Growing length and time scales in a suspension of athermal particles

Considerations on the relaxation time in shear-driven jamming

We study the jamming transition in a model of elastic particles under shear at zero temperature, with a focus on the relaxation time τ_{1}. This relaxation time is from two-step simulations where the first step is the ordinary shearing simulation and the second step is the relaxation of the energy after stopping the shearing. τ_{1} is determined from the final exponential decay of the energy. Such relaxations are done with many different starting configurations generated by a long shearing simulation in which the shear variable γ slowly increases. We study the correlations of both τ_{1}, determined from the decay, and the pressure, p_{1}, from the starting configurations as a function of the difference in γ. We find that the correlations of p_{1} are longer lived than the ones of τ_{1} and find that the reason for this is that the individual τ_{1} is controlled both by p_{1} of the starting configuration and a random contribution which depends on the relaxation path length-the average distance moved by the particles during the relaxation. We further conclude that it is γ_{τ}, determined from the correlations of τ_{1}, which is the relevant one when the aim is to generate data that may be used for determining the critical exponent that characterizes the jamming transition.

Strain-Controlled Critical Slowing Down in the Rheology of Disordered Networks

Networks and dense suspensions frequently reside near a boundary between soft (or fluidlike) and rigid (or solidlike) regimes. Transitions between these regimes can be driven by changes in structure, density, or applied stress or strain. In general, near the onset or loss of rigidity in these systems, dissipation-limiting heterogeneous nonaffine rearrangements dominate the macroscopic viscoelastic response, giving rise to diverging relaxation times and power-law rheology. Here, we describe a simple quantitative relationship between nonaffinity and the excess viscosity. We test this nonaffinity-viscosity relationship computationally and demonstrate its rheological consequences in simulations of strained filament networks and dense suspensions. We also predict critical signatures in the rheology of semiflexible and stiff biopolymer networks near the strain stiffening transition.

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.

Origin of Two Distinct Stress Relaxation Regimes in Shear Jammed Dense Suspensions

Many dense particulate suspensions show a stress induced transformation from a liquidlike state to a solidlike shear jammed (SJ) state. However, the underlying particle-scale dynamics leading to such striking, reversible transition of the bulk remains unknown. Here, we study transient stress relaxation behaviour of SJ states formed by a well-characterized dense suspension under a step strain perturbation. We observe a strongly nonexponential relaxation that develops a sharp discontinuous stress drop at short time for high enough peak-stress values. High resolution boundary imaging and normal stress measurements confirm that such stress discontinuity originates from the localized plastic events, whereas system spanning dilation controls the slower relaxation process. We also find an intriguing correlation between the nature of transient relaxation and the steady-state shear jamming phase diagram obtained from the Wyart-Cates model.

Arrested coalescence of multicellular aggregates

Multicellular aggregates are known to exhibit liquid-like properties. The fusion process of two cell aggregates is commonly studied as the coalescence of two viscous drops. However, tissues are complex materials and can exhibit viscoelastic behaviour. It is known that elastic effects can prevent the complete fusion of two drops, a phenomenon known as arrested coalescence. Here we study this phenomenon in stem cell aggregates and provide a theoretical framework which agrees with the experiments. In addition, agent-based simulations show that active cell fluctuations can control a solid-to-fluid phase transition, revealing that arrested coalescence can be found in the vicinity of an unjamming transition. By analysing the dynamics of the fusion process and combining it with nanoindentation measurements, we obtain the effective viscosity, shear modulus and surface tension of the aggregates. More generally, our work provides a simple, fast and inexpensive method to characterize the mechanical properties of viscoelastic materials.

Relaxation times, rheology, and finite size effects for non-Brownian disks in two dimensions

We carry out overdamped simulations in a simple model of jamming-a collection of bidisperse soft core frictionless disks in two dimensions-with the aim to explore the finite size dependence of different quantities, both the relaxation time obtained from the relaxation of the energy and the pressure equivalent of the shear viscosity. The motivation for the paper is the observation [Nishikawa et al., J. Stat. Phys. 182, 37 (2021)0022-471510.1007/s10955-021-02710-8] that there are finite size effects in the relaxation time, τ, that give problems in the determination of the critical divergence, and the claim that this is due to a finite size dependence, τ∼lnN, which makes τ an ill-defined quantity. Beside analyses to determine the relaxation time for the whole system we determine particle relaxation times which allow us to determine both histograms of particle relaxation times and the average particle relaxation times-two quantities that are very useful for the analyses. The starting configurations for the relaxation simulations are of two different kinds-completely random or taken from steady shearing simulations-and we find that the difference between these two cases are bigger than previously noted and that the observed problems in the determination of the critical divergence obtained when starting from random configurations are not present when instead starting the relaxations from shearing configurations. We also argue that the the effect that causes the lnN dependence is not as problematic as asserted. When it comes to the finite size dependence of the pressure equivalent of the shear viscosity we find that our data don't give support for the claimed strong finite size dependence, but also that the finite size dependence is at odds with what one would normally expect for a system with a diverging correlation length, and that this calls for an alternative understanding of the phenomenon of shear-driven jamming.

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.

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.

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: Particle rotations and orientational ordering

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 simple shear applied at a fixed volume and a fixed finite strain rate γ[over ̇]. Energy dissipation is via a viscous drag with respect to a uniformly sheared host fluid, giving a simple model for flow in a non-Brownian suspension with Newtonian rheology. Considering a range of packing fractions ϕ and particle asphericities α at small γ[over ̇], we study the angular rotation θ[over ̇]_{i} and the nematic orientational ordering S_{2} of the particles induced by the shear flow, finding a nonmonotonic behavior as the packing ϕ is varied. We interpret this nonmonotonic behavior as a crossover from dilute systems at small ϕ, where single-particle-like behavior occurs, to dense systems at large ϕ, where the geometry of the dense packing dominates and a random Poisson-like process for particle rotations results. We also argue that the finite nematic ordering S_{2} is a consequence of the shearing serving as an ordering field, rather than a result of long-range cooperative behavior among the particles. We arrive at these conclusions by consideration of (i) the distribution of waiting times for a particle to rotate by π, (ii) the behavior of the system under pure, as compared to simple, shearing, (iii) the relaxation of the nematic order parameter S_{2} when perturbed away from the steady state, and (iv) by construction, a numerical mean-field model for the rotational motion of a particle. Our results also help to explain the singular behavior observed when taking the α→0 limit approaching circular disks.

Universal Relaxation Dynamics of Sphere Packings below Jamming

We show that non-Brownian suspensions of repulsive spheres below jamming display a slow relaxational dynamics with a characteristic timescale that diverges at jamming. This slow timescale is fully encoded in the structure of the unjammed packing and can be readily measured via the vibrational density of states. We show that the corresponding dynamic critical exponent is the same for randomly generated and sheared packings. Our results show that a wide variety of physical situations, from suspension rheology to algorithmic studies of the jamming transition are controlled by a unique diverging timescale, with a universal critical exponent.

Coarsening and mechanics in the bubble model for wet foams

Aqueous foams are an important model system that displays coarsening dynamics. Coarsening in dispersions and foams is well understood in the dilute and dry limits, where the gas fraction tends to zero and one, respectively. However, foams are known to undergo a jamming transition from a fluidlike to a solidlike state at an intermediate gas fraction ϕ_{c}. Much less is known about coarsening dynamics in wet foams near jamming, and the link to mechanical response, if any, remains poorly understood. Here we probe coarsening and mechanical response using numerical simulations of a variant of the Durian bubble model for wet foams. As in other coarsening systems we find a steady state scaling regime with an associated particle size distribution. We relate the time rate of evolution of the coarsening process to the wetness of the foam and identify a characteristic coarsening time that diverges approaching jamming. We further probe mechanical response of the system to strain while undergoing coarsening. There are two competing timescales, namely the coarsening time and the mechanical relaxation time. We relate these to the evolution of the elastic response and the mechanical structure.

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.

On the apparent yield stress in non-Brownian magnetorheological fluids

We use simulations to probe the flow properties of dense two-dimensional magnetorheological fluids. Prior results from both experiments and simulations report that the shear stress σ scales with strain rate [small gamma, Greek, dot above] as σ ∼ [small gamma, Greek, dot above]^1-Δ, with values of the exponent ranging between 2/3 < Δ ≤ 1. However it remains unclear what properties of the system select the value of Δ, and in particular under what conditions the system displays a yield stress (Δ = 1). To address these questions, we perform simulations of a minimalistic model system in which particles interact via long ranged magnetic dipole forces, finite ranged elastic repulsion, and viscous damping. We find a surprising dependence of the apparent exponent Δ on the form of the viscous force law. For experimentally relevant values of the volume fraction ϕ and the dimensionless Mason number Mn (which quantifies the competition between viscous and magnetic stresses), models using a Stokes-like drag force show Δ ≈ 0.75 and no apparent yield stress. When dissipation occurs at the contact, however, a clear yield stress plateau is evident in the steady state flow curves. In either case, increasing ϕ towards the jamming transition suffices to induce a yield stress. We relate these qualitatively distinct flow curves to clustering mechanisms at the particle scale. For Stokes-like drag, the system builds up anisotropic, chain-like clusters as Mn tends to zero (vanishing strain rate and/or high field strength). For contact damping, by contrast, there is a second clustering mechanism due to inelastic collisions.

Stress relaxation in viscous soft spheres

We report the results of molecular dynamics simulations of stress relaxation tests in athermal viscous soft sphere packings close to their unjamming transition. By systematically and simultaneously varying both the amplitude of the applied strain step and the pressure of the initial condition, we access both linear and nonlinear response regimes and control the distance to jamming. Stress relaxation in viscoelastic solids is characterized by a relaxation time τ* that separates short time scales, where viscous loss is substantial, from long time scales, where elastic storage dominates and the response is essentially quasistatic. We identify two distinct plateaus in the strain dependence of the relaxation time, one each in the linear and nonlinear regimes. The height of both plateaus scales as an inverse power law with the distance to jamming. By probing the time evolution of particle velocities during relaxation, we further identify a correlation between mechanical relaxation in the bulk and the degree of non-affinity in the particle velocities on the micro scale.

Effect of particle collisions in dense suspension flows

We study nonlocal effects associated with particle collisions in dense suspension flows, in the context of the Affine Solvent Model, known to capture various aspects of the jamming transition. We show that an individual collision changes significantly the velocity field on a characteristic volume Ω_{c}∼1/δz that diverges as jamming is approached, where δz is the deficit in coordination number required to jam the system. Such an event also affects the contact forces between particles on that same volume Ω_{c}, but this change is modest in relative terms, of order f_{coll}∼f[over ¯]^{0.8}, where f[over ¯] is the typical contact force scale. We then show that the requirement that coordination is stationary (such that a collision has a finite probability to open one contact elsewhere in the system) yields the scaling of the viscosity (or equivalently the viscous number) with coordination deficit δz. The same scaling result was derived [E. DeGiuli, G. Düring, E. Lerner, and M. Wyart, Phys. Rev. E 91, 062206 (2015)PLEEE81539-375510.1103/PhysRevE.91.062206] via different arguments making an additional assumption. The present approach gives a mechanistic justification as to why the correct finite size scaling volume behaves as 1/δz and can be used to recover a marginality condition known to characterize the distributions of contact forces and gaps in jammed packings.

Beyond linear elasticity: jammed solids at finite shear strain and rate

The shear response of soft solids can be modeled with linear elasticity, provided the forcing is slow and weak. Both of these approximations must break down when the material loses rigidity, such as in foams and emulsions at their (un)jamming point - suggesting that the window of linear elastic response near jamming is exceedingly narrow. Yet precisely when and how this breakdown occurs remains unclear. To answer these questions, we perform computer simulations of stress relaxation and shear start-up tests in athermal soft sphere packings, the canonical model for jamming. By systematically varying the strain amplitude, strain rate, distance to jamming, and system size, we identify characteristic strain and time scales that quantify how and when the window of linear elasticity closes, and relate these scales to changes in the microscopic contact network.

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.

Collective motion of macroscopic spheres floating on capillary ripples: dynamic heterogeneity and dynamic criticality

When a densely packed monolayer of macroscopic spheres floats on chaotic capillary Faraday waves, a coexistence of large scale convective motion and caging dynamics typical for glassy systems is observed. We subtract the convective mean flow using a coarse graining (homogenization) method and reveal subdiffusion for the caging time scales followed by a diffusive regime at later times. We apply the methods developed to study dynamic heterogeneity and show that the typical time and length scales of the fluctuations due to rearrangements of observed particle groups significantly increase when the system approaches its largest experimentally accessible packing concentration. To connect the system to the dynamic criticality literature, we fit power laws to our results. The resultant critical exponents are consistent with those found in densely packed suspensions of colloids.

Length scales and self-organization in dense suspension flows

Dense non-Brownian suspension flows of hard particles display mystifying properties: As the jamming threshold is approached, the viscosity diverges, as well as a length scale that can be identified from velocity correlations. To unravel the microscopic mechanism governing dissipation and its connection to the observed correlation length, we develop an analogy between suspension flows and the rigidity transition occurring when floppy networks are pulled, a transition believed to be associated with the stress stiffening of certain gels. After deriving the critical properties near the rigidity transition, we show numerically that suspension flows lie close to it. We find that this proximity causes a decoupling between viscosity and the correlation length of velocities ξ, which scales as the length l(c) characterizing the response to a local perturbation, previously predicted to follow l(c)∼ 1/sqrt[z(c)-z] ∼ p(0.18), where p is the dimensionless particle pressure, z is the coordination of the contact network made by the particles, and z(c) is twice the spatial dimension. We confirm these predictions numerically and predict the existence of a larger length scale l(r)∼sqrt[p] with mild effects on velocity correlation and of a vanishing strain scale δγ ∼ 1/p that characterizes decorrelation in flow.

Dynamic critical response in damped random spring networks

The isostatic state plays a central role in organizing the response of many amorphous materials. We construct a diverging length scale in nearly isostatic spring networks that is defined both above and below isostaticity and at finite frequencies and relate the length scale to viscoelastic response. Numerical measurements verify that proximity to isostaticity controls the viscosity, shear modulus, and creep of random networks.

Avalanches in strained amorphous solids: does inertia destroy critical behavior?

Simulations are used to determine the effect of inertia on athermal shear of amorphous two-dimensional solids. In the quasistatic limit, shear occurs through a series of rapid avalanches. The distribution of avalanches is analyzed using finite-size scaling with thousands to millions of disks. Inertia takes the system to a new underdamped universality class rather than driving the system away from criticality as previously thought. Scaling exponents are determined for the underdamped and overdamped limits and a critical damping that separates the two regimes. Systems are in the overdamped universality class even when most vibrational modes are underdamped.

Bubble rearrangement duration in foams near the jamming point

We investigate the dynamics of bubble rearrangements in coarsening foams, using a time-resolved multiple light scattering technique. We measure the average duration of such events as a function of the foam confinement pressure. Rearrangements slow down as the pressure is decreased toward the jamming point. Our results are explained by a scaling law based on the balance of pressure and Darcy flow, highlighting an analogy between wet foams with mobile interfaces and suspensions of hard grains.

Inhomogeneous shear flows in soft jammed materials with tunable attractive forces

We perform molecular dynamics simulations to characterize the occurrence of inhomogeneous shear flows in soft jammed materials. We use rough walls to impose a simple shear flow and study the athermal motion of jammed assemblies of soft particles in two spatial dimensions, both for purely repulsive interactions and in the presence of an additional short-range attraction of varying strength. In steady state, pronounced flow inhomogeneities emerge for all systems when the shear rate becomes small. Deviations from linear flow are stronger in magnitude and become very long lived when the strength of the attraction increases, but differ from permanent shear bands. Flow inhomogeneities occur in a stress window bounded by the dynamic and static yield stress values. Attractive forces enhance the flow heterogeneities because they accelerate stress relaxation, thus effectively moving the system closer to the yield stress regime where inhomogeneities are most pronounced. The present scenario for understanding the effect of particle adhesion on shear localization, which is based on detailed molecular dynamics simulations with realistic particle interactions, differs qualitatively from previous qualitative explanations and ad hoc theoretical modeling.

Similarities between protein folding and granular jamming

Grains and glasses, widely different materials, arrest their motions upon decreasing temperature and external load, respectively, in common ways, leading to a universal jamming phase diagram conjecture. However, unified theories are lacking, mainly because of the disparate nature of the particle interactions. Here we demonstrate that folded proteins exhibit signatures common to both glassiness and jamming by using temperature- and force-unfolding molecular dynamics simulations. Upon folding, proteins develop a peak in the interatomic force distributions that falls on a universal curve with experimentally measured forces on jammed grains and droplets. Dynamical signatures are found as a dramatic slowdown of stress relaxation upon folding. Together with granular similarities, folding is tied not just to the jamming transition, but a more nuanced picture of anisotropy, preparation protocol and internal interactions emerges. Results have implications for designing stable polymers and can open avenues to link protein folding to jamming theory.

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 of shearing rheology at the jamming transition of soft-core frictionless disks

We perform numerical simulations to determine the shear stress and pressure of steady-state shear flow in a soft-disk model in two dimensions at zero temperature in the vicinity of the jamming transition ϕ{J}. We use critical point scaling analyses to determine the critical behavior at jamming, and we find that it is crucial to include corrections to scaling for a reliable analysis. We find that the relative size of these corrections are much smaller for pressure than for shear stress. We furthermore find a superlinear behavior for pressure and shear stress above ϕ{J}, both from the scaling analysis and from a direct analysis of pressure data extrapolated to the limit of vanishing shear rate.

Finite-size scaling at the jamming transition: corrections to scaling and the correlation-length critical exponent

We carry out a finite-size scaling analysis of the jamming transition in frictionless bidisperse soft core disks in two dimensions. We consider two different jamming protocols: (i) quench from random initial positions and (ii) quasistatic shearing. By considering the fraction of jammed states as a function of packing fraction for systems with different numbers of particles, we determine the spatial correlation length critical exponent ν ≈ 1 and show that corrections to scaling are crucial for analyzing the data. We show that earlier numerical results yielding ν < 1 are due to the improper neglect of these corrections.

Fluidization of wet granulates under shear

Small amounts of a wetting liquid render sand a stiff and moldable material. The cohesive forces between the sand grains are caused by capillary bridges at the points of contact. Due to the finite strength of these bridges wet sand undergoes a transition from an arrested (i.e., solidified) to a fluidized state under an externally applied shear force. The transition between these two dynamic states is studied in a MD-type simulation of a two-dimensional assembly of bidisperse frictionless disks under the action of a cosine force profile. In addition to soft core repulsion the disks interact through a hysteretic and short ranged attractive force modeling the effect of the capillary bridges. In this model the transition between the fluidized and the arrested state is discontinuous and hysteretic. The parameter dependence of the critical force for solidification is modeled by combining theoretical arguments with a detailed numerical exploration of the transition. We address a range of densities from slightly below close packing until slightly above densities where the system approaches a shear-banded state. Differences and similarities of the transition in wet granulates to the jamming transition are also addressed.

Fluctuations, jamming, and yielding for a driven probe particle in disordered disk assemblies

Using numerical simulations we examine the velocity fluctuations and velocity-force curve characteristics of a probe particle driven with constant force through a two-dimensional disordered assembly of disks which has a well-defined jamming point J at a density of ϕJ=0.843. As ϕ increases toward ϕJ, the average velocity of the probe particle decreases and the velocity fluctuations show an increasingly intermittent or avalanchelike behavior. When ϕ is within a few percent of the jamming density, the velocity distributions are exponential, while when ϕ is less than 1% away from jamming, the velocity distributions have a power-law character with exponents in agreement with recent experiments. The velocity power spectra exhibit a crossover from a Lorentzian form to a 1/f shape near jamming. We extract a correlation length exponent ν which is in good agreement with recent shear simulations. For ϕ>ϕJ, there is a critical threshold force F(c) that must be applied for the probe particle to move through the sample which increases with increasing ϕ. The velocity-force curves are linear below jamming, while at jamming they have a power-law form. The onset of the probe motion above ϕJ occurs via a local yielding of the particles around the probe particle which we term a local shear banding effect.

Mesoscopic length scale controls the rheology of dense suspensions

From the flow properties of dense granular suspensions on an inclined plane, we identify a mesoscopic length scale strongly increasing with volume fraction. When the flowing layer height is larger than this length scale, a diverging Newtonian viscosity is determined. However, when the flowing layer height drops below this scale, we evidence a nonlocal effective viscosity, decreasing as a power law of the flow height. We establish a scaling relation between this mesoscopic length scale and the suspension viscosity. These results support recent theoretical and numerical results implying collective and clustered granular motion when the jamming point is approached from below.

Model for the scaling of stresses and fluctuations in flows near jamming

We probe flows of soft, viscous spheres near the jamming point, which acts as a critical point for static soft spheres. Starting from energy considerations, we find nontrivial scaling of velocity fluctuations with strain rate. Combining this scaling with insights from jamming, we arrive at an analytical model that predicts four distinct regimes of flow, each characterized by rational-valued scaling exponents. Both the number of regimes and the values of the exponents depart from prior results. We validate predictions of the model with simulations.

Plastic deformations in crystal, polycrystal, and glass in binary mixtures under shear: collective yielding

Using molecular dynamics simulation, we examine the dynamics of crystal, polycrystal, and glass in a Lennard-Jones binary mixture composed of small and large particles in two dimensions. The crossovers occur among these states as the composition c is varied at fixed size ratio. Shear is applied to a system of 9000 particles in contact with moving boundary layers composed of 1800 particles. The particle configurations are visualized with a sixfold orientation angle αj(t) and a disorder variable Dj(t) defined for particle j, where the latter represents the deviation from hexagonal order. Fundamental plastic elements are classified into dislocation gliding and grain boundary sliding. At any c, large-scale yielding events occur on the acoustic time scale. Moreover, they multiply occur in narrow fragile areas, forming shear bands. The dynamics of plastic flow is highly hierarchical with a wide range of time scales for slow shearing. We also clarify the relationship between the shear stress averaged in the bulk region and the wall stress applied at the boundaries.

Scaling of the glassy dynamics of soft repulsive particles: a mode-coupling approach

We combine the hypernetted chain approximation of liquid state theory with the mode-coupling theory of the glass transition to analyze the structure and dynamics of soft spheres interacting via harmonic repulsion. We determine the locus of the fluid-glass dynamic transition in a temperature--volume fraction phase diagram. The zero-temperature (hard-sphere) glass transition influences the dynamics at finite temperatures in its vicinity. This directly implies a form of dynamic scaling for both the average relaxation time and dynamic susceptibilities quantifying dynamic heterogeneity. We discuss several qualitative disagreements between theory and existing simulations at equilibrium. Our theoretical results are, however, very similar to numerical results for the driven athermal dynamics of repulsive spheres, suggesting that "mean-field" mode-coupling approaches might be good starting points to describe these nonequilibrium dynamics.

Jamming of soft particles: geometry, mechanics, scaling and isostaticity

Amorphous materials as diverse as foams, emulsions, colloidal suspensions and granular media can jam into a rigid, disordered state where they withstand finite shear stresses before yielding. Here we review the current understanding of the transition to jamming and the nature of the jammed state for disordered packings of particles that act through repulsive contact interactions and are at zero temperature and zero shear stress. We first discuss the breakdown of affine assumptions that underlies the rich mechanics near jamming. We then extensively discuss jamming of frictionless soft spheres. At the jamming point, these systems are marginally stable (isostatic) in the sense of constraint counting, and many geometric and mechanical properties scale with distance to this jamming point. Finally, we discuss current explorations of jamming of frictional and non-spherical (ellipsoidal) particles. Both friction and asphericity tune the contact number at jamming away from the isostatic limit, but in opposite directions. This allows one to disentangle the distance to jamming and the distance to isostaticity. The picture that emerges is that most quantities are governed by the contact number and scale with the distance to isostaticity, while the contact number itself scales with the distance to jamming.

Jamming at zero temperature, zero friction, and finite applied shear stress

Shear stress, as temperature, induces particle motion, and affects the jamming transition from a fluid to a disordered solid state. Here, we show that at finite shear stress, the jamming transition is characterized by the presence of hysteresis, as in a given range of control parameters a flowing or a jammed state can be found, depending on whether the system is prepared coming from the fluid or the jammed phase. At small shear stress, where the hysteresis is negligible, the jamming transition has a mixed first-order second-order character close to that found at the glass transition of thermal systems, with discontinuities in the asymptotic values of two time quantities such as the self-intermediate scattering function.

Critical behaviors of sheared frictionless granular materials near the jamming transition

Critical behaviors of sheared dense and frictionless granular materials in the vicinity of the jamming transition are numerically investigated. From the extensive molecular dynamics simulation, we verify the validity of the scaling theory near the jamming transition proposed by Otsuki and Hayakawa [Prog. Theor. Phys. 121, 647 (2009)]]. We also clarify the critical behaviors of the shear viscosity and the pair correlation function based on both a mean field theory and the simulation.