Session B52: Granular Media: Flow and Structure

Simulations of Granular Particles Under Cyclic Shear

We perform molecular dynamics (MD) simulations of spherical grains subjected to cyclic, quasi-static shear in a 3D parallelepiped shear cell. This virtual shear cell is constructed out of rough, bumpy walls in order to minimize wall-induced ordering and has an open top surface to allow the packing to readily dilate or compact. Using a standard routine for MD simulations of frictional grains, we simulate over 1000 shear cycles, measuring grain displacements, the local packing density and changes in the contact network. Varying the shear amplitude and the friction coefficient between grains, we map out a phase diagram for the different types of behavior exhibited by these sheared grains. With low friction and high enough shear, the grains can spontaneously order into densely packed crystals. With low shear and increasing friction the packing remains disordered, yet the grains arrange themselves into configurations which exhibit limit cycles where all grains return to the same position after each full shear cycle. At higher shear and friction there is a transition to a diffusive state, where grains continue rearrange and move throughout the shear cell.   

A Continuous Time Random Walk Description of Monodisperse, Hard-Sphere Colloids below the Ordering Transition

Diffusive transport is a ubiquitous process that is typically understood in terms of a classical random walk of non-interacting particles. Here we present the results for a model of hard-sphere colloids in a Newtonian incompressible solvent at various volume fractions below the ordering transition ($\sim $50{\%}). We numerically simulate the colloidal systems via Fast Lubrication Dynamics -- a Brownian Dynamics approach with corrected mean-field hydrodynamic interactions. Colloid-colloid interactions are also included so that we effectively solve a system of interacting Langevin equations. The results of the simulations are analyzed in terms of the diffusion coefficient as a function of time with the early and late time diffusion coefficients comparing well with experimental results. An interpretation of the full time dependent behavior of the diffusion coefficient and mean-squared displacement is given in terms of a continuous time random walk. Therefore, the deterministic, continuum diffusion equation which arises from the discrete, interacting random walkers is presented. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.   

Shear failure of granular materials

Connecting the macroscopic behavior of granular materials with the microstructure remains a great challenge. Recent work connects these scales with a discrete calculus [1]. In this work we generalize this formalism from monodisperse packings of disks to 2D assemblies of arbitrarily shaped grains. In particular, we derive Airy's expression for a symmetric, divergence-free stress tensor. Using these tools, we derive, from first-principles and in a mean-field approximation, the entropy of frictional force configurations in the Force Network Ensemble. As a macroscopic consequence of the Coulomb friction condition at contacts, we predict shear failure at a critical shear stress, in accordance with the Mohr-Coulomb failure condition well known in engineering. Results are compared with numerical simulations, and the dependence on the microscopic geometric configuration is discussed. \\[4pt] [1] E. DeGiuli \& J. McElwaine, PRE 2011. doi: 10.1103/PhysRevE.84.041310   

A simple analytic theory for the statistics of avalanches in sheared granular materials

Slowly sheared granular materials at high packing fractions deform via slip avalanches with a broad range of sizes. Conventional continuum descriptions are not expected to apply to such highly inhomogeneous, intermittent deformations. Here, we show that it is possible to analytically compute the dynamics using a simple model that is inherently discrete. This model predicts quantities such as the avalanche size distribution, power spectra and temporal avalanche profiles as functions of the grain number fraction and the frictional weakening. A dynamical phase diagram emerges with quasi-static avalanches at high number fractions, and more regular, fluid-like flow at lower number fractions. The predictions agree with experiments and simulations for different granular materials, motivate future experiments and provide a fresh approach to data analysis. The simplicity of the model reveals quantitative connections to plasticity and earthquake statistics. (Reference: K.A. Dahmen, Y. Ben-Zion, J.T. Uhl, Nature Physics 7, 554-557 (2011).)   

Simulation of 2D Granular Hopper Flow

Jamming and intermittent granular flow are big problems in industry, and the vertical hopper is a canonical example of these difficulties. We simulate gravity driven flow and jamming of 2D disks in a vertical hopper and compare with identical companion experiments presented in this session. We measure and compare the flow rate and probability for jamming as a function of particle properties and geometry. We evaluate the ability of standard Hertz-Mindlin contact mode to quantitatively predict the experimental flow.   

Forces on intruders in granular media

We measure the forces acting on intruders moving in different directions in a granular medium consisting of mono-disperse spherical glass beads. We present the dependence of the drag force on the intruder's geometry and surface roughness, bead size, dragging speed and immersion depth. We also determine the distribution of the forces on the intruder's surfaces. We compare our results with lithostatic pressure (p = $\rho$ gz).   

2D Granular Impact Dynamics with Photoelasic Particles

What is the response of a granular material to a high speed impact from a foreign object? To answer this question, we use a large 2D granular system which is impacted from above by an intruder. Using photoelastic discs and a high-speed camera (frame rates at 7,000-775,00 fps at varied resolution, typically 40,000 fps at 584x256 pixels), we are able to observe the dynamics in this process in a way which has not been done previously. Data consists of the trajectory of the intruder, as well as either particle positions or interparticle force information. High frame rates allow observation of complex acoustic waves during the impact process. We examine the effects of varying the initial velocity, density, shape, and size of the intruder, with the goal of extracting the grain-scale mechanisms responsible for the dissipation of the intruder's kinetic energy. In comparing our data to macroscopic frictional models used in past work, we observe good agreement with the low-frequency behavior in our experiments, but we also observe large high-frequency fluctuations in the acceleration which are inherently granular, and not captured by these models. The large fluctuations are well correlated to the emission of localized intermittent stress pulses, seen in the photoelastic response.   

Cooperative rotations of 2d frictional disks under oscillatory shear

We explore the dynamics of the contact network under cyclic shear, with a particular focus on cooperative rolling and sliding contacts, using a molecular dynamics (MD) simulation approach and external fixed pressure. We systematically study the formation and persistance of clusters of cooperatively rolling grains for a range of reversal amplitudes. The propensity for cooperative rotation dictates structural properties of the contact network: loop configurations of even numbers of grains are able to rotate without sliding, while odd numbers of grains must have at least one sliding contract. We report on the statistics of loop structures in the contact network, as well as their relationships to cooperatively rotating grains. Finally, we demonstrate a characteristic scale over which grains can cooperatively rotate as well as the dependence on friction parameters.   

High-speed measurement of axial grain transport in a rotating drum

Over short timescales granular mixtures separate by size when tumbled in a partially filled horizontal drum. The smaller grains move toward the axis of rotation to form a central core; undulations in this core gradually increase in amplitude until they grow into axial bands. Using non-invasive high-speed synchrotron x-ray particle tracking, we investigate the axial transport properties of tracer particles traveling amongst glass spheres. This new technique allows us to gather data on time scales not previously possible. When the tracers are present in larger proportions the mixtures we used should have different tendencies to segregate axially according to size ratio; one of our findings, however, is that when the tracer concentration is low the single-particle dynamics of these mixtures do not depend on the relative particle sizes in any appreciable way. This implies that the potential for a mixture to axially segregate cannot be inferred from the microscopic dynamics of individual small particles. A second finding is that while the slope of the mean-squared displacement is close to that expected from diffusive transport, as determined from the single-particle dynamics, more detailed analyses indicate anomalous transport.   

Jamming of Cylindrical Grains in Vertical Channels

We study jamming of low aspect-ratio cylindrical Delrin grains in a vertical channel. These cylindrical grains resemble antacid tablets, poker chips, or coins since their height is less than their diameter. Grains are allowed to fall through a vertical channel with a square cross section where the channel width is greater than the diameter of a grain and constant throughout the length of the channel with no obstructions or constrictions. Within this channel, grains are sometimes observed to form jams, stable structures supported by the channel walls with no support beneath them. The probability of jam occurrence and the strength or robustness of a jam is effected by the grain dimensions and channel size. We will present experimental measurements of the jamming probability and jam strength in this system and discuss the relationship of these results to other experiments and theories.   

Ordered and disordered granular sphere packings obtained by epitaxial growth

We study granular packings obtained by depositing spheres on a substrate under the influence of gravity. By exploiting the direct particle tracking enabled by X-ray tomography, the nature of the order and disorder is investigated using statistical measures including density pair correlation function, and the orientational order parameter. We find that by using a low deposition rate, impinging particles with sufficient energy can overcome friction and come to rest in a potential minimum of a periodic substrate, giving rise to ordered face-centered cubic structures. However, impinging particles with large kinetic energy can dislodge particles in the substrate leading to disorder as mobile particles cooperatively form arches while they come to rest. Thus, a wide range of volume fractions and packing structures is accessed by simply controlling the nature of the substrate and deposition rate and energy, along with the shape of the impinging particles. We contrast the ordered and disordered phases observed as a function of packing fraction with our previous study with cyclically sheared packings. In that study, compaction, nucleation and growth of face centered cubic and hexagonal close packed crystalline order was observed after hundreds of thousands of shear cycles.   

Tunable acoustic switching and rectification in one-dimensional granular crystals

We study a new mechanism for tunable acoustic switching and rectification, which we experimentally demonstrate in a one-dimensional granular crystal. The granular crystal is composed of an array of statically compressed elastic spherical particles that interact nonlinearly via Hertzian contact. The granular crystal is uniform except for a single light-mass defect placed near one boundary of the crystal. Because of the interplay of the periodicity, nonlinearity, dissipation, and asymmetry of the granular crystal, vibrations applied near the defect position cause the system response to bifurcate from periodic non-transmitting states to quasiperiodic and chaotic transmitting states with broadband frequency content. We illustrate the nature of this bifurcation using numerical simulations and compare these results to experimental observations. Because the bifurcation causes a sharp transition between states, this mechanism can lead to phononic switching and sensing. Furthermore, as switches and rectification devices are fundamental components used for controlling the flow of energy in numerous applications, we envision that this mechanism could more generally enable the design of advanced photonic, thermal, and acoustic materials and devices.   

Granular compaction under confinement

A granular pack that is vertically vibrated undergoes rearrangements and often progresses to more dense configurations. The experiments presented here study the role of dilation in this granular compaction process. By applying a confining force to the granular pack during vibration, the dilation is inhibited and the compaction is greatly reduced. In general, systems with different accelerations during vibration will compact differently. However, these systems will compact in the same manner if the confining force is tuned to result in the same amount of dilation. Under large confining forces, there is very little dilation. In this regime, the compaction is significantly slowed and may approach a steady state packing fraction of approximately 0.60, consistent with ideas of a critical packing fraction for the onset of dilation.   

Densest columnar structures of hard spheres from sequential deposition

The rich variety of densest columnar structures of identical hard spheres inside a cylinder can surprisingly be constructed from a simple and computationally fast sequential deposition of cylinder-touching spheres, if the cylinder-to-sphere diameter ratio D is within [1,2.7013]. This provides a direction for theoretically deriving all these densest structures and for constructing such densest packings with nano-, micro-, colloidal or charged particles, which all self-assemble like hard spheres [\textit{Rapid Communication}, Physical Review E (in press)].   

Discrete Calculus as a Bridge between Scales

Understanding how continuum descriptions of disordered media emerge from the microscopic scale is a fundamental challenge in condensed matter physics. In many systems, it is necessary to coarse-grain balance equations at the microscopic scale to obtain macroscopic equations. We report development of an exact, discrete calculus, which allows identification of discrete microscopic equations with their continuum equivalent [1]. This allows the application of powerful techniques of calculus, such as the Helmholtz decomposition, the Divergence Theorem, and Stokes' Theorem. We illustrate our results with granular materials. In particular, we show how Newton's laws for a single grain reproduce their continuum equivalent in the calculus. This allows introduction of a discrete Airy stress function, exactly as in the continuum. As an application of the formalism, we show how these results give the natural mean-field variation of discrete quantities, in agreement with numerical simulations. The discrete calculus thus acts as a bridge between discrete microscale quantities and continuous macroscale quantities. \\[4pt] [1] E. DeGiuli \& J. McElwaine, PRE 2011. doi: 10.1103/PhysRevE.84.041310