Self-diffusion in dense granular shear flows

Micromechanical origin of heat transfer to granular flow

Heat transfer across a granular flow is comprised of two resistances in series : near the wall and within the bulk particle bed, neither of which is well understood due to the lack of experimental probes to separate their respective contribution. Here, we use a frequency modulated photothermal technique to separately quantify the thermal resistances in the near-wall and the bulk bed regions of particles in flowing states. Compared to the stationary state, the flowing leads to a higher near-wall resistance and a lower thermal conductivity of bulk beds. Coupled with discrete element method simulation, we show that the near-wall resistance can be explained by particle diffusion in granular flows.

Percolation of a fine particle in static granular beds

We study the percolation of a fine spherical particle under gravity in static randomly packed large-particle beds with different packing densities ϕ and large to fine particle size ratios R ranging from 4 to 7.5 using discrete element method simulations. The particle size ratio at the geometrical trapping threshold, defined by three touching large particles, R_{t}=sqrt[3]/(2-sqrt[3])=6.464, divides percolation behavior into passing and trapping regimes. However, the mean percolation velocity and diffusion of untrapped fine particles, which depend on both R and ϕ, are similar in both regimes and can be collapsed over a range of R and ϕ with the appropriate scaling. An empirical relationship for the local percolation velocity based on the local pore throat to fine particle size ratio and packing density is obtained, which is valid for the full range of size ratio and packing density we study. Similarly, in the trapping regime, the probability for a fine particle to reach a given depth is well described by a simple statistical model. Finally, the percolation velocity and fine particle diffusion are found to decrease with increasing restitution coefficient.

Self-diffusion in inhomogeneous granular shearing flows

In this Letter, we discuss how flow inhomogeneity affects the self-diffusion behavior in granular flows. Whereas self-diffusion scalings have been well characterized in the past for homogeneous shearing, the effect of shear localization and nonlocality of the flow has not been studied. We, therefore, present measurements of self-diffusion coefficients in discrete numerical simulations of steady, inhomogeneous, and collisional shearing flows of nearly identical, frictional, and inelastic spheres. We focus on a wide range of dense solid volume fractions, that correspond to geophysical and industrial shearing flows that are dominated by collisional interactions. We compare the measured values first with a scaling based on shear rate and, then, on a scaling based on the granular temperature. We find that the latter does much better than the former in collapsing the data. The results lay the foundations of diffusion models for inhomogeneous shearing flows, which should be useful in treating problems of mixing and segregation.

Segregation patterns in three-dimensional granular flows

Flow of size-bidisperse particle mixtures in a spherical tumbler rotating alternately about two perpendicular axes produces segregation patterns that track the location of nonmixing islands predicted by a dynamical systems approach. To better understand the paradoxical accumulation of large particles in regions defined by barriers to transport, we perform discrete element method (DEM) simulations to visualize the three-dimensional structure of the segregation patterns and track individual particles. Our DEM simulations and modeling results indicate that segregation pattern formation in the biaxial spherical tumbler is due to the interaction of size-driven radial segregation with the weak spanwise component of the advective surface flow. Specifically, we find that after large particles segregate to the surface, slow axial drift in the flowing layer, which is inherent to spherical tumblers, is sufficient to drive large particles across nominal transport barriers and into nonmixing islands predicted by an advective flow model in the absence of axial drift. Axial drift alters the periodic dynamics of nonmixing islands, turning them into "sinks" where large particles accumulate even in the presence of collisional diffusion. Overall, our results indicate that weak perturbation of chaotic flow has the potential to alter key dynamical system features (e.g., transport barriers), which ultimately can result in unexpected physical phenomena.

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.

Self-diffusion of spherocylindrical particles flowing under non-uniform shear rate

This work is devoted to study numerically the self-diffusion of spherocylindrical particles flowing down an inclined plane, using the discrete element method (DEM). This system is challenging due to particles being non-spherical and because they are subjected to a non-uniform shear rate. We performed simulations for several aspect ratios and inclination angles, tracking individual particle trajectories. Using the simulation data, we computed the diffusion coefficients D, and a coarse-graining methodology allowed accessing the shear rate spatial profiles (z). This data enabled us to identify the spatial regions where the diffusivity strongly correlates with the local shear rate. Introducing an effective particle size d_⊥, we proposed a well-rationalized scaling law between D and . Our findings also identified specific locations where the diffusivity does not correlate with the shear rate. This observation corresponds to zones where  has non-linear spatial variation, and the velocity probability density distributions exhibit asymmetric shapes.

Shear-induced diffusion in dense granular fluids

Granular materials are comprised of solid, athermal grains. Whilst immune to thermal motion, these grains move and diffuse when they undergo shear deformation. Here we introduce this process of shear-induced diffusion with a focus on dense flows. The goal is to present the established scaling laws for continuum diffusivity and to relate them to the micro-mechanisms of a granular random walk. We then suggest how this knowledge may help advance our understanding of granular rheology and diffusion in other soft-materials.

Self-diffusion scalings in dense granular flows

We report on measurements of self-diffusion coefficients in discrete numerical simulations of steady, homogeneous, collisional shearing flows of nearly identical, frictional, inelastic spheres. We focus on a range of relatively high solid volume fractions that are important in those terrestrial gravitational shearing flows that are dominated by collisional interactions. Diffusion over this range of solid fraction has not been well characterized in previous studies. We first compare the measured values with an empirical scaling based on shear rate previously proposed in the literature, and highlight the presence of anisotropy and the solid fraction dependence. We then compare the numerical measurements with those predicted by the kinetic theory for shearing flows of inelastic spheres and offer an explanation for why the measured and predicted values differ.

Shear-driven density segregation: an experimental study

Granular materials can segregate spontaneously due to differences in particle properties when subjected to vibrations, shear strain or because of the equipment geometries. Although the difference in particle size is the most critical factor that drives segregation, the effects of large density difference may also be detrimental for a lot of industries. In this work, we experimentally investigate density-driven segregation in bi-disperse mixtures of particles having the same size but different density when subjected to non-uniform shear rates. We found that the features of the segregation process are related to the density ratio as well as to the dimensionless loaded mass. The experimental outcomes are then compared with the solution of a simple density-driven segregation model. The model can successfully capture the main features of segregation driven by density for a range of density ratios.

Shear-induced diffusion: the role of granular clusters

This paper is concerned with the physical mechanisms controlling shear-induced diffusion in dense granular flows. The starting point is that of the granular random walk occurring in diluted granular flows, which underpins Bagnold’s scaling relating the coefficient of self-diffusion to the grain size and shear rate. By means of DEM simulations of plane shear flows, we measure some deviations from this scaling in dense granular flows with and without contact adhesion. We propose to relate these deviations to the development of correlated motion of grains in these flows, which impacts the magnitude of grain velocity fluctuations and their time persistence.

Boltzmann Distribution of Sediment Transport

The coupling of sediment transport with the flow that drives it allows rivers to shape their own bed. Cross-stream fluxes of sediment play a crucial, yet poorly understood, role in this process. Here, we track particles in a laboratory flume to relate their statistical behavior to the self-organization of the granular bed they make up. As they travel downstream, the transported grains wander randomly across the bed's surface, thus inducing cross-stream diffusion. The balance of diffusion and gravity results in a peculiar Boltzmann distribution, in which the bed's roughness plays the role of thermal fluctuations, while its surface forms the potential well that confines the sediment flux.

Modeling Segregation in Granular Flows

Accurate continuum models of flow and segregation of dense granular flows are now possible. This is the result of extensive comparisons, over the last several years, of computer simulations of increasing accuracy and scale, experiments, and continuum models, in a variety of flows and for a variety of mixtures. Computer simulations-discrete element methods (DEM)-yield remarkably detailed views of granular flow and segregation. Conti-nuum models, however, offer the best possibility for parametric studies of outcomes in what could be a prohibitively large space resulting from the competition between three distinct driving mechanisms: advection, diffusion, and segregation. We present a continuum transport equation-based framework, informed by phenomenological constitutive equations, that accurately predicts segregation in many settings, both industrial and natural. Three-way comparisons among experiments, DEM, and theory are offered wherever possible to validate the approach. In addition to the flows and mixtures described here, many straightforward extensions of the framework appear possible.

Diffusion of size bidisperse spheres in dense granular shear flow

Diffusion is an important particle behavior in granular flow. Although granular diffusion has been studied for decades, the diffusion of size bidisperse particles has not been well understood. In this paper, discrete element method simulations with the Lees-Edwards boundary condition are performed to quantify the relation between the diffusion coefficient (D) and flow parameters for size bidisperse spheres in dense granular flow. The influences of the shear rate (γ[over ̇]), the solids fraction (f), and the diameter ratio (D_{LS}) of particles on diffusion are studied. The effects of the friction coefficient (μ) and the restitution coefficient (e) are also investigated. The results indicate that while small particles diffuse faster than large particles in a binary system the volume weighted average diffusion coefficient is proportional to the shear rate and the square of the volume weighted average particle diameter, d^{2}, and it is inversely proportional to the solids fraction. The quantified relation is given as D=k_{d}γ[over ̇]d^{2}, where k_{d}=0.0186/f, and this relation is not sensitive to the diameter ratio for D_{LS}≤3. The diffusion coefficient is not sensitive to the friction coefficient except for the extreme condition where μ<0.1, and it is also not sensitive to the restitution coefficient between 0.3 and 0.9.

Langevin equations from experimental data: The case of rotational diffusion in granular media

A model has two main aims: predicting the behavior of a physical system and understanding its nature, that is how it works, at some desired level of abstraction. A promising recent approach to model building consists in deriving a Langevin-type stochastic equation from a time series of empirical data. Even if the protocol is based upon the introduction of drift and diffusion terms in stochastic differential equations, its implementation involves subtle conceptual problems and, most importantly, requires some prior theoretical knowledge about the system. Here we apply this approach to the data obtained in a rotational granular diffusion experiment, showing the power of this method and the theoretical issues behind its limits. A crucial point emerged in the dense liquid regime, where the data reveal a complex multiscale scenario with at least one fast and one slow variable. Identifying the latter is a major problem within the Langevin derivation procedure and led us to introduce innovative ideas for its solution.

Surface flow profiles for dry and wet granular materials by Particle Tracking Velocimetry; the effect of wall roughness

Two-dimensional Particle Tracking Velocimetry (PTV) is a promising technique to study the behaviour of granular flows. The aim is to experimentally determine the free surface width and position of the shear band from the velocity profile to validate simulations in a split-bottom shear cell geometry. The position and velocities of scattered tracer particles are tracked as they move with the bulk flow by analyzing images. We then use a new technique to extract the continuum velocity field, applying coarse-graining with the postprocessing toolbox MercuryCG on the discrete experimental PTV data. For intermediate filling heights, the dependence of the shear (or angular) velocity on the radial coordinate at the free surface is well fitted by an error function. From the error function, we get the width and the centre position of the shear band. We investigate the dependence of these shear band properties on filling height and rotation frequencies of the shear cell for dry glass beads for rough and smooth wall surfaces. For rough surfaces, the data agrees with the existing experimental results and theoretical scaling predictions. For smooth surfaces, particle-wall slippage is significant and the data deviates from the predictions. We further study the effect of cohesion on the shear band properties by using small amount of silicon oil and glycerol as interstitial liquids with the glass beads. While silicon oil does not lead to big changes, glycerol changes the shear band properties considerably. The shear band gets wider and is situated further inward with increasing liquid saturation, due to the correspondingly increasing trend of particles to stick together.

Energy Fluctuations in Slowly Sheared Granular Materials

Here we show the first experimental measurement of the particle-scale energy fluctuations ΔE in a slowly sheared layer of photoelastic disks. Starting from an isotropically jammed state, applying shear causes the shear-induced stochastic strengthening and weakening of particle-scale energies, whose statistics and dynamics govern the evolution of the macroscopic stress-strain curve. We find that the ΔE behave as a temperaturelike noise field, showing a novel, Boltzmann-type, double-exponential distribution at any given shear strain γ. Following the framework of the soft glassy rheology theory, we extract an effective temperature χ from the statistics of the energy fluctuations to interpret the slow startup shear (shear starts from an isotropically jammed state) of granular materials as an "aging" process: Starting below one, χ gradually approaches one as γ increases, similar to those of spin glasses, thermal glasses, and bulk metallic glasses.

Persistent structures in a three-dimensional dynamical system with flowing and non-flowing regions

Mixing of fluids and mixing of solids are both relatively mature fields. In contrast, mixing in systems where flowing and non-flowing regions coexist remains largely unexplored and little understood. Here we report remarkably persistent mixing and non-mixing regions in a three-dimensional dynamical system where randomness is expected. A spherical shell half-filled with dry non-cohesive particles and periodically rotated about two horizontal axes generates complex structures that vary non-trivially with the rotation angles. They result from the interplay between fluid-like mixing by stretching-and-folding, and solids mixing by cutting-and-shuffling. In the experiments, larger non-mixing regions predicted by a cutting-and-shuffling model alone can persist for a range of protocols despite the presence of stretching-and-folding flows and particle-collision-driven diffusion. By uncovering the synergy of simultaneous fluid and solid mixing, we point the way to a more fundamental understanding of advection driven mixing in materials with coexisting flowing and non-flowing regions.

Effect of pressure on segregation in granular shear flows

The effect of confining pressure (overburden) on segregation of granular material is studied in discrete element method (DEM) simulations of horizontal planar shear flow. To mitigate changes to the shear rate due to the changing overburden, a linear with depth variation in the streamwise velocity component is imposed using a simple feedback scheme. Under these conditions, both the rate of segregation and the ultimate degree of segregation in size bidisperse and density bidisperse granular flows decrease with increasing overburden pressure and scale with the overburden pressure normalized by the lithostatic pressure of the particle bed. At overburdens greater than approximately 20 times the lithostatic pressure at the bottom of the bed, the density segregation rate is zero while the size segregation rate is small but nonzero, suggesting that different physical mechanisms drive the two types of segregation. The segregation rate scales close to linearly with the inertial number for both size bidisperse and density bidisperse mixtures under various flow conditions, leading to a proposed pressure dependence term for existing segregation velocity correlations. Surprisingly, particle stiffness has only a minor effect on segregation, although it significantly affects the packing density.

Continuum modelling of segregating tridisperse granular chute flow

Segregation and mixing of size multidisperse granular materials remain challenging problems in many industrial applications. In this paper, we apply a continuum-based model that captures the effects of segregation, diffusion and advection for size tridisperse granular flow in quasi-two-dimensional chute flow. The model uses the kinematics of the flow and other physical parameters such as the diffusion coefficient and the percolation length scale, quantities that can be determined directly from experiment, simulation or theory and that are not arbitrarily adjustable. The predictions from the model are consistent with experimentally validated discrete element method (DEM) simulations over a wide range of flow conditions and particle sizes. The degree of segregation depends on the Péclet number, Pe, defined as the ratio of the segregation rate to the diffusion rate, the relative segregation strength κ_ij between particle species i and j, and a characteristic length L, which is determined by the strength of segregation between smallest and largest particles. A parametric study of particle size, κ_ij , Pe and L demonstrates how particle segregation patterns depend on the interplay of advection, segregation and diffusion. Finally, the segregation pattern is also affected by the velocity profile and the degree of basal slip at the chute surface. The model is applicable to different flow geometries, and should be easily adapted to segregation driven by other particle properties such as density and shape.

Vortices Enhance Diffusion in Dense Granular Flows

This Letter introduces unexpected diffusion properties in dense granular flows and shows that they result from the development of partially jammed clusters of grains, or granular vortices. Transverse diffusion coefficients D and average vortex sizes ℓ are systematically measured in simulated plane shear flows at differing inertial numbers I revealing (i) a strong deviation from the expected scaling D∝d^{2}γ[over ˙] involving the grain size d and shear rate γ[over ˙] and (ii) an increase in average vortex size ℓ at low I, following ℓ∝dI^{-1/2} but limited by the system size. A general scaling D∝ℓdγ[over ˙] is introduced that captures all the measurements and highlights the key role of vortex size. This leads to establishing a scaling for the diffusivity in dense granular flow as D∝d^{2}sqrt[γ[over ˙]/t_{i}] involving the geometric average of shear time 1/γ[over ˙] and inertial time t_{i} as the relevant time scale. Analysis of grain trajectories is further evidence that this diffusion process arises from a vortex-driven random walk.

Planar granular shear flow under external vibration

We present results from a planar shear experiment in which a two-dimensional horizontal granular assembly of pentagonal particles sheared between two parallel walls is subjected to external vibration. Particle tracking and photoelastic measurements are used to quantify both grain scale motion and interparticle stresses with and without imposed vibrations. We characterize the particle motion in planar shear and find that flow of these strongly interlocking particles consists of transient vortex motion with a mean flow given by the sum of exponential profiles imposed by the shearing walls. Vibration is applied either through the shearing surface or as bulk vertical vibration of the entire shearing region with dimensionless accelerations Γ=A(2πf)^{2}/g≈0-2. In both cases, increasing amplitude of vibration A at fixed frequency f leads to failure of the force network, reduction in mean stress, and a corresponding reduction in imposed strain. Vibration of the shearing surface is shown to induce the preferential slipping of large-angle force chains. These effects are insensitive to changes in frequency in the range studied (f=30-120 Hz), as sufficiently large displacements are required to relieve the geometrical frustration of the jammed states.

Photoelastic force measurements in granular materials

Photoelastic techniques are used to make both qualitative and quantitative measurements of the forces within idealized granular materials. The method is based on placing a birefringent granular material between a pair of polarizing filters, so that each region of the material rotates the polarization of light according to the amount of local stress. In this review paper, we summarize the past work using the technique, describe the optics underlying the technique, and illustrate how it can be used to quantitatively determine the vector contact forces between particles in a 2D granular system. We provide a description of software resources available to perform this task, as well as key techniques and resources for building an experimental apparatus.

Granular core phenomenon induced by convection in a vertically vibrated cylindrical container

A mixture of 13X molecular sieve (13XMS) particles and glass particles with identical diameters is placed in a cylindrical container. Under vertical vibration, heavier glass particles tend to cluster and are wrapped inside the convection of 13XMS particles, resulting in the granular core phenomenon. The vibration frequency f strongly influences particle convection and particle cluster modes. By contrast, the effect of the dimensionless acceleration amplitude Γ can be neglected. For different f ranges, the granular core is classified as center-type and ring-type cores. For the center-type core, heavy particles are distributed as an approximate zeroth-order Bessel function of the first kind in the radial direction and an exponential function in the height direction. For the ring-type core, the concentration of heavy particles follows the power-series function in the radial direction. A granular transport model is then established based on heavy-particle movements under steady state to analyze the effect of vibration parameters and granular convection on density segregation.

Modelling density segregation in flowing bidisperse granular materials

Preventing segregation in flowing granular mixtures is an ongoing challenge for industrial processes that involve the handling of bulk solids. A recent continuum-based modelling approach accurately predicts spatial concentration fields in a variety of flow geometries for mixtures varying in particle size. This approach captures the interplay between advection, diffusion and segregation using kinematic information obtained from experiments and/or discrete element method (DEM) simulations combined with an empirically determined relation for the segregation velocity. Here, we extend the model to include density-driven segregation, thereby validating the approach for the two important cases of practical interest. DEM simulations of density bidisperse flows of mono-sized particles in a quasi-two-dimensional-bounded heap were performed to determine the dependence of the density-driven segregation velocity on local shear rate and particle concentration. The model yields theoretical predictions of segregation patterns that quantitatively match the DEM simulations over a range of density ratios and flow rates. Matching experiments reproduce the segregation patterns and quantitative segregation profiles obtained in both the simulations and the model, thereby demonstrating that the modelling approach captures the essential physics of density-driven segregation in granular heap flow.

Modifying self-assembly and species separation in three-dimensional systems of shape-anisotropic particles

The behaviors of large, dynamic assemblies of macroscopic particles are of direct relevance to geophysical and industrial processes and may also be used as easily studied analogs to micro- or nano-scale systems, or model systems for microbiological, zoological, and even anthropological phenomena. We study vibrated mixtures of elongated particles, demonstrating that the inclusion of differing particle "species" may profoundly alter a system's dynamics and physical structure in various diverse manners. The phase behavior observed suggests that our system, despite its athermal nature, obeys a minimum free energy principle analogous to that observed for thermodynamic systems. We demonstrate that systems of exclusively spherical objects, which form the basis of numerous theoretical frameworks in many scientific disciplines, represent only a narrow region of a wide, multidimensional phase space. Thus, our results raise significant questions as to whether such models can accurately describe the behaviors of systems outside this highly specialized case.

An itinerant oscillator model with cage inertia for mesorheological granular experiments

Recent experiments with a rotating probe immersed in weakly fluidized granular materials show a complex behavior on a wide range of time scales. Viscous-like relaxation at high frequency is accompanied by an almost harmonic dynamical trapping at intermediate times, with possibly anomalous long time behavior in the form of super-diffusion. Inspired by the itinerant oscillator model for diffusion in molecular liquids, and other models with coupled thermostats acting at different time scales, here we discuss a new model able to account for fast viscous relaxation, dynamical trapping, and super-diffusion at long times. The main difference with respect to liquids is a non-negligible cage inertia for the surrounding (granular) fluid, which allows it to sustain a slow but persistent motion for long times. The computed velocity power density spectra and mean-squared displacement qualitatively reproduce the experimental findings. We also discuss the linear response to external perturbations and the tail of the distribution of persistency time, which is associated with superdiffusion, and whose cut-off time is determined by cage inertia.

Shear-Rate-Independent Diffusion in Granular Flows

We computationally study the behavior of the diffusion coefficient D in granular flows of monodisperse and bidisperse particles spanning regions of relatively high and low shear rate in open and closed laterally confined heaps. Measurements of D at various flow rates, streamwise positions, and depths collapse onto a single curve when plotted as a function of γd2, where d is the local mean particle diameter and γ is the local shear rate. When γ is large, D is proportional to γd2, as in previous studies. However, for γd2 below a critical value, D is independent of γd2. The acceleration due to gravity g and particle stiffness (or, equivalently, the binary collision time t(c)) together determine the transition in D between regimes. This suggests that while shear rate and particle size determine diffusion at relatively high shear rates in surface-driven flows, diffusion at low shear rates is an elastic phenomenon with time and length scales dependent on gravity (sqrt d/g) and particle stiffness (t(c)sqrt(dg), respectively.

Fluctuations of particle motion in granular avalanches - from the microscopic to the macroscopic scales

In this study, we have investigated the fluctuations of particle motion, i.e. the non-affine motion, during the avalanche process, discovering a rich dynamic from the microscopic to the macroscopic scales. We find that there is a strong correlation between the magnitude of the velocity fluctuation and the velocity magnitude in the spatial and temporal domains. The possible connection between this finding and the theory of the shear transformation zones is discussed based on the direct measurement of the T1 events. In addition, the velocity magnitude of the system and the stress fluctuations of the system are strongly temporally correlated. Our finding will pose challenges to the development of more rigorous theories to describe avalanche dynamics based on the microscopic approach. Moreover, our finding presents a plausible mechanism of particle entrainment in a simple system.

Cages and anomalous diffusion in vibrated dense granular media

A vertically shaken granular medium hosts a blade rotating around a fixed vertical axis, which acts as a mesorheological probe. At high densities, independently of the shaking intensity, the blade's dynamics shows strong caging effects, marked by transient subdiffusion and a maximum in the velocity power density spectrum, at a resonant frequency ~10 Hz. Interpreting the data through a diffusing harmonic cage model allows us to retrieve the elastic constant of the granular medium and its collective diffusion coefficient. For high frequencies f, a tail ~1/f in the velocity power density spectrum reveals nontrivial correlations in the intracage microdynamics. At very long times (larger than 10 s), a superdiffusive behavior emerges, ballistic in the most extreme cases. Consistently, the distribution of slow velocity inversion times τ displays a power-law decay, likely due to persistent collective fluctuations of the host medium.