The permeability of pillar arrays in microfluidic devices: an application of Brinkman's theory towards wall friction.

Darcy's law describes the flow of Newtonian fluids through bulk porous media as the product of the applied pressure difference, the fluid's viscosity and the medium's permeability. Brinkman extended Darcy's law with a viscous stress term, thereby enabling boundary conditions to the flow field at the surface of the medium. The validity of Brinkman's term, and the value of its effective viscosity, have been heavily debated since their introduction nearly 75 years ago. We use experiments and Multibody Dissipative Particle Dynamics (MDPD) simulations to study flows through ordered and disordered pillar arrays in microfluidic channels of limited height. We find that the simulated velocity profiles are well described by an expedient interpretation of Brinkman's theory. Depending on the solid volume fraction and pillar arrangement, the effective viscosity varies between two and three times the bulk fluid viscosity. The calculated effective permeabilities of the flow devices, combining the flow resistances due to the pillars and the walls by Brinkman's theory, agree well with the experimental data. This approach enables fast and accurate estimates of the effective permeability of micropillared chips. The simulated force distributions over the walls and pillars require an effective viscosity equal to the bulk viscosity and an elevation-dependent permeability of the pillar array.

Bulk modulus along jamming transition lines of bidisperse granular packings.

We present three-dimensional discrete element method simulations of bidisperse granular packings to investigate their jamming densities ϕ_{J} and dimensionless bulk moduli K as functions of the size ratio δ and the concentration of small particles X_{S}. We determine the partial and total bulk moduli for packings near their jamming densities, including a second transition that occurs for sufficiently small δ and X_{S} when the system is compressed beyond its first jamming transition. While the first transition is sharp, exclusively with large-large contacts, the second is rather smooth, carried by small-large interactions at densities much higher than the monodisperse random packing baseline, ϕ_{J}^{mono}≈0.64. When only nonrattlers are considered, all the effective transition densities are reduced, and the density of the second transition emerges rather close to the reduced baseline, ϕ[over ̃]_{J}^{mono}≈0.61, due to its smooth nature. At size ratios δ≤0.22 a concentration X_{S}^{*} divides the diagram-either with most small particles nonjammed or jammed jointly with large ones. For X_{S}

Understanding slow compression and decompression of frictionless soft granular matter by network analysis.

We consider dense granular systems in three spatial dimensions exposed to slow compression and decompression, below, during, above and well above jamming. The evolution of granular systems under slow deformation is non-trivial and involves smooth, continuous, reversible (de)compression periods, interrupted by fast, discontinuous, irreversible transition events. These events are often, but not always, associated with rearrangements of particles and of the contact network. How many particles are involved in these transitions between two states can range from few to almost all in the system. An analysis of the force network that is built on top of the contact network is carried out using the tools of persistent homology. Results involve the observation that kinetic energy is correlated with the intensity of rearrangements, while the evolution of global mechanical measures, such as pressure, is strongly correlated with the evolution of the topological measures quantifying loops in the force network. Surprisingly, some transitions are clearly detected by persistent homology even though motion/rearrangement of particles is much weaker, i.e., much harder to detect or, in some cases, not observed at all.

Stochastic Model for Energy Propagation in Disordered Granular Chains.

Energy transfer is one of the essentials of mechanical wave propagation (along with momentum transport). Here, it is studied in disordered one-dimensional model systems mimicking force-chains in real systems. The pre-stressed random masses (other types of disorder lead to qualitatively similar behavior) interact through (linearized) Hertzian repulsive forces, which allows solving the deterministic problem analytically. The main goal, a simpler, faster stochastic model for energy propagation, is presented in the second part, after the basic equations are re-visited and the phenomenology of pulse propagation in disordered granular chains is reviewed. First, the propagation of energy in space is studied. With increasing disorder (quantified by the standard deviation of the random mass distribution), the attenuation of pulsed signals increases, transiting from ballistic propagation (in ordered systems) towards diffusive-like characteristics, due to energy localization at the source. Second, the evolution of energy in time by transfer across wavenumbers is examined, using the standing wave initial conditions of all wavenumbers. Again, the decay of energy (both the rate and amount) increases with disorder, as well as with the wavenumber. The dispersive ballistic transport in ordered systems transits to low-pass filtering, due to disorder, where localization of energy occurs at the lowest masses in the chain. Instead of dealing with the too many degrees of freedom or only with the lowest of all the many eigenmodes of the system, we propose a stochastic master equation approach with reduced complexity, where all frequencies/energies are grouped into bands. The mean field stochastic model, the matrix of energy-transfer probabilities between bands, is calibrated from the deterministic analytical solutions by ensemble averaging various band-to-band transfer situations for short times, as well as considering the basis energy levels (decaying with the wavenumber increasing) that are not transferred. Finally, the propagation of energy in the wavenumber space at transient times validates the stochastic model, suggesting applications in wave analysis for non-destructive testing, underground resource exploration, etc.

Elastic waves in particulate glass-rubber mixtures.

We investigate the propagation of waves in dense static granular packings made of soft and stiff particles subjected to hydrostatic stress. Physical experiments in a triaxial cell equipped with broadband piezoelectric wave transducers have been performed at ultrasound frequencies. The time of flight is measured in order to study the combined effect of applied stress and rubber content on the elastic properties of the mixtures. The bulk stiffness deduced from the wave speed is nonlinear and non-monotonic with the increasing percentage of rubber with a more prominent effect at higher pressures. Moreover, in the frequency domain, a spectral analysis gives insights on the transition from a glass- to a rubber-dominated regime and the influence of rubber particles on the energy dissipation. Mixtures with rubber content below 30% show enhanced damping properties, associated with slightly higher stiffness and lighter weight.

Additional Transition Line in Jammed Asymmetric Bidisperse Granular Packings.

We present numerical evidence for an additional discontinuous transition, upon compression, inside the jammed regime for an asymmetric bidisperse granular packing. This additional transition line separates jammed states with networks of predominantly large particles from jammed networks formed by both large and small particles, and the transition is indicated by a discontinuity in the number of particles contributing to the jammed network. The additional transition line emerges from the curves of jamming transitions and terminates in an end point where the discontinuity vanishes. The additional line is starting at a size ratio around δ=0.22 and grows longer for smaller δ. For δ→0, the additional transition line approaches a limit that can be derived analytically. The observed jamming scenarios are reminiscent of glass-glass transitions found in colloidal glasses.

Cohesion-driven mixing and segregation of dry granular media.

Granular segregation is a common, yet still puzzling, phenomenon encountered in many natural and engineering processes. Here, we experimentally investigate the effect of particles cohesion on segregation in dry monodisperse and bidisperse systems using a rotating drum mixer. Chemical silanization, glass surface functionalization via a Silane coupling agent, is used to produce cohesive dry glass particles. The cohesive force between the particles is controlled by varying the reaction duration of the silanization process, and is measured using an in-house device specifically designed for this study. The effects of the cohesive force on flow and segregation are then explored and discussed. For monosized particulate systems, while cohesionless particles perfectly mix when tumbled, highly cohesive particles segregate. For bidisperse mixtures of particles, an adequate cohesion-tuning reduces segregation and enhances mixing. Based on these results, a simple scheme is proposed to describe the system's mixing behaviour with important implications for the control of segregation or mixing in particulate industrial processes.

Correction: Transition rates for slip-avalanches in soft athermal disks under quasi-static simple shear deformations.

Correction for 'Transition rates for slip-avalanches in soft athermal disks under quasi-static simple shear deformations' by Kuniyasu Saitoh et al., Soft Matter, 2019, DOI: 10.1039/c8sm01966e.

Transition rates for slip-avalanches in soft athermal disks under quasi-static simple shear deformations.

We study slip-avalanches in two-dimensional soft athermal disks by quasi-static simulations of simple shear deformations. Sharp drops in shear stress, or slip-avalanches, are observed intermittently during steady state. Such stress drops are caused by restructuring of the contact networks, accompanied by drastic changes of the interaction forces, Δf. The changes of the forces happen heterogeneously in space, indicating that collective non-affine motions of the disks are most pronounced when slip-avalanches occur. We analyze and predict the statistics for the force changes, Δf, by transition rates of the contact forces and angles, where slip-avalanches are characterized by wide power-law tails. We find that the transition rates are described by a q-Gaussian distribution regardless of the area fraction of the disks. Because the transition rates quantify structural changes of the force-chains, our findings are an important step towards linking macroscopic observations to a microscopic theory of slip-avalanches in the experimentally accessible quasi-static regime.

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.

Rotational sound in disordered granular materials.

We numerically study the evolution of elastic standing waves in disordered disk systems with a focus on the dispersion relations of rotational sound. As on a lattice, the rotational mode exhibits an optical-like dispersion relation in the high frequency regime, representing a shoulder in the vibrational density of states and fast oscillations of the autocorrelations of rotational velocities. If tangential stiffness between the disks is large enough, a lattice-based model perfectly describes the dispersion relation of the rotational mode. If it is comparable to or smaller than the normal stiffness, the model fails for short wavelengths. However, the dispersion relation then follows the model prediction for the transverse mode, implying that the fast oscillations of disks' rotations switch to acousticlike behavior. We evidence such a transition from rotational to transverse modes by analyzing their respective participation of different degrees of freedom to the eigenvectors.

Merging fluid and solid granular behavior.

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

From the granular Leidenfrost state to buoyancy-driven convection.

Grains inside a vertically vibrated box undergo a transition from a density-inverted and horizontally homogeneous state, referred to as the granular Leidenfrost state, to a buoyancy-driven convective state. We perform a simulational study of the precursors of such a transition and quantify their dynamics as the bed of grains is progressively fluidized. The transition is preceded by transient convective states, which increase their correlation time as the transition point is approached. Increasingly correlated convective flows lead to density fluctuations, as quantified by the structure factor, that also shows critical behavior near the transition point. The amplitude of the modulations in the vertical velocity field are seen to be best described by a quintic supercritical amplitude equation with an additive noise term. The validity of such an amplitude equation, and previously observed collective semiperiodic oscillations of the bed of grains, suggests a new interpretation of the transition analogous to a coupled chain of vertically vibrated damped oscillators. Increasing the size of the container shows metastability of convective states, as well as an overall invariant critical behavior close to the transition.

A master equation for the probability distribution functions of forces in soft particle packings.

We study the microscopic response of force-chain networks in jammed soft particles to quasi-static isotropic (de)compressions by molecular dynamics simulations. We show that not only contacts but also interparticle gaps between the nearest neighbors must be considered for the stochastic evolution of the probability distribution functions (PDFs) of forces, where the mutual exchange of contacts and interparticle gaps, i.e. opening and closing contacts, are also crucial to the incremental system behavior. By numerically determining the transition rates for all changes of contacts and gaps, we formulate a Master equation for the PDFs of forces, where the insight one gets from the transition rates is striking: the mean change of forces reflects non-affine system responses, while their fluctuations obey uncorrelated Gaussian statistics. In contrast, interparticle gaps react mostly affine in average, but imply multi-scale correlations according to a much wider stable distribution function.

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.

Effect of cohesion on shear banding in quasistatic granular materials.

Cohesive powders have widely different bulk behavior due to their peculiar interactions. We use discrete element simulations to investigate the effect of contact cohesion on the steady state flow of dense powders in a slowly sheared split-bottom Couette cell, which imposes a wide stable shear band. The intensity of cohesive forces can be quantified by the granular Bond number (Bo), namely the ratio between maximum attractive force and average force due to external compression. We find that the shear banding phenomenon is almost independent of cohesion for Bond numbers Bo<1, however for Bo≥1 cohesive forces start to play an important role, as both width and center position of the band increase. Inside the shear band, the mean normal contact force is independent of cohesion and depends only on the confining stress. In contrast, when the behavior is analyzed focusing on the eigendirections of the local strain rate tensor, a dependence on cohesion shows up. Forces carried by contacts along the compressive and tensile directions are symmetric about the mean force (larger and smaller respectively), while the force along the third, neutral direction follows the mean force. This anisotropy of the force network increases with cohesion, just like the heterogeneity in all (compressive, tensile and neutral) directions.

Communication: Structure characterization of hard sphere packings in amorphous and crystalline states.

The channel size distribution in hard sphere systems, based on the local neighbor correlation of four particle positions, is investigated for all volume fractions up to jamming. For each particle, all three particle combinations of neighbors define channels, which are relevant for the concept of caging. The analysis of the channel size distribution is shown to be very useful in distinguishing between gaseous, liquid, partially and fully crystallized, and glassy (random) jammed states. A common microstructural feature of four coplanar particles is observed in crystalline and glassy jammed states, suggesting the presence of "hidden" two-dimensional order in three-dimensional random close packings.

Micro-macro correlations and anisotropy in granular assemblies under uniaxial loading and unloading.

The influence of contact friction on the behavior of dense, polydisperse granular assemblies under uniaxial (oedometric) loading and unloading deformation is studied using discrete element simulations. Even though the uniaxial deformation protocol is one of the "simplest" element tests possible, the evolution of the structural anisotropy necessitates its careful analysis and understanding, since it is the source of interesting and unexpected observations. On the macroscopic, homogenized, continuum scale, the deviatoric stress ratio and the deviatoric fabric, i.e., the microstructure behave in a different fashion during uniaxial loading and unloading. The maximal stress ratio and strain increase with increasing contact friction. In contrast, the deviatoric fabric reaches its maximum at a unique strain level independent of friction, with the maximal value decreasing with friction. For unloading, both stress and fabric respond to unloading strain with a friction-dependent delay but at different strains. On the micro-level, a friction-dependent non-symmetry of the proportion of weak (strong) and sliding (sticking) contacts with respect to the total contacts during loading and unloading is observed. Coupled to this, from the directional probability distribution, the "memory" and history-dependent behavior of granular systems is confirmed. Surprisingly, while a rank-2 tensor is sufficient to describe the evolution of the normal force directions, a sixth order harmonic approximation is necessary to describe the probability distribution of contacts, tangential force, and mobilized friction. We conclude that the simple uniaxial deformation activates microscopic phenomena not only in the active Cartesian directions, but also at intermediate orientations, with the tilt angle being dependent on friction, so that this microstructural features cause the interesting, nontrivial macroscopic behavior.

A constitutive framework for the non-Newtonian pressure tensor of a simple fluid under planar flows.

Non-equilibrium molecular dynamics simulations of an atomic fluid under shear flow, planar elongational flow, and a combination of shear and elongational flow are unified consistently with a tensorial model over a wide range of strain rates. A model is presented that predicts the pressure tensor for a non-Newtonian bulk fluid under a homogeneous planar flow field. The model provides a quantitative description of the strain-thinning viscosity, pressure dilatancy, deviatoric viscoelastic lagging, and out-of-flow-plane pressure anisotropy. The non-equilibrium pressure tensor is completely described through these four quantities and can be calculated as a function of the equilibrium material constants and the velocity gradient. This constitutive framework in terms of invariants of the pressure tensor departs from the conventional description that deals with an orientation-dependent description of shear stresses and normal stresses. The present model makes it possible to predict the full pressure tensor for a simple fluid under various types of flows without having to produce these flow types explicitly in a simulation or experiment.