We have produced loose packings of cohesionless, frictional spheres by sequential deposition of highly-spherical, monodisperse particles through a fluid. By varying the properties of the fluid and the particles, we have identified the Stokes number (St) - rather than the buoyancy of the particles in the fluid - as the parameter controlling the approach to the loose packing limit. The loose packing limit is attained at a threshold value of St at which the kinetic energy of a particle impinging on the packing is fully dissipated by the fluid. Thus, for cohesionless particles, the dynamics of the deposition process, rather than the stability of the static packing, defines the random loose packing limit. We have made direct measurements of the interparticle friction in the fluid, and present an experimental measurement of the loose packing volume fraction, ϕ_{RLP}, as a function of the friction coefficient μ_s.
We study static packings of frictionless and frictional spheres in three dimensions, obtained via molecular dynamics simulations, in which we vary particle hardness, friction coefficient, and coefficient of restitution. Although frictionless packings of hard-spheres are always isostatic (with six contacts) regardless of construction history and restitution coefficient, frictional packings achieve a multitude of hyperstatic packings that depend on system parameters and construction history. Instead of immediately dropping to four, the coordination number reduces smoothly from $z=6$ as the frict…
We show that in slowly generated 2d packings of frictional spheres, a significant fraction of the friction forces lies 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 j…
Particle tracking and displacement covariance matrix techniques are employed to investigate the phonon dispersion relations of two-dimensional colloidal glasses composed of soft, thermoresponsive microgel particles whose temperature-sensitive size permits \textit{in situ} variation of particle packing fraction. Bulk, $B$, and shear, $G$, moduli of the colloidal glasses are extracted from the dispersion relations as a function of packing fraction, and variation of the ratio $G/B$ with packing fraction is found to agree quantitatively with predictions for jammed packings of frictional soft parti…
We develop a theoretical description for mechanically stable frictional packings in terms of the difference between the total number of contacts required for isostatic packings of frictionless disks and the number of contacts in frictional packings, $m=N_c^0-N_c$. The saddle order $m$ represents the number of unconstrained degrees of freedom that a static packing would possess if friction were removed. Using a novel numerical method that allows us to enumerate disk packings for each $m$, we show that the probability to obtain a packing with saddle order $m$ at a given static friction coefficie…
We analyze the local structure of two dimensional packings of frictional disks numerically. We focus on the fractions x_i of particles that are in contact with i neighbors, and systematically vary the confining pressure p and friction coefficient μ. We find that for all μ, the fractions x_i exhibit powerlaw scaling with p, which allows us to obtain an accurate estimate for x_i at zero pressure. We uncover how these zero pressure fractions x_i vary with μ, and introduce a simple model that captures most of this variation. We also probe the correlations between the contact numbers of neighboring…
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…
The role of friction coefficient, $μ$, on the jamming properties of disordered, particle packings is studied using computer simulations. Compressed, soft-sphere packings are brought towards the jamming transition - the point where a packing loses mechanical stability - by decreasing the packing fraction. The values of the packing fraction at the jamming transition, $ϕ^μ_{c}$, gradually decrease from the random close packing point for zero friction, to a value coincident with random loose packing as the friction coefficient is increased over several orders of magnitude. This is accompanied by a…
We explore adhesive loose packings of dry small spherical particles of micrometer size using 3D discrete-element simulations with adhesive contact mechanics. A dimensionless adhesion parameter ($Ad$) successfully combines the effects of particle velocities, sizes and the work of adhesion, identifying a universal regime of adhesive packings for $Ad>1$. The structural properties of the packings in this regime are well described by an ensemble approach based on a coarse-grained volume function that includes correlations between bulk and contact spheres. Our theoretical and numerical results predi…
Stress transmission inside three dimensional granular packings is investigated using computer simulations. Localized force perturbation techniques are implemented for frictionless and frictional shallow, ordered, granular arrays confined by solid boundaries for a range of system sizes. Stress response profiles for frictional packings agree well with the predictions for the semi-infinite half plane of classical isotropic elasticity theory down to boxes of linear dimensions of approximately forty particle diameters and over several orders of magnitude in the applied force. The response profiles …
We show how rigidity emerges in experiments of sheared frictional granular materials by using generalizations of two methods for identifying rigid structures. Both approaches, the force-based dynamical matrix and the topology-based rigidity percolation, agree with each other and identify similar rigid structures. As the system becomes jammed, at a contact number of $z=2.4\pm 0.1$, a rigid backbone interspersed with floppy, particle-filled holes of a broad range of sizes emerges, creating a sponge-like morphology. We also find that the pressure within rigid structures always exceeds the pressur…
The mechanical response of static, unconfined, overcompressed face centred cubic, granular arrays is studied using large-scale, discrete element method simulations. Specifically, the stress response due to the application of a localised force perturbation - the Green function technique - is obtained in granular packings generated over several orders of magnitude in both the particle friction coefficient and the applied forcing. We observe crossover behaviour in the mechanical state of the system characterised by the changing nature of the resulting stress response. The transition between aniso…
The equation of state of a hard sphere fluid at high density should exhibit a simple pole at the random close packing limit. Here we show that trying to obtain a compact semi-empirical equation of state simultaneously compatible with that asymptotic behaviour and with the known virial coefficients raises an analytical difficulty which can be solved if a glassy transition occurs in the disordered metastable phase at a density intermediate between the freezing point and the random close packing limit, numerically close to the melting point. The estimated value for the transition point, which is …
We prepare packings of frictional tetrahedra with volume fractions ϕ ranging from 0.469 to 0.622 using three different experimental protocols under isobaric conditions. Analysis via X-ray micro-tomography reveals that the contact number Z grows with ϕ, but does depend on the preparation protocol. While there exist four different types of contacts in tetrahedra packings, our analysis shows that the edge-to-face contacts contribute about 50% of the total increase in Z. The number of constraints per particle C increases also with ϕ and even the loosest packings are strongly hyperstatic i.e. mecha…
We have obtained the random loose packing fraction of the parking lot model (PLM) by taking the limit of infinite compactivity in the two-variable statistical description of Tarjus and Viot for the PLM. The PLM is a stochastic model of adsorption and desorption of particles on a substrate that have been used as a model for compaction of granular materials. An order parameter $ρ$ is introduced to characterize how far from a steady state situation the model is. Thus, configurations with $ρ<1$ age. We propose that $ρ$ can be a starting point in order to stablish a connection between Edwards' stat…
The hyperstatic nature of granular packings of perfectly rigid disks is analyzed algebraically and through numerical simulation. The elementary loops of grains emerge as a fundamental element in addressing hyperstaticity. Loops consisting of an odd number of grains behave differently than those with an even number. For odd loops, the latent stresses are exterior and are characterized by the sum of frictional forces around each loop. For even loops, the latent stresses are interior and are characterized by the alternating sum of frictional forces around each loop. The statistics of these two ty…
With a novel 3D discrete-element method specially developed with adhesive contact mechanics, random loose packings of uniform spherical micron-sized particles are fully investigated. The results show that large velocity, large size or weak adhesion can produce a relatively dense packing when other parameters are fixed, and these combined effects can be characterized by a dimensionless adhesion parameter ( $Ad=ω/2ρ_pU^2_0R$). Four regimes are identified based on the value of $Ad$: RCP regime with $Ad<\sim 0.01$; RLP regime with $\sim 0.01<Ad<1$; adhesion regime with $1<Ad<20$ and an asymptotic …
Using sedimentation to obtain precisely controlled packings of noncohesive spheres, we find that the volume fraction $ϕ_{\rm RLP}$ of the loosest mechanically stable packing is in an operational sense well defined by a limit process. This random loose packing volume fraction decreases with decreasing pressure $p$ and increasing interparticle friction coefficient $μ$. Using X-ray tomography to correct for a container boundary effect that depends on particle size, we find for rough particles in the limit $p \to 0$ a new lower bound, $ϕ_{\rm RLP} = 0.550 \pm 0.001$.
Static packings of frictional rigid particles are investigated by means of discrete element simulations. We explore the ensemble of allowed force realizations in the space of contact forces for a given packing structure. We estimate the extent of force indeterminacy with different methods. The indeterminacy exhibits a nonmonotonic dependence on the interparticle friction coefficient. We verify directly that larger force-indeterminacy is accompanied by a more robust behavior against local perturbations. We also investigate the local indeterminacy of individual contact forces. The probability di…
We measure shear response in packings of glass beads by pulling a thin, rough, metal plate vertically through a bed of volume fraction phi, which is set, before the plate is pulled, in the range 0.575 to 0.628. The yield stress is velocity independent over 4 decades and increases exponentially with phi, with a transition at phi approximately 0.595. An analysis of the measured force fluctuations indicates that the shear modulus is significantly smaller than the bulk modulus.
We systematically generate a large set of random micro-particle packings over a wide range of adhesion and friction by means of adhesive contact dynamics simulation. The ensemble of generated packings covers a range of volume fraction $ϕ$ from $0.135 \pm 0.007$ to $0.639 \pm 0.004$, and of coordination number $Z$ from $2.11 \pm 0.03$ to $6.40 \pm 0.06$. We determine $ϕ$ and $Z$ at four limits (random close packing, random loose packing, adhesive close packing, and adhesive loose packing), and find a universal equation of state $ϕ(Z)$ to describe packings with arbitrary adhesion and friction. F…
We introduce and simulate a two dimensional probabilistic model of granular matter at vanishing pressure. The model exhibits a perfectly sharp random loose packing density, a phenomenon that should be verifiable for real granular matter.
We study the response to shear deformations of packings of long spherocylindrical particles that interact via frictional forces with friction coefficient $μ$. The packings are produced and deformed with the help of molecular dynamics simulations combined with minimization techniques performed on a GPU. We calculate the linear shear modulus $g_\infty$, which is orders of magnitude larger than the modulus $g_0$ in the corresponding frictionless system. The motion of the particles responsible for these large frictional forces is governed by and increases with the length $\ell$ of the spherocylind…
The packing and flow of aspherical frictional particles are studied using discrete element simulations. Particles are superballs with shape $|x|^{s}+|y|^{s}+|z|^{s} = 1$ that varies from sphere ($s=2$) to cube ($s=\infty$), constructed with an overlapping-sphere model. Both packing fraction, $ϕ$, and coordination number, $z$, decrease monotonically with microscopic friction $μ$, for all shapes. However, this decrease is more dramatic for larger $s$ due to a reduction in the fraction of face-face contacts with increasing friction. For flowing grains, the dynamic friction $\tildeμ$ - the ratio o…
Intuition tells us that a rolling or spinning sphere will eventually stop due to the presence of friction and other dissipative interactions. The resistance to rolling and spinning/twisting torque that stops a sphere also changes the microstructure of a granular packing of frictional spheres by increasing the number of constraints on the degrees of freedom of motion. We perform discrete element modeling simulations to construct sphere packings implementing a range of frictional constraints under a pressure-controlled protocol. Mechanically stable packings are achievable at volume fractions and…
We present a theoretical prediction on random close packing factor ϕ_RCP^b of binary granular packings based on the hard-sphere fluid theory. An unexplored regime is unravelled, where the packing fraction ϕ_RCP^b is smaller than that of the mono-sized one ϕ_RCP^m, i.e., the so-called loose jamming state. This is against our common perception that binary packings should always reach a denser packing than mono-sized packings at the jamming state. Numerical evidence further supports this prediction and confirms the regime location in the size ratio and mole fraction (R_r-X_s) space, where the siz…
In particulate soft matter systems the average number of contacts $Z$ of a particle is an important predictor of the mechanical properties of the system. Using X-ray tomography, we analyze packings of frictional, oblate ellipsoids of various aspect ratios $α$, prepared at different global volume fractions $ϕ_g$. We find that $Z$ is a monotonously increasing function of $ϕ_g$ for all $α$. We demonstrate that this functional dependence can be explained by a local analysis where each particle is described by its local volume fraction $ϕ_l$ computed from a Voronoi tessellation. $Z$ can be expresse…
Simulated granular packings with different particle friction coefficient mu are examined. The distribution of the particle-particle and particle-wall normal and tangential contact forces P(f) are computed and compared with existing experimental data. Here f equivalent to F/F-bar is the contact force F normalized by the average value F-bar. P(f) exhibits exponential-like decay at large forces, a plateau/peak near f = 1, with additional features at forces smaller than the average that depend on mu. Computations of the force-force spatial distribution function and the contact point radial distrib…
Dense, stabilized, frictional particulate suspensions in a viscous liquid undergo increasingly strong continuous shear thickening (CST) as the solid packing fraction, $ϕ$, increases above a critical volume fraction, and discontinuous shear thickening (DST) is observed for even higher packing fractions. Recent studies have related shear thickening to a transition from mostly lubricated to predominantly frictional contacts with the increase in stress. The rheology and networks of frictional forces from two and three-dimensional simulations of shear-thickening suspensions are studied. These are a…
The stress profile and reorientation of grains, in response to a point force applied to a preloaded two dimensional granular system, are calculated in the context of a continuum theory that incorporates the texture of the packing. When high friction prevents slip at the inter-grain contacts, an anisotropic packing propagates stress along two peaks which amalgamate into a single peak as the packing is disordered into a less anisotropic structure; this single peak may be wider or narrower than in the isotropic case, depending on the preparation of the packing. At lower frictions, an effective tr…
We use 2D numerical simulations to study dense suspensions of non-Brownian hard particles using the Critical Load Model (CLM) under constant confining pressures. This simple model shows discontinuous shear thickening (DST) as the tangential forces get activated upon increased shear stresses. By parameterizing a simple binary system of frictional and non-frictional particles of different proportions we show that the jamming packing fraction, at which the viscosity diverges, is controlled by the fraction of frictional contacts. The viscosity of dense suspensions can thereby be expressed as a fun…
We study the effect of freeze-thaw cycling on the packing fraction of equal spheres immersed in water. The water located between the grains experiences a dilatation during freezing and a contraction during melting. After several cycles, the packing fraction converges to a particular value $η_{\infty} = 0.595$ independently of its initial value $η_0$. This behavior is well reproduced by numerical simulations. Moreover, the numerical results allow to analyze the packing structural configuration. With a Voronoï partition analysis, we show that the piles are fully random during the whole process a…
The formation of self-organised structures that resist shear deformation have been discussed in the context of shear jamming and thickening[1-3], with frictional forces playing a key role. However, shear induces geometric features necessary for jamming even in frictionless packings[4]. We analyse conditions for jamming in such assemblies by solving force and torque balance conditions for their contact geometry. We demonstrate, and validate with frictional simulations, that the isostatic condition for mean contact number Z = D + 1 (for spatial dimension D = 2, 3) holds at jamming for both finit…
An incremental evolution equation, i.e. a Master equation in statistical mechanics, is introduced for force distributions in polydisperse frictional particle packings. As basic ingredients of the Master equation, the conditional probability distributions of particle overlaps are determined by molecular dynamics simulations. Interestingly, tails of the distributions become much narrower in the case of frictional particles than frictionless particles, implying that correlations of overlaps are strongly reduced by microscopic friction. Comparing different size distributions, we find that the tail…
In stark contrast to the jamming of frictionless hard-sphere (hard-disk if in two dimensions) packings, the critical jamming of frictional packings is much more elusive and still under intense debate. Here we show that frictional hard-disk packings self-organize near a critical jamming state when subjected to quasi-static steady-state cyclic shear, displaying scale-free fluctuations in particle velocity fields as characterized by the power spectra $E(k)\propto k^{-α}$ for both longitudinal and transverse modes in wavevector space, with $α\approx2.0\pm0.15$, and the nearly flat spectra $E(ω)\pr…
The transmission rules of interparticle forces between granular particles were analyzed, and a discrete packing model was proposed to calculate the static pressure at the bottom of granular material confined in a vertical column. The mechanical analysis and numerical simulation results indicate that the silo effect is caused by the frictional contacts between border particles and inner walls, the static pressure at the bottom depends on the external load initially, and then tends to a saturation pressure (Pn) in an exponential form. The saturation pressure (Pn) is positive linear related to th…
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…
Amorphous packings of frictionless, spherical particles are isostatic at jamming onset, with the number of constraints (contacts) equal to the number of degrees of freedom. Their structural and mechanical properties are controlled by the interparticle contact network. In contrast, amorphous packings of frictional particles are typically hyperstatic at jamming onset. We perform extensive numerical simulations in two dimensions of the geometrical asperity (GA) model for static friction, to further investigate the role of isostaticity. In the GA model, interparticle forces are obtained by summing…
Packings of macroscopic granular chains capture some of the essential aspects of molecular polymer systems and have been suggested as a paradigm to understand the physics on a molecular scale. However, here we demonstrate that the interparticle friction $μ$ in granular chain packings, which has no counterpart in polymer systems, leads to a nontrivial yielding and rheological response. Based on discrete element simulations we study the nonlinear rheology of random packings of granular chains under large amplitude oscillatory shear. We find that the maximum stress and the penetration depth of th…
The packing of charged micron-sized particles was investigated using discrete element simulations based on adhesive contact dynamic model. The formation process and the final obtained structures of ballistic packings are studied to show the effect of interparticle Coulomb force. It was found that increasing the charge on particles causes a remarkable decrease of the packing volume fraction ϕand the average coordination number Z, indicating a looser and chainlike structure. Force-scaling analysis shows that the long-range Coulomb interaction changes packing structures through its influence on p…
Superellipse sector particles (SeSPs) are segments of superelliptical curves that form a tunable set of hard-particle shapes for granular and colloidal systems. SeSPs allow for continuous parameterization of corner sharpness, aspect ratio, and particle curvature; rods, circles, rectangles, and staples are examples of shapes SeSPs can model. We compare three computational processes: pair-wise Monte Carlo simulations that look only at particle-particle geometric constraints, Monte Carlo simulations that look at how these geometric constraints play out over extended dispersions of many particles,…
Dense granular systems subjected to an imposed shear stress undergo stick-slip dynamics with systematic patterns of dilation-compaction. During each stick phase, as the frictional strength builds up, the granular system dilates to accommodate shear strain, developing stronger force networks. During each slip event, when the stored energy is released, particles experience large rearrangements and the granular network can significantly change. Here, we use numerical simulations of 3D, sheared frictional packings to show that the mean betweenness centrality -- a property of network of interpartic…
It is shown that vibrated packings of frictional disks self-organize cooperatively onto a rotational-transport state where the long-time angular velocity $\barω_i$ of each disk $i$ is nonzero. Steady rotation is mediated by the spontaneous breaking of local reflection symmetry, arising when the cages in which disks are constrained by their neighbors acquire quenched disorder at large packing densities. Experiments and numerical simulation of this unexpected phenomenon show excellent agreement with each other, revealing two rotational phases as a function of excitation intensity, respectively t…
By generalizing a geometric argument for frictionless spheres, a model is proposed for the jamming density $ϕ_J$ of mechanically stable packings of bidisperse, frictional spheres. The monodisperse, $μ_s$-dependent jamming density $ϕ_J^{\mathrm{mono}}(μ_s)$ is the only input required in the model, where $μ_s$ is the coefficient of friction. The predictions of the model are validated by robust estimates of $ϕ_J$ obtained from computer simulations of up to $10^7$ particles for a wide range of $μ_s$, and size ratios up to 40:1. Although $ϕ_J$ varies nonmonotonically with the volume fraction of sma…
We combine advanced image reconstruction techniques from computed X-ray micro tomography (XCT) with state-of-the-art discrete element method simulations (DEM) to study granular materials. This "virtual-laboratory" platform allows us to access quantities, such as frictional forces, which would be otherwise experimentally immeasurable.
At low volume fraction, disordered arrangements of frictionless spheres are found in un--jammed states unable to support applied stresses, while at high volume fraction they are found in jammed states with mechanical strength. Here we show, focusing on the hard sphere zero pressure limit, that the transition between un-jammed and jammed states does not occur at a single value of the volume fraction, but in a whole volume fraction range. This result is obtained via the direct numerical construction of disordered jammed states with a volume fraction varying between two limits, $0.636$ and $0.646…
Loose granular structures stabilized against gravity by an effective cohesive force are investigated on a microscopic basis using contact dynamics. We study the influence of the granular Bond number on the density profiles and the generation process of packings, generated by ballistic deposition under gravity. The internal compaction occurs discontinuously in small avalanches and we study their size distribution. We also develop a model explaining the final density profiles based on insight about the collapse of a packing under changes of the Bond number.
A random packing of hard particles represents a fundamental model for granular matter. Despite its importance, analytical modeling of random packings remains difficult due to the existence of strong correlations which preclude the development of a simple theory. Here, we take inspiration from liquid theories for the $n$-particle angular correlation function to develop a formalism of random packings of hard particles from the bottom-up. A progressive expansion into a shell of particles converges in the large layer limit under a Kirkwood-like approximation of higher-order correlations. We apply …
Shearing granular materials induces non-affine displacements. Such non-affine displacements have been studied extensively, and are known to correlate with plasticity and other mechanical features of amorphous packings. A well known example is shear transformation zones as captured by the local deviation from affine deformation, $D^2_{min}$, and their relevance to failure and stress fluctuations. We analyze sheared frictional athermal disk packings and show that there exists at least one additional mesoscopic transport mechanism that superimposes itself on top of local diffusive motion. We evid…
Convective Flows in Sheared Packings of Spherical Particles
Institute for Advanced Studies in Basic Sciences (IASBS)（基础科学高等研究院）；Heinrich Heine Universität Düsseldorf（杜塞尔多夫大学）；Saarland University（萨尔兰大学）；University of Pennsylvania（宾夕法尼亚大学）；University of Northern Iowa（北爱荷华大学）
A load applied to a jammed frictional granular system will be localized into a network of force chains making inter-particle connections throughout the system. Because such systems are typically under-constrained, the observed force network is not unique to a given particle configuration, but instead varies upon repeated formation. In this paper, we examine the ensemble of force chain configurations created under repeated assembly in order to develop tools to statistically forecast the observed force network. In experiments on a gently suspended 2D layer of photoelastic particles, we subject t…
We present a multiscale simulation algorithm for amorphous materials, which we illustrate and validate in a canonical case of dense granular flow. Our algorithm is based on the recently proposed Spot Model, where particles in a dense random packing undergo chain-like collective displacements in response to diffusing "spots" of influence, carrying a slight excess of interstitial free volume. We reconstruct the microscopic dynamics of particles from the "coarse grained" dynamics of spots by introducing a localized particle relaxation step after each spot-induced block displacement, simply to enf…
The force distribution of jammed disordered packings has always been considered a central object in the physics of granular materials. However, many of its features are poorly understood. In particular, analytic relations to other key macroscopic properties of jammed matter, such as the contact network and its coordination number, are still lacking. Here we develop a mean-field theory for this problem, based on the consideration of the contact network as a random graph where the force transmission becomes a constraint optimization problem. We can thus use the cavity method developed in the las…
