boundary element analysis of a rough surface of sediment represented as a homogeneous fluid. By a careful modeling of the constitutive behavior for plane seafloors, it is possible to partly reproduce some features of known experimental dependencies for acoustical pressure. However, accounting for roughness appears to be more important. Accordingly, the authors present a detailed numerical analysis of penetration into rough sediments using the boundary element method. The simulation results support conclusions reached by other investigators and demonstrate how local surface irregularities violate the evanescence condition that holds for a plane interface at subcritical incidence, thus considerably increasing penetration. The results apply to the frequency range 0.5-50 kHz and grazing angles larger than approximately 6 degrees -8 degrees at 10-50 kHz. For lower frequencies, when diffraction becomes important, the lowest possible grazing angle strongly depends on the range covered by the incident beam and is, in general, considerably larger. The authors provide several characteristic examples with frequencies 5 and 15 kHz and grazing angles 15 degrees -30 degrees illustrating the impact of roughness on penetration.

Long-wavelength structural anomalies in jammed systems

The structural properties of static, jammed packings of monodisperse spheres in the vicinity of the jamming transition are investigated using large-scale computer simulations. At small wave number k , we argue that the anomalous behavior in the static structure factor, S(k) approximately k , is consequential of an excess of low-frequency, collective excitations seen in the vibrational spectrum. This anomalous feature becomes more pronounced closest to the jamming transition, such that S(0)-->0 at the transition point. We introduce an appropriate dispersion relation that accounts for these phenomena that leads us to relate these structural features to characteristic length scales associated with the low-frequency vibrational modes of these systems. When the particles are frictional, this anomalous behavior is suppressed providing yet more evidence that the jamming transitions for frictional spheres lie at lower packing fractions than for frictionless spheres. These results suggest that the mechanical properties of jammed and glassy media may therefore be inferred from measurements of both the static and dynamical structure factors.

Jamming versus glass transitions

Recent ideas based on the properties of assemblies of frictionless particles in mechanical equilibrium provide a perspective of amorphous systems different from that offered by the traditional approach originating in liquid theory. The relation, if any, between these two points of view, and the relevance of the former to the glass phase, has been difficult to ascertain. In this Letter, we introduce a model for which both theories apply strictly: it exhibits on the one hand an ideal glass transition and on the other "jamming" features (fragility, soft modes) virtually identical to that of real systems. This allows us to disentangle the two physical phenomena.

Critical behaviors of sheared frictionless granular materials near the jamming transition

Critical behaviors of sheared dense and frictionless granular materials in the vicinity of the jamming transition are numerically investigated. From the extensive molecular dynamics simulation, we verify the validity of the scaling theory near the jamming transition proposed by Otsuki and Hayakawa [Prog. Theor. Phys. 121, 647 (2009)]]. We also clarify the critical behaviors of the shear viscosity and the pair correlation function based on both a mean field theory and the simulation.

Statistical mechanics framework for static granular matter

The physical properties of granular materials have been extensively studied in recent years. So far, however, there exists no theoretical framework which can explain the observations in a unified manner beyond the phenomenological jamming diagram. This work focuses on the case of static granular matter, where we have constructed a statistical ensemble which mirrors equilibrium statistical mechanics. This ensemble, which is based on the conservation properties of the stress tensor, is distinct from the original Edwards ensemble and applies to packings of deformable grains. We combine it with a field theoretical analysis of the packings, where the field is the Airy stress function derived from the force and torque balance conditions. In this framework, Point J characterized by a diverging stiffness of the pressure fluctuations. Separately, we present a phenomenological mean-field theory of the jamming transition, which incorporates the mean contact number as a variable. We link both approaches in the context of the marginal rigidity picture proposed by Wyart and others.

Jamming transition as probed by quasistatic shear flow

We study the rheology of amorphous packings of soft, frictionless particles close to jamming. Implementing a quasistatic simulation method we generate a well-defined ensemble of states that directly samples the system at its yield stress. A continuous jamming transition from a freely flowing state to a yield-stress situation takes place at a well-defined packing fraction, where the scaling laws characteristic of isostatic solids are observed. We propose that long-range correlations observed below the transition are dominated by this isostatic point, while those that are observed above the transition are characteristic of dense, disordered elastic media.

Growing length and time scales in a suspension of athermal particles

We simulate a relaxation process of non-Brownian particles in a sheared viscous medium; the system is subject to the small shear strain and then undergoes relaxation. We estimate the exponents with which the relaxation time and the correlation length diverge as the density approaches the jamming density from below. In particular, the dynamic critical exponent is estimated as 4.6(2). It is also found that shear stress undergoes power-law decay at the jamming density, which is reminiscent of critical slowing down.

Critical scaling and aging in cooling systems near the jamming transition

We conduct athermal simulations of freely cooling, viscous soft spheres around the jamming transition density varphi(J) and find evidence for a growing length xi(t) that governs relaxation to mechanical equilibrium. xi(t) is manifest in both the velocity correlation function and the spatial correlations in a scalar measure of local force balance which we define. Data for different densities varphi can be collapsed onto two master curves by scaling xi(t) and t by powers of |varphi-varphi(J)|, indicative of critical scaling. Furthermore, particle transport for varphi>varphi(J) exhibits aging and superdiffusion similar to a range of soft matter experiments, suggesting a common origin. Finally, we explain how xi(t) at late times maps onto known behavior away from varphi(J).

Normal modes in model jammed systems in three dimensions

Vibrational spectra and normal modes of mechanically stable particle packings in three dimensions are analyzed over a range of compressions, from near the jamming transition, where the packings lose their rigidity, to far above it. At high frequency, the normal modes are localized at all compressions. At low frequency, the nature of the modes depends somewhat on compression. At large compressions, far from the transition, the lowest-frequency normal modes have some plane-wave character, though less than one would expect for a crystalline or isotropic solid. At low compressions near the jamming transition, the lowest-frequency modes are neither plane-wave-like nor localized. We characterize these differences, highlighting the unusual dispersion behavior that emerges for marginally jammed solids.

Energy transport in jammed sphere packings

We calculate the normal modes of vibration in jammed sphere packings to obtain the energy diffusivity, a spectral measure of transport. At the boson peak frequency, we find an Ioffe-Regel crossover from a diffusivity that drops rapidly with frequency to one that is nearly frequency independent. This crossover frequency shifts to zero as the system is decompressed towards the jamming transition, providing unambiguous evidence of a regime in frequency of nearly constant diffusivity. Such a regime, postulated to exist in glasses to explain the temperature dependence of the thermal conductivity, therefore appears to arise from properties of the jamming transition.

Elasticity of floppy and stiff random networks

We study the linear and nonlinear elastic behavior of amorphous systems using a two-dimensional random network of harmonic springs as a model system. A natural characterization of these systems arises in terms of the network coordination (average number of springs per node) relative to that of a marginally rigid network deltaz: a floppy network has deltaz<0, while a stiff network has deltaz>0. Under the influence of an externally applied load, we observe that the response of both floppy and stiff networks is controlled by the critical point corresponding to the onset of rigidity. We use numerical simulations to compute the exponents which characterize the shear modulus, the heterogeneity of the response, and the network stiffening as a function of deltaz and derive these theoretically, thus allowing us to predict aspects of the mechanical response of glasses and fibrous networks.

Evidence of rayleigh-hertz surface waves and shear stiffness anomaly in granular media

Using the nonlinear dependence of sound propagation speed with pressure, we evidence the anomalous elastic softness of a granular packing in the vicinity of the jamming transition. Under gravity and close to a free surface, the acoustic propagation is only possible through surface modes guided by the stiffness gradient. These Rayleigh-Hertz modes are evidenced in a controlled laboratory experiment. The shape and the dispersion relation of both transverse and sagittal modes are compared to the prediction of nonlinear elasticity including finite size effects. These results allow one to access the elastic properties of the packing under vanishing confining pressure.

Acoustic probing of the jamming transition in an unconsolidated granular medium

Experimentally determined dispersion relations for acoustic waves guided along the mechanically free surface of an unconsolidated granular packed structure provide information on the elasticity of granular media at very low pressures that are naturally controlled by the gravitational acceleration and the depth beneath the surface. The experiments confirm recent theoretical predictions that relaxation of the disordered granular packing through nonaffine motion leads to a peculiar scaling of shear rigidity with pressure near the jamming transition corresponding to zero pressure.

Critical scaling of shear viscosity at the jamming transition

We carry out numerical simulations to study transport behavior about the jamming transition of a model granular material in two dimensions at zero temperature. Shear viscosity eta is computed as a function of particle volume density rho and applied shear stress sigma, for diffusively moving particles with a soft core interaction. We find an excellent scaling collapse of our data as a function of the scaling variable sigma/|rho(c)-rho|(Delta), where rho(c) is the critical density at sigma=0 ("point J"), and Delta is the crossover scaling critical exponent. We define a correlation length xi from velocity correlations in the driven steady state and show that it diverges at point J. Our results support the assertion that jamming is a true second-order critical phenomenon.

Underconstrained jammed packings of nonspherical hard particles: ellipses and ellipsoids

Continuing on recent computational and experimental work on jammed packings of hard ellipsoids [Donev, Science 303, 990 (2004)] we consider jamming in packings of smooth strictly convex nonspherical hard particles. We explain why an isocounting conjecture, which states that for large disordered jammed packings the average contact number per particle is twice the number of degrees of freedom per particle (Z[over]=2d{f}) , does not apply to nonspherical particles. We develop first- and second-order conditions for jamming and demonstrate that packings of nonspherical particles can be jammed even though they are underconstrained (hypoconstrained, Z[over]<2d{f}). We apply an algorithm using these conditions to computer-generated hypoconstrained ellipsoid and ellipse packings and demonstrate that our algorithm does produce jammed packings, even close to the sphere point. We also consider packings that are nearly jammed and draw connections to packings of deformable (but stiff) particles. Finally, we consider the jamming conditions for nearly spherical particles and explain quantitatively the behavior we observe in the vicinity of the sphere point.

Well defined transition to gel-like aggregates of attractive athermal particles

In an attempt to extend the range of model jamming transitions, we simulate systems of athermal particles which attract when slightly overlapping. Following from recent work on purely repulsive systems, dynamics are neglected and relaxation performed via a potential energy minimisation algorithm. Our central finding is of a transition to a low-density tensile solid which is sharp in the limit of infinite system size. The critical density depends on the range of the attractive regime in the pair-potential. Furthermore, solidity is shown to be related to the coordination number of the packing according to the approximate constraint-counting scheme known as Maxwell counting, although more corrections need to be considered than with the repulsive-only case, as explained. We finish by discussing how the numerical difficulties encountered in this work could be overcome in future studies.

Critical and noncritical jamming of frictional grains

We probe the nature of the jamming transition of frictional granular media by studying their vibrational properties as a function of the applied pressure p and friction coefficient mu. The density of vibrational states exhibits a crossover from a plateau at frequencies omega > or similar to omega*(p,mu) to a linear growth for omega < or similar to omega*(p,mu). We show that omega* is proportional to Deltaz, the excess number of contacts per grain relative to the minimally allowed, isostatic value. For zero and infinitely large friction, typical packings at the jamming threshold have Deltaz-->0, and then exhibit critical scaling. We study the nature of the soft modes in these two limits, and find that the ratio of elastic moduli is governed by the distance from isostaticity.

Force mobilization and generalized isostaticity in jammed packings of frictional grains

We show that in slowly generated two-dimensional packings of frictional spheres, a significant fraction of the friction forces lie at the Coulomb threshold-for small pressure p and friction coefficient mu , about half of the contacts. Interpreting these contacts as constrained leads to a generalized concept of isostaticity, which relates the maximal fraction of fully mobilized contacts and contact number. For p-->0 , our frictional packings approximately satisfy this relation over the full range of mu . This is in agreement with a previous conjecture that gently built packings should be marginal solids at jamming. In addition, the contact numbers and packing densities scale with both p and mu .

Surface elastic waves in granular media under gravity and their relation to booming avalanches

Due to the nonlinearity of Hertzian contacts, the speed of sound c in granular matter is expected to increase with pressure as P(1/6). A static layer of grains under gravity is thus stratified so that the bulk waves are refracted toward the surface. The reflection at the surface being total, there is a discrete number of modes (both in the sagittal plane and transverse to it) localized close to the free surface. The shape of these modes and the corresponding dispersion relation are investigated in the framework of an elastic description taking into account the main features of granular matter: Nonlinearity between stress and strain and the existence of a yield transition. We show in this context that the surface modes localized at the free surface exhibit a waveguide effect related to the nonlinear Hertz contact. Recent results about the song of dunes are reinterpreted in light of the theoretical results. The predicted propagation speed is compared with measurements performed in the field. Taking into account the finite depth effects, we show that the booming instability threshold can be explained quantitatively by a waveguide cutoff frequency below which no sound can propagate. Therefore, we propose another look at a recent controversy, confirming that the song of dunes can well originate from a coupling between avalanching grains and surface elastic waves once the specificity of surface waves (we baptized Rayleigh-Hertz) is correctly taken into account.

Critical scaling in linear response of frictionless granular packings near jamming

We study the origin of the scaling behavior in frictionless granular media above the jamming transition by analyzing their linear response. The response to local forcing is non-self-averaging and fluctuates over a length scale that diverges at the jamming transition. The response to global forcing becomes increasingly nonaffine near the jamming transition. This is due to the proximity of floppy modes, the influence of which we characterize by the local linear response. We show that the local response also governs the anomalous scaling of elastic constants and contact number.

Force heterogeneities in particle assemblies: from order to disorder

The effect of increasing structural disorder on the distribution of contact forces P(f) inside three dimensional particle assemblies is systematically studied using computer simulations of model granular packings. Starting from a face-centered-cubic array, where all contact forces are identical, an increasing number of defects is introduced into the assembly, after which the system is then allowed to relax into a different mechanically stable state. Three distinct protocols for imposing disorder are compared. A quantitative measure of the disorder is obtained from distributions of the coordination number and three-particle contact angle. The distribution of normal contact forces show dramatic qualitative changes with increasing disorder. In the regime where the disorder is relatively weak, the pressure and the lowest normal mode frequency scale approximately linearly in the coordination number, with distance from the crystalline state. These results for P(f) are discussed in the context of jamming phenomena in glassy and granular materials.