Critical behaviors of sheared frictionless granular materials near the jamming transition

Nonmonotonic Constitutive Curves and Shear Banding in Dry and Wet Granular Flows

We use particle simulations to map comprehensively the shear rheology of dry and wet granular matter comprising particles of finite stiffness, in both fixed pressure and fixed volume protocols. At fixed pressure we find nonmonotonic constitutive curves that are shear thinning, whereas at fixed volume we find nonmonotonic constitutive curves that are shear thickening. We show that the presence of one nonmonotonicity does not imply the other. Instead, there exists a signature in the volume fraction measured under fixed pressure that, when present, ensures nonmonotonic constitutive curves at fixed volume. In the context of dry granular flow we show that gradient and vorticity bands arise under fixed pressure and volume, respectively, as implied by the constitutive curves. For wet systems our results are consistent with a recent experimental observation of shear thinning at fixed pressure. We furthermore predict discontinuous shear thickening in the absence of critical load friction.

Rheology of Suspensions of Non-Brownian Soft Spheres across the Jamming and Viscous-to-Inertial Transitions

The rheology of suspensions of non-Brownian soft spheres is studied across jamming but also across the viscous and inertial regimes using a custom pressure- and volume-imposed rheometer. The study shows that the granular rheology found for suspensions of hard spheres can be extended to a soft granular rheology (SGranR) by renormalizing the critical volume fraction and friction coefficient to pressure-dependent values and using the addition of the viscous and inertial stress scales. This SGranR encompasses rheological behaviors on both sides of the jamming transition, resulting in an approximate collapse of the rheological data into two branches when scaled with the distance to jamming, as observed for soft colloids. This research suggests that suspensions of soft particles across flow regimes can be described by a unified SGranR framework around the jamming transition.

Unified Understanding of Nonlinear Rheology near the Jamming Transition Point

When slowly sheared, jammed packings respond elastically before yielding. This linear elastic regime becomes progressively narrower as the jamming transition point is approached, and rich nonlinear rheologies such as shear softening and hardening or melting emerge. However, the physical mechanism of these nonlinear rheologies remains elusive. To clarify this, we numerically study jammed packings of athermal frictionless soft particles under quasistatic shear γ. We find the universal scaling behavior for the ratio of the shear stress σ and the pressure P, independent of the preparation protocol of the initial configurations. In particular, we reveal shear softening σ/P∼γ^{1/2} over an unprecedentedly wide range of strain up to the yielding point, which a simple scaling argument can rationalize.

Comparison of compression versus shearing near jamming, for a simple model of athermal frictionless disks in suspension

Using a simplified model for a non-Brownian suspension, we numerically study the response of athermal, overdamped, frictionless disks in two dimensions to isotropic and uniaxial compression, as well as to pure and simple shearing, all at finite constant strain rates ε[over ̇]. We show that isotropic and uniaxial compression result in the same jamming packing fraction ϕ_{J}, while pure-shear- and simple-shear-induced jamming occurs at a slightly higher ϕ_{J}^{*}, consistent with that found previously for simple shearing. A critical scaling analysis of pure shearing gives critical exponents consistent with those previously found for both isotropic compression and simple shearing. Using orientational order parameters for contact bond directions, we compare the anisotropy of the force and contact networks at both lowest nematic order, as well as higher 2n-fold order.

Universality of stress-anisotropic and stress-isotropic jamming of frictionless spheres in three dimensions: Uniaxial versus isotropic compression

We numerically study a three-dimensional system of athermal, overdamped, frictionless spheres, using a simplified model for a non-Brownian suspension. We compute the bulk viscosity under both uniaxial and isotropic compression as a means to address the question of whether stress-anisotropic and stress-isotropic jamming are in the same critical universality class. Carrying out a critical scaling analysis of the system pressure p, shear stress σ, and macroscopic friction μ=σ/p, as functions of particle packing fraction ϕ and compression rate ε[over ̇], we find good agreement for all critical parameters comparing the isotropic and anisotropic cases. In particular, we determine that the bulk viscosity diverges as p/ε[over ̇]∼(ϕ_{J}-ϕ)^{-β}, with β=3.36±0.09, as jamming is approached from below. We further demonstrate that the average contact number per particle Z can also be written in a scaling form as a function of ϕ and ε[over ̇]. Once again, we find good agreement between the uniaxial and isotropic cases. We compare our results to prior simulations and theoretical predictions.

Critical scaling of compression-driven jamming of athermal frictionless spheres in suspension

We study numerically a system of athermal, overdamped, frictionless spheres, as in a non-Brownian suspension, in two and three dimensions. Compressing the system isotropically at a fixed rate ε[over ̇], we investigate the critical behavior at the jamming transition. The finite compression rate introduces a control timescale, which allows one to probe the critical timescale associated with jamming. As was found previously for steady-state shear-driven jamming, we find for compression-driven jamming that pressure obeys a critical scaling relation as a function of packing fraction ϕ and compression rate ε[over ̇], and that the bulk viscosity p/ε[over ̇] diverges upon jamming. A scaling analysis determines the critical exponents associated with the compression-driven jamming transition. Our results suggest that stress-isotropic, compression-driven jamming may be in the same universality class as stress-anisotropic, shear-driven jamming.

Dilatancy of frictional granular materials under oscillatory shear with constant pressure

We perform numerical simulations of a two-dimensional frictional granular system under oscillatory shear confined by constant pressure. We found that the system undergoes dilatancy as the strain increases. We confirmed that compaction also takes place at an intermediate strain amplitude for a small mutual friction coefficient between particles. We also found that compaction depends on the confinement pressure while dilatancy little depends on the pressure.

Dynamic length scales in athermal, shear-driven jamming of frictionless disks in two dimensions

We carry out numerical simulations of athermally sheared, bidisperse, frictionless disks in two dimensions. From an appropriately defined velocity correlation function, we determine that there are two diverging length scales, ξ and ℓ, as the jamming transition is approached. We analyze our results using a critical scaling ansatz for the correlation function and argue that the more divergent length ℓ is a consequence of a dangerous irrelevant scaling variable and that it is ξ, which is the correlation length that determines the divergence of the system viscosity as jamming is approached from below in the liquid phase. We find that ξ∼(ϕ_{J}-ϕ)^{-ν} diverges with the critical exponent ν=1. We provide evidence that ξ measures the length scale of fluctuations in the rotation of the particle velocity field, while ℓ measures the length scale of fluctuations in the divergence of the velocity field.

Translational and rotational velocities in shear-driven jamming of ellipsoidal particles

We study shear-driven jamming of ellipsoidal particles at zero temperature with a focus on the microscopic dynamics. We find that a change from spherical particles to ellipsoids with aspect ratio α=1.02 gives dramatic changes of the microscopic dynamics with much lower translational velocities and a new role for the rotations. Whereas the velocity difference at contacts-and thereby the dissipation-in collections of spheres is dominated by the translational velocities and reduced by the rotations, the same quantity is in collections of ellipsoids instead totally dominated by the rotational velocities. By also examining the effect of different aspect ratios we find that the examined quantities show either a peak or a change in slope at α≈1.2, which thus gives evidence for a crossover between different regions of low and high aspect ratio.

Scaling laws for frictional granular materials confined by constant pressure under oscillatory shear

Herein we numerically study the rheology of a two-dimensional frictional granular system confined by constant pressure under oscillatory shear. Several scaling laws for the storage and loss moduli against the scaled strain amplitude have been found. The scaling laws in plastic regime for large strain amplitude can be understood by the angular distributions of the contact force. The scaling exponents are estimated by considering the physical mechanism.

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.

Rheological similarities between dense self-propelled and sheared particulate systems

Different from previous modeling of self-propelled particles, we develop a method to propel particles with a constant average velocity instead of a constant force. This constant propulsion velocity (CPV) approach is validated by its agreement with the conventional constant propulsion force (CPF) approach in the flowing regime. However, the CPV approach shows its advantage of accessing quasistatic flows of yield stress fluids with a vanishing propulsion velocity, while the CPF approach is usually unable to because of finite system size. Taking this advantage, we realize cyclic self-propulsion and study the evolution of the propulsion force with the propelled particle displacement, both in the quasistatic flow regime. By mapping the shear stress and shear rate to the propulsion force and propulsion velocity, we find similar rheological behaviors of self-propelled systems to sheared systems, including the yield force gap between the CPF and CPV approaches, propulsion force overshoot, reversible-irreversible transition under cyclic propulsion, and propulsion bands in plastic flows. These similarities suggest underlying connections between self-propulsion and shear, although they act on systems in different ways.

Stress Relaxation above and below the Jamming Transition

We numerically investigate stress relaxation in soft athermal disks to reveal critical slowing down when the system approaches the jamming point. The exponents describing the divergence of the relaxation time differ dramatically depending on whether the transition is approached from the jammed or unjammed phase. This contrasts sharply with conventional dynamic critical scaling scenarios, where a single exponent characterizes both sides. We explain this surprising difference in terms of the vibrational density of states, which is a key ingredient of linear viscoelastic theory. The vibrational density of states exhibits an extra slow mode that emerges below jamming, which we utilize to demonstrate the anomalous exponent below jamming.

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.

Laboratory experiment and discrete-element-method simulation of granular-heap flows under vertical vibration

Granular flow dynamics on a vertically vibrated pile is studied by means of both laboratory experiments and numerical simulations. As already revealed, the depth-averaged velocity of a fully fluidized granular pile under strong vibration, which is measured by a high-speed laser profiler in the experiment, can be explained by the nonlinear diffusion transport model proposed by our previous paper [Tsuji et al., Phys. Rev. Lett. 120, 128001 (2018)PRLTAO0031-900710.1103/PhysRevLett.120.128001]. In this paper, we report that a similar transport model can be applied to the relation between the surface velocity and slope in the experiment. These facts are also reproduced by particle-scale numerical simulations based on the discrete element method. In addition, using these numerical results, the velocity profile inside the fluidized pile is measured. As a result, we show that the flow velocity decreases exponentially with depth from the surface of the pile, which means that a clearly fluidized region, also known as shear band structure, is localized around the surface. However, its thickness grows proportionally with the local height of the pile, i.e., the shear band does not consist of a fluidized layer with a constant thickness. From these features, we finally demonstrate that the integration of this exponentially decreasing velocity profile is consistent with the depth-averaged velocity predicted by the nonlinear diffusion transport model.

Nonlocal Effects in Inhomogeneous Flows of Soft Athermal Disks

We numerically investigate nonlocal effects on inhomogeneous flows of soft athermal disks close to but below their jamming transition. We employ molecular dynamics to simulate Kolmogorov flows, in which a sinusoidal flow profile with fixed wave number is externally imposed, resulting in a spatially inhomogeneous shear rate. We find that the resulting rheology is strongly wave-number-dependent, and that particle migration, while present, is not sufficient to describe the resulting stress profiles within a conventional local model. We show that, instead, stress profiles can be captured with nonlocal constitutive relations that account for gradients to fourth order. Unlike nonlocal flow in yield stress fluids, we find no evidence of a diverging length scale.

Dimensionality and Viscosity Exponent in Shear-driven Jamming

Collections of bidisperse frictionless particles at zero temperature in three dimensions are simulated with a shear-driven dynamics with the aim to compare with the behavior in two dimensions. Contrary to the prevailing picture, and in contrast to results from isotropic jamming from compression or quench, we find that the critical exponents in three dimensions are different from those in two dimensions and conclude that shear-driven jamming in two and three dimensions belong to different universality classes.

Viscous forces and bulk viscoelasticity near jamming

When weakly jammed packings of soft, viscous, non-Brownian spheres are probed mechanically, they respond with a complex admixture of elastic and viscous effects. While many of these effects are understood for specific, approximate models of the particles' interactions, there are a number of proposed force laws in the literature, especially for viscous interactions. We numerically measure the complex shear modulus G* of jammed packings for various viscous force laws that damp relative velocities between pairs of contacting particles or between a particle and the continuous fluid phase. We find a surprising sensitive dependence of G* on the viscous force law: the system may or may not display dynamic critical scaling, and the exponents describing how G* scales with frequency can change. We show that this sensitivity is closely linked to manner in which viscous damping couples to floppy-like, non-affine motion, which is prominent near jamming.

Anisotropic decay of the energy spectrum in two-dimensional dense granular flows

We study anisotropic collective motions of two-dimensional granular particles under simple shear deformations. Employing molecular-dynamics simulations of large system sizes, we find that anisotropic fluidized bands develop in the system yielding under quasistatic deformations, where the spectrum of nonaffine velocities, which is associated with the energy spectrum for turbulent flows, exhibits a quadrupole structure. To explain theoretically the anisotropic spectrum, we derive hydrodynamic modes from a continuum model of dense granular materials, where we find that fluidized bands are caused by long-lived hydrodynamic fluctuations characterized by compressibility.

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}.

Slow creep in soft granular packings

Transient creep mechanisms in soft granular packings are studied numerically using a constant pressure and constant stress simulation method. Rapid compression followed by slow dilation is predicted on the basis of a logarithmic creep phenomenon. Characteristic scales of creep strain and time exhibit a power-law dependence on jamming pressure, and they diverge at the jamming point. Microscopic analysis indicates the existence of a correlation between rheology and nonaffine fluctuations. Localized regions of large strain appear during creep and grow in magnitude and size at short times. At long times, the spatial structure of highly correlated local deformation becomes time-invariant. Finally, a microscale connection between local rheology and local fluctuations is demonstrated in the form of a linear scaling between granular fluidity and nonaffine velocity.

Characterizing rare fluctuations in soft particulate flows

Soft particulate media include a wide range of systems involving athermal dissipative particles both in non-living and biological materials. Characterization of flows of particulate media is of great practical and theoretical importance. A fascinating feature of these systems is the existence of a critical rigidity transition in the dense regime dominated by highly intermittent fluctuations that severely affects the flow properties. Here, we unveil the underlying mechanisms of rare fluctuations in soft particulate flows. We find that rare fluctuations have different origins above and below the critical jamming density and become suppressed near the jamming transition. We then conjecture a time-independent local fluctuation relation, which we verify numerically, and that gives rise to an effective temperature. We discuss similarities and differences between our proposed effective temperature with the conventional kinetic temperature in the system by means of a universal scaling collapse.

Soft particulate flows such as granular media are prone to fluctuations like jamming and avalanches. Here Rahbari et al. consider the statistics of rare fluctuations to identify an effective temperature which, unlike previous ones, is valid for packing fractions both near and far from the jamming point.

Effect of collisional elasticity on the Bagnold rheology of sheared frictionless two-dimensional disks

We carry out constant volume simulations of steady-state, shear-driven flow in a simple model of athermal, bidisperse, soft-core, frictionless disks in two dimensions, using a dissipation law that gives rise to Bagnoldian rheology. Focusing on the small strain rate limit, we map out the rheological behavior as a function of particle packing fraction ϕ and a parameter Q that measures the elasticity of binary particle collisions. We find a Q^{*}(ϕ) that marks the clear crossover from a region characteristic of strongly inelastic collisions, Q<Q^{*}, to a region characteristic of weakly inelastic collisions, Q>Q^{*}, and give evidence that Q^{*}(ϕ) diverges as ϕ→ϕ_{J}, the shear-driven jamming transition. We thus conclude that the jamming transition at any value of Q behaves the same as the strongly inelastic case, provided one is sufficiently close to ϕ_{J}. We further characterize the differing nature of collisions in the strongly inelastic vs weakly inelastic regions, and recast our results into the constitutive equation form commonly used in discussions of hard granular matter.

Merging fluid and solid granular behavior

Simple homogeneous shear flows of frictionless, deformable particles are studied by particle simulations at large shear rates and for differently soft, deformable particles. Particle stiffness sets a time-scale that can be used to scale the physical quantities; thus the dimensionless shear rate, i.e. the inertial number I (inversely proportional to pressure), can alternatively be expressed as inversely proportional to the square root of particle stiffness. Asymptotic scaling relations for the field variables pressure, shear stress and granular temperature are inferred from simulations in both fluid and solid regimes, corresponding to unjammed and jammed conditions. Then the limit cases are merged to unique constitutive relations that cover also the transition zone in the proximity of jamming. By exploiting the diverging behavior of the scaling laws at the jamming density, we arrive at continuous and differentiable phenomenological constitutive relations for stresses and granular temperature as functions of the volume fraction, shear rate, particle stiffness and distance from jamming. In contrast to steady shear flows of hard particles the (shear) stress ratio μ does not collapse as a function of the inertial number, indicating the need for an additional control parameter. In the range of particle stiffnesses investigated, in the solid regime, only pressure is rate independent, whereas shear stress exhibits a slight shear rate- and stiffness-dependency.

Enstrophy cascades in two-dimensional dense granular flows

Employing two-dimensional molecular dynamics simulations of dense granular materials under simple shear deformations, we investigate vortex structures of particle rearrangements. Introducing vorticity fields as a measure of local spinning motions of the particles, we observe their heterogeneous distributions, where statistics of vorticity fields exhibit the highly non-Gaussian behavior and typical domain sizes of vorticity fields significantly increase if the system is yielding under quasistatic deformations. In such dense granular flows, a power-law decay of vorticity spectra can be observed at mesoscopic scale, implying anomalous local structures of kinetic energy dissipation. We explain the power-law decay, or enstrophy cascades in dense granular materials, by a dimensional analysis, where the dependence of vorticity spectra not only on the wave number, but also on the shear rate, is well explained. From our dimensional analyses, the scaling of granular temperature and rotational kinetic energy is also predicted.

Critical scaling of Bagnold rheology at the jamming transition of frictionless two-dimensional disks

We carry out constant volume simulations of steady-state shear-driven rheology in a simple model of bidisperse soft-core frictionless disks in two dimensions, using a dissipation law that gives rise to Bagnoldian rheology. We discuss in detail the critical scaling ansatz for the shear-driven jamming transition and carry out a detailed scaling analysis of our resulting data for pressure p and shear stress σ. Our analysis determines the critical exponent β that describes the algebraic divergence of the Bagnold transport coefficients lim_{γ[over ̇]→0}p/γ[over ̇]^{2},σ/γ[over ̇]^{2}∼(ϕ_{J}-ϕ)^{-β} as the jamming transition ϕ_{J} is approached from below. For the low strain rates considered in this work, we show that it is still necessary to consider the leading correction-to-scaling term in order to achieve a self-consistent analysis of our data, in which the critical parameters become independent of the size of the window of data used in the analysis. We compare our resulting value β≈5.0±0.4 against previous numerical results and competing theoretical models. Our results confirm that the shear-driven jamming transition in Bagnoldian systems is well described by a critical scaling theory and we relate this scaling theory to the phenomenological constituent laws for dilatancy and friction.

Anomalous energy cascades in dense granular materials yielding under simple shear deformations

By using molecular dynamics (MD) simulations of dense granular particles in two dimensions, we study turbulent-like structures of their non-affine velocities under simple shear deformations. We find that the spectrum of non-affine velocities, introduced as an analog of the energy spectrum for turbulent flows, exhibits the power-law decay if the system is yielding in a quasi-static regime, where large-scale collective motions and inelastic interactions of granular particles are crucial for the anomalous cascade of kinetic energy. Based on hydrodynamic equations of dense granular materials, which include both kinetic and contact contributions in constitutive relations, we derive a theoretical expression for the spectrum, where a good agreement between the result of MD simulations and theoretical prediction is established over a wide range of length scales.

Free volume under shear

Using an athermal quasistatic simulation protocol, we study the distribution of free volumes in sheared hard-particle packings close to, but below, the random-close packing threshold. We show that under shear, and independent of volume fraction, the free volumes develop features similar to close-packed systems - particles self-organize in a manner as to mimick the isotropically jammed state. We compare athermally sheared packings with thermalized packings and show that thermalization leads to an erasure of these structural features. The temporal evolution in particular the opening-up and the closing of free-volume patches is associated with the single-particle dynamics, showing a crossover from ballistic to diffusive behavior.

Divergence of Viscosity in Jammed Granular Materials: A Theoretical Approach

A theory for jammed granular materials is developed with the aid of a nonequilibrium steady-state distribution function. The approximate nonequilibrium steady-state distribution function is explicitly given in the weak dissipation regime by means of the relaxation time. The theory quantitatively agrees with the results of the molecular dynamics simulation on the critical behavior of the viscosity below the jamming point without introducing any fitting parameter.

Universal rescaling of flow curves for yield-stress fluids close to jamming

The experimental flow curves of four different yield-stress fluids with different interparticle interactions are studied near the jamming concentration. By appropriate scaling with the distance to jamming all rheology data can be collapsed onto master curves below and above jamming that meet in the shear-thinning regime and satisfy the Herschel-Bulkley and Cross equations, respectively. In spite of differing interactions in the different systems, master curves characterized by universal scaling exponents are found for the four systems. A two-state microscopic theory of heterogeneous dynamics is presented to rationalize the observed transition from Herschel-Bulkley to Cross behavior and to connect the rheological exponents to microscopic exponents for the divergence of the length and time scales of the heterogeneous dynamics. The experimental data and the microscopic theory are compared with much of the available literature data for yield-stress systems.

Relaxation times and rheology in dense athermal suspensions

We study the jamming transition in a model of elastic particles under shear at zero temperature. The key quantity is the relaxation time τ which is obtained by stopping the shearing and letting energy and pressure decay to zero. At many different densities and initial shear rates we do several such relaxations to determine the average τ. We establish that τ diverges with the same exponent as the viscosity and determine another exponent from the relation between τ and the coordination number. Though most of the simulations are done for the model with dissipation due to the motion of particles relative to an affinely shearing substrate, we also examine a model, where the dissipation is instead due to velocity differences of disks in contact, and confirm that the above-mentioned exponent is the same for these two models. We also consider finite size effects on both τ and the coordination number.

Unified theory of inertial granular flows and non-Brownian suspensions

Rheological properties of dense flows of hard particles are singular as one approaches the jamming threshold where flow ceases both for aerial granular flows dominated by inertia and for over-damped suspensions. Concomitantly, the length scale characterizing velocity correlations appears to diverge at jamming. Here we introduce a theoretical framework that proposes a tentative, but potentially complete, scaling description of stationary flows. Our analysis, which focuses on frictionless particles, applies both to suspensions and inertial flows of hard particles. We compare our predictions with the empirical literature, as well as with novel numerical data. Overall, we find a very good agreement between theory and observations, except for frictional inertial flows whose scaling properties clearly differ from frictionless systems. For overdamped flows, more observations are needed to decide if friction is a relevant perturbation. Our analysis makes several new predictions on microscopic dynamical quantities that should be accessible experimentally.

Avalanche contribution to shear modulus of granular materials

Shear modulus of frictionless granular materials near the jamming transition under oscillatory shear is numerically investigated. It is found that the shear modulus G satisfies a scaling law to interpolate between G∼(ϕ-ϕJ)(1/2) and G∼γ0(-1/2)(ϕ-ϕJ) for a linear spring model of the elastic interaction between contacting grains, where ϕ, ϕJ, and γ0 are, respectively, the volume fraction of grains, the fraction at the jamming point, and the amplitude of the oscillatory shear. The linear relation between the shear modulus and ϕ-ϕJ can be understood by slip avalanches.

Rheology of cohesive granular materials across multiple dense-flow regimes

We investigate the dense-flow rheology of cohesive granular materials through discrete element simulations of homogeneous, simple shear flows of frictional, cohesive, spherical particles. Dense shear flows of noncohesive granular materials exhibit three regimes: quasistatic, inertial, and intermediate, which persist for cohesive materials as well. It is found that cohesion results in bifurcation of the inertial regime into two regimes: (a) a new rate-independent regime and (b) an inertial regime. Transition from rate-independent cohesive regime to inertial regime occurs when the kinetic energy supplied by shearing is sufficient to overcome the cohesive energy. Simulations reveal that inhomogeneous shear band forms in the vicinity of this transition, which is more pronounced at lower particle volume fractions. We propose a rheological model for cohesive systems that captures the simulation results across all four regimes.

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.

Linear and nonlinear rheology of dense emulsions across the glass and the jamming regimes

We discuss the linear and nonlinear rheology of concentrated microscale emulsions, amorphous disordered solids composed of repulsive and deformable soft colloidal spheres. Based on recent results from simulation and theory, we derive quantitative predictions for the dependences of the elastic shear modulus and the yield stress on the droplet volume fraction. The remarkable agreement with experiments we observe supports the scenario that the repulsive glass and the jammed state can be clearly identified in the rheology of soft spheres at finite temperature while crossing continuously from a liquid to a highly compressed yet disordered solid.

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.

Rheology across the zero-temperature jamming transition

Many soft-matter systems show a transition between fluidlike and mechanically solidlike states when the volume fraction of the material, e.g., particles, drops, or bubbles is increased. Using an emulsion as a model system with a precisely controllable volume fraction, we show that the entire mechanical behavior in the vicinity of the jamming point can be understood if the mechanical transition is assumed to be analogous to a phase transition. We find power-law scalings in the distance to the jamming point, in which the parameters and exponents connect the behavior above and below jamming. We propose a simple two-state model with heterogeneous dynamics to describe the transition between jammed and mobile states. The model reproduces the steady-state and creep rheology and relates the power-law exponents to diverging microscopic time scales.

Herschel-Bulkley shearing rheology near the athermal jamming transition

We consider the rheology of soft-core frictionless disks in two dimensions in the neighborhood of the athermal jamming transition. From numerical simulations of bidisperse, overdamped particles, we argue that the divergence of the viscosity below jamming is characteristic of the hard-core limit, independent of the particular soft-core interaction. We develop a mapping from soft-core to hard-core particles that recovers all the critical behavior found in earlier scaling analyses. Using this mapping we derive a relation that gives the exponent of the nonlinear Herschel-Bulkley rheology above jamming in terms of the exponent of the diverging viscosity below jamming.

Critical scaling of a jammed system after a quench of temperature

Critical behavior of soft repulsive particles after quench of temperature near the jamming transition is numerically investigated. It is found that the plateau of the mean-square displacement of tracer particles and the pressure satisfy critical scaling laws. The critical density for the jamming transition depends on the protocol to prepare the system, while the values of the critical exponents which are consistent with the prediction of a phenomenology are independent of the protocol.

Phase transition in peristaltic transport of frictionless granular particles

Flows of dissipative particles driven by the peristaltic motion of a tube are numerically studied. A transition from a slow "unjammed" flow to a fast "jammed" flow is found through the observation of the flow rate at a critical width of the bottleneck of a peristaltic tube. It is also found that the average and fluctuation of the transition time, and the peak value of the second moment of the flow rate exhibit power-law divergence near the critical point and that these variables satisfy scaling relationships near the critical point. The dependence of the critical width and exponents on the peristaltic speed and the density is also discussed.

Bridging the rheology of granular flows in three regimes

We investigate the rheology of granular materials via molecular dynamics simulations of homogeneous, simple shear flows of soft, frictional, noncohesive spheres. In agreement with previous results for frictionless particles, we observe three flow regimes existing in different domains of particle volume fraction and shear rate, with all stress data collapsing upon scaling by powers of the distance to the jamming point. Though this jamming point is a function of the interparticle friction coefficient, the relation between pressure and strain rate at this point is found to be independent of friction. We also propose a rheological model that blends the asymptotic relations in each regime to obtain a general description for these flows. Finally, we show that departure from inertial number scalings is a direct result of particle softness, with a dimensionless shear rate characterizing the transition.

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.

Critical scaling near jamming transition for frictional granular particles

The critical rheology of sheared frictional granular materials near jamming transition is numerically investigated. It is confirmed that there exists a true critical density which characterizes the onset of the yield stress and two fictitious critical densities which characterize the scaling laws of rheological properties. We find the existence of a hysteresis loop between two of the critical densities for each friction coefficient. It is noteworthy that the critical scaling law for frictionless jamming transition seems to be still valid even for frictional jamming despite using fictitious critical density values.