Packings of deformable spheres

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

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

Onset of glassiness in two-dimensional ring polymers: Interplay of stiffness and crowding

The effect of ring stiffness and pressure on the glassy dynamics of a thermal assembly of two-dimensional ring polymers is investigated using extensive coarse-grained molecular dynamics simulations. In all cases, dynamical slowing down is observed with increasing pressure, and thereby, a phase space for equilibrium dynamics is identified in the plane of the obtained monomer density and ring stiffness. When the rings are highly flexible, i.e., have low ring stiffness, glassiness sets in via the crowding of crumpled polymers, which take on a globular form. In contrast, at large ring stiffness, when the rings tend to have large asphericity under compaction, we observe the emergence of local domains having orientational ordering at high pressures. Therefore, our simulations highlight how varying the deformability of rings leads to contrasting mechanisms in driving the system toward the glassy regime.

Compacting an assembly of soft balls far beyond the jammed state: Insights from three-dimensional imaging

Very soft grain assemblies have unique shape-changing capabilities that allow them to be compressed far beyond the rigid jammed state by filling void spaces more effectively. However, accurately following the formation of these systems by monitoring the creation of new contacts, monitoring the changes in grain shape, and measuring grain-scale stresses is challenging. We developed an experimental method that overcomes these challenges and connects their microscale behavior to their macroscopic response. By tracking the local strain energy during compression, we reveal a transition from granular-like to continuous-like material. Mean contact geometry is shown to vary linearly with the packing fraction, which is supported by a mean field approximation. We also validate a theoretical framework which describes the compaction from a local view. Our experimental framework provides insights into the granular micromechanisms and opens perspectives for rheological analysis of highly deformable grain assemblies in various fields ranging from biology to engineering.

Effect of ring stiffness and ambient pressure on the dynamical slowdown in ring polymers

Using extensive molecular dynamics simulations, we investigate the slowdown of dynamics in a 3D system of ring polymers by varying the ambient pressure and the stiffness of the rings. Our study demonstrates that the stiffness of the rings determines the dynamics of the ring polymers, leading to glassiness at lower pressures for stiffer rings. The threading of the ring polymers, a unique feature that emerges only due to the topological nature of such polymers in three dimensions, is shown to be the determinant feature of dynamical slowdown, albeit only in a certain stiffness range. Our results suggest a possible framework for exploring the phase space spanned by ring stiffness and pressure to obtain spontaneously emerging topologically constrained polymer glasses.

Three-dimensional compaction of soft granular packings

This paper analyzes the compaction behavior of assemblies composed of soft (elastic) spherical particles beyond the jammed state, using three-dimensional non-smooth contact dynamic simulations. The assemblies of particles are characterized using the evolution of the packing fraction, the coordination number, and the von Misses stress distribution within the particles as the confining stress increases. The packing fraction increases and tends toward a maximum value close to 1, and the mean coordination number increases as a square root of the packing fraction. As the confining stress increases, a transition is observed from a granular-like material with exponential tails of the shear stress distributions to a continuous-like material characterized by Gaussian-like distributions of the shear stresses. We develop an equation that describes the evolution of the packing fraction as a function of the applied pressure. This equation, based on the micromechanical expression of the granular stress tensor, the limit of the Hertz contact law for small deformation, and the power-law relation between the packing fraction and the coordination of the particles, provides good predictions from the jamming point up to very high densities without the need for tuning any parameters.

Structural characterization and statistical properties of jammed soft ellipsoid packing

The jamming transition and jammed packing structures of hydrogel soft ellipsoids are studied using magnetic resonance imaging techniques. As the packing fraction increases, the fluctuation of local free volume decreases and the fluctuation of particle deformation increases. Effective thermodynamic quantities are obtained by characterizing these fluctuations using k-gamma distributions based on an underlying statistical model for granular materials. Surprisingly, the two granular temperatures measuring the relative fluctuations of both free volume and particle deformation remain basically unchanged as the packing fraction increases. The total configurational entropy is also approximately constant for packing with different packing fractions. The significantly different behaviors of these effective thermodynamic quantities compared with hard sphere systems are further attributed to a statistically affine structural transformation of the packing structures along with particle deformations when the packing fraction changes.

Under pressure: Hydrogel swelling in a granular medium

Hydrogels hold promise in agriculture as reservoirs of water in dry soil, potentially alleviating the burden of irrigation. However, confinement in soil can markedly reduce the ability of hydrogels to absorb water and swell, limiting their widespread adoption. Unfortunately, the underlying reason remains unknown. By directly visualizing the swelling of hydrogels confined in three-dimensional granular media, we demonstrate that the extent of hydrogel swelling is determined by the competition between the force exerted by the hydrogel due to osmotic swelling and the confining force transmitted by the surrounding grains. Furthermore, the medium can itself be restructured by hydrogel swelling, as set by the balance between the osmotic swelling force, the confining force, and intergrain friction. Together, our results provide quantitative principles to predict how hydrogels behave in confinement, potentially improving their use in agriculture as well as informing other applications such as oil recovery, construction, mechanobiology, and filtration.

Soft granular particles sheared at a controlled volume: rate-dependent dynamics and the solid-fluid transition

We study the responses of fluid-immersed soft hydrogel spheres that are sheared under controlled volume fractions. Slippery, deformable particles along with the density-matched interstitial fluid are sandwiched between two opposing rough cones, allowing studies for a wide range of volume fraction φ both above and below the jamming of granular suspension. We utilize sudden cessations of shearing, accompanied by refraction-matched internal imaging, to supplement the conventional flow-curve measurements. At sufficiently high volume fractions, the settling of particles after the cessations exhibits a continuous yet distinct transition over the change of the shear rate. Such changes back out the qualitative difference in the state of flowing prior to the cessations: the quasi-static yielding of a tightly packed network, as opposed to the rapid sliding of particles mediated by the interstitial fluid whose dynamics depends on the driving rate. In addition, we determine the solid-fluid transition using two independent methods: the extrapolation of stress residues and the estimated yield stress from high values of φ, and the settling of particles upon shear cessations as φ goes across the transition. We also verify the power law on values of characteristic stress with respect to the distance from jamming φ - φc, with an exponent close to 2. These results demonstrate a multitude of relaxation timescales behind the dynamics of soft particles, and raise questions on how we extend the existing paradigms of the flow of a densely packed system when the softness is actively involved.

Scaling Law for Cracking in Shrinkable Granular Packings

Hydrated granular packings often crack into discrete clusters of grains when dried. Despite its ubiquity, an accurate prediction of cracking remains elusive. Here, we elucidate the previously overlooked role of individual grain shrinkage-a feature common to many materials-in determining crack patterning using both experiments and simulations. By extending classical Griffith crack theory, we obtain a scaling law that quantifies how cluster size depends on the interplay between grain shrinkage, stiffness, and size-applicable to a diverse array of shrinkable granular packings.

Crack formation and self-closing in shrinkable, granular packings

Many clays, soils, biological tissues, foods, and coatings are shrinkable, granular materials: they are composed of packed, hydrated grains that shrink when dried. In many cases, these packings crack during drying, critically hindering applications. However, while cracking has been widely studied for bulk gels and packings of non-shrinkable grains, little is known about how packings of shrinkable grains crack. Here, we elucidate how grain shrinkage alters cracking during drying. Using experiments with model shrinkable hydrogel beads, we show that differential shrinkage can dramatically alter crack evolution during drying-in some cases, even causing cracks to spontaneously "self-close". In other cases, packings shrink without cracking or crack irreversibly. We developed both granular and continuum models to quantify the interplay between grain shrinkage, poromechanics, packing size, drying rate, capillarity, and substrate friction on cracking. Guided by the theory, we also found that cracking can be completely altered by varying the spatial profile of drying. Our work elucidates the rich physics underlying cracking in shrinkable, granular packings, and yields new strategies for controlling crack evolution.

Molecular insights into the irreversible mechanical behavior of sickle hemoglobin

Sickle cell disease is caused by the amino acid substitution of glutamic acid to valine, which leads to the polymerization of deoxygenated sickle hemoglobin (HbS) into long strands. These strands are responsible for the sickling of red blood cells (RBCs), making blood hyper-coagulable leading to an increased chance of vaso-occlusive crisis. The conformational changes in sickled RBCs traveling through narrow blood vessels in a highly viscous fluid are critical in understanding; however, there are few studies that investigate the origins of the molecular mechanical behavior of sickled RBCs. In this work, we investigate the molecular mechanical properties of HbS molecules. A mechanical model was used to estimate the directional stiffness of an HbS molecule and the results were compared to adult human hemoglobin (HbA). The comparison shows a significant difference in strength between HbS and HbA, as well as anisotropic behavior of the hemoglobin molecules. The results also indicated that the HbS molecule experienced more irreversible mechanical behavior than HbA under compression. Further, we have characterized the elastic and compressive properties of a double stranded sickle fiber using six HbS molecules, and it shows that the HbS molecules are bound to each other through strong inter-molecular forces.

Note: Eliminating stripe artifacts in light-sheet fluorescence imaging

We report two techniques to mitigate stripe artifacts in light-sheet fluorescence imaging. The first uses an image processing algorithm called the multidirectional stripe remover method to filter stripes from an existing image. The second uses an elliptical holographic diffuser with strong scattering anisotropy to prevent stripe formation during image acquisition. These techniques facilitate accurate interpretation of image data, especially in denser samples. They are also facile and cost-effective.

Force distribution in a granular medium under dynamic loading

Force distribution in a granular medium subjected to an impulse loading is investigated in experiment and computer simulations. An experimental technique is developed to measure forces acting on individual grains at the bottom of the granular sample consisting of steel balls. Discrete element method simulation also is performed under conditions mimicking those in experiment. Both theory and experiment display exponentially decaying maximum force distributions at the bottom of the sample in the range of large forces. In addition, the simulations also reveal exponential force distribution throughout the sample and uncover correlation properties of the interparticle forces during dynamic loading of the granular samples. Simulated time dependence of coordination number, orientational order parameter, correlation radius, and force distribution clearly demonstrates the nonequilibrium character of the deformation process in a granular medium under impulse loading.

X-ray fluorescence at nanoscale resolution for multicomponent layered structures: a solar cell case study

The study of a multilayered and multicomponent system by spatially resolved X-ray fluorescence microscopy poses unique challenges in achieving accurate quantification of elemental distributions. This is particularly true for the quantification of materials with high X-ray attenuation coefficients, depth-dependent composition variations and thickness variations. A widely applicable procedure for use after spectrum fitting and quantification is described. This procedure corrects the elemental distribution from the measured fluorescence signal, taking into account attenuation of the incident beam and generated fluorescence from multiple layers, and accounts for sample thickness variations. Deriving from Beer-Lambert's law, formulae are presented in a general integral form and numerically applicable framework. The procedure is applied using experimental data from a solar cell with a Cu(In,Ga)Se₂ absorber layer, measured at two separate synchrotron beamlines with varied measurement geometries. This example shows the importance of these corrections in real material systems, which can change the interpretation of the measured distributions dramatically.

Effect of long-range repulsive Coulomb interactions on packing structure of adhesive particles

The packing of charged micron-sized particles is investigated using discrete element simulations based on adhesive contact dynamic model. The formation process and the final obtained structures of ballistic packings are studied to show the effect of interparticle Coulomb force. It is found that increasing the charge on particles causes a remarkable decrease of the packing volume fraction ϕ and the average coordination number 〈Z〉, indicating a looser and chainlike structure. Force-scaling analysis shows that the long-range Coulomb interaction changes packing structures through its influence on particle inertia before they are bonded into the force networks. Once contact networks are formed, the expansion effect caused by repulsive Coulomb forces are dominated by short-range adhesion. Based on abundant results from simulations, a dimensionless adhesion parameter Ad*, which combines the effects of the particle inertia, the short-range adhesion and the long-range Coulomb interaction, is proposed and successfully scales the packing results for micron-sized particles within the latest derived adhesive loose packing (ALP) regime. The structural properties of our packings follow well the recent theoretical prediction which is described by an ensemble approach based on a coarse-grained volume function, indicating some kind of universality in the low packing density regime of the phase diagram regardless of adhesion or particle charge. Based on the comprehensive consideration of the complicated inter-particle interactions, our findings provide insight into the roles of short-range adhesion and repulsive Coulomb force during packing formation and should be useful for further design of packings.

Slow crystallisation of a monodisperse foam stabilised against coarsening

The evolution of a three-dimensional monodisperse foam was investigated using X-ray tomography over the course of seven days. The coarsening of the sample was inhibited through the use of perfluorohexane gas. The internal configuration of bubbles is seen to change markedly, evolving from a disordered arrangement towards a more ordered state. We chart this ordering process through the use of the coordination number, the bond orientational order parameter (BOOP) and the translational order parameter.

Extraction of force-chain network architecture in granular materials using community detection

Force chains form heterogeneous physical structures that can constrain the mechanical stability and acoustic transmission of granular media. However, despite their relevance for predicting bulk properties of materials, there is no agreement on a quantitative description of force chains. Consequently, it is difficult to compare the force-chain structures in different materials or experimental conditions. To address this challenge, we treat granular materials as spatially-embedded networks in which the nodes (particles) are connected by weighted edges that represent contact forces. We use techniques from community detection, which is a type of clustering, to find sets of closely connected particles. By using a geographical null model that is constrained by the particles' contact network, we extract chain-like structures that are reminiscent of force chains. We propose three diagnostics to measure these chain-like structures, and we demonstrate the utility of these diagnostics for identifying and characterizing classes of force-chain network architectures in various materials. To illustrate our methods, we describe how force-chain architecture depends on pressure for two very different types of packings: (1) ones derived from laboratory experiments and (2) ones derived from idealized, numerically-generated frictionless packings. By resolving individual force chains, we quantify statistical properties of force-chain shape and strength, which are potentially crucial diagnostics of bulk properties (including material stability). These methods facilitate quantitative comparisons between different particulate systems, regardless of whether they are measured experimentally or numerically.

Self-assembly in a near-frictionless granular material: conformational structures and transitions in uniaxial cyclic compression of hydrogel spheres

We use a Markov transition matrix-based analysis to explore the structures and structural transitions in a three-dimensional assembly of hydrogel spheres under cyclic uniaxial compression. We apply these methods on experimental data obtained from a packing of nearly frictionless hydrogel balls. This allows an exploration of the emergence and evolution of mesoscale internal structures - a key micromechanical property that governs self-assembly and self-organization in dense granular media. To probe the mesoscopic force network structure, we consider two structural state spaces: (i) a particle and its contacting neighbours, and (ii) a particle's local minimal cycle topology summarized by a cycle vector. In both spaces, our analysis of the transition dynamics reveals which structures and which sets of structures are most prevalent and most likely to transform into each other during the compression/decompression of the material. In compressed states, structures rich in 3-cycle or triangle topologies form in abundance. In contrast, in uncompressed states, transitions comprising poorly connected structures are dominant. An almost-invariant transition set within the cycle vector space is discovered that identifies an intermediate set of structures crucial to the material's transition from weakly jammed to strongly jammed, and vice versa. Preferred transition pathways are also highlighted and discussed with respect to thermo-micro-mechanical constitutive formulations.

Spanning the scales of granular materials through microscopic force imaging

If you walk on sand, it supports your weight. How do the disordered forces between particles in sand organize, to keep you from sinking? This simple question is surprisingly difficult to answer experimentally: measuring forces in three dimensions, between deeply buried grains, is challenging. Here we describe experiments in which we have succeeded in measuring forces inside a granular packing subject to controlled deformations. We connect the measured micro-scale forces to the macro-scale packing force response with an averaging, mean field calculation. This calculation explains how the combination of packing structure and contact deformations produce the observed nontrivial mechanical response of the packing, revealing a surprising microscopic particle deformation enhancement mechanism.

One major challenge for granular materials is how their microscopic contact forces respond to applied stress or strains at the macroscopic scale. Here, Brodu et al. map the contact forces of deformable hydrogel particles in three-dimensions and identify nonlinear stiffening of packing under compression.

Many-body contact repulsion of deformable disks

We use a spring-and-plaquette network model to analyze the repulsion between elastic disks in contact. By studying various 2D geometries, we find that as disks approach the incompressibility limit the many-body effects become dominant and the disk-disk interaction is not pairwise additive. Upon compression, the disks undergo a transition from the localized to the distributed deformation regime accompanied by a steep increase of energy consistent with the onset of a hard core. These results shed new light on the structures formed by deformable objects such as soft nanocolloids.

Equilibrating temperaturelike variables in jammed granular subsystems

Although jammed granular systems are athermal, several thermodynamiclike descriptions have been proposed which make quantitative predictions about the distribution of volume and stress within a system and provide a corresponding temperaturelike variable. We perform experiments with an apparatus designed to generate a large number of independent, jammed, two-dimensional configurations. Each configuration consists of a single layer of photoelastic disks supported by a gentle layer of air. New configurations are generated by cyclically dilating, mixing, and then recompacting the system through a series of boundary displacements. Within each configuration, a bath of particles surrounds a smaller subsystem of particles with a different interparticle friction coefficient than the bath. The use of photoelastic particles permits us to find all particle positions as well as the vector forces at each interparticle contact. By comparing the temperaturelike quantities in both systems, we find compactivity (conjugate to the volume) does not equilibrate between the systems, while the angoricity (conjugate to the stress) does. Both independent components of the angoricity are linearly dependent on the hydrostatic pressure, in agreement with predictions of the stress ensemble.

Influence of network topology on sound propagation in granular materials

Granular media, whose features range from the particle scale to the force-chain scale and the bulk scale, are usually modeled as either particulate or continuum materials. In contrast with each of these approaches, network representations are natural for the simultaneous examination of microscopic, mesoscopic, and macroscopic features. In this paper, we treat granular materials as spatially embedded networks in which the nodes (particles) are connected by weighted edges obtained from contact forces. We test a variety of network measures to determine their utility in helping to describe sound propagation in granular networks and find that network diagnostics can be used to probe particle-, curve-, domain-, and system-scale structures in granular media. In particular, diagnostics of mesoscale network structure are reproducible across experiments, are correlated with sound propagation in this medium, and can be used to identify potentially interesting size scales. We also demonstrate that the sensitivity of network diagnostics depends on the phase of sound propagation. In the injection phase, the signal propagates systemically, as indicated by correlations with the network diagnostic of global efficiency. In the scattering phase, however, the signal is better predicted by mesoscale community structure, suggesting that the acoustic signal scatters over local geographic neighborhoods. Collectively, our results demonstrate how the force network of a granular system is imprinted on transmitted waves.

Invited Article: Refractive index matched scanning of dense granular materials

We review an experimental method that allows to probe the time-dependent structure of fully three-dimensional densely packed granular materials and suspensions by means of particle recognition. The method relies on submersing a granular medium in a refractive index matched fluid. This makes the resulting suspension transparent. The granular medium is then visualized by exciting, layer by layer, the fluorescent dye in the fluid phase. We collect references and unreported experimental know-how to provide a solid background for future development of the technique, both for new and experienced users.