Isostaticity at frictional jamming

The yielding of granular matter is marginally stable and critical

Amorphous materials undergo a transition from liquid-like to solid-like states through processes like rapid quenching or densification. Under external loads, they exhibit yielding, with minimal structural changes compared to crystals. However, these universal characteristics are rarely explored comprehensively in a single granular experiment due to the added complexity of inherent friction. The discernible differences between static configurations before and after yielding are largely unaddressed, and a comprehensive examination from both statistical physics and mechanical perspectives is lacking. To address these gaps, we conducted experiments using photoelastic disks, simultaneously tracking particles and measuring forces. Our findings reveal that the yielding transition demonstrates critical behavior from a statistical physics standpoint and marginal stability from a mechanical perspective, akin to the isotropic jamming transition. This criticality differs significantly from spinodal criticality in frictionless amorphous solids, highlighting unique characteristics of granular yielding. Furthermore, our analysis confirms the marginal stability of granular yielding by assessing the contact number and evaluating the balance between weak forces and small gaps. These factors serve as structural indicators for configurations before and after yielding. Our results not only contribute to advancing our understanding of the fundamental physics of granular materials but also bear significant implications for practical applications in various fields.

Evident structural anisotropies arising from near-zero particle asphericity in granular spherocylinder packings

With magnetic resonance imaging experiments, we study packings of granular spherocylinders with merely 2% asphericity. Evident structural anisotropies across all length scales are identified. Most interestingly, the global nematic order decreases with increasing packing fraction, while the local contact anisotropy shows an opposing trend. We attribute this counterintuitive phenomenon to a competition between gravity-driven ordering aided by frictional contacts and a geometric frustration effect at the marginally jammed state. It is also surprising to notice that such slight particle asphericity can trigger non-negligible correlations between contact-level and mesoscale structures, manifested in drastically different nonaffine structural rearrangements upon compaction from that of granular spheres. These observations can help improve statistical mechanical models for the orientational order transformation of nonspherical granular particle packings, which involves complex interplays between particle shape, frictional contacts, and external force field.

From creep to flow: Granular materials under cyclic shear

When unperturbed, granular materials form stable structures that resemble the ones of other amorphous solids like metallic or colloidal glasses. Whether or not granular materials under shear have an elastic response is not known, and also the influence of particle surface roughness on the yielding transition has so far remained elusive. Here we use X-ray tomography to determine the three-dimensional microscopic dynamics of two granular systems that have different roughness and that are driven by cyclic shear. Both systems, and for all shear amplitudes Γ considered, show a cross-over from creep to diffusive dynamics, indicating that rough granular materials have no elastic response and always yield, in stark contrast to simple glasses. For the system with small roughness, we observe a clear dynamic change at Γ ≈ 0.1, accompanied by a pronounced slowing down and dynamical heterogeneity. For the large roughness system, the dynamics evolves instead continuously as a function of Γ. We rationalize this roughness dependence using the potential energy landscape of the systems: The roughness induces to this landscape a micro-corrugation with a new length scale, whose ratio over the particle size is the relevant parameter. Our results reveal the unexpected richness in relaxation mechanisms for real granular materials.

Jamming on convex deformable surfaces

Jamming is a fundamental transition that governs the behavior of particulate media, including sand, foams and dense suspensions. Upon compression, such media change from freely flowing to a disordered, marginally stable solid that exhibits non-Hookean elasticity. While the jamming process is well established for fixed geometries, the nature and dynamics of jamming for a diverse class of soft materials and deformable substrates, including emulsions and biological matter, remains unknown. Here we propose a new scenario, metric jamming, where rigidification occurs on a surface that has been deformed from its ground state. Unlike classical jamming processes that exhibit discrete mechanical transitions, surprisingly we find that metric jammed states possess mechanical properties continuously tunable between those of classically jammed and conventional elastic media. The compact and curved geometry significantly alters the vibrational spectra of the structures relative to jamming in flat Euclidean space, and metric jammed systems also possess new types of vibrational mode that couple particle and shape degrees of freedom. Our work provides a theoretical framework that unifies our understanding of solidification processes that take place on deformable media and lays the groundwork to exploit jamming for the control and stabilization of shape in self-assembly processes.

Influence of friction on the packing efficiency and short-to-intermediate range structure of hard-sphere systems

Using particle-resolved computer simulations, we investigate the effect of friction on the packing structure of hard-sphere mixtures with two kinds of particles under external compression. We first show that increasing friction between the particles results in a more disordered and less efficient packing of the local structure on the nearest neighbor scale. It is also found that standard two-point correlation functions, i.e., radial distribution function and static structure factor, show basically no detectable changes beyond short-range distances upon varying inter-particle friction. Further analysis of the structure using a four-point correlation method reveals that these systems have on the intermediate-range scale a three-dimensional structure with an icosahedral/dodecahedral symmetry that exhibits a pronounced dependence on friction: small friction gives rise to an orientational order that extends to larger distances. Our results also demonstrate that composition plays a role in that the degree of structural order and the structural correlation length are mainly affected by the friction coefficients associated with the more abundant species.

Complex Memory Formation in Frictional Granular Media

Using numerical simulations it is shown that a jammed, random pack of soft frictional grains can store an arbitrary waveform that is applied as a small time-dependent shear while the system is slowly compressed. When the system is decompressed at a later time, an approximation of the input waveform is recalled in time-reversed order as shear stresses on the system boundaries. This effect depends on friction between the grains, and is independent of some aspects of the friction model. This type of memory could potentially be observable in other types of random media that form internal contacts when compressed.

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}.

How surface roughness affects the interparticle interactions at a liquid interface

HYPOTHESIS

Colloidal particles can be trapped at a liquid interface, which reduces the energetically costly interfacial area. Once at an interface, colloids undergo various self-assemblies and structural transitions due to shape-dependent interparticle interactions. Particles with rough surfaces receive increasing attention and have been applied in material design, such as Pickering emulsions and shear-thickening materials. However, the roughness effects on the interactions at a liquid interface remain less understood.

EXPERIMENTS

Experimentally, particles with four surface roughnesses were designed and compared via isotherm measurements upon a uniaxial compression. At each stage of the compression, micrographic observations were conducted via the Blodgett method. Numerically, the compression of monolayer was simulated by using Langevin dynamics. Rough colloids were modelled as particles with capillary attraction and tangential constraints.

FINDINGS

Sufficiently rough systems exhibit a non-trivial intermediate state between a gas-like state and a close-packed jamming state. This state is understood as a gel state due to roughness-induced capillary attraction. Roughness-induced friction lowers the jamming point. Furthermore, the tangential contact force owing to surface asperities can cause a gradual off-plane collapse of the compressed monolayer.

Frictional Weakening of Vibrated Granular Flows

We computationally study the frictional properties of sheared granular media subjected to harmonic vibration applied at the boundary. Such vibrations are thought to play an important role in weakening flows, yet the independent effects of amplitude, frequency, and pressure on the process have remained unclear. Based on a dimensional analysis and DEM simulations, we show that, in addition to a previously proposed criterion for peak acceleration that leads to breaking of contacts, weakening requires the absolute amplitude squared of the displacement to be sufficiently large relative to the confining pressure. The analysis provides a basis for predicting flows subjected to arbitrary external vibration and demonstrates that a previously unrecognized second process that is dependent on dissipation contributes to shear weakening under vibrations.

Development of a biaxial apparatus for jamming profiles of photoelastic granular media

We describe a two-dimensional biaxial apparatus that is used to conduct the experimental study of the jamming of granular media. The setup is designed based on the photoelastic imaging technique, which allows us to detect force-bearing contacts among particles, calculate the pressure on each particle according to the mean squared intensity gradient method, and compute contact forces on each particle [T. S. Majmudar and R. P. Behringer, Nature 435, 1079-1082 (2005)]. Particles float in a density-matched solution to avoid basal friction during experiments. We can compress (uniaxially or biaxially) or shear the granular system by an entangled comb geometry by moving the paired boundary walls independently. A novel design for the corner of each pair of perpendicular walls is described, which allows for independent motion. We control the system using a Raspberry Pi with Python code. Three typical experiments are described briefly. Furthermore, more complicated experiment protocols can be implemented to achieve specific granular materials research goals.

Hopper flows of deformable particles

Numerous experimental and computational studies show that continuous hopper flows of granular materials obey the Beverloo equation that relates the volume flow rate Q and the orifice width w: Q ∼ (w/σ_avg - k)^β, where σ_avg is the average particle diameter, kσ_avg is an offset where Q ∼ 0, the power-law scaling exponent β = d - 1/2, and d is the spatial dimension. Recent studies of hopper flows of deformable particles in different background fluids suggest that the particle stiffness and dissipation mechanism can also strongly affect the power-law scaling exponent β. We carry out computational studies of hopper flows of deformable particles with both kinetic friction and background fluid dissipation in two and three dimensions. We show that the exponent β varies continuously with the ratio of the viscous drag to the kinetic friction coefficient, λ = ζ/μ. β = d - 1/2 in the λ → 0 limit and d - 3/2 in the λ → ∞ limit, with a midpoint λ_c that depends on the hopper opening angle θ_w. We also characterize the spatial structure of the flows and associate changes in spatial structure of the hopper flows to changes in the exponent β. The offset k increases with particle stiffness until k ∼ k_max in the hard-particle limit, where k_max ∼ 3.5 is larger for λ → ∞ compared to that for λ → 0. Finally, we show that the simulations of hopper flows of deformable particles in the λ → ∞ limit recapitulate the experimental results for quasi-2D hopper flows of oil droplets in water.

Gardner-like crossover from variable to persistent force contacts in granular crystals

We report experimental evidence of a Gardner-like crossover from variable to persistent force contacts in a two-dimensional bidisperse granular crystal by analyzing the variability of both particle positions and force networks formed under uniaxial compression. Starting from densities just above the freezing transition and for variable amounts of additional compression, we compare configurations to both their own initial state and to an ensemble of equivalent reinitialized states. This protocol shows that force contacts are largely undetermined when the density is below a Gardner-like crossover, after which they gradually transition to being persistent, being fully so only above the jamming point. We associate the disorder that underlies this effect with the size of the microscopic asperities of the photoelastic disks used, by analogy to other mechanisms that have been previously predicted theoretically.

Jammed solids with pins: Thresholds, force networks, and elasticity

The role of fixed degrees of freedom in soft or granular matter systems has broad applicability and theoretical interest. Here we address questions of the geometrical role that a scaffolding of fixed particles plays in tuning the threshold volume fraction and force network in the vicinity of jamming. Our two-dimensional simulated system consists of soft particles and fixed "pins," both of which harmonically repel overlaps. On the one hand, we find that many of the critical scalings associated with jamming in the absence of pins continue to hold in the presence of even dense pin latices. On the other hand, the presence of pins lowers the jamming threshold in a universal way at low pin densities and a geometry-dependent manner at high pin densities, producing packings with lower densities and fewer contacts between particles. The onset of strong lattice dependence coincides with the development of bond-orientational order. Furthermore, the presence of pins dramatically modifies the network of forces, with both unusually weak and unusually strong forces becoming more abundant. The spatial organization of this force network depends on pin geometry and is described in detail. Using persistent homology, we demonstrate that pins modify the topology of the network. Finally, we observe clear signatures of this developing bond-orientational order and broad force distribution in the elastic moduli which characterize the linear response of these packings to strain.

The effect of grain shape and material on the nonlocal rheology of dense granular flows

Nonlocal rheologies allow for the modeling of granular flows from the creeping to intermediate flow regimes, using a small number of parameters. In this paper, we report on experiments testing how particle properties affect the model parameters used in the Kamrin & Koval cooperative nonlocal model, using particles of three different shapes (circles, ellipses, and pentagons) and three different materials, including one which allows for the measurement of stresses via photoelasticity. Our experiments are performed on a quasi-2D annular shear cell with a rotating inner wall and a fixed outer wall. Each type of particle is found to exhibit flows which are well-fit by nonlocal rheology, with each particle having a distinct triad of the local, nonlocal, and frictional parameters. While the local parameter b is always approximately unity, the nonlocal parameter A depends sensitively on both the particle shape and material. The critical stress ratio μ_s, above which Coulomb failure occurs, varies for particles with the same material but different shape, indicating that geometric friction can dominate over material friction.

Aspects of bulk properties of amorphous jammed disks under isotopic compression

By investigating the bidisperse disks under isotropic compression, we show the importance of non-affine deformation on the bulk properties of jammed disordered matter and how the mechanical properties are affected by the variation of microscopic quantities with the excess volume density [Formula: see text] and the friction coefficient [Formula: see text]. In theory, we derive a simple formula for the pressure of disk packings which sets up a bridge between the pressure and other statistical quantities like the contact number density and the average normal force. This pressure formula is used to derive the reduced pressure [Formula: see text] and the reduced bulk modulus [Formula: see text] for disk packings with linear interactions and under affine compression without new contacts. Combining theoretical formulae with Discrete Element Method (DEM) simulations, we investigate the average contact number [Formula: see text] and the average reduced overlap [Formula: see text] and give the analysis on how [Formula: see text] and [Formula: see text] are affected by the variation of Z and [Formula: see text]. For frictionless disk packings, we find that the affine assumption causes large deviation on Z and [Formula: see text] relative to those of non-affine compression and therefore fails to predict the quantitative results of [Formula: see text]. For packings with a fixed [Formula: see text], due to the non-affine deformation, [Formula: see text] varies approximately linear with the increasing [Formula: see text] and Z increases sharply near the jamming point and then approaches a saturation value. With a fixed [Formula: see text] and the increasing [Formula: see text], [Formula: see text] changes by a small amount while Z presents obvious decrease. The decrease of Z causes the decrease of the slope of function [Formula: see text] and the value of [Formula: see text] at a fixed [Formula: see text].

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.

The structural, vibrational, and mechanical properties of jammed packings of deformable particles in three dimensions

We investigate the structural, vibrational, and mechanical properties of jammed packings of deformable particles with shape degrees of freedom in three dimensions (3D). Each 3D deformable particle is modeled as a surface-triangulated polyhedron, with spherical vertices whose positions are determined by a shape-energy function with terms that constrain the particle surface area, volume, and curvature, and prevent interparticle overlap. We show that jammed packings of deformable particles without bending energy possess low-frequency, quartic vibrational modes, whose number decreases with increasing asphericity and matches the number of missing contacts relative to the isostatic value. In contrast, jammed packings of deformable particles with non-zero bending energy are isostatic in 3D, with no quartic modes. We find that the contributions to the eigenmodes of the dynamical matrix from the shape degrees of freedom are significant over the full range of frequency and shape parameters for particles with zero bending energy. We further show that the ensemble-averaged shear modulus 〈G〉 scales with pressure P as 〈G〉 ∼ Pβ, with β ≈ 0.75 for jammed packings of deformable particles with zero bending energy. In contrast, β ≈ 0.5 for packings of deformable particles with non-zero bending energy, which matches the value for jammed packings of soft, spherical particles with fixed shape. These studies underscore the importance of incorporating particle deformability and shape change when modeling the properties of jammed soft materials.

Disorder perturbation expansion for athermal crystals

We introduce a perturbation expansion for athermal systems that allows an exact determination of displacement fields away from the crystalline state as a response to disorder. We show that the displacement fields in energy-minimized configurations of particles interacting through central potentials with microscopic disorder can be obtained as a series expansion in the strength of the disorder. We introduce a hierarchy of force-balance equations that allows an order-by-order determination of the displacement fields, with the solutions at lower orders providing sources for the higher-order solutions. This allows the simultaneous force-balance equations to be solved, within a hierarchical perturbation expansion to arbitrary accuracy. We present exact results for an isotropic defect introduced into the crystalline ground state at linear order and second order in our expansion. We show that the displacement fields produced by the defect display interesting self-similar properties at every order. We derive a |δr|∼1/r and |δf|∼1/r^{2} decay for the displacement fields and excess interparticle forces at large distances r away from the defect. Finally, we derive nonlinear corrections introduced by the interactions between defects at second order in our expansion. We verify our exact results with displacement fields obtained from energy-minimized configurations of soft disks.

Mechanical response of packings of nonspherical particles: A case study of two-dimensional packings of circulo-lines

We investigate the mechanical response of jammed packings of circulo-lines in two spatial dimensions, interacting via purely repulsive, linear spring forces, as a function of pressure P during athermal, quasistatic isotropic compression. The surface of a circulo-line is defined as the collection of points that is equidistant to a line; circulo-lines are composed of a rectangular central shaft with two semicircular end caps. Prior work has shown that the ensemble-averaged shear modulus for jammed disk packings scales as a power law, 〈G(P)〉∼P^{β}, with β∼0.5, over a wide range of pressure. For packings of circulo-lines, we also find robust power-law scaling of 〈G(P)〉 over the same range of pressure for aspect ratios R≳1.2. However, the power-law scaling exponent β∼0.8-0.9 is much larger than that for jammed disk packings. To understand the origin of this behavior, we decompose 〈G〉 into separate contributions from geometrical families, G_{f}, and from changes in the interparticle contact network, G_{r}, such that 〈G〉=〈G_{f}〉+〈G_{r}〉. We show that the shear modulus for low-pressure geometrical families for jammed packings of circulo-lines can both increase and decrease with pressure, whereas the shear modulus for low-pressure geometrical families for jammed disk packings only decreases with pressure. For this reason, the geometrical family contribution 〈G_{f}〉 is much larger for jammed packings of circulo-lines than for jammed disk packings at finite pressure, causing the increase in the power-law scaling exponent for 〈G(P)〉.

Experimental Test of the Edwards Volume Ensemble for Tapped Granular Packings

Using x-ray tomography, we experimentally investigate granular packings subject to mechanical tapping for three types of beads with different friction coefficients. We validate the Edwards volume ensemble in these three-dimensional granular systems and establish a granular version of thermodynamic zeroth law. Within the Edwards framework, we also explicitly clarify how friction influences granular statistical mechanics by modifying the density of states, which allows us to determine the entropy as a function of packing fraction and friction. Additionally, we obtain a granular jamming phase diagram based on geometric coordination number and packing fraction.

Spongelike Rigid Structures in Frictional Granular Packings

We show how rigidity emerges in experiments on sheared two-dimensional frictional granular materials by using generalizations of two methods for identifying rigid structures. Both approaches, the force-based dynamical matrix and the topology-based rigidity percolation, agree with each other and identify similar rigid structures. As the system becomes jammed, at a critical contact number z_{c}=2.4±0.1, a rigid backbone interspersed with floppy, particle-filled holes of a broad range of sizes emerges, creating a spongelike morphology. While the pressure within rigid structures always exceeds the pressure outside the rigid structures, they are not identified with the force chains of shear jamming. These findings highlight the need to focus on mechanical stability arising through arch structures and hinges at the mesoscale.

Friction-Controlled Entropy-Stability Competition in Granular Systems

Using cyclic shear to drive a two-dimensional granular system, we determine the structural characteristics for different interparticle friction coefficients. These characteristics are the result of a competition between mechanical stability and entropy, with the latter's effect increasing with friction. We show that a parameter-free maximum-entropy argument alone predicts an exponential cell order distribution, with excellent agreement with the experimental observation. We show that friction only tunes the mean cell order and, consequently, the exponential decay rate and the packing fraction. We further show that cells, which can be very large in such systems, are short-lived, implying that our systems are liquidlike rather than glassy.

Critical point of jamming transition in two-dimensional monodisperse systems

The existence of amorphous packings in two-dimensional monodisperse system is a classical unsolved problem. We get the energy minimum state by the energy minimization method of enthalpy under constant pressure conditions. Firstly, we find that there are two peaks in the experiment, which demonstrate the interesting features of the coexistence of crystals and amorphous crystals. And then, we confirm the critical point of jamming transition of the two-dimensional monodisperse is [Formula: see text]. Finally, we prove that the jamming scaling is still satisfied in two-dimensional monodispersed system: [Formula: see text] and vanishes as [Formula: see text], and the boson peak shifts to lower frequencies for less compressed systems.

Jamming with Tunable Roughness

We introduce a new model to study the effect of surface roughness on the jamming transition. By performing numerical simulations, we show that for a smooth surface, the jamming transition density and the contact number at the transition point both increase upon increasing asphericity, as for ellipsoids and spherocylinders. Conversely, for a rough surface, both quantities decrease, in quantitative agreement with the behavior of frictional particles. Furthermore, in the limit corresponding to the Coulomb friction law, the model satisfies a generalized isostaticity criterion proposed in previous studies. We introduce a counting argument that justifies this criterion and interprets it geometrically. Finally, we propose a simple theory to predict the contact number at finite friction from the knowledge of the force distribution in the infinite friction limit.

Oscillatory Instabilities in Frictional Granular Matter

Frictional granular matter is shown to be fundamentally different in its plastic responses to external strains from generic glasses and amorphous solids without friction. While regular glasses exhibit plastic instabilities due to the vanishing of a real eigenvalue of the Hessian matrix, frictional granular materials can exhibit a previously unnoticed additional mechanism for instabilities, i.e., the appearance of a pair of complex eigenvalues leading to oscillatory exponential growth of perturbations that are tamed by dynamical nonlinearities. This fundamental difference appears crucial for the understanding of plasticity and failure in frictional granular materials. The possible relevance to earthquake physics is discussed.

Spatial strain correlations, machine learning, and deformation history in crystal plasticity

Systems far from equilibrium respond to probes in a history-dependent manner. The prediction of the system response depends on either knowing the details of that history or being able to characterize all the current system properties. In crystal plasticity, various processing routes contribute to a history dependence that may manifest itself through complex microstructural deformation features with large strain gradients. However, the complete spatial strain correlations may provide further predictive information. In this paper, we demonstrate an explicit example where spatial strain correlations can be used in a statistical manner to infer and classify prior deformation history at various strain levels. The statistical inference is provided by machine-learning techniques. As source data, we consider uniaxially compressed crystalline thin films generated by two dimensional discrete dislocation plasticity simulations, after prior compression at various levels. Crystalline thin films at the nanoscale demonstrate yield-strength size effects with very noisy mechanical responses that produce a serious challenge to learning techniques. We discuss the influence of size effects and structural uncertainty to the ability of our approach to distinguish different plasticity regimes.

Jamming of Deformable Polygons

We introduce the deformable particle (DP) model for cells, foams, emulsions, and other soft particulate materials, which adds to the benefits and eliminates deficiencies of existing models. The DP model combines the ability to model individual soft particles with the shape-energy function of the vertex model, and adds arbitrary particle deformations. We focus on 2D deformable polygons with a shape-energy function that is minimized for area a_{0} and perimeter p_{0} and repulsive interparticle forces. We study the onset of jamming versus particle asphericity, A=p_{0}^{2}/4πa_{0}, and find that the packing fraction grows with A until reaching A^{*}=1.16 of the underlying Voronoi cells at confluence. We find that DP packings above and below A^{*} are solidlike, which helps explain the solid-to-fluid transition at A^{*} in the vertex model as a transition from tension- to compression-dominated regimes.

Skinny emulsions take on granular matter

Our understanding of the structural features of foams and emulsions has advanced significantly over the last 20 years. However, with a search for "super-stable" liquid dispersions, foam and emulsion science employs increasingly complex formulations which create solid-like visco-elastic layers at the bubble/drop surfaces. These lead to elastic, adhesive and frictional forces between bubbles/drops, impacting strongly how they pack and deform against each other, asking for an adaptation of the currently available structural description. The possibility to modify systematically the interfacial properties makes these dispersions ideal systems for the exploration of soft granular materials with complex interactions. We present here a first systematic analysis of the structural features of such a system using a model silicone emulsion containing millimetre-sized polyethylene glycol drops (PEG). Solid-like drop surfaces are obtained by polymeric cross-linking reactions at the PEG-silicone interface. Using a novel droplet-micromanipulator, we highlight the presence of elastic, adhesive and frictional interactions between two drops. We then provide for the first time a full tomographic analysis of the structural features of these emulsions. An in-depth analysis of the angle of repose, local volume fraction distributions, pair correlation functions and the drop deformations for different skin formulations allow us to put in evidence the striking difference with "ordinary" emulsions having fluid-like drop surfaces. While strong analogies with frictional hard-sphere systems can be drawn, these systems display a set of unique features due to the high deformability of the drops which await systematic exploration.

Symmetry-reversals in chiral active matter

We perform experiments on an active granular material composed of individually-driven, spinning disks confined within a circular arena. Small bumps at the outer edges of the disks provide a variable amount of interparticle coupling in the form of geometric friction. The disks each spin counter-clockwise, but undergo a transition in their collective circulation around the center of the arena, from a clockwise orbit to a counter-clockwise orbit, as a function of packing fraction φ. We identify that, unlike for vibrated granular gases, the particles' velocity distributions are Gaussian over a large range of φ. By fitting the speed distribution to a Maxwell-Boltzmann distribution, we identify a temperature-like parameter which is a universal function of φ; this parameter is also equal to the mean translational energy of the particles. We quantify the collective circulation via its solid-body-like rotation rate, and find that this is a universal function centered around a critical packing fraction. In addition, the ratio of orbital kinetic energy to spin kinetic energy is also a universal function for non-zero geometric friction. These findings highlight the important role of both the type of driving and the interparticle interactions (here, geometric friction) in controlling the collective behavior of active granular systems.

Hypostatic jammed packings of frictionless nonspherical particles

We perform computational studies of static packings of a variety of nonspherical particles including circulo-lines, circulo-polygons, ellipses, asymmetric dimers, dumbbells, and others to determine which shapes form packings with fewer contacts than degrees of freedom (hypostatic packings) and which have equal numbers of contacts and degrees of freedom (isostatic packings), and to understand why hypostatic packings of nonspherical particles can be mechanically stable despite having fewer contacts than that predicted from naive constraint counting. To generate highly accurate force- and torque-balanced packings of circulo-lines and cir-polygons, we developed an interparticle potential that gives continuous forces and torques as a function of the particle coordinates. We show that the packing fraction and coordination number at jamming onset obey a masterlike form for all of the nonspherical particle packings we studied when plotted versus the particle asphericity A, which is proportional to the ratio of the squared perimeter to the area of the particle. Further, the eigenvalue spectra of the dynamical matrix for packings of different particle shapes collapse when plotted at the same A. For hypostatic packings of nonspherical particles, we verify that the number of "quartic" modes along which the potential energy increases as the fourth power of the perturbation amplitude matches the number of missing contacts relative to the isostatic value. We show that the fourth derivatives of the total potential energy in the directions of the quartic modes remain nonzero as the pressure of the packings is decreased to zero. In addition, we calculate the principal curvatures of the inequality constraints for each contact in circulo-line packings and identify specific types of contacts with inequality constraints that possess convex curvature. These contacts can constrain multiple degrees of freedom and allow hypostatic packings of nonspherical particles to be mechanically stable.

Packing in protein cores

Proteins are biological polymers that underlie all cellular functions. The first high-resolution protein structures were determined by x-ray crystallography in the 1960s. Since then, there has been continued interest in understanding and predicting protein structure and stability. It is well-established that a large contribution to protein stability originates from the sequestration from solvent of hydrophobic residues in the protein core. How are such hydrophobic residues arranged in the core; how can one best model the packing of these residues, and are residues loosely packed with multiple allowed side chain conformations or densely packed with a single allowed side chain conformation? Here we show that to properly model the packing of residues in protein cores it is essential that amino acids are represented by appropriately calibrated atom sizes, and that hydrogen atoms are explicitly included. We show that protein cores possess a packing fraction of [Formula: see text], which is significantly less than the typically quoted value of 0.74 obtained using the extended atom representation. We also compare the results for the packing of amino acids in protein cores to results obtained for jammed packings from discrete element simulations of spheres, elongated particles, and composite particles with bumpy surfaces. We show that amino acids in protein cores pack as densely as disordered jammed packings of particles with similar values for the aspect ratio and bumpiness as found for amino acids. Knowing the structural properties of protein cores is of both fundamental and practical importance. Practically, it enables the assessment of changes in the structure and stability of proteins arising from amino acid mutations (such as those identified as a result of the massive human genome sequencing efforts) and the design of new folded, stable proteins and protein-protein interactions with tunable specificity and affinity.

Pore configuration landscape of granular crystallization

Uncovering grain-scale mechanisms that underlie the disorder–order transition in assemblies of dissipative, athermal particles is a fundamental problem with technological relevance. To date, the study of granular crystallization has mainly focussed on the symmetry of crystalline patterns while their emergence and growth from irregular clusters of grains remains largely unexplored. Here crystallization of three-dimensional packings of frictional spheres is studied at the grain-scale using X-ray tomography and persistent homology. The latter produces a map of the topological configurations of grains within static partially crystallized packings. Using numerical simulations, we show that similar maps are measured dynamically during the melting of a perfect crystal. This map encodes new information on the formation process of tetrahedral and octahedral pores, the building blocks of perfect crystals. Four key formation mechanisms of these pores reproduce the main changes of the map during crystallization and provide continuous deformation pathways representative of the crystallization dynamics.

Emergence and growth of crystalline domains in granular media remains under-explored. Here, the authors analyse tomographic snapshots from partially recrystallized packings of spheres using persistent homology and find agreement with proposed transitions based on continuous deformation of octahedral and tetrahedral voids.

Equation of state for random sphere packings with arbitrary adhesion and friction

We systematically generate a large set of random micro-particle packings over a wide range of adhesion and friction by means of adhesive contact dynamics simulation. The ensemble of generated packings covers a range of volume fractions ϕ from 0.135 ± 0.007 to 0.639 ± 0.004, and of coordination numbers Z from 2.11 ± 0.03 to 6.40 ± 0.06. We determine ϕ and Z at four limits (random close packing, random loose packing, adhesive close packing, and adhesive loose packing), and find a universal equation of state ϕ(Z) to describe packings with arbitrary adhesion and friction. From a mechanical equilibrium analysis, we determine the critical friction coefficient μ_f,c: when the friction coefficient μ_f is below μ_f,c, particles' rearrangements are dominated by sliding, otherwise they are dominated by rolling. Because of this reason, both ϕ(μ_f) and Z(μ_f) change sharply across μ_f,c. Finally, we generalize the Maxwell counting argument to micro-particle packings, and show that the loosest packing, i.e., adhesive loose packing, satisfies the isostatic condition at Z = 2.

Shearing a glass and the role of pinning delay in models of interface depinning

When a disordered solid is sheared, yielding is followed by the onset of intermittent response that is characterized by slip in local regions usually labeled shear-transformation zones. Such intermittent response resembles the behavior of earthquakes or contact depinning, where a well-defined landscape of pinning disorder prohibits the deformation of an elastic medium. Nevertheless, a disordered solid is evidently different in that pinning barriers of particles are due to neighbors that are also subject to motion. Microscopic yielding leads to destruction of the local microstructure and local heating. It is natural to assume that locally a liquid emerges for a finite timescale before cooling down to a transformed configuration. For including this characteristic transient in glass depinning models, we propose a general mechanism that involves a "pinning delay" time T(pd), during which each region that slipped evolves as a fluid. The new timescale can be as small as a single avalanche time step. This is a local, effective, and dynamical in nature mechanism that may be thought as dynamical softening. We demonstrate that the inclusion of this mechanism causes a drift of the critical exponents toward higher values for the slip sizes τ, until a transition to permanent shear-banding behavior happens causing almost oscillatory, stick-slip response. Moreover, it leads to a proliferation of large events that are highly inhomogeneous and resemble sharp slip band formation.

Structural and mechanical features of the order-disorder transition in experimental hard-sphere packings

Here we present an experimental and numerical investigation on the grain-scale geometrical and mechanical properties of partially crystallized structures made of macroscopic frictional grains. Crystallization is inevitable in arrangements of monosized hard spheres with packing densities exceeding Bernal's limiting density ϕ(Bernal)≈0.64. We study packings of monosized hard spheres whose density spans over a wide range (0.59<ϕ<0.72). These experiments harness x-ray computed tomography, three-dimensional image analysis, and numerical simulations to access precisely the geometry and the 3D structure of internal forces within the sphere packings. We show that clear geometrical transitions coincide with modifications of the mechanical backbone of the packing both at the grain and global scale. Notably, two transitions are identified at ϕ(Bernal)≈0.64 and ϕ(c)≈0.68. These results provide insights on how geometrical and mechanical features at the grain scale conspire to yield partially crystallized structures that are mechanically stable.

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.

Cavity method for force transmission in jammed disordered packings of hard particles

The force distribution of jammed disordered packings has always been considered a central object in the physics of granular materials. However, many of its features are poorly understood. In particular, analytic relations to other key macroscopic properties of jammed matter, such as the contact network and its coordination number, are still lacking. Here we develop a mean-field theory for this problem, based on the consideration of the contact network as a random graph where the force transmission becomes a constraint satisfaction problem. We can thus use the cavity method developed in the past few decades within the statistical physics of spin glasses and hard computer science problems. This method allows us to compute the force distribution P(f) for random packings of hard particles of any shape, with or without friction. We find a new signature of jamming in the small force behavior P(f) ∼ f(θ), whose exponent has attracted recent active interest: we find a finite value for P(f = 0), along with θ = 0. Furthermore, we relate the force distribution to a lower bound of the average coordination number z[combining macron](μ) of jammed packings of frictional spheres with coefficient μ. This bridges the gap between the two known isostatic limits z[combining macron]c (μ = 0) = 2D (in dimension D) and z[combining macron]c(μ → ∞) = D + 1 by extending the naive Maxwell's counting argument to frictional spheres. The theoretical framework describes different types of systems, such as non-spherical objects in arbitrary dimensions, providing a common mean-field scenario to investigate force transmission, contact networks and coordination numbers of jammed disordered packings.

Mechanical characterization of partially crystallized sphere packings

We study grain-scale mechanical and geometrical features of partially crystallized packings of frictional spheres, produced experimentally by a vibrational protocol. By combining x-ray computed tomography, 3D image analysis, and discrete element method simulations, we have access to the 3D structure of internal forces. We investigate how the network of mechanical contacts and intergranular forces change when the packing structure evolves from amorphous to near perfect crystalline arrangements. We compare the behavior of the geometrical neighbors (quasicontracts) of a grain to the evolution of the mechanical contacts. The mechanical coordination number Z(m) is a key parameter characterizing the crystallization onset. The high fluctuation level of Z(m) and of the force distribution in highly crystallized packings reveals that a geometrically ordered structure still possesses a highly random mechanical backbone similar to that of amorphous packings.

Statistics of frictional families

We develop a theoretical description for mechanically stable frictional packings in terms of the difference between the total number of contacts required for isostatic packings of frictionless disks and the number of contacts in frictional packings, m=Nc0 - Nc. The saddle order m represents the number of unconstrained degrees of freedom that a static packing would possess if friction were removed. Using a novel numerical method that allows us to enumerate disk packings for each m, we show that the probability to obtain a packing with saddle order m at a given static friction coefficient μ, Pm(μ), can be expressed as a power series in μ. Using this form for Pm(μ), we quantitatively describe the dependence of the average contact number on the friction coefficient for static disk packings obtained from direct simulations of the Cundall-Strack model for all μ and N.

Jamming in finite systems: stability, anisotropy, fluctuations, and scaling

Athermal packings of soft repulsive spheres exhibit a sharp jamming transition in the thermodynamic limit. Upon further compression, various structural and mechanical properties display clean power-law behavior over many decades in pressure. As with any phase transition, the rounding of such behavior in finite systems close to the transition plays an important role in understanding the nature of the transition itself. The situation for jamming is surprisingly rich: the assumption that jammed packings are isotropic is only strictly true in the large-size limit, and finite-size has a profound effect on the very meaning of jamming. Here, we provide a comprehensive numerical study of finite-size effects in sphere packings above the jamming transition, focusing on stability as well as the scaling of the contact number and the elastic response.

Phonon dispersion and elastic moduli of two-dimensional disordered colloidal packings of soft particles with frictional interactions

Particle tracking and displacement covariance matrix techniques are employed to investigate the phonon dispersion relations of two-dimensional colloidal glasses composed of soft, thermoresponsive microgel particles whose temperature-sensitive size permits in situ variation of particle packing fraction. Bulk, B, and shear, G, moduli of the colloidal glasses are extracted from the dispersion relations as a function of packing fraction, and variation of the ratio G/B with packing fraction is found to agree quantitatively with predictions for jammed packings of frictional soft particles. In addition, G and B individually agree with numerical predictions for frictional particles. This remarkable level of agreement enabled us to extract an energy scale for the interparticle interaction from the individual elastic constants and to derive an approximate estimate for the interparticle friction coefficient.

Particle-scale reversibility in athermal particulate media below jamming

We perform numerical simulations of repulsive, frictionless athermal disks in two and three spatial dimensions undergoing cyclic quasistatic simple shear to investigate particle-scale reversible motion. We identify three classes of steady-state dynamics as a function of packing fraction φ and maximum strain amplitude per cycle γ(max). Point-reversible states, where particles do not collide and exactly retrace their intracycle trajectories, occur at low φ and γ(max). Particles in loop-reversible states undergo numerous collisions and execute complex trajectories but return to their initial positions at the end of each cycle. For sufficiently large φ and γ(max), systems display irreversible dynamics with nonzero self-diffusion. Loop-reversible dynamics enables the reliable preparation of configurations with specified structural and mechanical properties over a broad range of φ.