Modeling the Transition between Localized and Extended Deposition in Flow Networks through Packings of Glass Beads.

We use a theoretical model to explore how fluid dynamics, in particular, the pressure gradient and wall shear stress in a channel, affect the deposition of particles flowing in a microfluidic network. Experiments on transport of colloidal particles in pressure-driven systems of packed beads have shown that at lower pressure drop, particles deposit locally at the inlet, while at higher pressure drop, they deposit uniformly along the direction of flow. We develop a mathematical model and use agent-based simulations to capture these essential qualitative features observed in experiments. We explore the deposition profile over a two-dimensional phase diagram defined in terms of the pressure and shear stress threshold, and show that two distinct phases exist. We explain this apparent phase transition by drawing an analogy to simple one-dimensional mass-aggregation models in which the phase transition is calculated analytically.

Stress-stress correlations reveal force chains in gels.

We investigate the spatial correlations of microscopic stresses in soft particulate gels using 2D and 3D numerical simulations. We use a recently developed theoretical framework predicting the analytical form of stress-stress correlations in amorphous assemblies of athermal grains that acquire rigidity under an external load. These correlations exhibit a pinch-point singularity in Fourier space. This leads to long-range correlations and strong anisotropy in real space, which are at the origin of force-chains in granular solids. Our analysis of the model particulate gels at low particle volume fractions demonstrates that stress-stress correlations in these soft materials have characteristics very similar to those in granular solids and can be used to identify force chains. We show that the stress-stress correlations can distinguish floppy from rigid gel networks and that the intensity patterns reflect changes in shear moduli and network topology, due to the emergence of rigid structures during solidification.

Tensor electromagnetism and emergent elasticity in jammed solids.

The theory of mechanical response and stress transmission in disordered, jammed solids poses several open questions of how nonperiodic networks-apparently indistinguishable from a snapshot of a fluid-sustain shear. We present a stress-only theory of emergent elasticity for a nonthermal amorphous assembly of grains in a jammed solid, where each grain is subjected to mechanical constraints of force and torque balance. These grain-level constraints lead to the Gauss's law of an emergent U(1) tensor electromagnetism, which then accounts for the mechanical response of such solids. This formulation of amorphous elasticity has several immediate consequences. The mechanical response maps exactly to the static, dielectric response of this tensorial electromagnetism with the polarizability of the medium mapping to emergent elastic moduli. External forces act as vector electric charges, whereas the tensorial magnetic fields are sourced by momentum density. The dynamics in the electric and magnetic sectors naturally translate into the dynamics of the rigid jammed network and ballistic particle motion, respectively. The theoretical predictions for both stress-stress correlations and responses are borne out by the results of numerical simulations of frictionless granular packings in the static limit of the theory in both 2D and 3D.

Dilatancy, shear jamming, and a generalized jamming phase diagram of frictionless sphere packings.

Granular packings display the remarkable phenomenon of dilatancy, wherein their volume increases upon shear deformation. Conventional wisdom and previous results suggest that dilatancy, also being the related phenomenon of shear-induced jamming, requires frictional interactions. Here, we show that the occurrence of isotropic jamming densities φj above the minimal density (or the J-point density) φJ leads both to the emergence of shear-induced jamming and dilatancy in frictionless packings. Under constant pressure shear, the system evolves into a steady-state at sufficiently large strains, whose density only depends on the pressure and is insensitive to the initial jamming density φj. In the limit of vanishing pressure, the steady-state exhibits critical behavior at φJ. While packings with different φj values display equivalent scaling properties under compression, they exhibit striking differences in rheological behaviour under shear. The yield stress under constant volume shear increases discontinuously with density when φj > φJ, contrary to the continuous behaviour in generic packings that jam at φJ. Our results thus lead to a more coherent, generalised picture of jamming in frictionless packings, which also have important implications on how dilatancy is understood in the context of frictional granular matter.

Emergent Elasticity in Amorphous Solids.

The mechanical response of naturally abundant amorphous solids such as gels, jammed grains, and biological tissues are not described by the conventional paradigm of broken symmetry that defines crystalline elasticity. In contrast, the response of such athermal solids are governed by local conditions of mechanical equilibrium, i.e., force and torque balance of its constituents. Here we show that these constraints have the mathematical structure of a generalized electromagnetism, where the electrostatic limit successfully captures the anisotropic elasticity of amorphous solids. The emergence of elasticity from local mechanical constraints offers a new paradigm for systems with no broken symmetry, analogous to emergent gauge theories of quantum spin liquids. Specifically, our U(1) rank-2 symmetric tensor gauge theory of elasticity translates to the electromagnetism of fractonic phases of matter with the stress mapped to electric displacement and forces to vector charges. We corroborate our theoretical results with numerical simulations of soft frictionless disks in both two and three dimensions, and experiments on frictional disks in two dimensions. We also present experimental evidence indicating that force chains in granular media are subdimensional excitations of amorphous elasticity similar to fractons.

Athermal Fluctuations in Disordered Crystals.

We analyze the fluctuations in particle positions and interparticle forces in disordered crystals composed of jammed soft particles in the limit of weak disorder. We demonstrate that such athermal systems are fundamentally different from their thermal counterparts, characterized by constrained fluctuations of forces perpendicular to the lattice directions. We develop a disorder perturbation expansion in polydispersity about the crystalline state, which we use to derive exact results to linear order. We show that constrained fluctuations result as a consequence of local force balance conditions, and are characterized by non-Gaussian distributions, which we derive exactly. We analytically predict several properties of such systems, including the scaling of the average coordination with polydispersity and packing fraction, which we verify with numerical simulations using soft disks with one-sided harmonic interactions.

Motion of active tracer in a lattice gas with cross-shaped particles.

We analyze the dynamics of an active tracer particle embedded in a thermal lattice gas. All particles are subject to exclusion up to third nearest neighbors on the square lattice, which leads to slow dynamics at high densities. For the case with no rotational diffusion of the tracer, we derive an analytical expression for the resulting drift velocity v of the tracer in terms of non-equilibrium density correlations involving the tracer particle and its neighbors, which we verify using numerical simulations. We show that the properties of the passive system alone do not adequately describe even this simple system of a single non-rotating active tracer. For large activity and low density, we develop an approximation for v. For the case where the tracer undergoes rotational diffusion independent of its neighbors, we relate its diffusion coefficient to the thermal diffusion coefficient and v. Finally, we study dynamics where the rotation of the tracer is limited by the presence of neighboring particles. We find that the effect of this rotational locking may be quantitatively described in terms of a reduction in the rotation rate.

Stress fluctuations in transient active networks.

Inspired by experiments on dynamic extensile gels of biofilaments and motors, we propose a model of a network of linear springs with kinetics consisting of growth at a prescribed rate, death after a lifetime drawn from a distribution, and birth at a randomly chosen node. The model captures features such as the build-up of self-stress, that are not easily incorporated into hydrodynamic theories. We study the model numerically and show that our observations can largely be understood through a stochastic effective-medium model. The resulting dynamically extending force-dipole network displays many features of yielded plastic solids, and offers a way to incorporate strongly non-affine effects into theories of active solids. A rather distinctive form for the stress distribution, and a Herschel-Bulkley dependence of stress on activity, are our major predictions.

The physics of jamming for granular materials: a review.

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

Microscopic Origin of Frictional Rheology in Dense Suspensions: Correlations in Force Space.

We develop a statistical framework for the rheology of dense, non-Brownian suspensions, based on correlations in a space representing forces, which is dual to position space. Working with the ensemble of steady state configurations obtained from simulations of suspensions in two dimensions, we find that the anisotropy of the pair correlation function in force space changes with confining shear stress (σ_{xy}) and packing fraction (ϕ). Using these microscopic correlations, we build a statistical theory for the macroscopic friction coefficient: the anisotropy of the stress tensor, µ=σ_{xy}/P. We find that µ decreases (i) as ϕ is increased and (ii) as σ_{xy} is increased. Using a new constitutive relation between µ and viscosity for dense suspensions that generalizes the rate-independent one, we show that our theory predicts a discontinuous shear thickening flow diagram that is in good agreement with numerical simulations, and the qualitative features of µ that lead to the generic flow diagram of a discontinuous shear thickening fluid observed in experiments.

Ergodicity breaking dynamics of arch collapse.

Flows in hoppers and silos are susceptible to clogging due to the formation of arches at the exit. The failure of these arches is the key to reinitiation of flow, yet the physical mechanism of failure is not well understood. Experiments on vibrated hoppers exhibit a broad distribution of the duration of clogs. Using numerical simulations of a hopper in two dimensions, we show that arches become trapped in locally stable shapes that are explored dynamically under vibrations. The shape dynamics, preceding failure, break ergodicity and can be modeled as a continuous-time random walk with a broad distribution of waiting, or trapping, times. We argue that arch failure occurs as a result of this random walk crossing a stability boundary, which is a first-passage process that naturally gives rise to a broad distribution of unclogging times.

Scaling Theory for the Frictionless Unjamming Transition.

We develop a scaling theory of the unjamming transition of soft frictionless disks in two dimensions by defining local areas, which can be uniquely assigned to each contact. These serve to define local order parameters, whose distribution exhibits divergences as the unjamming transition is approached. We derive scaling forms for these divergences from a mean-field approach that treats the local areas as noninteracting entities, and demonstrate that these results agree remarkably well with numerical simulations. We find that the asymptotic behavior of the scaling functions arises from the geometrical structure of the packing while the overall scaling with the compression energy depends on the force law. We use the scaling forms of the distributions to determine the scaling of the total grain area A_{G} and the total number of contacts N_{C}.

Gaps between avalanches in one-dimensional random-field Ising models.

We analyze the statistics of gaps (ΔH) between successive avalanches in one-dimensional random-field Ising models (RFIMs) in an external field H at zero temperature. In the first part of the paper we study the nearest-neighbor ferromagnetic RFIM. We map the sequence of avalanches in this system to a nonhomogeneous Poisson process with an H-dependent rate ρ(H). We use this to analytically compute the distribution of gaps P(ΔH) between avalanches as the field is increased monotonically from -∞ to +∞. We show that P(ΔH) tends to a constant C(R) as ΔH→0^{+}, which displays a nontrivial behavior with the strength of disorder R. We verify our predictions with numerical simulations. In the second part of the paper, motivated by avalanche gap distributions in driven disordered amorphous solids, we study a long-range antiferromagnetic RFIM. This model displays a gapped behavior P(ΔH)=0 up to a system size dependent offset value ΔH_{off}, and P(ΔH)∼(ΔH-ΔH_{off})^{θ} as ΔH→H_{off}^{+}. We perform numerical simulations on this model and determine θ≈0.95(5). We also discuss mechanisms which would lead to a nonzero exponent θ for general spin models with quenched random fields.

Theory of microphase separation in bidisperse chiral membranes.

We present a Ginzburg-Landau theory of microphase separation in a bidisperse chiral membrane consisting of rods of opposite handedness. This model system undergoes a phase transition from an equilibrium state where the two components are completely phase separated to a state composed of microdomains of a finite size comparable to the twist penetration depth. Characterizing the phenomenology using linear stability analysis and numerical studies, we trace the origin of the discontinuous change in microdomain size that occurs during this phase transition to a competition between the cost of creating an interface and the gain in twist energy for small microdomains in which the twist penetrates deep into the center of the domain.

Shear-induced rigidity of frictional particles: Analysis of emergent order in stress space.

Solids are distinguished from fluids by their ability to resist shear. In equilibrium systems, the resistance to shear is associated with the emergence of broken translational symmetry as exhibited by a nonuniform density pattern that is persistent, which in turn results from minimizing the free energy. In this work, we focus on a class of systems where this paradigm is challenged. We show that shear-driven jamming in dry granular materials is a collective process controlled by the constraints of mechanical equilibrium. We argue that these constraints can lead to a persistent pattern in a dual space that encodes the statistics of contact forces and the topology of the contact network. The shear-jamming transition is marked by the appearance of this persistent pattern. We investigate the structure and behavior of patterns both in real space and the dual space as the system evolves through the rigidity transition for a range of packing fractions and in two different shear protocols. We show that, in the protocol that creates homogeneous jammed states without shear bands, measures of shear jamming do not depend on strain and packing fraction independently but obey a scaling form with a packing-fraction-dependent characteristic strain that goes to zero at the isotropic jamming point ϕ_{J}. We demonstrate that it is possible to define a protocol-independent order parameter in this dual space, which provides a quantitative measure of the rigidity of shear-jammed states.

Protocol dependence of the jamming transition.

We propose a theoretical framework for predicting the protocol dependence of the jamming transition for frictionless spherical particles that interact via repulsive contact forces. We study isostatic jammed disk packings obtained via two protocols: isotropic compression and simple shear. We show that for frictionless systems, all jammed packings can be obtained via either protocol. However, the probability to obtain a particular jammed packing depends on the packing-generation protocol. We predict the average shear strain required to jam initially unjammed isotropically compressed packings from the density of jammed packings, shape of their basins of attraction, and path traversed in configuration space. We compare our predictions to simulations of shear strain-induced jamming and find quantitative agreement. We also show that the packing fraction range, over which shear strain-induced jamming occurs, tends to zero in the large system limit for frictionless packings with overdamped dynamics.

Shear-induced rigidity in athermal materials: A unified statistical framework.

Recent studies of athermal systems such as dry grains and dense, non-Brownian suspensions have shown that shear can lead to solidification through the process of shear jamming in grains and discontinuous shear thickening in suspensions. The similarities observed between these two distinct phenomena suggest that the physical processes leading to shear-induced rigidity in athermal materials are universal. We present a nonequilibrium statistical mechanics model, which exhibits the phenomenology of these shear-driven transitions, shear jamming and discontinuous shear thickening, in different regions of the predicted phase diagram. Our analysis identifies the crucial physical processes underlying shear-driven rigidity transitions, and clarifies the distinct roles played by shearing forces and the packing fraction of grains.

Synchronization patterns in geometrically frustrated rings of relaxation oscillators.

Diffusively coupled chemical oscillators can exhibit a wide variety of complex spatial patterns. In this paper, we show that a ring of relaxation oscillators diffusively coupled through the inhibitory species leads to remarkable spatiotemporal patterns in the regime where there is a large separation of time scales between the activator and the inhibitor dynamics. The origin of these complex patterns can be traced back to a preponderance of antiphase synchronized states in the space of attractors. We provide an analytical explanation for the existence and stability of the antiphase synchronized states by examining the limit of extreme time scale separation. Numerical results on rings with small numbers of oscillators show that an explosion of patterns occurs for a ring with five oscillators.

Origin of rigidity in dry granular solids.

Solids are distinguished from fluids by their ability to resist shear. In traditional solids, the resistance to shear is associated with the emergence of broken translational symmetry as exhibited by a nonuniform density pattern. In this work, we focus on the emergence of shear rigidity in a class of solids where this paradigm is challenged. Dry granular materials have no energetically or entropically preferred density modulations. We show that, in contrast to traditional solids, the emergence of shear rigidity in these granular solids is a collective process, which is controlled solely by boundary forces, the constraints of force and torque balance, and the positivity of the contact forces. We develop a theoretical framework based on these constraints, which connects rigidity to broken translational symmetry in the space of forces, not positions of grains. We apply our theory to experimentally generated shear-jammed states and show that these states are indeed characterized by a persistent, non-uniform density modulation in force space, which emerges at the shear-jamming transition.

Structural relaxation in dense liquids composed of anisotropic particles.

We perform extensive molecular dynamics simulations of dense liquids composed of bidisperse dimer- and ellipse-shaped particles in two dimensions that interact via purely repulsive contact forces. We measure the structural relaxation times obtained from the long-time α decay of the self part of the intermediate scattering function for the translational and rotational degrees of freedom (DOF) as a function of packing fraction φ, temperature T, and aspect ratio α. We are able to collapse the packing-fraction and temperature-dependent structural relaxation times for disks, and dimers and ellipses over a wide range of α, onto a universal scaling function F(±)(|φ-φ(0)|,T,α), which is similar to that employed in previous studies of dense liquids composed of purely repulsive spherical particles in three dimensions. F(±) for both the translational and rotational DOF are characterized by the α-dependent scaling exponents µ and δ and packing fraction φ(0)(α) that signals the crossover in the scaling form F(±) from hard-particle dynamics to super-Arrhenius behavior for each aspect ratio. We find that the fragility of structural relaxation at φ(0), m(φ(0)), decreases monotonically with increasing aspect ratio for both ellipses and dimers. For α>α(p), where α(p) is the location of the peak in the packing fraction φ(J) at jamming onset, the rotational DOF are strongly coupled to the translational DOF, and the dynamic scaling exponents and φ(0) are similar for the rotational and translational DOF. For 1<α<α(p), the translational DOF become frozen at higher temperatures than the rotational DOF, and φ(0) for the rotational degrees of freedom increases above φ(J). Moreover, the results for the slow dynamics of dense liquids composed of dimer- and ellipse-shaped particles are qualitatively the same, despite the fact that zero-temperature static packings of dimers are isostatic, while static packings of ellipses are hypostatic. Thus, zero-temperature contact counting arguments do not apply to structural relaxation of dense liquids of anisotropic particles near the glass transition.