Contact changes near jamming

Structural Measures as Guides to Ultrastable States in Overjammed Packings

Jammed, disordered packings of given sets of particles possess a multitude of equilibrium states with different mechanical properties. Identifying and constructing desired states, e.g., of superior stability, is a complex task. Here, we show that in two-dimensional particle packings the energy of all metastable states (inherent structures) is reliably classified by simple scalar measures of local steric packing. These structural measures are insensitive to the particle interaction potential and so robust that they can be used to guide a modified swap algorithm that anneals polydisperse packings toward low-energy metastable states exceptionally fast. The low-energy states are extraordinarily stable against applied shear, so that the approach also efficiently identifies ultrastable packings.

Softening and Residual Loss Modulus of Jammed Grains under Oscillatory Shear in an Absorbing State

From a theoretical study of the mechanical response of jammed materials comprising frictionless and overdamped particles under oscillatory shear, we find that the material becomes soft, and the loss modulus remains nonzero even in an absorbing state where any irreversible plastic deformation does not exist. The trajectories of the particles in this region exhibit hysteresis loops. We succeed in clarifying the origin of the softening of the material and the residual loss modulus with the aid of Fourier analysis. We also clarify the roles of the yielding point in the softening to distinguish the plastic deformation from reversible deformation in the absorbing state.

Correlation of plastic events with local structure in jammed packings across spatial dimensions

In frictionless jammed packings, existing evidence suggests a picture in which localized physics dominates in low spatial dimensions, d = 2, 3, but quickly loses relevance as d rises, replaced by spatially extended mean-field behavior. For example, quasilocalized low-energy vibrational modes and low-coordination particles associated with deviation from mean-field behavior (rattlers and bucklers) all vanish rapidly with increasing d. These results suggest that localized rearrangements, which are associated with low-energy vibrational modes, correlated with local structure, and dominant in low dimensions, should give way in higher d to extended rearrangements uncorrelated with local structure. Here, we use machine learning to analyze simulations of jammed packings under athermal, quasistatic shear, identifying a local structural variable, softness, that correlates with rearrangements in dimensions d = 2 to d = 5. We find that softness—and even just the local coordination number Z—is essentially equally predictive of rearrangements in all d studied. This result provides direct evidence that local structure plays an important role in higher d, suggesting a modified picture for the dimensional cross-over to mean-field theory.

Avalanche dynamics in sheared athermal particle packings occurs via localized bursts predicted by unstable linear response

Under applied shear strain, granular and amorphous materials deform via particle rearrangements, which can be small and localized or organized into system-spanning avalanches. While the statistical properties of avalanches under quasi-static shear are well-studied, the dynamics during avalanches is not. In numerical simulations of sheared soft spheres, we find that avalanches can be decomposed into bursts of localized deformations, which we identify using an extension of persistent homology methods. We also study the linear response of unstable systems during an avalanche, demonstrating that eigenvalue dynamics are highly complex during such events, and that the most unstable eigenvector is a poor predictor of avalanche dynamics. Instead, we modify existing tools that identify localized excitations in stable systems, and apply them to these unstable systems with non-positive definite Hessians, quantifying the evolution of such excitations during avalanches. We find that bursts of localized deformations in the avalanche almost always occur at localized excitations identified using the linear spectrum. These new tools will provide an improved framework for validating and extending mesoscale elastoplastic models that are commonly used to explain avalanche statistics in glasses and granular matter.

Marginally jammed states of hard disks in a one-dimensional channel

We have studied a class of marginally jammed states in a system of hard disks confined in a narrow channel-a quasi-one-dimensional system-whose exponents are not those predicted by theories valid in the infinite dimensional limit. The exponent γ which describes the distribution of small gaps takes the value 1 rather than the infinite dimensional value 0.41269⋯. Our work shows that there exist jammed states not found within the tiling approach of Ashwin and Bowles. The most dense of these marginal states is an unusual state of matter that is asymptotically crystalline.

Stress Relaxation above and below the Jamming Transition

We numerically investigate stress relaxation in soft athermal disks to reveal critical slowing down when the system approaches the jamming point. The exponents describing the divergence of the relaxation time differ dramatically depending on whether the transition is approached from the jammed or unjammed phase. This contrasts sharply with conventional dynamic critical scaling scenarios, where a single exponent characterizes both sides. We explain this surprising difference in terms of the vibrational density of states, which is a key ingredient of linear viscoelastic theory. The vibrational density of states exhibits an extra slow mode that emerges below jamming, which we utilize to demonstrate the anomalous exponent below jamming.

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

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

Coarsening and mechanics in the bubble model for wet foams

Aqueous foams are an important model system that displays coarsening dynamics. Coarsening in dispersions and foams is well understood in the dilute and dry limits, where the gas fraction tends to zero and one, respectively. However, foams are known to undergo a jamming transition from a fluidlike to a solidlike state at an intermediate gas fraction ϕ_{c}. Much less is known about coarsening dynamics in wet foams near jamming, and the link to mechanical response, if any, remains poorly understood. Here we probe coarsening and mechanical response using numerical simulations of a variant of the Durian bubble model for wet foams. As in other coarsening systems we find a steady state scaling regime with an associated particle size distribution. We relate the time rate of evolution of the coarsening process to the wetness of the foam and identify a characteristic coarsening time that diverges approaching jamming. We further probe mechanical response of the system to strain while undergoing coarsening. There are two competing timescales, namely the coarsening time and the mechanical relaxation time. We relate these to the evolution of the elastic response and the mechanical structure.

Softening and yielding of soft glassy materials

Solids deform and fluids flow, but soft glassy materials, such as emulsions, foams, suspensions, and pastes, exhibit an intricate mix of solid- and liquid-like behavior. While much progress has been made to understand their elastic (small strain) and flow (infinite strain) properties, such understanding is lacking for the softening and yielding phenomena that connect these asymptotic regimes. Here we present a comprehensive framework for softening and yielding of soft glassy materials, based on extensive numerical simulations of oscillatory rheological tests, and show that two distinct scenarios unfold depending on the material's packing density. For dense systems, there is a single, pressure-independent strain where the elastic modulus drops and the particle motion becomes diffusive. In contrast, for weakly jammed systems, a two-step process arises: at an intermediate softening strain, the elastic and loss moduli both drop down and then reach a new plateau value, whereas the particle motion becomes diffusive at the distinctly larger yield strain. We show that softening is associated with an extensive number of microscopic contact changes leading to a non-analytic rheological signature. Moreover, the scaling of the softening strain with pressure suggest the existence of a novel pressure scale above which softening and yielding coincide, and we verify the existence of this crossover scale numerically. Our findings thus evidence the existence of two distinct classes of soft glassy materials - jamming dominated and dense - and show how these can be distinguished by their rheological fingerprint.

Response of jammed packings to thermal fluctuations

We focus on the response of mechanically stable (MS) packings of frictionless, bidisperse disks to thermal fluctuations, with the aim of quantifying how nonlinearities affect system properties at finite temperature. In contrast, numerous prior studies characterized the structural and mechanical properties of MS packings of frictionless spherical particles at zero temperature. Packings of disks with purely repulsive contact interactions possess two main types of nonlinearities, one from the form of the interaction potential (e.g., either linear or Hertzian spring interactions) and one from the breaking (or forming) of interparticle contacts. To identify the temperature regime at which the contact-breaking nonlinearities begin to contribute, we first calculated the minimum temperatures T_{cb} required to break a single contact in the MS packing for both single- and multiple-eigenmode perturbations of the T=0 MS packing. We find that the temperature required to break a single contact for equal velocity-amplitude perturbations involving all eigenmodes approaches the minimum value obtained for a perturbation in the direction connecting disk pairs with the smallest overlap. We then studied deviations in the constant volume specific heat C[over ¯]_{V} and deviations of the average disk positions Δr from their T=0 values in the temperature regime T_{C[over ¯]_{V}}<T<T_{r}, where T_{r} is the temperature beyond which the system samples the basin of a new MS packing. We find that the deviation in the specific heat per particle ΔC[over ¯]_{V}^{0}/C[over ¯]_{V}^{0} relative to the zero-temperature value C[over ¯]_{V}^{0} can grow rapidly above T_{cb}; however, the deviation ΔC[over ¯]_{V}^{0}/C[over ¯]_{V}^{0} decreases as N^{-1} with increasing system size. To characterize the relative strength of contact-breaking versus form nonlinearities, we measured the ratio of the average position deviations Δr^{ss}/Δr^{ds} for single- and double-sided linear and nonlinear spring interactions. We find that Δr^{ss}/Δr^{ds}>100 for linear spring interactions is independent of system size. This result emphasizes that contact-breaking nonlinearities are dominant over form nonlinearities in the low-temperature range T_{cb}<T<T_{r} for model jammed systems.

Stress relaxation in viscous soft spheres

We report the results of molecular dynamics simulations of stress relaxation tests in athermal viscous soft sphere packings close to their unjamming transition. By systematically and simultaneously varying both the amplitude of the applied strain step and the pressure of the initial condition, we access both linear and nonlinear response regimes and control the distance to jamming. Stress relaxation in viscoelastic solids is characterized by a relaxation time τ* that separates short time scales, where viscous loss is substantial, from long time scales, where elastic storage dominates and the response is essentially quasistatic. We identify two distinct plateaus in the strain dependence of the relaxation time, one each in the linear and nonlinear regimes. The height of both plateaus scales as an inverse power law with the distance to jamming. By probing the time evolution of particle velocities during relaxation, we further identify a correlation between mechanical relaxation in the bulk and the degree of non-affinity in the particle velocities on the micro scale.

Disentangling defects and sound modes in disordered solids

We develop a new method to isolate localized defects from extended vibrational modes in disordered solids. This method augments particle interactions with an artificial potential that acts as a high-pass filter: it preserves small-scale structures while pushing extended vibrational modes to higher frequencies. The low-frequency modes that remain are "bare" defects; they are exponentially localized without the quadrupolar tails associated with elastic interactions. We demonstrate that these localized excitations are excellent predictors of plastic rearrangements in the solid. We characterize several of the properties of these defects that appear in mesoscopic theory of plasticity, including their distribution of energy barriers, number density, and size, which is a first step in testing and revising continuum models for plasticity in disordered solids.

Discontinuous change of shear modulus for frictional jammed granular materials

The shear modulus of jammed frictional granular materials with harmonic repulsive interaction under an oscillatory shear is numerically investigated. It is confirmed that the storage modulus, the real part of the shear modulus, for frictional grains with sufficiently small strain amplitude γ_{0} discontinuously emerges at the jamming transition point. The storage modulus for small γ_{0} differs from that of frictionless grains even in the zero friction limit, whereas they are almost identical with each other for sufficiently large γ_{0}, where the transition becomes continuous. The stress-strain curve exhibits a hysteresis loop even for a small strain, which connects a linear region for sufficiently small strain to another linear region for larger strain. We propose a scaling law to interpolate between the states of small and large γ_{0}.

Slow creep in soft granular packings

Transient creep mechanisms in soft granular packings are studied numerically using a constant pressure and constant stress simulation method. Rapid compression followed by slow dilation is predicted on the basis of a logarithmic creep phenomenon. Characteristic scales of creep strain and time exhibit a power-law dependence on jamming pressure, and they diverge at the jamming point. Microscopic analysis indicates the existence of a correlation between rheology and nonaffine fluctuations. Localized regions of large strain appear during creep and grow in magnitude and size at short times. At long times, the spatial structure of highly correlated local deformation becomes time-invariant. Finally, a microscale connection between local rheology and local fluctuations is demonstrated in the form of a linear scaling between granular fluidity and nonaffine velocity.

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}.

Nonlocal Elasticity near Jamming in Frictionless Soft Spheres

We use simulations of frictionless soft sphere packings to identify novel constitutive relations for linear elasticity near the jamming transition. By forcing packings at varying wavelengths, we directly access their transverse and longitudinal compliances. These are found to be wavelength dependent, in violation of conventional (local) linear elasticity. Crossovers in the compliances select characteristic length scales, which signify the appearance of nonlocal effects. Two of these length scales diverge as the pressure vanishes, indicating that critical effects near jamming control the breakdown of local elasticity. We expect these nonlocal constitutive relations to be applicable to a wide range of weakly jammed solids, including emulsions, foams, and granulates.

Excitation of vibrational soft modes in disordered systems using active oscillation

We propose a new method to characterize the spatial distribution of particles' vibrations in solids with much lower computational costs compared to the usual normal mode analysis. We excite the specific vibrational mode in a two dimensional athermal jammed system by giving a small amplitude of active oscillation to each particle's size with an identical driving frequency. The response is then obtained as the real time displacements of the particles. We show remarkable correlations between the real time displacements and the eigen vectors obtained from conventional normal mode analysis. More importantly, from these real time displacements, we can measure the participation ratio and spatial polarization of particles' vibrations. From these measurements, we find three distinct frequency regimes which characterize the spatial distribution and correlation of particles' vibrations in jammed amorphous solids. Furthermore, we can possibly apply this method to a much larger system to examine the low frequency behaviour of amorphous solids with a much higher resolution of the frequency space.

Contact changes of sheared systems: Scaling, correlations, and mechanisms

We probe the onset and effect of contact changes in two-dimensional soft harmonic particle packings which are sheared quasistatically under controlled strain. First, we show that, in the majority of cases, the first contact changes correspond to the creation or breaking of contacts on a single particle, with contact breaking overwhelmingly likely for low pressures and/or small systems, and contact making and breaking equally likely for large pressures and in the thermodynamic limit. The statistics of the corresponding strains are near-Poissonian, in particular for large-enough systems. The mean characteristic strains exhibit scaling with the number of particles N and pressure P and reveal the existence of finite-size effects akin to those seen for linear response quantities [C. P. Goodrich et al., Phys. Rev. Lett. 109, 095704 (2012)PRLTAO0031-900710.1103/PhysRevLett.109.095704; C. P. Goodrich et al., Phys. Rev. E 90, 022138 (2014)].PLEEE81539-375510.1103/PhysRevE.90.022138 Second, we show that linear response accurately predicts the strains of the first contact changes, which allows us to accurately study the scaling of the characteristic strains of making and breaking contacts separately. Both of these show finite-size scaling, and we formulate scaling arguments that are consistent with the observed behavior. Third, we probe the effect of the first contact change on the shear modulus G and show in detail how the variation of G remains smooth and bounded in the large-system-size limit: Even though contact changes occur then at vanishingly small strains, their cumulative effect, even at a fixed value of the strain, are limited, so, effectively, linear response remains well defined. Fourth, we explore multiple contact changes under shear and find strong and surprising correlations between alternating making and breaking events. Fifth, we show that by making a link with extremal statistics, our data are consistent with a very slow crossover to self-averaging with system size, so the thermodynamic limit is reached much more slowly than expected based on finite-size scaling of elastic quantities or contact breaking strains.

Attraction-induced jamming in the flow of foam through a channel

We study the flow of a pressure-driven foam through a straight channel using numerical simulations, and examine the effects of a tuneable attractive potential between bubbles. We show that the effect of an attractive potential is to introduce a regime of jamming and stick-slip flow in a channel, and report on the behaviour resulting from varying the strength of the attraction. We find that there is a force threshold below which the flow jams, and upon further increasing the driving force, a crossover from intermittent (stick-slip) to smooth flow is observed. This threshold force below which the foam jams increases linearly with the strength of the attractive potential. By examining the spectra of energy fluctuations, we show that stick-slip flow is characterized by low frequency rearrangements and strongly local behaviour, whereas steady flow shows a broad spectrum of energy drop events and collective behaviour. Our work suggests that the stick-slip and the jamming regimes occur due to the increased stabilization of contact networks by the attractive potential - as the strength of attraction is increased, bubbles are increasingly trapped within networks, and there is a decrease in the number of contact changes.

Scaling ansatz for the jamming transition

We propose a Widom-like scaling ansatz for the critical jamming transition. Our ansatz for the elastic energy shows that the scaling of the energy, compressive strain, shear strain, system size, pressure, shear stress, bulk modulus, and shear modulus are all related to each other via scaling relations, with only three independent scaling exponents. We extract the values of these exponents from already known numerical or theoretical results, and we numerically verify the resulting predictions of the scaling theory for the energy and residual shear stress. We also derive a scaling relation between pressure and residual shear stress that yields insight into why the shear and bulk moduli scale differently. Our theory shows that the jamming transition exhibits an emergent scale invariance, setting the stage for the potential development of a renormalization group theory for jamming.

Critical behaviour in the nonlinear elastic response of hydrogels

In this paper we study the elastic response of synthetic hydrogels to an applied shear stress. The hydrogels studied here have previously been shown to mimic the behaviour of biopolymer networks when they are sufficiently far above the gel point. We show that near the gel point they exhibit an elastic response that is consistent with the predicted critical behaviour of networks near or below the isostatic point of marginal stability. This point separates rigid and floppy states, distinguished by the presence or absence of finite linear elastic moduli. Recent theoretical work has also focused on the response of such networks to finite or large deformations, both near and below the isostatic point. Despite this interest, experimental evidence for the existence of criticality in such networks has been lacking. Using computer simulations, we identify critical signatures in the mechanical response of sub-isostatic networks as a function of applied shear stress. We also present experimental evidence consistent with these predictions. Furthermore, our results show the existence of two distinct critical regimes, one of which arises from the nonlinear stretch response of semi-flexible polymers.

Beyond linear elasticity: jammed solids at finite shear strain and rate

The shear response of soft solids can be modeled with linear elasticity, provided the forcing is slow and weak. Both of these approximations must break down when the material loses rigidity, such as in foams and emulsions at their (un)jamming point - suggesting that the window of linear elastic response near jamming is exceedingly narrow. Yet precisely when and how this breakdown occurs remains unclear. To answer these questions, we perform computer simulations of stress relaxation and shear start-up tests in athermal soft sphere packings, the canonical model for jamming. By systematically varying the strain amplitude, strain rate, distance to jamming, and system size, we identify characteristic strain and time scales that quantify how and when the window of linear elasticity closes, and relate these scales to changes in the microscopic contact network.

Critical scaling of Bagnold rheology at the jamming transition of frictionless two-dimensional disks

We carry out constant volume simulations of steady-state shear-driven rheology in a simple model of bidisperse soft-core frictionless disks in two dimensions, using a dissipation law that gives rise to Bagnoldian rheology. We discuss in detail the critical scaling ansatz for the shear-driven jamming transition and carry out a detailed scaling analysis of our resulting data for pressure p and shear stress σ. Our analysis determines the critical exponent β that describes the algebraic divergence of the Bagnold transport coefficients lim_{γ[over ̇]→0}p/γ[over ̇]^{2},σ/γ[over ̇]^{2}∼(ϕ_{J}-ϕ)^{-β} as the jamming transition ϕ_{J} is approached from below. For the low strain rates considered in this work, we show that it is still necessary to consider the leading correction-to-scaling term in order to achieve a self-consistent analysis of our data, in which the critical parameters become independent of the size of the window of data used in the analysis. We compare our resulting value β≈5.0±0.4 against previous numerical results and competing theoretical models. Our results confirm that the shear-driven jamming transition in Bagnoldian systems is well described by a critical scaling theory and we relate this scaling theory to the phenomenological constituent laws for dilatancy and friction.

Fluctuations in flows near jamming

Bubbles, droplets or particles in flowing complex media such as foams, emulsions or suspensions follow highly complex paths, with the relative motion of the constituents setting the energy dissipation rate. What is their dynamics, and how is this connected to the global rheology? To address these questions, we probe the statistics and spatio-temporal organization of the local particle motion and energy dissipation in a model for sheared disordered materials. We find that the fluctuations in the local dissipation vary from nearly Gaussian and homogeneous at low densities and fast flows, to strongly intermittent for large densities and slow flows. The higher order moments of the relative particle velocities reveal strong evidence for a qualitative difference between two distinct regimes which are nevertheless connected by a smooth crossover. In the critical regime, the higher order moments are related by novel multiscaling relations. In the plastic regime the relations between these moments take on a different form, with higher moments diverging rapidly when the flow rate vanishes. As these velocity differences govern the energy dissipation, we can distinguish two qualitatively different types of flow: an intermediate density, critical regime related to jamming, and a large density, plastic regime.

Helical inner-wall texture prevents jamming in granular pipe flows

Granular pipe flows are characterized by intermittent behavior and large, potentially destructive solid fraction variations in the transport direction. By means of particle-based numerical simulations of gravity-driven flows in vertical pipes, we show that it is possible to obtain steady material transport by adding a helical texture to the inner-wall of the pipe. The helical texture leads to a more homogeneous mass flux along the pipe, prevents the emergence of large density waves and substantially reduces the probability of plug formation thus avoiding jamming of the particulate flow. We show that the granular mass flux Q through a pipe of diameter D with a helical texture of wavelength λ follows the equation Q = Q0·{1 - B sin[arctan(2πD/λ)]}, where Q0 is the flow without helix, predicted from the well-known Beverloo equation. Our new expression yields, thus, a modification of the Beverloo equation with only one additional fit parameter, B, and describes the particle mass flux with the helical texture with excellent quantitative agreement with simulation results. Future application of the method proposed here has the potential to improve granular pipe flows in a broad range of processes without the need for energy input from any external source.

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.

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

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