Session J53: Disordered Systems: Jamming

The contact percolation transition

Typical quasistatic compression algorithms for generating jammed packings of athermal, purely repulsive particles begin with dilute configurations and then apply successive compressions with relaxation of the elastic energy allowed between each compression step. It is well-known that during isotropic compression athermal systems with purely repulsive interactions undergo a jamming transition at packing fraction $\phi_J$ from an unjammed state with zero pressure to a jammed, rigid state with nonzero pressure. Using extensive computer simulations, we show that a novel second-order-like, contact percolation, which signals the formation of a system-spanning cluster of mutually contacting particles, occurs at $\phi_P < \phi_J$, preceding the jamming transition. By measuring the number of non-floppy modes of the dynamical matrix, the displacement field between successive compression steps, and the overlap between the adjacency matrix, which represents the network of contacting grains, at $\phi$ and $\phi_J$, we find that the contact percolation transition also heralds the onset of nontrivial response to applied stress. Highly heterogeneous, cooperative, and non-affine particle motion occurs in unjammed systems significantly below the jamming transition for $\phi_P < \phi < \phi_J$,   

Constraint counting for frictional jamming

While the frictionless jamming transition has been intensely studied in recent years, more realistic frictional packings are less well understood. In frictionless sphere packings, the transition is predicted by a simple mean-field constraint counting argument, the isostaticity argument. For frictional packings, a modified constraint counting argument, which includes slipping contacts at the Coulomb threshold, has had limited success in accounting for the transition. We propose that the frictional jamming transition is not mean field and is triggered by the nucleation of unstable regions, which are themselves dynamical objects due to the Coulomb criterion. We create frictional packings using MD simulations and test for the presence and shape of rigid clusters with the pebble game to identify the partition of the packing into stable and unstable regions. To understand the dynamics of these unstable regions we follow perturbations at contacts crucial to the stability of the ``frictional house of cards.''   

Decoupling of Rotational and Translational Diffusion in a 2D Granular Experiment

We experimentally study the rotation and diffusion of granular clusters in a 2D binary granular system. Our apparatus vibrates a 2D system of densely packed granular bidisperse disks (to avoid crystallization) containing trackable 3-particle clusters. We use this system to mimic hard-sphere fluids and the clusters probe the system's local translational and rotational dynamics. As the area fraction of the bidisperse disks is increased, diffusion within the sample becomes slower, and above a critical area fraction, the sample behaves as a granular glass. We analyze the rotational and translational motions of the clusters to determine whether they decouple with changing area fraction of the system. As we approach the glass transition, we observe a decoupling between the two motions.   

Characterizing the clogging transition by individual grain behavior

Granular media clogs as it flows out of a hopper when the exit hole is appropriately small. However, when the hole enlarged, the grains will never clog: there exists a well-defined transition between these two regimes at a particular critical hole size. To understand the origin of this clogging transition we follow the behavior of individual grains in the bulk. Using a quasi-2D hopper and a fast CCD, we can measure the positions of grains to sub-pixel accuracy. We then use this information to determine various local and global properties, including strain rate, velocity correlations, and dynamical heterogeneity time scales. We study how these quantities depend on the distance to the clogging transition, defined as the difference between the hole size and the critical hole size. This helps to explain the clogging transition and its relationship to the jamming transition.   

Jamming around fixed obstacles

Lattices of obstacles, such as microfluidic arrays, are capable of filtering or sorting particles like emulsion droplets, colloidal particles, and even cells. We study the jamming of soft, bidisperse discs placed within a lattice of fixed obstacles. Such obstacles provide a supporting structure for the jammed configuration, and their ability to alter a jamming threshold is of interest. Conjugate gradient methods are used to find minimum energy configurations, both with and without fixed obstacles. The likelihood of jamming as a function of disc volume fraction is calculated. If a configuration is jammed, the coordination number, energy, pressure, and other parameters of interest are calculated as a function of the obstacle size, number density, and configuration (hexagonal vs. square vs. random lattices).   

Frictional Jammed Packings: Classification, Protocol Dependence and the Phase Diagram

We probe the nature of the jamming transition in systems of frictional disks, where static friction is modeled geometrically using ``bumpy-particles" with uniform circular asperities on the disks' surface. First, we enumerate and classify the mechanically stable (MS) packings in small systems using exhaustive numerical simulations. We explicitly show that finite friction stabilizes packings that are unstable for frictionless particles, which causes the number of MS packings to increase strongly with the friction coefficient. MS packings for frictional particles are organized into low-dimensional geometric families in configuration space. We then calculate the critical behavior of the structural and mechanical properties near the jamming transition for frictional particles and as a function of protocol and show that friction drastically alters the nature of the transition.   

Jammed 2D circle packing reconsidered as a jigsaw puzzle

Athermal random packings are inherently non-equilibrium structures. For a bidisperse jammed packing of N disks the global packing structure can be thought of as composed of N jigsaw pieces, each representing the local structure around a disk. We show that we can assign a unique identifier, termed a jigsaw number, to each local packing structure. We find that as the number of disks grows to infinity the number of different jigsaw numbers present in a packing remains finite. We report on the distribution of jigsaw numbers and find that certain local packing structures are more common than others, demonstrating that the non-equilibrium packing structure is incompatible with a flat measure over all configurations. We further report on the correlations present between jigsaw pieces.   

An experimental test of equilibration of temperature-like variables in jammed granular materials

Although jammed granular systems are athermal, a number of thermodynamic-like descriptions have been proposed which make predictions about the distributions of volume and stress fluctuations. We perform experiments with an apparatus designed to generate a large number of jammed two-dimensional configurations, which consists of a single layer of photoelastic disks supported by a layer of air driven through a microporous membrane. New configurations are automatically generated by alternately dilating the system (permitting large-scale rearrangements) and compressing it biaxially until a desired volume or pressure is reached. Within each configuration, a bath of $\ga 10^3$ particles surrounds a smaller subsystem of particles with either the same or a different inter-particle friction coefficient than the bath.The use of photoelastic particles permits us to find all particle positions, and to numerically solve for the vector forces at each inter-particle contact. By comparing temperature-like quantities between subsystems, we test whether equilibration is observed under several proposed volume and stress ensembles.   

Energy decay of freely cooling granular gases in three dimensions

Freely cooling granular gases, wherein a dilute system of macroscopic particles with uncorrelated initial velocities lose energy through inelastic collisions, have been extensively studied both as a simple model for granular systems as well as a nonequilibrium system showing nontrivial coarsening at late times. As the system cools, inelasticity induces clustering, making the system inhomogeneous. While the form of energy decay ($E(t)\sim t^{-\theta}$) in the initial homogeneous regime is well established by Haff's law ($\theta=2$), the energy decay in the clustered regime is still unresolved in higher dimensions. Within mean field theory, $\theta=2 d/(d+2)$ (where $d$ is the spatial dimension), while a correspondence to Burgers equation implies an exponent $\theta= 2/3 (d=1), d/2 (d>1)$. In one and two dimensions, the two formulae predict the same exponents. By performing extensive event driven molecular dynamics simulations, we show that in three dimensions, the energy decays asymptotically with a power $\approx 1.2$, for all coefficients of restitution $r<1$, consistent with the mean field exponent. However, we argue that the mean field arguments fail due to non local interactions between mass clusters.   

The Influence of Topology on Signal Propagation in Granular Force Networks

Granular materials exhibit numerous rich and complex behaviours, which have been investigated from both continuum and particulate perspectives. In particular, sound propagation through granular materials is both heterogeneous and complicated, and understanding its features is important not only from the perspective of fundamental physics but also for practical applications such as the characterization and non-destructive testing of such materials. Unfortunately, continuum models of sound propagation have been unable to explain the full range of observed behaviours. Here we represent granular materials as spatially-embedded networks composed of nodes (particles) and weighted edges (contact forces between particles) located in Euclidean space, and we use network science to provide fundamental insights into how sound propagates. Using photoelastic particles, we quantitatively characterise the internal force structure and show that its meso-scale network structure plays a crucial role in sound propagation. These results might help to explain the failure of previous physical models, and illustrate that contact topology alone is insufficient to understand signal propagation in granular materials.   

Jamming of Brownian disks in a channel with a constriction

We investigate jamming dynamics of an externally driven system of Brownian particles in a two-dimensional channel with an abrupt constriction. Our numerical simulations reveal a rich dynamical behavior that results from an interplay between external driving forces, thermal excitations, and geometrical constraints due to confinement. In particular, we have observed that channel blockage arising from particle accumulation at the constriction entrance alternates with sudden unjamming events originating from thermal fluctuations. We have also found that the dependence of the average particle flux on the channel width is non-monotonic as a result of strong spatial particle-wall correlations. Under some conditions there exist spontaneous particle ordering and dynamic switching between phases with square and hexagonal symmetry. We expect that similar phenomena can be observed in confined granular flows and in suspension flows in microchannels.   

To be or not to be jammed

When are packings of soft athermal spheres jammed? Any experimentally relevant definition must at least require a jammed packing to resist compression and shear. Numerical algorithms usually rely on a global compression monitored by a parameter (like pressure) that signals whether the packing is jammed or not. Here we show that compression is not sufficient to ensure properly jammed packings : some of those packings have positive pressures and bulk moduli, but negative shear moduli, and even for large systems, the number of these ``bad apples'' diverges as the jamming point is approached. We will discuss how to understand this situation and propose as a remedy the boundary relaxation, that is including the boundary shape parameters as variables in the equilibration process; finally we will compare the distribution of shear moduli obtained for both methods.   

Jamming, Yielding and Rheology of Weakly Vibrated Granular Media

We establish that the rheological curve of dry granular media is nonmonotonic, both in the presence and absence of external mechanical agitations. In the absence of vibrations, the nonmonotonic flow curve governs the yielding behavior of granular media. In the presence of weak vibrations, the nonmonotonic flow curves govern a hysteretic transition between slow but steady and fast, inertial flows. For large agitations, the transition becomes non-hysteretic. We probe the fluctuations near the point where the 1st order transition becomes of 2nd order.   

Dynamics near shear-jamming for a dense granular system

This talk will present several systematic experimental studies of a two-dimensional, frictional dense granular system subjected to simple shear deformation. The first experiment consists of linear shear for densities smaller than the isotropic jamming point, and examines both the evolution of the average stress and the evolution of force network. These measures reveal three distinguishable regimes of the granular system with increasing shear strain: unjammed, fragile, and shear-jammed regimes. The second experiment uses small amplitude cyclic shear to probe the dynamical response of the states from the first experiment. For fragile or jammed regimes, cyclic shear drives the system through transient states that evolve towards relatively stable forces networks and system-averaged stress. The timescale of the transient increases rapidly as the system moves deeper into the fragile, or shear-jammed regimes. These experiments also involve particle tracking (displacements and rotations) to search for and characterize non-affine motion and spatial heterogeneity. There is a clear increase in particle diffusion with increasing density and shear strain amplitude, even when the system is still unjammed and experiences only minimal stress. When the system is fragile or jammed, the heterogeneity of particle displacements reveals subtle correlations with the force network.   

Tricriticality in constraint percolation

Constraint percolation goes beyond ordinary percolation to include constraints on the occupation of sites/bonds. For instance, $k$-core site percolation implements a geometric constraint requiring each occupied vertex on a network have at least $k$ occupied neighboring vertices. It turns out that the percolation transition in such a model is essentially equivalent to the study of a dynamical glass transition in the Fredrickson-Andersen model, one of the models underlying the kinetically-constrained approach to the glass transition. We study hetereogenous $k$-core bond percolation on a random network with $f$ denoting the probability of a $k=2$-core vertex and $1-f$ the probability of a $k=3$-core vertex. This model corresponds to a hetereogeneous extension of the Frederickson-Anderson model. For $f=1$, the percolation transition is continuous, while for $f=0$, it is discontinuous. Using a master equation approach, we show that there exists a tricritical point at $f=1/2$ with a new order parameter exponent of unity. Our results are consistent with other mean field results obtained via a different method. We also look for tricriticality beyond mean field by investigating another constraint percolation model dubbed force-balance percolation.