Jamming phase diagram for frictional particles

Fabric-based jamming phase diagram for frictional granular materials

A jamming phase diagram maps the phase states of granular materials to their intensive properties such as shear stress and density (or packing fraction). We investigate how different phases in a jamming phase diagram of granular materials are related to their fabric structure via three-dimensional discrete element method simulations. Constant-volume quasi-static simple shear tests ensuring uniform shear strain field are conducted on bi-disperse spherical frictional particles. Specimens with different initial solid fractions are sheared until reaching steady state at a large shear strain (200%). The jamming threshold in terms of stress, non-rattler fraction, and coordination numbers (Z's) of different contact networks is discussed. The evolution of fabric anisotropy (F) of each contact network during shearing is also examined. By plotting the fabric data in the F-Z space, a unique critical fabric surface (CFS) becomes apparent across all specimens, irrespective of their initial phase states. Through the correlation of this CFS with fabric signals corresponding to jamming transitions, we introduce a novel jamming phase diagram in the fabric F-Z space, offering a convenient approach to distinguish the various phases of granular materials solely through the direct observation of geometrical arrangements of particles. This jamming phase diagram underscores the importance of the microstructure underlying the conventional jamming phenomenon and introduces a novel standpoint for interpreting the phase transitions of granular materials that have been exposed to processes such as compaction, shearing, and other complex loading histories.

Theory of rigidity and numerical analysis of density of states of two-dimensional amorphous solids with dispersed frictional grains in the linear response regime

Using the Jacobian matrix, we obtain a theoretical expression of rigidity and the density of states of two-dimensional amorphous solids consisting of frictional grains in the linear response to an infinitesimal strain, in which we ignore the dynamical friction caused by the slip processes of contact points. The theoretical rigidity agrees with that obtained by molecular dynamics simulations. We confirm that the rigidity is smoothly connected to the value in the frictionless limit. We find that there are two modes in the density of states for sufficiently small k_{T}/k_{N}, which is the ratio of the tangential to normal stiffness. Rotational modes exist at low frequencies or small eigenvalues, whereas translational modes exist at high frequencies or large eigenvalues. The location of the rotational band shifts to the high-frequency region with an increase in k_{T}/k_{N} and becomes indistinguishable from the translational band for large k_{T}/k_{N}.

Granular piston-probing in microgravity: powder compression, from densification to jamming

The macroscopic response of granular solids is determined by the microscopic fabric of force chains, which, in turn, is intimately linked to the history of the solid. To query the influence of gravity on powder flow behavior, a granular material is subjected to compression by a piston in a closed container, on-ground and in microgravity (on parabolic flights). Results show that piston-probing densifies the packing, eventually leading to jamming of the material compressed by the piston, regardless of the gravitational environment. The onset of jamming is found to appear at lower packing fraction in microgravity ([Formula: see text]) than on-ground ([Formula: see text]). We interpret these findings as the manifestation of a granular fabric altered by the gravitational force field: in absence of a secondary load (due to gravitational acceleration) to stimulate reorganization in a different direction to the major compression stress, the particles' configuration becomes stable at lower density, as the particles have no external drive to promote reorganization into a denser packing. This is coupled with a change in interparticular force balance which takes place under low gravity, as cohesive interactions become predominant. We propose a combination of microscopic and continuum arguments to rationalize our results.

Universal scaling law of glass rheology

The similarity in atomic/molecular structure between liquids and glasses has stimulated a long-standing hypothesis that the nature of glasses may be more fluid-like, rather than the apparent solid. In principle, the nature of glasses can be characterized by the dynamic response of their rheology in a wide rate range, but this has not been realized experimentally, to the best of our knowledge. Here we report the dynamic response of shear stress to the shear strain rate of metallic glasses over a timescale of nine orders of magnitude, equivalent to hundreds of years, by broadband stress relaxation experiments. The dynamic response of the metallic glasses, together with other 'glasses', follows a universal scaling law within the framework of fluid dynamics. The universal scaling law provides comprehensive validation of the conjecture on the jamming (dynamic) phase diagram by which the dynamic behaviours of a wide variety of 'glasses' can be unified under one rubric parameterized by the thermodynamic variables of temperature, volume and stress in the trajectory space.

Jammed disks of two sizes and weights in a channel: Alternating sequences

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 microstates into overlapping tiles from which we construct a small number of species of statistically interacting particles. Jammed macrostates depend on dimensionless control parameters inferred from ratios between measures of expansion work against the pistons, gravitational potential energy, and intensity of random agitations. These control parameters enter the configurational statistics via the activation energies prior to jamming of the particles. The range of disk weights naturally divides into regimes where qualitatively different features come into play. We sketch a path toward generalizations that include random sequences under a modified jamming protocol.

Flow and arrest in stressed granular materials

Flowing granular materials often abruptly arrest if not driven by sufficient applied stresses. Such abrupt cessation of motion can be economically expensive in industrial materials handling and processing, and is significantly consequential in intermittent geophysical phenomena such as landslides and earthquakes. Using discrete element simulations, we calculate states of steady flow and arrest for granular materials under the conditions of constant applied pressure and shear stress, which are also most relevant in practice. Here the material can dilate or compact, and flow or arrest, in response to the applied stress. Our simulations highlight that under external stress, the intrinsic response of granular materials is characterized by uniquely-defined steady states of flow or arrest, which are highly sensitive to interparticle friction. While the flowing states can be equivalently characterized by volume fraction, coordination number or internal stress ratio, to characterize the states of shear arrest, one needs to also consider the structural anisotropy in the contact network. We highlight the role of dilation in the flow-arrest transition, and discuss our findings in the context of rheological transitions in granular materials.

Unconventional rheological properties in systems of deformable particles

We demonstrate the existence of unconventional rheological and memory properties in systems of soft-deformable particles whose energy depends on their shape, via numerical simulations. At large strains, these systems experience an unconventional shear weakening transition characterized by an increase in the mechanical energy and a drastic drop in shear stress, which stems from the emergence of short-ranged tetratic order. In these weakened states, the contact network evolves reversibly under strain reversal, keeping memory of its initial state, while the microscopic dynamics is irreversible.

Interplay between hysteresis and nonlocality during onset and arrest of flow in granular materials

The jamming transition in granular materials is well-known for exhibiting hysteresis, wherein the level of shear stress required to trigger flow is larger than that below which flow stops. Although such behavior is typically modeled as a simple non-monotonic flow rule, the rheology of granular materials is also nonlocal due to cooperativity at the grain scale, leading for instance to increased strengthening of the flow threshold as system size is reduced. We investigate how these two effects - hysteresis and nonlocality - couple with each other by incorporating non-monotonicity of the flow rule into the nonlocal granular fluidity (NGF) model, a nonlocal constitutive model for granular flows. By artificially tuning the strength of nonlocal diffusion, we demonstrate that both ingredients are key to explaining certain features of the hysteretic transition between flow and arrest. Finally, we assess the ability of the NGF model to quantitatively predict material behavior both around the transition and in the flowing regime, through stress-driven discrete element method (DEM) simulations of flow onset and arrest in various geometries. Along the way, we develop a new methodology to compare deterministic model predictions with the stochastic behavior exhibited by the DEM simulations around the jamming transition.

Effects of interparticle friction on the response of 3D cyclically compressed granular material

We numerically study the effect of inter-particle friction coefficient on the response to cyclical pure shear of spherical particles in three dimensions. We focus on the rotations and translations of grains and look at the spatial distribution of these displacements as well as their probability distribution functions. We find that with increasing friction, the shear band becomes thinner and more pronounced. At low friction, the amplitude of particle rotations is homogeneously distributed in the system and is therefore mostly independent from both the affine and non-affine particle translations. In contrast, at high friction, the rotations are strongly localized in the shear zone. This work shows the importance of studying the effects of inter-particle friction on the response of granular materials to cyclic forcing, both for a better understanding of how rotations correlate to translations in sheared granular systems, and due to the relevance of cyclic forcing for most real-world applications in planetary science and industry.

Shear modulus and reversible particle trajectories of frictional granular materials under oscillatory shear

In this study, we numerically investigated the mechanical responses and trajectories of frictional granular particles under oscillatory shear in the reversible phase where particle trajectories form closed loops below the yielding point. When the friction coefficient is small, the storage modulus exhibits softening, and the loss modulus remains finite in the quasi-static limit. As the friction coefficient increases, the softening and residual loss modulus are suppressed. The storage and loss moduli satisfy scaling laws if they are plotted as functions of the areas of the loop trajectories divided by the strain amplitude and diameter of grains, at least for small values of the areas.

Evolution of granular materials under isochoric cyclic simple shearing

By means of 3D particle dynamics simulations, we analyze the microstructure of granular materials subjected to isochoric (constant volume) cyclic shearing, which drives the system towards a liquefaction state characterized by loops of jamming-unjamming transition with periodic loss of strength and irreversible accumulation of shear strain. We first show that the macroscopic response obtained by these simulations agrees well with the most salient features of the well-known cyclic behavior of granular materials both before and after liquefaction. Then we investigate the evolution of particle connectivity, force transmission, and anisotropies of contact and force networks. The onset of liquefaction is marked by partial collapse of the force-bearing network with rapid drop of the coordination number and nonrattler fraction of particles, and significant broadening of the contact force probability density function, which begins in the preliquefaction period. We find that the jamming transition in each cycle occurs for a critical value of the coordination number that can be interpreted as the percolation threshold of the contact network and appears to be independent of the initial mean stress, void ratio, and cyclic shear amplitude. We show that upon unjamming in each cycle an isotropic loss of contacts occurs and is followed by the development of high contact anisotropy and a large proportion of particles with only two or three contacts. The higher mobility of the particles also involves a lower degree of frustration of particle rotations and thus lower friction mobilization and tangential force anisotropy. These findings are relevant to both undrained cyclic deformations of saturated soils and rheology of dense non-Brownian suspensions where volume change is coupled with pore liquid drainage conditions.

A jamming plane of sphere packings

The concept of jamming has attracted great research interest due to its broad relevance in soft-matter, such as liquids, glasses, colloids, foams, and granular materials, and its deep connection to sphere packing and optimization problems. Here, we show that the domain of amorphous jammed states of frictionless spheres can be significantly extended, from the well-known jamming-point at a fixed density, to a jamming-plane that spans the density and shear strain axes. We explore the jamming-plane, via athermal and thermal simulations of compression and shear jamming, with initial equilibrium configurations prepared by an efficient swap algorithm. The jamming-plane can be divided into reversible-jamming and irreversible-jamming regimes, based on the reversibility of the route from the initial configuration to jamming. Our results suggest that the irreversible-jamming behavior reflects an escape from the metastable glass basin to which the initial configuration belongs to or the absence of such basins. All jammed states, either compression- or shear-jammed, are isostatic and exhibit jamming criticality of the same universality class. However, the anisotropy of contact networks nontrivially depends on the jamming density and strain. Among all state points on the jamming-plane, the jamming-point is a unique one with the minimum jamming density and the maximum randomness. For crystalline packings, the jamming-plane shrinks into a single shear jamming-line that is independent of initial configurations. Our study paves the way for solving the long-standing random close-packing problem and provides a more complete framework to understand jamming.

Shear response of granular packings compressed above jamming onset

We investigate the mechanical response of jammed packings of repulsive, frictionless spherical particles undergoing isotropic compression. Prior simulations of the soft-particle model, where the repulsive interactions scale as a power law in the interparticle overlap with exponent α, have found that the ensemble-averaged shear modulus 〈G(P)〉 increases with pressure P as ∼P^{(α-3/2)/(α-1)} at large pressures. 〈G〉 has two key contributions: (1) continuous variations as a function of pressure along geometrical families, for which the interparticle contact network does not change, and (2) discontinuous jumps during compression that arise from changes in the contact network. Using numerical simulations, we show that the form of the shear modulus G^{f} for jammed packings within near-isostatic geometrical families is largely determined by the affine response G^{f}∼G_{a}^{f}, where G_{a}^{f}/G_{a0}=(P/P_{0})^{(α-2)/(α-1)}-P/P_{0}, P_{0}∼N^{-2(α-1)} is the characteristic pressure at which G_{a}^{f}=0, G_{a0} is a constant that sets the scale of the shear modulus, and N is the number of particles. For near-isostatic geometrical families that persist to large pressures, deviations from this form are caused by significant nonaffine particle motion. We further show that the ensemble-averaged shear modulus 〈G(P)〉 is not simply a sum of two power laws, but 〈G(P)〉∼(P/P_{c})^{a}, where a≈(α-2)/(α-1) in the P→0 limit and 〈G(P)〉∼(P/P_{c})^{b}, where b≳(α-3/2)/(α-1), above a characteristic pressure that scales as P_{c}∼N^{-2(α-1)}.

Inference of chromosome 3D structures from GAM data by a physics computational approach

The combination of modelling and experimental advances can provide deep insights for understanding chromatin 3D organization and ultimately its underlying mechanisms. In particular, models of polymer physics can help comprehend the complexity of genomic contact maps, as those emerging from technologies such as Hi-C, GAM or SPRITE. Here we discuss a method to reconstruct 3D structures from Genome Architecture Mapping (GAM) data, based on PRISMR, a computational approach introduced to find the minimal polymer model best describing Hi-C input data from only polymer physics. After recapitulating the PRISMR procedure, we describe how we extended it for treating GAM data. We successfully test the method on a 6 Mb region around the Sox9 gene and, at a lower resolution, on the whole chromosome 7 in mouse embryonic stem cells. The PRISMR derived 3D structures from GAM co-segregation data are finally validated against independent Hi-C contact maps. The method results to be versatile and robust, hinting that it can be similarly applied to different experimental data, such as SPRITE or microscopy distance data.

Frictional active Brownian particles

Frictional forces affect the rheology of hard-sphere colloids, at high shear rate. Here we demonstrate, via numerical simulations, that they also affect the dynamics of active Brownian particles and their motility-induced phase separation. Frictional forces increase the angular diffusivity of the particles, in the dilute phase, and prevent colliding particles from resolving their collision by sliding one past to the other. This leads to qualitatively changes of motility-induced phase diagram in the volume-fraction motility plane. While frictionless systems become unstable towards phase separation as the motility increases only if their volume fraction overcomes a threshold, frictional systems become unstable regardless of their volume fraction. These results suggest the possibility of controlling the motility-induced phase diagram by tuning the roughness of the particles.

Shear jamming, discontinuous shear thickening, and fragile states in dry granular materials under oscillatory shear

We numerically study the linear response of two-dimensional frictional granular materials under oscillatory shear. The storage modulus G^{'} and the loss modulus G^{''} in the zero strain rate limit depend on the initial strain amplitude of the oscillatory shear before measurement. The shear jammed state (satisfying G^{'}>0) can be observed at an amplitude greater than a critical initial strain amplitude. The fragile state is defined by the emergence of liquid-like and solid-like states depending on the form of the initial shear. In this state, the observed G^{'} after the reduction of the strain amplitude depends on the phase of the external shear strain. The loss modulus G^{''} exhibits a discontinuous jump corresponding to discontinuous shear thickening in the fragile state.

A Dynamic Folded Hairpin Conformation Is Associated with α-Globin Activation in Erythroid Cells

We investigate the three-dimensional (3D) conformations of the α-globin locus at the single-allele level in murine embryonic stem cells (ESCs) and erythroid cells, combining polymer physics models and high-resolution Capture-C data. Model predictions are validated against independent fluorescence in situ hybridization (FISH) data measuring pairwise distances, and Tri-C data identifying three-way contacts. The architecture is rearranged during the transition from ESCs to erythroid cells, associated with the activation of the globin genes. We find that in ESCs, the spatial organization conforms to a highly intermingled 3D structure involving non-specific contacts, whereas in erythroid cells the α-globin genes and their enhancers form a self-contained domain, arranged in a folded hairpin conformation, separated from intermingling flanking regions by a thermodynamic mechanism of micro-phase separation. The flanking regions are rich in convergent CTCF sites, which only marginally participate in the erythroid-specific gene-enhancer contacts, suggesting that beyond the interaction of CTCF sites, multiple molecular mechanisms cooperate to form an interacting domain.

Rheology in dense assemblies of spherocylinders: Frictional vs. frictionless

Using molecular dynamics simulations, we study the steady shear flow of dense assemblies of anisotropic spherocylindrical particles of varying aspect ratios. Comparing frictionless and frictional particles we discuss the specific role of frictional inter-particle forces for the rheological properties of the system. In the frictional system we evidence a shear-thickening regime, similar to that for spherical particles. Furthermore, friction suppresses the alignment of the spherocylinders along the flow direction. Finally, the jamming density in frictional systems is rather insensitive to variations in aspect ratio, quite contrary to what is known from frictionless systems.

Shear-driven flow of athermal, frictionless, spherocylinder suspensions in two dimensions: Stress, jamming, and contacts

We use numerical simulations to study the flow of a bidisperse mixture of athermal, frictionless, soft-core two-dimensional spherocylinders driven by a uniform steady-state shear strain applied at a fixed finite rate. Energy dissipation occurs via a viscous drag with respect to a uniformly sheared host fluid, giving a simple model for flow in a non-Brownian suspension and resulting in a Newtonian rheology. We study the resulting pressure p and deviatoric shear stress σ of the interacting spherocylinders as a function of packing fraction ϕ, strain rate γ[over ̇], and a parameter α that measures the asphericity of the particles; α is varied to consider the range from nearly circular disks to elongated rods. We consider the direction of anisotropy of the stress tensor, the macroscopic friction μ=σ/p, and the divergence of the transport coefficient η_{p}=p/γ[over ̇] as ϕ is increased to the jamming transition ϕ_{J}. From a phenomenological analysis of Herschel-Bulkley rheology above jamming, we estimate ϕ_{J} as a function of asphericity α and show that the variation of ϕ_{J} with α is the main cause for differences in rheology as α is varied; when plotted as ϕ/ϕ_{J}, rheological curves for different α qualitatively agree. However, a detailed scaling analysis of the divergence of η_{p} for our most elongated particles suggests that the jamming transition of spherocylinders may be in a different universality class than that of circular disks. We also compute the number of contacts per particle Z in the system and show that the value at jamming Z_{J} is a nonmonotonic function of α that is always smaller than the isostatic value. We measure the probability distribution of contacts per unit surface length P(ϑ) at polar angle ϑ with respect to the spherocylinder spine and find that as α→0 this distribution seems to diverge at ϑ=π/2, giving a finite limiting probability for contacts on the vanishingly small flat sides of the spherocylinder. Finally, we consider the variation of the average contact force as a function of location on the particle surface.

Oscillatory Instabilities in Frictional Granular Matter

Frictional granular matter is shown to be fundamentally different in its plastic responses to external strains from generic glasses and amorphous solids without friction. While regular glasses exhibit plastic instabilities due to the vanishing of a real eigenvalue of the Hessian matrix, frictional granular materials can exhibit a previously unnoticed additional mechanism for instabilities, i.e., the appearance of a pair of complex eigenvalues leading to oscillatory exponential growth of perturbations that are tamed by dynamical nonlinearities. This fundamental difference appears crucial for the understanding of plasticity and failure in frictional granular materials. The possible relevance to earthquake physics is discussed.

DEM simulation of the local ordering of tetrahedral granular matter

The formation and growth of local order clusters in a tetrahedral granular assembly driven by 3D mechanical vibration were captured in DEM (discrete element method) dynamic simulation using a multi-sphere model. Two important kinds of clusters, dimer and wagon wheel structures, were observed based on which the growth behavior and mechanism of each local cluster with different orientations/structures were investigated. The results show that during vibration, dimer clusters are formed first and then most of them grow into linear trimers and tetramers. Wagon wheel clusters are also frequently observed that grow into hexamers and, further, octamer and nonamer local clusters. Coordination number (CN) evolution indicates that the decrease of local mean CN can be regarded as the signal for the formation of local clusters in the tetrahedral particle packing system. Nematic order metric analysis shows that although the two basic structures (dimer and wagon wheel structures) grow into complex local clusters during packing densification, these local clusters are randomly distributed in the tetrahedral particle packing system. Stress analysis indicates that the dimer-based local clusters are mostly formed in the compaction state of the tetrahedral particle packing system during the vibrated packing densification process. In comparison, the wagon wheel-based local clusters need much stronger interaction forces from tetrahedral particles during vibrated packing densification.

Flow-Arrest Transitions in Frictional Granular Matter

The transition between shear-flowing and shear-arrested states of frictional granular matter is studied using constant-stress discrete element simulations. By subjecting a dilute system of frictional grains to a constant external shear stress and pressure, friction-dependent critical shear stress and density are clearly identified with both exhibiting a crossover between low and high friction. The critical shear stress bifurcates two nonequilibrium steady states: (i) steady state shear flow characterized by a constant deformation rate, and (ii) shear arrest characterized by temporally decaying creep to a statically stable state. The onset of arrest below critical shear stress occurs at a time t_{c} that exhibits a heavy-tailed distribution, whose mean and variance diverge as a power law at the critical shear stress with a friction-dependent exponent that also exhibits a crossover between low and high friction. These observations indicate that granular arrest near critical shear stress is highly unpredictable and is strongly influenced by interparticle friction.

Rattler-induced aging dynamics in jammed granular systems

Granular materials jam when developing a network of contact forces able to resist the applied stresses. Through numerical simulations of the dynamics of the jamming process, we show that the jamming transition does not occur when the kinetic energy vanishes. Rather, as the system jams, the kinetic energy becomes dominated by rattler particles, which scatter within their cages. The relaxation of the kinetic energy in the jammed configuration exhibits a double power-law decay, which we interpret in terms of the interplay between backbone and rattler particles.

Kinetic theory of shear thickening for a moderately dense gas-solid suspension: From discontinuous thickening to continuous thickening

The Enskog kinetic theory for moderately dense gas-solid suspensions under simple shear flow is considered as a model to analyze the rheological properties of the system. The influence of the environmental fluid on solid particles is modeled via a viscous drag force plus a stochastic Langevin-like term. The Enskog equation is solved by means of two independent but complementary routes: (i) Grad's moment method and (ii) event-driven Langevin simulation of hard spheres. Both approaches clearly show that the flow curve (stress-strain rate relation) depends significantly on the volume fraction of the solid particles. In particular, as the density increases, there is a transition from the discontinuous shear thickening (observed in dilute gases) to the continuous shear thickening for denser systems. The comparison between theory and simulations indicates that while the theoretical predictions for the kinetic temperature agree well with simulations for densities φ≲0.5, the agreement for the other rheological quantities (the viscosity, the stress ratio, and the normal stress differences) is limited to more moderate densities (φ≲0.3) if the inelasticity during collisions between particles is not large.

Discontinuous change of shear modulus for frictional jammed granular materials

The shear modulus of jammed frictional granular materials with harmonic repulsive interaction under an oscillatory shear is numerically investigated. It is confirmed that the storage modulus, the real part of the shear modulus, for frictional grains with sufficiently small strain amplitude γ_{0} discontinuously emerges at the jamming transition point. The storage modulus for small γ_{0} differs from that of frictionless grains even in the zero friction limit, whereas they are almost identical with each other for sufficiently large γ_{0}, where the transition becomes continuous. The stress-strain curve exhibits a hysteresis loop even for a small strain, which connects a linear region for sufficiently small strain to another linear region for larger strain. We propose a scaling law to interpolate between the states of small and large γ_{0}.

Oscillatory rheology of dense, athermal suspensions of nearly hard spheres below the jamming point

The viscosity of a dense suspension has contributions from hydrodynamics and particle interactions, both of which depend upon the flow-induced arrangement of particles into fragile structures. Here, we study the response of nearly hard sphere suspensions to oscillatory shear using simulations and experiments in the athermal, non-inertial limit. Three distinct regimes are observed as a function of the strain amplitude γ₀. For γ₀ < 10^-1, initially non-contacting particles remain separated and the suspension behaves similarly to a Newtonian fluid, with the shear stress proportional to the strain rate, and the normal stresses close to zero. For γ₀ > 10¹, the microstructure becomes well-established at the beginning of each shear cycle and the rheology is quasi-Newtonian: the shear stress varies with the rate, but flow-induced structures lead to non-zero normal stresses. At intermediate γ₀, particle-particle contacts break and reform across entire oscillatory cycles, and we probe a non-linear regime that reveals the fragility of the material. Guided by these features, we further show that oscillatory shear may serve as a diagnostic tool to isolate specific stress contributions in dense suspensions, and more generally in those materials whose rheology has contributions with both hydrodynamic and non-hydrodynamic origin.

Shear banding, discontinuous shear thickening, and rheological phase transitions in athermally sheared frictionless disks

We report on numerical simulations of simple models of athermal, bidisperse, soft-core, massive disks in two dimensions, as a function of packing fraction ϕ, inelasticity of collisions as measured by a parameter Q, and applied uniform shear strain rate γ[over ̇]. Our particles have contact interactions consisting of normally directed elastic repulsion and viscous dissipation, as well as tangentially directed viscous dissipation, but no interparticle Coulombic friction. Mapping the phase diagram in the (ϕ,Q) plane for small γ[over ̇], we find a sharp first-order rheological phase transition from a region with Bagnoldian rheology to a region with Newtonian rheology, and show that the system is always Newtonian at jamming. We consider the rotational motion of particles and demonstrate the crucial importance that the coupling between rotational and translational degrees of freedom has on the phase structure at small Q (strongly inelastic collisions). At small Q, we show that, upon increasing γ[over ̇], the sharp Bagnoldian-to-Newtonian transition becomes a coexistence region of finite width in the (ϕ,γ[over ̇]) plane, with coexisting Bagnoldian and Newtonian shear bands. Crossing this coexistence region by increasing γ[over ̇] at fixed ϕ, we find that discontinuous shear thickening can result if γ[over ̇] is varied too rapidly for the system to relax to the shear-banded steady state corresponding to the instantaneous value of γ[over ̇].

Phase diagram for inertial granular flows

Flows of hard granular materials depend strongly on the interparticle friction coefficient μ_{p} and on the inertial number I, which characterizes proximity to the jamming transition where flow stops. Guided by numerical simulations, we derive the phase diagram of dense inertial flow of spherical particles, finding three regimes for 10^{-4}≲I≲10^{-1}: frictionless, frictional sliding, and rolling. These are distinguished by the dominant means of energy dissipation, changing from collisional to sliding friction, and back to collisional, as μ_{p} increases from zero at constant I. The three regimes differ in their kinetics and rheology; in particular, the velocity fluctuations and the stress ratio both display nonmonotonic behavior with μ_{p}, corresponding to transitions between the three regimes of flow. We rationalize the phase boundaries between these regimes, show that energy balance yields scaling relations between microscopic properties in each of them, and derive the strain scale at which particles lose memory of their velocity. For the frictional sliding regime most relevant experimentally, we find for I≥10^{-2.5} that the growth of the macroscopic friction μ(I) with I is induced by an increase of collisional dissipation. This implies in that range that μ(I)-μ(0)∼I^{1-2b}, where b≈0.2 is an exponent that characterizes both the dimensionless velocity fluctuations L∼I^{-b} and the density of sliding contacts χ∼I^{b}.

Rheological chaos of frictional grains

A two-dimensional dense fluid of frictional grains is shown to exhibit time-chaotic, spatially heterogeneous flow in a range of stress values, σ, chosen in the unstable region of s-shaped flow curves. Stress-controlled simulations reveal a phase diagram with reentrant stationary flow for small and large stress σ. In between, no steady flow state can be reached, instead the system either jams or displays time-dependent heterogeneous strain rates γ(r,t). The results of simulations are in agreement with the stability analysis of a simple hydrodynamic model, coupling stress and microstructure which we tentatively associate with the frictional contact network.

Rigid Cluster Decomposition Reveals Criticality in Frictional Jamming

We study the nature of the frictional jamming transition within the framework of rigidity percolation theory. Slowly sheared frictional packings are decomposed into rigid clusters and floppy regions with a generalization of the pebble game including frictional contacts. Our method suggests a second-order transition controlled by the emergence of a system-spanning rigid cluster accompanied by a critical cluster size distribution. Rigid clusters also correlate with common measures of rigidity. We contrast this result with frictionless jamming, where the rigid cluster size distribution is noncritical.

Statistical mechanics of unsaturated porous media

We explore a mean-field theory of fluid imbibition and drainage through permeable porous solids. In the limit of vanishing inertial and viscous forces, the theory predicts the hysteretic "retention curves" relating the capillary pressure applied across a connected domain to its degree of saturation in wetting fluid in terms of known surface energies and void space geometry. To avoid complicated calculations, we adopt the simplest statistical mechanics, in which a pore interacts with its neighbors through narrow openings called "necks," while being either full or empty of wetting fluid. We show how the main retention curves can be calculated from the statistical distribution of two dimensionless parameters λ and α measuring the specific areas of, respectively, neck cross section and wettable pore surface relative to pore volume. The theory attributes hysteresis of these curves to collective first-order phase transitions. We illustrate predictions with a porous domain consisting of a random packing of spheres, show that hysteresis strength grows with λ and weakens as the distribution of α broadens, and reproduce the behavior of Haines jumps observed in recent experiments on an ordered pore network.

Steady shearing flows of deformable, inelastic spheres

We extend models for granular flows based on the kinetic theory beyond the critical volume fraction at which a rate-independent contribution to the stresses develops. This involves the incorporation of a measure of the duration of the particle interaction before and after this volume fraction. At volume fractions less than the critical, the stress components contain contributions from momentum exchanged in collisions that are influenced by the particle elasticity. At volume fractions greater than the critical, the stress components contain both static contributions from particle elasticity and dynamic contributions from the momentum transfer associated with the release of elastic energy by the breaking of force chains. A simple expression for the duration of a collision before and after the critical volume fraction permits a smooth transition between the two regimes and predictions for the components of the stress in steady, homogeneous shearing that are in good agreement with the results of numerical simulations. Application of the theory to steady, inhomogeneous flows reproduces the features of such flows seen in numerical simulations and physical experiments.

Jamming of frictional particles: a nonequilibrium first-order phase transition

We propose a phase diagram for the shear flow of dry granular particles in two dimensions based on simulations and a phenomenological Landau theory for a nonequilibrium first-order phase transition. Our approach incorporates both frictional as well as frictionless particles. The most important feature of the frictional phase diagram is reentrant flow and a critical jamming point at finite stress. In the frictionless limit the regime of reentrance vanishes and the jamming transition is continuous with a critical point at zero stress. The jamming phase diagrams derived from the model agree with the experiments of Bi et al. [Nature (London) 480, 355 (2011)] and brings together previously conflicting numerical results.

Rheology near jamming: the influence of lubrication forces

We study, by computer simulations, the roles of different dissipation forces in the rheological properties of highly dense particle-laden flows. In particular, we are interested in the close-packing limit (jamming) and the question of whether "universal" observables can be identified that do not depend on the details of the dissipation model. To this end, we define a simplified lubrication force and systematically vary the range h(c) of this interaction. For fixed h(c) a crossover is seen from a Newtonian flow regime at small strain rates to inertia-dominated flow at larger strain rates. The same crossover is observed as a function of the lubrication range h(c). At the same time, but only at high densities close to jamming, single-particle velocities as well as local density distributions are unaffected by changes in the lubrication range--they are candidates for universal behavior. At densities away from jamming, this invariance is lost: short-range lubrication forces lead to pronounced particle clustering, while longer-ranged lubrication does not. These findings highlight the importance of "geometric" packing constraints for particle motion--independent of the specific dissipation model. With the free volume vanishing at random close packing, particle motion is more and more constrained by the ever smaller amount of free space. On the other hand, macroscopic rheological observables as well as higher-order correlation functions retain the variability of the underlying dissipation model.

Universal features of the jamming phase diagram of wet granular materials

We investigate the influence of the shape of a particle on the structure of the jamming phase diagram of wet granular materials. We compute the jamming phase diagram of wet dimers (two fused disks) and compare it with that of the wet disks. Amplitude of the external force at solidification, i.e., the jamming force F(s), is computed as a function of the packing fraction ϕ, the capillary bridge energy ɛ, and the aspect ratio of dimers α. Based on data collapse, an equation for amplitude of the external force at solidification F(s)(ϕ,ɛ,α) is derived. F(s) has scaling and logarithmic relations with ϕ and ɛ, respectively, exactly the same type reported for wet disks earlier. Interestingly, F(s) does not depend on the aspect ratio of dimers α. The only difference is that wet dimers are found to be more stiffer than wet disks. However, the similarities of the equations describing F(s)(ϕ,ɛ,α) of wet dimers and disks imply that there exists, yet unknown, universal aspects of mechanical response of wet granular materials to the external forces, independent from the particle shape. In addition, we study local orientation of particles and its statistical properties.

Shear thickening in granular suspensions: interparticle friction and dynamically correlated clusters

We consider the shear rheology of concentrated suspensions of non-Brownian frictional particles. The key result of our study is the emergence of a pronounced shear-thickening regime, where frictionless particles would normally undergo shear thinning. We can clarify that shear thickening in our simulations is due to enhanced energy dissipation via frictional interparticle forces. Moreover, we evidence the formation of dynamically correlated particle clusters of size ξ, which contribute to shear thickening via an increase in viscous dissipation. A scaling argument gives for the associated viscosity η(v)~ξ(2), which is in very good agreement with the data.

Polydirectional stability of granular matter

We investigate jammed granular matter in a slowly rotating drum partially filled with granular material and find a state of polydirectional stability. In this state, the material responds elastically to small stresses in a wide angular interval while it responds by plastic deformation when subjected to small stresses outside this interval of directions. We describe the evolution of the granulate by means of a rate equation and find quantitative agreement with the experiment. The state of polydirectional stability complements the fragile state, where the material responds elastically to small applied stresses only in a certain direction but even very small stresses in any other direction would lead to plastic deformations. Similar to fragile matter, polydirectionally stable matter is created in a dynamic process by self-organization.

Microscopic mechanism for shear thickening of non-Brownian suspensions

We propose a simple model, supported by contact-dynamics simulations as well as rheology and friction measurements, that links the transition from continuous to discontinuous shear thickening in dense granular pastes to distinct lubrication regimes in the particle contacts. We identify a local Sommerfeld number that determines the transition from Newtonian to shear-thickening flows, and then show that the suspension's volume fraction and the boundary lubrication friction coefficient control the nature of the shear-thickening transition, both in simulations and experiments.

Penetration of projectiles into granular targets

Energetic collisions of subatomic particles with fixed or moving targets have been very valuable to penetrate into the mysteries of nature. But the mysteries are quite intriguing when projectiles and targets are macroscopically immense. We know that countless debris wandering in space impacted (and still do) large asteroids, moons and planets; and that millions of craters on their surfaces are traces of such collisions. By classifying and studying the morphology of such craters, geologists and astrophysicists obtain important clues to understand the origin and evolution of the Solar System. This review surveys knowledge about crater phenomena in the planetary science context, avoiding detailed descriptions already found in excellent papers on the subject. Then, it examines the most important results reported in the literature related to impact and penetration phenomena in granular targets obtained by doing simple experiments. The main goal is to discern whether both schools, one that takes into account the right ingredients (planetary bodies and very high energies) but cannot physically reproduce the collisions, and the other that easily carries out the collisions but uses laboratory ingredients (small projectiles and low energies), can arrive at a synergistic intersection point.

Melting a granular glass by cooling

Driven granular systems readily form glassy phases at high particle volume fractions and low driving amplitudes. We use computer simulations of a driven granular glass to evidence a reentrance melting transition into a fluid state, which, contrary to intuition, occurs by reducing the amplitude of the driving. This transition is accompanied by anomalous particle dynamics and superdiffusive behavior on intermediate time scales. We highlight the special role played by frictional interactions, which help particles to escape their glassy cages. Such an effect is in striking contrast to what friction is expected to do: reduce particle mobility by making them stick.