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

Jamming is a first-order transition with quenched disorder in amorphous materials sheared by cyclic quasistatic deformations

Jamming is an athermal transition between flowing and rigid states in amorphous systems such as granular matter, colloidal suspensions, complex fluids and cells. The jamming transition seems to display mixed aspects of a first-order transition, evidenced by a discontinuity in the coordination number, and a second-order transition, indicated by power-law scalings and diverging lengths. Here we demonstrate that jamming is a first-order transition with quenched disorder in cyclically sheared systems with quasistatic deformations, in two and three dimensions. Based on scaling analyses, we show that fluctuations of the jamming density in finite-sized systems have important consequences on the finite-size effects of various quantities, resulting in a square relationship between disconnected and connected susceptibilities, a key signature of the first-order transition with quenched disorder. This study puts the jamming transition into the category of a broad class of transitions in disordered systems where sample-to-sample fluctuations dominate over thermal fluctuations, suggesting that the nature and behavior of the jamming transition might be better understood within the developed theoretical framework of the athermally driven random-field Ising model.

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.

Slow and fast particles in shear-driven jamming: Critical behavior

We do extensive simulations of a simple model of shear-driven jamming in two dimensions to determine and analyze the velocity distribution at different densities ϕ around the jamming density ϕ_{J} and at different low shear strain rates, γ[over ̇]. We then find that the velocity distribution is made up of two parts which are related to two different physical processes which we call the slow process and the fast process as they are dominated by the slower and the faster particles, respectively. Earlier scaling analyses have shown that the shear viscosity η, which diverges as the jamming density is approached from below, consists of two different terms, and we present strong evidence that these terms are related to the two different processes: the leading divergence is due to the fast process, whereas the correction-to-scaling term is due to the slow process. The analysis of the slow process is possible thanks to the observation that the velocity distribution for different γ[over ̇] and ϕ at and around the shear-driven jamming transition has a peak at low velocities and that the distribution has a constant shape up to and slightly above this peak. We then find that it is possible to express the contribution to the shear viscosity due to the slow process in terms of height and position of the peak in the velocity distribution and find that this contribution matches the correction-to-scaling term, determined through a standard critical scaling analysis. A further observation is that the collective particle motion is dominated by the slow process. In contrast to the usual picture in critical phenomena with a direct link between the diverging correlation length and a diverging order parameter, we find that correlations and shear viscosity decouple since they are controlled by different sets of particles and that shear-driven jamming is thus an unusual kind of critical phenomenon.

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}.

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.

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.

Capability of the TFM Approach to Predict Fluidization of Cohesive Powders

The fluidization behavior of cohesive particles was investigated using an Euler–Euler approach. To do so, a two-fluid model (TFM) platform was developed to account for the cohesivity of particles. Specifically, the kinetic theory of granular flow (KTGF) was modified based on the solid rheology developed by Gu et al. J. Fluid Mech.2019. The results of our simulations demonstrated that the modified TFM approach can successfully predict the formation of particle agglomerates and clusters in the fluidized bed, induced by the negative (tensile-dominant) pressure. The formation of such granules and clusters highly depended on the particle Bond number and the tensile pressure prefactor. To evaluate fluidization regimes, a set of simulations was conducted for a wide range of particle cohesivity (e.g., Bond number and tensile pressure prefactor) at two different fluidization numbers of 2 and 5. Our simulation results reveal the formation of four different regimes of fluidization for cohesive particles: (i) bubbling, (ii) bubbling–clustering, (iii) bubble-less fluidization, and (iv) stagnant bed. Comprehensive analysis of the shear-to-yield ratio reveals that the observed regime map is attributed to the competition between the shear stress and yield stress acting on the particles. The obtained regime map can be extended to incorporate the effect of dimensionless velocity and dimensionless diameter as a comprehensive fluidization chart for cohesive particles. Such fluidization charts can facilitate the design of fluidized beds by predicting the conditions under which the formation of particle agglomeration and clustering is likely in fluidized beds.

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.

Colloidal Shear-Thickening Fluids Using Variable Functional Star-Shaped Particles: A Molecular Dynamics Study

Complex colloidal fluids, depending on constituent shapes and packing fractions, may have a wide range of shear-thinning and/or shear-thickening behaviors. An interesting way to transition between different types of such behavior is by infusing complex functional particles that can be manufactured using modern techniques such as 3D printing. In this paper, we perform 2D molecular dynamics simulations of such fluids with infused star-shaped functional particles, with a variable leg length and number of legs, as they are infused in a non-interacting fluid. We vary the packing fraction (ϕ) of the system, and for each different system, we apply shear at various strain rates, turning the fluid into a shear-thickened fluid and then, in jammed state, rising the apparent viscosity of the fluid and incipient stresses. We demonstrate the dependence of viscosity on the functional particles’ packing fraction and we show the role of shape and design dependence of the functional particles towards the transition to a shear-thickening fluid.

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.

Rheology finds distinct glass and jamming transitions in emulsions

We study the rheology of monodisperse and bidisperse emulsions with various droplet sizes (1-2 μm diameter). Above a critical volume fraction φc, these systems exhibit solid-like behavior and a yield stress can be detected. Previous experiments suggest that for small thermal particles, rheology will see a glass transition at φc = φg ≈ 0.58; for large athermal systems, rheology will see a jamming transition at φc = φJ ≈ 0.64. However, simulations point out that at the crossover of thermal and athermal regimes, the glass and jamming transitions may both be observed in the same sample. Here we conduct an experiment by shearing four oil-in-water emulsions with a rheometer. We observe both a glass and a jamming transition for our smaller diameter droplets, and only a jamming transition for our larger diameter droplets. The bidisperse sample behaves similarly to the small droplet sample, with two transitions observed. Our rheology data are well-fit by both the Herschel-Bulkley model and the three component model. Based on the fitting parameters, our raw rheological data would not collapse onto a master curve. Our results show that liquid-solid transitions in dispersions are not universal, but depend on particle size.

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.

Depletion forces in athermally sheared mixtures of frictionless disks and rods in two dimensions

We carry out numerical simulations to study the behavior of an athermal mixture of frictionless circular disks and elongated rods in two dimensions, under three different types of global linear deformation at a finite strain rate: (i) simple shearing, (ii) pure shearing, and (iii) isotropic compression. We find that the fluctuations induced by such deformations lead to depletion forces that cause rods to group in parallel oriented clusters for the cases of simple and pure shear, but not for isotropic compression. For simple shearing, we find that as the fraction of rods increases, this clustering increases, leading to an increase in the average rate of rotation of the rods, and a decrease in the magnitude of their nematic ordering.

Structured randomness: jamming of soft discs and pins

Simulations are used to find the zero temperature jamming threshold, φj, for soft, bidisperse disks in the presence of small fixed particles, or "pins", arranged in a lattice. The presence of pins leads, as one expects, to a decrease in φj. Structural properties of the system near the jamming threshold are calculated as a function of the pin density. While the correlation length exponent remains ν = 1/2 at low pin densities, the system is mechanically stable with more bonds, yet fewer contacts than the Maxwell criterion implies in the absence of pins. In addition, as pin density increases, novel bond orientational order and long-range spatial order appear, which are correlated with the square symmetry of the pin lattice.

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.

Shear-driven flow of athermal, frictionless, spherocylinder suspensions in two dimensions: Spatial structure and correlations

We use numerical simulations to study the flow 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. We study the resulting spatial structure of the sheared system, and compute correlation functions of the velocity, the particle density, the nematic order parameter, and the particle angular velocity. Correlations of density, nematic order, and angular velocity are shown to be short ranged both below and above jamming. We compare a system of size-bidisperse particles with a system of size-monodisperse particles, and argue how differences in spatial order as the packing increases lead to differences in the global nematic order parameter. We consider the effect of shearing on initially well ordered configurations, and show that in many cases the shearing acts to destroy the order, leading to the same steady-state ensemble as found when starting from random initial configurations.

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.

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.

Orientational Ordering in Athermally Sheared, Aspherical, Frictionless Particles

We numerically simulate the uniform athermal shearing of bidisperse, frictionless, two-dimensional spherocylinders and three-dimensional prolate ellipsoids. We focus on the orientational ordering of particles as an asphericity parameter α→0 and particles approach spherical. We find that the nematic order parameter S_{2} is nonmonotonic in the packing fraction ϕ and that, as α→0, S_{2} stays finite at jamming and above. The approach to spherical particles thus appears to be singular. We also find that sheared particles continue to rotate above jamming and that particle contacts preferentially lie along the narrowest width of the particles, even as α→0.

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.

The physics of jamming for granular materials: a review

Granular materials consist of macroscopic grains, interacting via contact forces, and unaffected by thermal fluctuations. They are one of a class systems that undergo jamming, i.e. a transition between fluid-like and disordered solid-like states. Roughly twenty years ago, proposals by Cates et al for the shear response of colloidal systems and by Liu and Nagel, for a universal jamming diagram in a parameter space of packing fraction, ϕ, shear stress, τ, and temperature, T raised key questions. Contemporaneously, experiments by Howell et al and numerical simulations by Radjai et al and by Luding et al helped provide a starting point to explore key insights into jamming for dry, cohesionless, granular materials. A recent experimental observation by Bi et al is that frictional granular materials have a a re-entrant region in their jamming diagram. In a range of ϕ, applying shear strain, γ, from an initially force/stress free state leads to fragile (in the sense of Cates et al), then anisotropic shear jammed states. Shear jamming at fixed ϕ is presumably conjugate to Reynolds dilatancy, involving dilation under shear against deformable boundaries. Numerical studies by Radjai and Roux showed that Reynolds dilatancy does not occur for frictionless systems. Recent numerical studies by several groups show that shear jamming occurs for finite, but not infinite, systems of frictionless grains. Shear jamming does not lead to known ordering in position space, but Sarkar et al showed that ordering occurs in a space of force tiles. Experimental studies seeking to understand random loose and random close packings (rlp and rcp) and dating back to Bernal have probed granular packings and their response to shear and intruder motion. These studies suggest that rlp's are anisotropic and shear-jammed-like, whereas rcp's are likely isotropically jammed states. Jammed states are inherently static, but the jamming diagram may provide a context for understanding rheology, i.e. dynamic shear in a variety of systems that include granular materials and suspensions.

On the universality of the flow properties of soft-particle glasses

We identify the minimal interparticle interactions necessary for a particle dynamics simulation to predict the structure and flow behaviour of soft particle glasses (SPGs). Generally, two kinds of forces between the particles must be accounted for in simulations of SPGs: viscous or frictional drag forces and elastic contact forces. Far field drag forces are required to dissipate energy in the simulations and capture the effect of the rheology of the suspending fluid. Elastic forces are found to be dominant compared to near-field drag or other forms of friction forces and are the most important component to compute the rheology. The shear stress, the first and second normal stress differences for different interparticle force laws collapse onto universal master curves of the Herschel-Bulkley form by non-dimensionalizing the stress with the yield stress and the shear rate with the viscosity of the suspending fluid divided by the low-frequency shear modulus. The Herschel-Bulkley exponents are close to 0.5 with a slight dependence on the repulsive pairwise elastic forces.

Critical scaling near the yielding transition in granular media

We show that the yielding transition in granular media displays second-order critical-point scaling behavior. We carry out discrete element simulations in the low-inertial-number limit for frictionless, purely repulsive spherical grains undergoing simple shear at fixed nondimensional shear stress Σ in two and three spatial dimensions. To find a mechanically stable (MS) packing that can support the applied Σ, isotropically prepared states with size L must undergo a total strain γ_{ms}(Σ,L). The number density of MS packings (∝γ_{ms}^{-1}) vanishes for Σ>Σ_{c}≈0.11 according to a critical scaling form with a length scale ξ∝|Σ-Σ_{c}|^{-ν}, where ν≈1.7-1.8. Above the yield stress (Σ>Σ_{c}), no MS packings that can support Σ exist in the large-system limit L/ξ≫1. MS packings generated via shear possess anisotropic force and contact networks, suggesting that Σ_{c} is associated with an upper limit in the degree to which these networks can be deformed away from those for isotropic packings.

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}.

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 ̇].

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.

Effect of friction on dense suspension flows of hard particles

We use numerical simulations to study the effect of particle friction on suspension flows of non-Brownian hard particles. By systematically varying the microscopic friction coefficient μ_{p} and the viscous number J, we build a phase diagram that identifies three regimes of flow: frictionless, frictional sliding, and rolling. Using energy balance in flow, we predict relations between kinetic observables, confirmed by numerical simulations. For realistic friction coefficients and small viscous numbers (below J∼10^{-3}), we show that the dominating dissipative mechanism is sliding of frictional contacts, and we characterize asymptotic behaviors as jamming is approached. Outside this regime, our observations support the idea that flow belongs to the universality class of frictionless particles. We discuss recent experiments in the context of our phase diagram.

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.

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}.

Dissipation and velocity distribution at the shear-driven jamming transition

We investigate energy dissipation and the distribution of particle velocities at the jamming transition for overdamped shear-driven frictionless disks in two dimensions at zero temperature. We find that the dissipation is caused by the fastest particles and that the fraction of particles responsible for the dissipation decreases towards zero as jamming is approached. These particles belong to an algebraic tail of the velocity distribution that approaches ∼v^{-3} as jamming is approached. We further find that different measures of the velocity diverge differently, which means that concepts such as typical velocity may no longer be used, a finding that should have implications for analytical approaches to shear-driven jamming.

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.

Viscoelastic properties of marginal networks in a solvent

Polymer networks at the margins of mechanical stability are known to be highly sensitive to applied forces and fields and to exhibit an anomalously large resistance to deformation. In this paper, we study the effects of hydrodynamic interactions on the behavior of marginal networks using a hybrid molecular dynamics and multiparticle collision dynamics simulation technique. We examine how the filament and solvent properties affect the response of marginal networks to shear. We find that the stiffening of the network shows a stronger dependence on the shear frequency when hydrodynamic interactions are present than when they are not. The network shear modulus scales as G'∼ω(α(c)), with a critical stiffening exponent α(c) that can be controlled by varying the relative concentrations of the network and the solvent. Our results show that this arises due to the solvent aiding the relaxation of the network and suppressing the network nonaffinity, with the system deforming more affinely when hydrodynamic interactions are maximized.

Fluctuations in flows near jamming

Bubbles, droplets or particles in flowing complex media such as foams, emulsions or suspensions follow highly complex paths, with the relative motion of the constituents setting the energy dissipation rate. What is their dynamics, and how is this connected to the global rheology? To address these questions, we probe the statistics and spatio-temporal organization of the local particle motion and energy dissipation in a model for sheared disordered materials. We find that the fluctuations in the local dissipation vary from nearly Gaussian and homogeneous at low densities and fast flows, to strongly intermittent for large densities and slow flows. The higher order moments of the relative particle velocities reveal strong evidence for a qualitative difference between two distinct regimes which are nevertheless connected by a smooth crossover. In the critical regime, the higher order moments are related by novel multiscaling relations. In the plastic regime the relations between these moments take on a different form, with higher moments diverging rapidly when the flow rate vanishes. As these velocity differences govern the energy dissipation, we can distinguish two qualitatively different types of flow: an intermediate density, critical regime related to jamming, and a large density, plastic regime.

Softening of stressed granular packings with resonant sound waves

We perform numerical simulations of a two-dimensional bidisperse granular packing subjected to both a static confining pressure and a sinusoidal dynamic forcing applied by a wall on one edge of the packing. We measure the response experienced by a wall on the opposite edge of the packing and obtain the resonant frequency of the packing as the static or dynamic pressures are varied. Under increasing static pressure, the resonant frequency increases, indicating a velocity increase of elastic waves propagating through the packing. In contrast, when the dynamic amplitude is increased for fixed static pressure, the resonant frequency decreases, indicating a decrease in the wave velocity. This occurs both for compressional and for shear dynamic forcing and is in agreement with experimental results. We find that the average contact number Zc at the resonant frequency decreases with increasing dynamic amplitude, indicating that the elastic softening of the packing is associated with a reduced number of grain-grain contacts through which the elastic waves can travel. We image the excitations created in the packing and show that there are localized disturbances or soft spots that become more prevalent with increasing dynamic amplitude. Our results are in agreement with experiments on glass bead packings and earth materials such as sandstone and granite and may be relevant to the decrease in elastic wave velocities that has been observed to occur near fault zones after strong earthquakes, in surficial sediments during strong ground motion, and in structures during earthquake excitation.

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.

Diverging viscosity and soft granular rheology in non-Brownian suspensions

We use large scale computer simulations and finite-size scaling analysis to study the shear rheology of dense three-dimensional suspensions of frictionless non-Brownian particles in the vicinity of the jamming transition. We perform simulations of soft repulsive particles at constant shear rate, constant pressure, and finite system size and carefully study the asymptotic limits of large system sizes and infinitely hard particle repulsion. We first focus on the asymptotic behavior of the shear viscosity in the hard particle limit. By measuring the viscosity increase over about 5 orders of magnitude, we are able to confirm its asymptotic power law divergence close to the jamming transition. However, a precise determination of the critical density and critical exponent is difficult due to the "multiscaling" behavior of the viscosity. Additionally, finite-size scaling analysis suggests that this divergence is accompanied by a growing correlation length scale, which also diverges algebraically. Finally, we study the effect of particle softness and propose a natural extension of the standard granular rheology, which we test against our simulation data. Close to the jamming transition, this "soft granular rheology" offers a detailed description of the nonlinear rheology of soft particles, which differs from earlier empirical scaling forms.

Universality of jamming criticality in overdamped shear-driven frictionless disks

We investigate the criticality of the jamming transition for overdamped shear-driven frictionless disks in two dimensions for two different models of energy dissipation: (i) Durian's bubble model with dissipation proportional to the velocity difference of particles in contact, and (ii) Durian's "mean-field" approximation to (i), with dissipation due to the velocity difference between the particle and the average uniform shear flow velocity. By considering the finite-size behavior of pressure, the pressure analog of viscosity, and the macroscopic friction σ/p, we argue that these two models share the same critical behavior.

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.

The jamming elasticity of emulsions stabilized by ionic surfactants

Oil-in-water emulsions composed of colloidal-scale droplets are often stabilized using ionic surfactants that provide a repulsive interaction between neighboring droplet interfaces, thereby inhibiting coalescence. If the droplet volume fraction is raised rapidly by applying an osmotic pressure, the droplets remain disordered, undergo an ergodic-nonergodic transition, and jam. If the applied osmotic pressure approaches the Laplace pressure of the droplets, then the jammed droplets also deform. Because solid friction and entanglements cannot play a role, as they might with solid particulate or microgel dispersions, the shear mechanical response of monodisperse emulsions can provide critical insight into the interplay of entropic, electrostatic, and interfacial forces. Here, we introduce a model that can be used to predict the plateau storage modulus and yield stress of a uniform charge-stabilized emulsion accurately if the droplet radius, interfacial tension, surface potential, Debye screening length, and droplet volume fraction are known.

Impact of attractive interactions on the rheology of dense athermal particles

Using numerical simulations, the rheological response of an athermal assembly of soft particles with tunable attractive interactions is studied in the vicinity of jamming. At small attractions, a fragile solid develops and a finite yield stress is measured. Moreover, the measured flow curves have unstable regimes, which lead to persistent shear banding. These features are rationalized by establishing a link between the rheology and the interparticle connectivity, which also provides a minimal model to describe the flow curves.

Rheology of soft colloids across the onset of rigidity: scaling behavior, thermal, and non-thermal responses

We study the rheological behavior of colloidal suspensions composed of soft sub-micron-size hydrogel particles across the liquid-solid transition. The measured stress and strain-rate data, when normalized by thermal stress and time scales, suggest our systems reside in a regime wherein thermal effects are important. In a different vein, critical point scaling predictions for the jamming transition, typical in athermal systems, are tested. Near dynamic arrest, the suspensions exhibit scaling exponents similar to those reported in Nordstrom et al., Phys. Rev. Lett., 2010, 105, 175701. The observation suggests that our system exhibits a glass transition near the onset of rigidity, but it also exhibits a jamming-like scaling further from the transition point. These observations are thought-provoking in light of recent theoretical and simulation findings, which show that suspension rheology across the full range of microgel particle experiments can exhibit both thermal and athermal mechanisms.

Physics in ordered and disordered colloidal matter composed of poly(N-isopropylacrylamide) microgel particles

This review collects and describes experiments that employ colloidal suspensions to probe physics in ordered and disordered solids and related complex fluids. The unifying feature of this body of work is its clever usage of poly(N-isopropylacrylamide) (PNIPAM) microgel particles. These temperature-sensitive colloidal particles provide experimenters with a 'knob' for in situ control of particle size, particle interaction and particle packing fraction that, in turn, influence the structural and dynamical behavior of the complex fluids and solids. A brief summary of PNIPAM particle synthesis and properties is given, followed by a synopsis of current activity in the field. The latter discussion describes a variety of soft matter investigations including those that explore formation and melting of crystals and clusters, and those that probe structure, rearrangement and rheology of disordered (jammed/glassy) and partially ordered matter. The review, therefore, provides a snapshot of a broad range of physics phenomenology which benefits from the unique properties of responsive microgel particles.

Dynamics of sheared ellipses and circular disks: effects of particle shape

Much recent effort has focused on glassy and jamming properties of spherical particles. Very little is known about such phenomena for nonspherical particles, and we take a first step by studying ellipses. We find important differences between the dynamical and structural properties of disks and two-dimensional ellipses subject to continuous Couette shear. In particular, ellipses show slow dynamical evolution, without a counterpart in disks, in the mean velocity, local density, orientational order, and local stress. Starting from an unjammed state, ellipses can first jam under shear, and then slowly unjam. The slow unjamming process is understood as a result of gradual changes in their orientations, leading to a denser packing. For disks, the rotation of particles only contributes to the relaxation of frictional forces, and hence, does not significantly cause structural changes. For the shear-jammed states, the global building up and relaxation of stress, which occurs in the form of stress avalanches, is qualitatively different for disks and ellipses, and is manifested by different forms of rate dependence for ellipses versus disks. Unlike the weak rate dependence typical for many granular systems, ellipses show power-law dependence on the shearing rate Ω.

Pressure distribution and critical exponent in statically jammed and shear-driven frictionless disks

We numerically study the distributions of global pressure that are found in ensembles of statically jammed and quasistatically sheared systems of bidisperse, frictionless disks at fixed packing fraction ϕ in two dimensions. We use these distributions to address the question of how pressure increases as ϕ increases above the jamming point ϕ(J), p ∼ |ϕ-ϕ(J)(y). For statically jammed ensembles, our results are consistent with the exponent y being simply related to the power law of the interparticle soft-core interaction. For sheared systems, however, the value of y is consistent with a nontrivial value, as found previously in rheological simulations.

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.

Athermal jamming versus thermalized glassiness in sheared frictionless particles

Numerical simulations of soft-core frictionless disks in two dimensions are carried out to study the behavior of a simple liquid as a function of temperature T, packing fraction φ, and uniform applied shear strain rate γ[over ·]. Inferring the hard-core limit from our soft-core results, we find that it depends on the two parameters φ and T/γ[over ·]. Here T/γ[over ·]→0 defines the athermal limit in which a shear-driven jamming transition occurs at a well defined φ(J) and T/γ[over ·]→∞ defines the thermalized limit where an equilibrium glass transition may take place at φ(G). This conclusion argues that athermal jamming and equilibrium glassy behavior are not controlled by the same critical point. Preliminary results suggest φ(G)<φ(J).

Rheology and shear band suppression in particle and chain mixtures

Using numerical simulations we consider an amorphous particle mixture which exhibits shear localization, and find that the addition of even a small fraction of chains strongly enhances the material strength, creating pronounced overshoot features in the stress-strain curves. The strengthening occurs in the case where the chains are initially perpendicular to the shear direction, leading to a suppression of the shear band. This also leads to stiffening effects that are typical of polymeric systems. For large strain, the chains migrate to the region where a shear band forms, resulting in a stress drop. For chains larger than the linear system size we find oscillatory behavior, which does not resemble polymeric systems since the second stress peak is larger than the first. Our results are also useful for providing insights into methods of controlling and strengthening granular materials against failure.