Packing of compressible granular materials

3D computer simulations and experiments are employed to study random packings of compressible spherical grains under external confining stress. In the rigid ball limit, we find a continuous transition in which the stress vanishes as (straight phi-straight phi(c))(beta), where straight phi is the (solid phase) volume density. The value of straight phi(c) depends on whether the grains interact via only normal forces (giving rise to random close packings) or by a combination of normal and friction generated transverse forces (producing random loose packings). In both cases, near the transition, the system's response is controlled by localized force chains.     

Disks vs. spheres: Contrasting properties of random packings

Collections of random packings of rigid disks and spheres have been generated by computer using a previously described concurrent algorithm. Particles begin as infinitesimal moving points, grow in size at a uniform rate, undergo energy-onconserving collisions, and eventually jam up. Periodic boundary conditions apply, and various numbers of particles have been considered (N2000 for disks,N8000 for spheres). The irregular disk packings thus formed are clearly polycrystalline with mean grain size dependent upon particle growth rate. By contrast, the sphere packings show a homogeneously amorphous texture substantially devoid of crystalline grains. This distinction strongly influences the respective results for packing pair correlation functions and for the distributions of particles by contact number. Rapidly grown disk packings display occasional vacancies within the crystalline grains; no comparable voids of such distinctive size have been found in the random sphere packings. Rattler particles free to move locally but imprisoned by jammed neighbors occur in both the disk and sphere packings.This paper is dedicated to Jerry Percus on the occasion of his 65th birthday.     

A model of binary assemblies of spheres

We propose a theoretical model of random binary assemblies of spheres at any packing fraction. We use the notion of geometrical neighborhood between grains that is defined through two generalizations of the Vorono? tessellation: the radical (or Laguerre) tessellation and the navigation map. The model is tested on different numerical packings. We find a weak local segregation for high packing fraction. We also find that the higher the size ratio of the particles, the more important the segregation. Received 19 February 2001 and Received in final form 27 June 2001     

Numerical study of the stress response of two-dimensional dense granular packings

We investigate the Green function of two-dimensional dense random packings of grains in order to discriminate between the different theories of stress transmission in granular materials. Our computer simulations allow for a detailed quantitative investigation of the dynamics which is difficult to obtain experimentally. We show that both hyperbolic and parabolic models of stress transmission fail to predict the correct stress distribution in the studied region of the parameters space. We demonstrate that the compressional and shear components of the stress compare very well with the predictions of isotropic elasticity for a wide range of pressures and porosities and for both frictional and frictionless packings. However, the states used in this study do not include the critical isostatic point for frictional particles, so that our results do not preclude the fact that corrections to elasticity may appear at the critical point of jamming, or for other sample preparation protocols, as discussed in the main text. We show that the agreement holds in the bulk of the packings as well as at the boundaries and we validate the linear dependence of the stress profile width with depth.     

Jamming III: Characterizing randomness via the entropy of jammed matter

The nature of randomness in disordered packings of frictional and frictionless spheres is investigated using theory and simulations of identical spherical grains. The entropy of the packings is defined through the force and volume ensemble of jammed matter and this is shown to be difficult to calculate analytically. A mesoscopic ensemble of isostatic states is then utilized in an effort to predict the entropy through the definition of a volume function that is dependent on the coordination number. Equations of state are obtained relating entropy, volume fraction and compactivity characterizing the different states of jammed matter, and elucidating the phase diagram for jammed granular matter. Analytical calculations are compared to numerical simulations using volume fluctuation analysis and graph theoretical methods, with reasonable agreement. The entropy of the jammed system reveals that random loose packings are more disordered than random close packings, allowing for an unambiguous interpretation of both limits. Ensemble calculations show that the entropy vanishes at random close packing (RCP), while numerical simulations show that a finite entropy remains in the microscopic states at RCP. The notion of a negative compactivity, which explores states with volume fractions below those achievable by existing simulation protocols, is also explored, expanding the equations of state. The mesoscopic theory reproduces the simulations results in shape well, though a difference in magnitude implies that the entire entropy of the packing may not be captured by the methods presented herein. We discuss possible extensions to the present mesoscopic approach describing packings from random loose packing (RLP) to RCP to the ordered branch of the equation of state in an effort to understand the entropy of jammed matter in the full range of densities from RLP to face-centered cubic (FCC) packing.     

Entropy and temperature of a static granular assembly: an ab initio approach

We construct a statistical framework for static assemblies of deformable grains which parallels that of equilibrium statistical mechanics but with a conservation principle based on the mechanical stress tensor. We define a state function that has all the attributes of entropy. In particular, maximizing this function leads to a well-defined granular temperature and the equivalent of the Boltzman distribution for ensembles of grain packings. Predictions of the ensemble are verified against simulated packings of frictionless, deformable disks.     

Elegance of disordered granular packings: a validation of Edward's hypothesis

We have found a way to analyze Edwards' density of states for static granular packings in the special case of round, rigid, frictionless grains assuming a constant coordination number. It obtains the most entropic density of single grain states, which predicts several observables including the distribution of contact forces. We compare these results against empirical data obtained in dynamic simulations of granular packings. The agreement is quite good, helping validate the use of statistical mechanics methods in granular physics. The differences between theory and empirics are mainly related to the coordination number, and when the empirical data are sorted by that number we obtain several insights that suggest an underlying elegance in the density of states.     

Relaxations and rheology near jamming

We determine the form of the complex shear modulus G* in soft sphere packings near jamming. Viscoelastic response at finite frequency is closely tied to a packing's intrinsic relaxational modes, which are distinct from the vibrational modes of undamped packings. We demonstrate and explain the appearance of an anomalous excess of slowly relaxing modes near jamming, reflected in a diverging relaxational density of states. From the density of states, we derive the dependence of G* on the frequency and distance to the jamming transition, which is confirmed by numerics.     

Scaling of the electrical conductivity of granular media

We derive the scaling properties of the dependence of the macroscopic electrical conductivity of granular media (e.g., sands) with a surface mechanism of electrical conduction on the grain size, when the medium is subjected to a given mechanical stress. In order to eliminate the influence of the inter-grain junction capacity, the direct electrical current is considered. We determine the applicability restrictions on the theory which disregards the ultimate crushing compression strength, adhesion, and the effect of charge carrier tunneling at grain junctions beyond the contact surface area. Solutions for several regular packings of grains are obtained as well.     

Rotating bearings in regular and irregular granular shear packings

For 2D regular dense packings of solid mono-size non-sliding disks there is a mechanism for bearing formation under shear that can be explained theoretically. There is, however, no easy way to extend this model to include random dense packings which would better describe natural packings. A numerical model that simulates shear deformation for both near-regular and irregular packings is used to demonstrate that rotating bearings appear roughly with the same density in random and regular packings. The main difference appears in the size distribution of the rotating clusters near the jamming threshold. The size distribution is well described by a scaling form with a large-size cut-off that seems to grow without bounds for regular packings at the jamming threshold, while it remains finite for irregular packings. At packing densities above the jamming transition there can be no shear, unless the disks are allowed to break. Breaking of disks induces a large number of small local bearings. Clusters of rotating particles may contribute to e.g. pre-rupture yielding in landslides, snow avalanches and to the formation of aseismic gaps in tectonic fault zones.     

Phase diagram and structural diversity of the densest binary sphere packings

The densest binary sphere packings have historically been very difficult to determine. The only rigorously known packings in the α-x plane of sphere radius ratio α and relative concentration x are at the Kepler limit α=1, where packings are monodisperse. Utilizing an implementation of the Torquato-Jiao sphere-packing algorithm [S. Torquato and Y. Jiao, Phys. Rev. E 82, 061302 (2010)], we present the most comprehensive determination to date of the phase diagram in (α,x) for the densest binary sphere packings. Unexpectedly, we find many distinct new densest packings.     

Geometric properties of random disk packings

Random packings ofN2000 rigid disks in the plane, subject to periodic boundary conditions on a square primitive cell, have been generated by a concurrent construction which treats all disks on an equal footing, as opposed to previously investigated sequential constructions. The particles start with random positions and velocities, and as they move about they grow uniformly in size, from points to jammed disks. The collection of packings displays several striking geometric features. These include (for largeN) typically polycrystalline textures with irregular grain boundaries and linear shear fractures. The packings occasionally contain monovacancies and trapped but unjammed rattler disks. The latter appear to be confined to the grain boundaries. The linear shear fractures preserve bond orientational order, but disrupt translational order, within the crystalline grains. A new efficient event-driven simulation algorithm is employed to generate the histories of colliding and jamming disks. On a computer which can process one million floating-point instructions per second the algorithm processes more than one million pairwise collisions per hour.     

Fluctuations in granular media

Dense slowly evolving or static granular materials exhibit strong force fluctuations even though the spatial disorder of the grains is relatively weak. Typically, forces are carried preferentially along a network of "force chains." These consist of linearly aligned grains with larger-than-average force. A growing body of work has explored the nature of these fluctuations. We first briefly review recent work concerning stress fluctuations. We then focus on a series of experiments in both two- and three-dimension [(2D) and (3D)] to characterize force fluctuations in slowly sheared systems. Both sets of experiments show strong temporal fluctuations in the local stress/force; the length scales of these fluctuations extend up to 10(2) grains. In 2D, we use photoelastic disks that permit visualization of the internal force structure. From this we can make comparisons to recent models and calculations that predict the distributions of forces. Typically, these models indicate that the distributions should fall off exponentially at large force. We find in the experiments that the force distributions change systematically as we change the mean packing fraction, gamma. For gamma's typical of dense packings of nondeformable grains, we see distributions that are consistent with an exponential decrease at large forces. For both lower and higher gamma, the observed force distributions appear to differ from this prediction, with a more Gaussian distribution at larger gamma and perhaps a power law at lower gamma. For high gamma, the distributions differ from this prediction because the grains begin to deform, allowing more grains to carry the applied force, and causing the distributions to have a local maximum at nonzero force. It is less clear why the distributions differ from the models at lower gamma. An exploration in gamma has led to the discovery of an interesting continuous or "critical" transition (the strengthening/softening transition) in which the mean stress is the order parameter, and the mean packing fraction, gamma, must be adjusted to a value gamma(c) to reach the "critical point." We also follow the motion of individual disks and obtain detailed statistical information on the kinematics, including velocities and particle rotations or spin. Distributions for the azimuthal velocity, V(theta), and spin, S, of the particles are nearly rate invariant, which is consistent with conventional wisdom. Near gamma(c), the grain motion becomes intermittent causing the mean velocity of grains to slow down. Also, the length of stress chains grows as gamma-->gamma(c). The 3D experiments show statistical rate invariance for the stress in the sense that when the power spectra and spectral frequencies of the stress time series are appropriately scaled by the shear rate, Omega, all spectra collapse onto a single curve for given particle and sample sizes. The frequency dependence of the spectra can be characterized by two different power laws, P proportional, variant omega(-alpha), in the high and low frequency regimes: alpha approximately 2 at high omega; alpha<2 at low omega. The force distributions computed from the 3D stress time series are at least qualitatively consistent with exponential fall-off at large stresses. (c) 1999 American Institute of Physics.     

Entropy of jammed matter

We investigate the nature of randomness in disordered packings of frictional spheres. We calculate the entropy of 3D packings through the force and volume ensemble of jammed matter, a mesoscopic ensemble and numerical simulations using volume fluctuation analysis and graph theoretical methods. Equations of state are obtained relating entropy, volume fraction and compactivity characterizing the different states of jammed matter. At the mesoscopic level the entropy vanishes at random close packing, while the microscopic states contribute to a finite entropy. The entropy of the jammed system reveals that the random loose packings are more disordered than random close packings, allowing for an unambiguous interpretation of both limits.     

Influence of particle size on the breaking of aluminum particle shells

Rupturing the alumina shell (shell-breaking) is a prerequisite for releasing energy from aluminum powder. Thermal stress overload in a high-temperature environment is an important factor in the rupture of the alumina shell. COMSOL Multiphysics was used to simulate and analyze the shell-breaking response of micron-scale aluminum particles with different particle sizes at 650 ℃ in vacuum. The simulation results show that the thermal stability time and shell-breaking response time of 10 μm-100 μm aluminum particles are 0.15 μs-11.44 μs and 0.08 μs-3.94 μs, respectively. They also reveal the direct causes of shell breaking for aluminum particles with different particle sizes. When the particle size is less than 80 μm, the shell-breaking response is a direct result of compressive stress overload. When the particle size is between 80 μm and 100 μm, the shell-breaking response is a direct result of tensile stress overload. This article provides useful guidance for research into the energy release of aluminum powder.     

Tapped granular packings described as complex networks

Abstract

We characterize the structure of simulated two-dimensional granular packings using concepts from complex networks theory. The packings are generated by a simulated tapping protocol, which allows us to obtain states in mechanical equilibrium in a wide range of densities. We show that our characterization method is able to discriminate non-equivalent states that have the same density. We do this by examining differences in the topological structure of the contact network of the packings. In particular, we find that the polygons of the network are specially sensitive probes for the contact structure. Additionally, we compare the network properties obtained in two different scenarios: the tapped and a compressed system.     

Jamming and crystallization in athermal polymer packings

Dense packings of chains of hard spheres possess characteristic features that do not have a counterpart in corresponding packings of monomeric spheres especially near the maximally random jammed (MRJ) state. From the modelling perspective the additional requirement that spheres keep their connectivity while maximizing the occupied volume fraction imposes severe constraints on generation algorithms of dense chain configurations. The extremely sluggish dynamics imposed by the uncrossability of chains precludes the use of deterministic or stochastic dynamics to generate all but dilute polymer packings. As a viable alternative, especially tailored chain-connectivity-altering Monte Carlo (MC) algorithms have been developed that bypass this kinetic hindrance and have actually been able to produce packings of hard-sphere chains in a volume fraction range spanning from infinite dilution up to the MRJ state. Such very dense athermal polymer packings share a number of structural features with packings of monomeric hard spheres, but also display unique characteristics due to the constraints imposed by connectivity. We give an overview of the most relevant results of our recent modeling work on packings of freely-jointed chains of tangent hard spheres about the MRJ state, local structure, chain dimensions and their scaling with density, topological constraints in the form of entanglements and knots, contact network at jamming, and entropically driven crystallization.     

Dense and nearly jammed random packings of freely jointed chains of tangent hard spheres

Dense packings of freely jointed chains of tangent hard spheres are produced by a novel Monte Carlo method. Within statistical uncertainty, chains reach a maximally random jammed (MRJ) state at the same volume fraction as packings of single hard spheres. A structural analysis shows that as the MRJ state is approached (i) the radial distribution function for chains remains distinct from but approaches that of single hard sphere packings quite closely, (ii) chains undergo progressive collapse, and (iii) a small but increasing fraction of sites possess highly ordered first coordination shells.     

Gravity driven instabilities in miscible non-Newtonian fluid displacements in porous media

Gravity driven instabilities in model porous packings of 1 mm diameter spheres are studied by comparing the broadening of the displacement front between fluids of slightly different densities in stable and unstable configurations. Water, water–glycerol and water–polymer solutions are used to vary independently viscosity and molecular diffusion and study the influence of shear-thinning properties. Both injected and displaced solutions are identical but for a different concentration of NaNO₃ salt used as an ionic tracer and to introduce the density contrast. Dispersivity in stable configuration increases with polymer concentration – as already reported for double porosity packings of porous grains. Gravity-induced instabilities are shown to develop below a same threshold Péclet number Pe for water and water–glycerol solutions of different viscosities and result in considerable increases of the dispersivity. Measured threshold Pe values decrease markedly on the contrary with polymer concentration. The quantitative analysis demonstrates that the development of the instabilities is controlled by viscosity through a characteristic gravity number G (ratio between hydrostatic and viscous pressure gradients). A single threshold value of G accounts for results obtained on Newtonian and non-Newtonian solutions.