A granular-matter model is exactly solved, where disks of two sizes and weights in alternating sequence are confined to a narrow channel. The axis of the channel is horizontal and its plane vertical. Disk sizes and channel width are such that under jamming no disks remain loose and all disks touch one wall. Jammed microstates are characterized via statistically interacting particles constructed out of two-disk tiles. Jammed macrostates depend on measures of expansion work, gravitational potential energy, and intensity of random agitations before jamming. The dependence of configurational entropy on excess volume exhibits a critical point.
Disks of two sizes and weights in alternating sequence are confined to a long and narrow channel. The axis of the channel is horizontal and its plane vertical. The channel is closed off by pistons that freeze jammed microstates out of loose disk configurations subject to moderate pressure, gravity, and random agitations. Disk sizes and channel width are such that under jamming no disk remains loose and all disks touch one wall. We present exact results for the characterization of jammed macrostates including volume and entropy. The rigorous analysis divides the disk sequences of jammed microst…
Granular packings of hard discs are investigated by means of contact dynamics which is an appropriate technique to explore the allowed force-realizations in the space of contact forces. Configurations are generated for given values of the friction coefficient, and then an ensemble of equilibrium forces is found for fixed contacts. We study the force fluctuations within this ensemble. In the limit of zero friction the fluctuations vanish in accordance with the isostaticity of the packing. The magnitude of the fluctuations has a non-monotonous friction dependence. The increase for small friction…
We investigate the mechanical response of jammed packings of circulo-lines, interacting via purely repulsive, linear spring forces, as a function of pressure $P$ during athermal, quasistatic isotropic compression. Prior work has shown that the ensemble-averaged shear modulus for jammed disk packings scales as a power-law, $\langle G(P) \rangle \sim P^β$, with $β\sim 0.5$, over a wide range of pressure. For packings of circulo-lines, we also find robust power-law scaling of $\langle G(P)\rangle$ over the same range of pressure for aspect ratios ${\cal R} \gtrsim 1.2$. However, the power-law sca…
We numerically study the distributions of global pressure that are found in ensembles of statically jammed and quasistatically sheared systems of bidisperse, frictionless, disks at fixed packing fraction $ϕ$ in two dimensions. We use these distributions to address the question of how pressure increases as $ϕ$ increases above the jamming point $ϕ_J$, $p\sim |ϕ- ϕ_J|^y$. For statically jammed ensembles, our results are consistent with the exponent $y$ being simply related to the power law of the interparticle soft-core interaction. For sheared systems, however, the value of $y$ is consistent wit…
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 $0.41269\dots$. 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.
Dynamics of jammed packings of soft athermal disks under finite-rate shear are studied by means of molecular dynamics simulations. Particularly, we investigate the spatial structures of stress drop events, which are expected to provide information about plasticity. Investigating the displacement fields during stress drop events, we show that there are qualitatively different two types of events in the low rate limit: localized ones and extended ones. We further investigate the time evolution of events and clarify that both types of events are due to oscillatory motion of the stress, which is u…
Numerous soft materials jam into an amorphous solid at high packing fraction. This non-equilibrium phase transition is best understood in the context of a model system in which particles repel elastically when they overlap. Recently, however, it was shown that introducing any finite amount of attraction between particles changes the universality class of the transition. The properties of this new ``sticky jamming'' class remain almost entirely unexplored. We use molecular dynamics simulations and scaling analysis to determine the shear modulus, bulk modulus, and coordination of marginal solids…
We investigate the jamming transition in a quasi-2D granular material composed of regular pentagons or disks subjected to quasistatic uniaxial compression. We report six major findings based on experiments with monodisperse photoelastic particles with static friction coefficient $μ\approx 1$. (1) For both pentagons and disks, the onset of rigidity occurs when the average coordination number of non-rattlers, $Z_{nr}$ , reaches 3, and the dependence of $Z_{nr}$ on the packing fraction $ϕ$ changes again when $Z_{nr}$ reaches 4. (2) Though the packing fractions $ϕ_{c1}$ and $ϕ_{c2}$ at these trans…
The inherent structure landscape for a system of hard spheres confined to a hard cylindrical channel, such that spheres can only contact their first and second neighbours, is studied using an analytical model that extends previous results [Phys. Rev. Lett. 115, 025702 (2015)] to provide a comprehensive picture of jammed packings over a range of packing densities. In the model, a packing is described as an arrangement of $k$ helical sections, separated by defects, that have alternating helical twist directions and where all spheres satisfy local jamming constraints. The structure of each helica…
We perform extensive computational studies of two-dimensional static bidisperse disk packings using two distinct packing-generation protocols. The first involves thermally quenching equilibrated liquid configurations to zero temperature over a range of thermal quench rates $r$ and initial packing fractions followed by compression and decompression in small steps to reach packing fractions $ϕ_J$ at jamming onset. For the second, we seed the system with initial configurations that promote micro- and macrophase-separated packings followed by compression and decompression to $ϕ_J$. We find that am…
We create collectively jammed (CJ) packings of 50-50 bidisperse mixtures of smooth disks in 2d using an algorithm in which we successively compress or expand soft particles and minimize the total energy at each step until the particles are just at contact. We focus on small systems in 2d and thus are able to find nearly all of the collectively jammed states at each system size. We decompose the probability $P(ϕ)$ for obtaining a collectively jammed state at a particular packing fraction $ϕ$ into two composite functions: 1) the density of CJ packing fractions $ρ(ϕ)$, which only depends on geome…
We compare the structural and mechanical properties of mechanically stable (MS) packings of frictional disks in two spatial dimensions (2D) generated with isotropic compression and simple shear protocols from discrete element modeling (DEM) simulations. We find that the average contact number and packing fraction at jamming onset are similar (with relative deviations $< 0.5\%$) for MS packings generated via compression and shear. In contrast, the average stress anisotropy $\langle {\hat Σ}_{xy} \rangle = 0$ for MS packings generated via isotropic compression, whereas $\langle {\hat Σ}_{xy} \ra…
Contact breaking and Hertzian interactions between grains can both give rise to nonlinear vibrational response of static granular packings. We perform molecular dynamics simulations at constant energy in 2D of frictionless bidisperse disks that interact via Hertzian spring potentials as a function of energy and measure directly the vibrational response from the Fourier transform of the velocity autocorrelation function. We compare the measured vibrational response of static packings near jamming onset to that obtained from the eigenvalues of the dynamical matrix to determine the temperature ab…
We investigate the short, medium, and long-range structure of soft disk configurations for a wide range of area fractions and simulation protocols by converting the real-space spectrum of volume fraction fluctuations for windows of width $L$ to the distance $h(L)$ from the window boundary over which fluctuations occur. Rapidly quenched unjammed configurations exhibit size-dependent super-Poissonian long-range features that, surprisingly, approach the totally-random limit even close to jamming. Above and just below jamming, the spectra exhibit a plateau, $h(L)=h_e$, for $L$ larger than particle…
We relate the structure factor $S(\mathbf{k} \to \mathbf{0})$ in a system of jammed hard spheres of number density $ρ$ to its complexity per particle $Σ(ρ)$ by the formula $S(\mathbf{k} \to \mathbf{0})=-1/ [ρ^2Σ''(ρ)+2ρΣ'(ρ)]$. We have verified this formula for the case of jammed disks in a narrow channel, for which it is possible to find $Σ(ρ)$ and $S(\mathbf{k})$ analytically. Hyperuniformity, which is the vanishing of $S(\mathbf{k} \to \mathbf{0})$, will therefore not occur if the complexity is nonzero. An example is given of a jammed state of hard disks in a narrow channel which is hyperun…
We perform computational studies of repulsive, frictionless disks to investigate the development of stress anisotropy in mechanically stable (MS) packings. We focus on two protocols for generating MS packings: 1) isotropic compression and 2) applied simple or pure shear strain $γ$ at fixed packing fraction $ϕ$. MS packings of frictionless disks occur as geometric families (i.e. parabolic segments with positive curvature) in the $ϕ$-$γ$ plane. MS packings from protocol 1 populate parabolic segments with both signs of the slope, $dϕ/dγ>0$ and $dϕ/dγ<0$. In contrast, MS packings from protocol 2 p…
We have explored isotropically jammed states of semi-2D granular materials through cyclic compression. In each compression cycle, systems of either identical ellipses or bi-disperse disks, transition between jammed and unjammed states. We determine the evolution of the average pressure, P, and structure through consecutive jammed states. We observe a transition point, ϕm, above which P persists over many cycles; below ϕm, P relaxes slowly. The relaxation time scale associated with P increases with packing fraction, while the relaxation time scale for collective particle motion remains constant…
The shear response of soft solids can be modeled with linear elasticity, provided the forcing is slow and weak. Both of these approximations must break down when the material loses rigidity, such as in foams and emulsions at their (un)jamming point -- suggesting that the window of linear elastic response near jamming is exceedingly narrow. Yet precisely when and how this breakdown occurs remains unclear. To answer these questions, we perform computer simulations of stress relaxation and shear startup experiments in athermal soft sphere packings, the canonical model for jamming. By systematical…
It is well known that a binary system of non-active disks that experience driving in opposite directions exhibits jammed, phase separated, disordered, and laning states. In active matter systems, such as a crowd of pedestrians, driving in opposite directions is common and relevant, especially in conditions which are characterized by high pedestrian density and emergency. In such cases, the transition from laning to disordered states may be associated with the onset of a panic state. We simulate a laning system containing active disks that obey run-and-tumble dynamics, and we measure the drift …
We numerically study the dynamics and the phases of self-propelled disk-shaped particles of different sizes with soft repulsive potential in two dimensions. Size diversity is introduced by the polydispersity index (PDI) $ε$, which is the width of the uniform distribution of the particle's radius. The self-propulsion speed of the particles controls the activity $v$. We observe enhanced dynamics for large size diversity among the particles. We calculate the effective diffusion coefficient $D_{eff}$ in the steady-state. The system exhibits four distinct phases, jammed phase with small $D_{eff}$ f…
Although jammed granular systems are athermal, several thermodynamic-like descriptions have been proposed which make quantitative predictions about the distribution of volume and stress within a system and provide a corresponding temperature-like variable. We perform experiments with an apparatus designed to generate a large number of independent, jammed, two-dimensional configurations. Each configuration consists of a single layer of photoelastic disks supported by a gentle layer of air. New configurations are generated by alternately dilating and re-compacting the system through a series of …
We focus on the response of mechanically stable (MS) packings of frictionless, bidisperse disks to thermal fluctuations, with the aim of quantifying how nonlinearities affect system properties at finite temperature. Packings of disks with purely repulsive contact interactions possess two main types of nonlinearities, one from the form of the interaction potential and one from the breaking (or forming) of interparticle contacts. To identify the temperature regime at which the contact-breaking nonlinearities begin to contribute, we first calculated the minimum temperatures $T_{cb}$ required to b…
We numerically examine clogging transitions for bidisperse disks flowing through a two dimensional periodic obstacle array. We show that clogging is a probabilistic event that occurs through a transition from a homogeneous flowing state to a heterogeneous or phase separated jammed state where the disks form dense connected clusters. The probability for clogging to occur during a fixed time increases with increasing particle packing and obstacle number. For driving at different angles with respect to the symmetry direction of the obstacle array, we show that certain directions have a higher clo…
We propose a theoretical framework for predicting the protocol dependence of the jamming transition for frictionless spherical particles that interact via purely repulsive contact forces. We study isostatic jammed disk packings obtained via two protocols: isotropic compression and simple shear. We show that for frictionless systems, all jammed packings can be obtained via either protocol. However, the probability to obtain a particular jammed packing depends on the packing-generation protocol. We predict the average shear strain required to induce jamming in initially unjammed packings from th…
The pressure components of "soft" disks in a two dimensional narrow channel are analyzed in the dilute gas regime using the Mayer cluster expansion and molecular dynamics. Channels with either periodic or reflecting boundaries are considered. It is found that when the two-body potential, u(r), is singular at some distance r_0, the dependence of the pressure components on the channel width exhibits a singularity at one or more channel widths which are simply related to r_0. In channels with periodic boundary conditions and for potentials which are discontinuous at r_0, the transverse and longit…
Using simulations we examine a two-dimensional disk system in which two disk species are driven in opposite directions. We measure the average velocity of one of the species versus the applied driving force and identify four phases as function of drive and disk density: a jammed state, a completely phase separated state, a continuously mixing phase, and a laning phase. The transitions between these phases are correlated with jumps in the velocity-force curves that are similar to the behavior observed at dynamical phase transitions in driven particle systems with quenched disorder such as vorti…
We numerically simulate mechanically stable packings of soft-core, frictionless, bidisperse disks in two dimensions, above the jamming packing fraction $ϕ_J$. For configurations with a fixed isotropic global stress tensor, we compute the averages, variances, and correlations of conserved quantities (stress $Γ_{\cal C}$, force-tile area $A_{\cal C}$, Voronoi volume $V_{\cal C}$, number of particles $N_{\cal C}$, and number of small particles $N_{s{\cal C}}$) on compact subclusters of particles ${\cal C}$, as a function of the cluster size and the global system stress. We find several significan…
Positional ordering of a two-dimensional fluid of hard disks is examined in such narrow tubes where only the nearest-neighbor interactions take place. Using the exact transfer-matrix method the transverse and longitudinal pressure components and the correlation function are determined numerically. Fluid-solid phase transition does not occur even in the widest tube, where the method just loses its exactness, but the appearance of the dramatic change in the equation of state and the longitudinal correlation function shows that the system undergoes a structural change from a fluid to a solid-like…
We study the effect of the presence of a regular substrate pattern on the irreversible adsorption of nanosized and colloid particles. Deposition of disks of radius $r_0$ is considered, with the allowed regions for their center attachment at the planar surface consisting of square cells arranged in a square lattice pattern. We study the jammed state properties of a generalized version of the random sequential adsorption model for different values of the cell size, $a$, and cell-cell separation, $b$. The model shows a surprisingly rich behavior in the space of the two dimensionless parameters $α…
The granular jamming transition is experimentally investigated in a two-dimensional system of frictional, bi-dispersed disks subject to quasi-static, uniaxial compression at zero granular temperature. Currently accepted results show the jamming transition occurs at a critical packing fraction $ϕ_c$. In contrast, we observe the first compression cycle exhibits {\it fragility} - metastable configuration with simultaneous jammed and un-jammed clusters - over a small interval in packing fraction ($ϕ_1 < ϕ< ϕ_2$). The fragile state separates the two conditions that define $ϕ_c$ with an exponential …
We numerically examine ballistic active disks driven through a random obstacle array. Formation of a pinned or clogged state occurs at much lower obstacle densities for the active disks than for passive disks. As a function of obstacle density we identify several distinct phases including a depinned fluctuating cluster state, a pinned single cluster or jammed state, a pinned multicluster state, a pinned gel state, and a pinned disordered state. At lower active disk densities, a drifting uniform liquid forms in the absence of obstacles, but when even a small number of obstacles are introduced, …
We propose a `phase diagram' for particulate systems that interact via purely repulsive contact forces, such as granular media and colloidal suspensions. We identify and characterize two distinct classes of behavior as a function of the input kinetic energy per degree of freedom $T_0$ and packing fraction deviation above and below jamming onset $Δϕ=ϕ- ϕ_J$ using numerical simulations of purely repulsive frictionless disks. Iso-coordinated solids (ICS) only occur above jamming for $Δϕ> Δϕ_c(T_0)$; they possess average coordination number equal to the isostatic value ($< z> = z_{\rm iso}$) requi…
We investigate the existence of random close and random loose packing limits in two-dimensional packings of monodisperse hard disks. A statistical mechanics approach-- based on several approximations to predict the probability distribution of volumes-- suggests the existence of the limiting densities of the jammed packings according to their coordination number and compactivity. This result has implications for the understanding of disordered states in the disk packing problem as well as the existence of a putative glass transition in two dimensional systems.
We numerically simulate mechanically stable packings of soft-core, frictionless, bidisperse disks in two dimensions, above the jamming packing fraction phi_J. For configurations with a fixed isotropic global stress tensor, we investigate the fluctuations of the local packing fraction phi(r) to test whether such configurations display the hyperuniformity that has been claimed to exist exactly at phi_J. For our configurations, generated by a rapid quench protocol, we find that hyperuniformity persists only out to a finite length scale, and that this length scale appears to remain finite as the s…
Jammed soft disks exhibit avalanches of particle rearrangements under quasistatic shear. We follow the avalanches using steepest descent to decompose them into individual localized rearrangements. We characterize the local structural environment of each particle by a machine-learned quantity, softness, designed to be highly correlated with rearrangements, and analyze the interplay between softness, rearrangements and strain. Our findings form the foundation of an augmented elastoplastic model that includes local structure.
We use an exact transfer-matrix approach to compute the equilibrium properties of a system of hard disks of diameter $σ$ confined to a two-dimensional channel of width $1.95\,σ$ at constant longitudinal applied force. At this channel width, which is sufficient for next-nearest-neighbor disks to interact, the system is known to have a great many jammed states. Our calculations show that the longitudinal force (pressure) extrapolates to infinity at a well-defined packing fraction $ϕ_K$ that is less than the maximum possible $ϕ_{\rm max}$, the latter corresponding to a buckled crystal. In this qu…
Jammed soft matter systems are often modelled as dense packings of overlapping soft spheres, thus ignoring particle deformation. For 2D (and 3D) soft disks packings, close to the critical packing fraction $φ_c$, this results in an increase of the average contact number $Z$ with a square root in $φ-φ_c$. Using the program PLAT, we find that in the case of idealised two-dimensional foams, close to the wet limit, $Z$ increases linearly with $φ-φ_c$, where $φ$ is the gas fraction. This result is consistent with the different distributions of separations for soft disks and foams at the critical pac…
The jamming of bi-disperse soft core disks is considered, using a variety of different protocols to produce the jammed state. In agreement with other works, we find that cooling and compression can lead to a broad range of jamming packing fractions $ϕ_J$, depending on cooling rate and initial configuration; the larger the degree of big particle clustering in the initial configuration, the larger will be the value of $ϕ_J$. In contrast, we find that shearing disrupts particle clustering, leading to a much narrower range of $ϕ_J$ as the shear strain rate varies. In the limit of vanishingly small…
We carry out a finite size scaling analysis of the jamming transition in frictionless bi-disperse 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 $ν\approx 1$, 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 imp…
The thermodynamic properties of disks moving in a channel sufficiently narrow that they can collide only with their nearest neighbors can be solved exactly by determining the eigenvalues and eigenfunctions of an integral equation. Using it we have determined the correlation length $ξ$ of this system. We have developed an approximate solution which becomes exact in the high density limit. It describes the system in terms of defects in the regular zigzag arrangement of disks found in the high-density limit. The correlation length is then effectively the spacing between the defects. The time scal…
We investigate the unjamming response of disordered packings of frictional hard disks with the help of computer simulations. First, we generate jammed configurations of the disks and then force them to move again by local perturbations. We study the spatial distribution of the stress and displacement response and find long range effects of the perturbation in both cases. We record the penetration depth of the displacements and the critical force that is needed to make the system yield. These quantities are tested in two types of systems: in ideal homogeneous packings in zero gravity and in pac…
We simulate via a Discrete Element Method the tapping of a narrow column of disk under gravity. For frictionless disks, this system has a simple analytic expression for the density of states in the Edwards volume ensemble. We compare the predictions of the ensemble at constant compactivity against the results for the steady states obtained in the simulations. We show that the steady states cannot be properly described since the microstates sampled are not in correspondence with the predicted distributions, suggesting that the postulates of flat measure and ergodicity are, either or both, inval…
Two drastically different theories predict the marginal criticality of jamming. The full replica symmetry breaking (fullRSB) theory [1-4] predicts the power-law distributions of weak contact forces and small inter-particle gaps in infinite-dimensional hard-sphere glass, with two nontrivial exponents $θ_f=0.42311...$ and $γ=0.41269...$, respectively. While the marginal mechanical stability (MMS) analysis [5-8] predicts that the isostatic random packings of hard frictionless spheres under external stress are marginally stable and provides inequality relationships for the exponents of the weak-fo…
The diffusion of particles trapped in long narrow channels occurs predominantly in one dimension. Here, molecular dynamics simulation is used to study the inertial dynamics of two-dimensional hard disks, confined to long, narrow, structureless channels with hard walls in the no-passing regime. We show that the diffusion coefficient obtained from the mean squared displacement can be mapped onto the exact results for the diffusion of the strictly one-dimensional hard rod system through an effective occupied volume fraction obtained from either the equation of state or a geometric projection of t…
Contrary to the theoretical predictions that all waves in two-dimensional disordered materials are localized, Anderson localization is observed only for sufficiently high frequencies in an isotropically jammed two-dimensional disordered granular packing of photoelastic disks. More specifically, we have performed an experiment in analyzing the level statistics of normal mode vibrations. We observe delocalized modes in the low-frequency boson-peak regime and localized modes in the high frequency regime with the crossover frequency just below the Debye frequency. We find that the level-distance d…
