Experimental signatures of a nonequilibrium phase transition governing the yielding of a soft glass

Absorbing-state transitions in particulate systems under spatially varying driving

Non-equilibrium transitions into absorbing states are widespread, and amorphous materials under cyclic shear have emerged as useful model systems in which to study their properties. Recent work has focused on homogeneous driving in which the shear amplitude is uniform in space, despite most real world flows involving spatially inhomogeneous conditions that are known to produce qualitatively distinct phenomenology. Here we study the absorbing state transition under inhomogeneous driving using a modified random organization model. For smoothly varying driving the steady state results map onto the homogeneous absorbing state phase diagram, with the position of the boundary between absorbing and diffusive states being insensitive to the driving wavelength. The phenomenology is well-described by a one-dimensional generalized continuum model that we pose. For discontinuously varying driving the position of the absorbing phase boundary and the exponent characterising the fraction of active particles are altered relative to the homogeneous case.

Anomalous Softness in Amorphous Matter in the Reversible Plastic Regime

We study an elastoplastic model of an amorphous solid subject to athermal quasistatic cyclic shear strain. We focus on cycling amplitudes in the so-called reversible-plastic regime where, after a transient, the system locks into a hysteretic limit cycle and returns to the same microscopic configuration after one or more strain cycles. We show that the ground state energy of the terminal limit cycle decreases with increasing cycling amplitude. In analogy to an annealed alloy or an aged colloidal glass, one would expect the states with lower energy to be mechanically harder and to require larger stresses and strains to trigger microscopic rearrangements. However, we show the opposite result: the systems with lower energy cycled at higher strain amplitude are mechanically softer and begin to exhibit plastic rearrangements at smaller stresses and strains within the cycle. We explain this anomaly quantitatively in terms of Eshelby inclusion theory where an inclusion is subjected to a particular negative stress value after it undergoes a yielding event. These results point the way toward measurements to be conducted in experiments and particle-based computer simulations on cyclically sheared amorphous solids.

Two-dimensional squishy glass: yielding under oscillatory shear

The yielding response to an imposed oscillatory shear is investigated for a model two-dimensional dense glass composed of bidisperse, deformable polymer rings, with the ring stiffness being the control parameter. In the quiescent glassy state, the more flexible rings exhibit a broader spectrum of shape fluctuations, which becomes increasingly constrained with increasing ring stiffness. Under shear, the highly packed rings yield, i.e. the thermal assembly loses rigidity, with the threshold yield strain increasing significantly with decreasing ring stiffness. Further, the rings display significant deviations in their shape compared to their unsheared counterparts. This study provides insights into the interplay between shape changes and translational rearrangements under shear, thus contributing to the understanding of yielding transition in densely packed, deformable polymer systems.

Linking structural and rheological memory in disordered soft materials

Linking the macroscopic flow properties and nanoscopic structure is a fundamental challenge to understanding, predicting, and designing disordered soft materials. Under small stresses, these materials are soft solids, while larger loads can lead to yielding and the acquisition of plastic strain, which adds complexity to the task. In this work, we connect the transient structure and rheological memory of a colloidal gel under cyclic shearing across a range of amplitudes via a generalized memory function using rheo-X-ray photon correlation spectroscopy (rheo-XPCS). Our rheo-XPCS data show that the nanometer scale aggregate-level structure recorrelates whenever the change in recoverable strain over some interval is zero. The macroscopic recoverable strain is therefore a measure of the nano-scale structural memory. We further show that yielding in disordered colloidal materials is strongly heterogeneous and that memories of prior deformation can exist even after the material has been subjected to flow.

Interplay between an Absorbing Phase Transition and Synchronization in a Driven Granular System

Absorbing phase transitions (APTs) are widespread in nonequilibrium systems, spanning condensed matter, epidemics, earthquakes, ecology, and chemical reactions. APTs feature an absorbing state in which the system becomes entrapped, along with a transition, either continuous or discontinuous, to an active state. Understanding which physical mechanisms determine the order of these transitions represents a challenging open problem in nonequilibrium statistical mechanics. Here, by numerical simulations and mean-field analysis, we show that a quasi-2D vibrofluidized granular system exhibits a novel form of APT. The absorbing phase is observed in the horizontal dynamics below a critical packing fraction, and can be continuous or discontinuous based on the emergent degree of synchronization in the vertical motion. Our results provide a direct representation of a feasible experimental scenario, showcasing a surprising interplay between dynamic phase transition and synchronization.

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.

Slow Fatigue and Highly Delayed Yielding via Shear Banding in Oscillatory Shear

We study theoretically the dynamical process of yielding in cyclically sheared amorphous materials, within a thermal elastoplastic model and the soft glassy rheology model. Within both models we find an initially slow accumulation, over many cycles after the inception of shear, of low levels of damage in the form strain heterogeneity across the sample. This slow fatigue then suddenly gives way to catastrophic yielding and material failure. Strong strain localization in the form of shear banding is key to the failure mechanism. We characterize in detail the dependence of the number of cycles N^{*} before failure on the amplitude of imposed strain, the working temperature, and the degree to which the sample is annealed prior to shear. We discuss our finding with reference to existing experiments and particle simulations, and suggest new ones to test our predictions.

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}.

An exact expression of three-body system for the complex shear modulus of frictional granular materials

We propose a simple model comprising three particles to study the nonlinear mechanical response of jammed frictional granular materials under oscillatory shear. Owing to the introduction of the simple model, we obtain an exact analytical expression of the complex shear modulus for a system including many monodispersed disks, which satisfies a scaling law in the vicinity of the jamming point. These expressions perfectly reproduce the shear modulus of the many-body system with low strain amplitudes and friction coefficients. Even for disordered many-body systems, the model reproduces the results by introducing a single fitting parameter.

Eigenvalue analysis of stress-strain curve of two-dimensional amorphous solids of dispersed frictional grains with finite shear strain

The stress-strain curve of two-dimensional frictional dispersed grains interacting with a harmonic potential without considering the dynamical slip under a finite strain is determined by using eigenvalue analysis of the Hessian matrix. After the configuration of grains is obtained, the stress-strain curve based on the eigenvalue analysis is in almost perfect agreement with that obtained by the simulation, even if there are plastic deformations caused by stress avalanches. Unlike the naive expectation, the eigenvalues in our model do not indicate any precursors to the stress-drop events.

Mechanical annealing and memories in a disordered solid

Shearing a disordered or amorphous solid for many cycles with a constant strain amplitude can anneal it, relaxing a sample to a steady state that encodes a memory of that amplitude. This steady state also features a remarkable stability to amplitude variations that allows one to read the memory. Here, we shed light on both annealing and memory by considering how to mechanically anneal a sample to have as little memory content as possible. In experiments, we show that a "ring-down" protocol reaches a comparable steady state but with no discernible memories and minimal structural anisotropy. We introduce a method to characterize the population of rearrangements within a sample and show how it connects with the response to amplitude variation and the size of annealing steps. These techniques can be generalized to other forms of glassy matter and a wide array of disordered solids, especially those that yield by flowing homogeneously.

Inter-particle adhesion induced strong mechanical memory in a dense granular suspension

Repeated/cyclic shearing can drive amorphous solids to a steady state encoding a memory of the applied strain amplitude. However, recent experiments find that the effect of such memory formation on the mechanical properties of the bulk material is rather weak. Here, we study the memory effect in a yield stress solid formed by a dense suspension of cornstarch particles in paraffin oil. Under cyclic shear, the system evolves toward a steady state showing training-induced strain stiffening and plasticity. A readout reveals that the system encodes a strong memory of the training amplitude ( γ T) as indicated by a large change in the differential shear modulus. We observe that memory can be encoded for a wide range of γ T values both above and below the yielding albeit the strength of the memory decreases with increasing γ T. In situ boundary imaging shows strain localization close to the shearing boundaries, while the bulk of the sample moves like a solid plug. In the steady state, the average particle velocity [Formula: see text] inside the solid-like region slows down with respect to the moving plate as γ approaches γ T; however, as the readout strain crosses γ T, [Formula: see text] suddenly increases. We demonstrate that inter-particle adhesive interaction is crucial for such a strong memory effect. Interestingly, our system can also remember more than one input only if the training strain with smaller amplitude is applied last.

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.

Cooperative effects driving the multi-periodic dynamics of cyclically sheared amorphous solids

When subject to cyclic forcing, amorphous solids can reach periodic, repetitive states, where the system behaves plastically, but the particles return to their initial positions after one or more forcing cycles, where the latter response is called multi-periodic. It is known that plasticity in amorphous materials is mediated by local rearrangements called ``soft spots' or ``shear transformation zones'.Experiments and simulations indicate that soft spots can be modeled as hysteretic two-state entities interacting via quadrupolar displacement fields generated when they switch states and that these interactions can give rise to multi-periodic behavior. However, how interactions facilitate multi-periodicity is unknown. Here we show, using a model of random interacting two-state systems and molecular dynamics simulations, that multi-periodicity arises from oscillations in the magnitudes of the switching field of soft spots which cause soft spots to be active during some forcing cycles and idle during others. We demonstrate that these oscillations result from cooperative effects facilitated by the frustrated interactions between the soft spots. The presence of such mechanisms has implications for manipulating memory in frustrated hysteretic systems.

Absorbing phase transitions in systems with mediated interactions

Experiments of periodically sheared colloidal suspensions or soft amorphous solids display a transition from reversible to irreversible particle motion that, when analyzed stroboscopically in time, is interpreted as an absorbing phase transition with infinitely many absorbing states. In these systems, interactions mediated by hydrodynamics or elasticity are present, causing passive regions to be affected by nearby active ones. We show that mediated interactions induce a universality class of absorbing phase transitions distinct from conserved directed percolation, and we obtain the corresponding critical exponents. We do so with large-scale numerical simulations of a minimal model for the stroboscopic dynamics of sheared soft materials and we derive the minimal field theoretical description.

Critical behavior of nonequilibrium depinning transitions for vortices driven by current and vortex density

We study the critical dynamics of vortices associated with dynamic disordering near the depinning transitions driven by dc force (dc current I) and vortex density (magnetic field B). Independent of the driving parameters, I and B, we observe the critical behavior of the depinning transitions, not only on the moving side, but also on the pinned side of the transition, which is the first convincing verification of the theoretical prediction. Relaxation times, [Formula: see text] and [Formula: see text], to reach either the moving or pinned state, plotted against I and B, respectively, exhibit a power-law divergence at the depinning thresholds. The critical exponents of both transitions are, within errors, identical to each other, which are in agreement with the values expected for an absorbing phase transition in the two-dimensional directed-percolation universality class. With an increase in B under constant I, the depinning transition at low B is replaced by the repinning transition at high B in the peak-effect regime. We find a trend that the critical exponents in the peak-effect regime are slightly smaller than those in the low-B regime and the theoretical one, which is attributed to the slight difference in the depinning mechanism in the peak-effect regime.

Activity waves and freestanding vortices in populations of subcritical Quincke rollers

Virtually all of the many active matter systems studied so far are made of units (biofilaments, cells, colloidal particles, robots, animals, etc.) that move even when they are alone or isolated. Their collective properties continue to fascinate, and we now understand better how they are unique to the bulk transduction of energy into work. Here we demonstrate that systems in which isolated but potentially active particles do not move can exhibit specific and remarkable collective properties. Combining experiments, theory, and numerical simulations, we show that such subcritical active matter can be realized with Quincke rollers, that is, dielectric colloidal particles immersed in a conducting fluid subjected to a vertical DC electric field. Working below the threshold field value marking the onset of motion for a single colloid, we find fast activity waves, reminiscent of excitable systems, and stable, arbitrarily large self-standing vortices made of thousands of particles moving at the same speed. Our theoretical model accounts for these phenomena and shows how they can arise in the absence of confining boundaries and individual chirality. We argue that our findings imply that a faithful description of the collective properties of Quincke rollers need to consider the fluid surrounding particles.

Critical behavior of density-driven and shear-driven reversible-irreversible transitions in cyclically sheared vortices

Random assemblies of particles subjected to cyclic shear undergo a reversible–irreversible transition (RIT) with increasing a shear amplitude d or particle density n, while the latter type of RIT has not been verified experimentally. Here, we measure the time-dependent velocity of cyclically sheared vortices and observe the critical behavior of RIT driven by vortex density B as well as d. At the critical point of each RIT, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{\mathrm {c}}$$\end{document}Bc and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{\mathrm {c}}$$\end{document}dc, the relaxation time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}τ to reach the steady state shows a power-law divergence. The critical exponent for B-driven RIT is in agreement with that for d-driven RIT and both types of RIT fall into the same universality class as the absorbing transition in the two-dimensional directed-percolation universality class. As d is decreased to the average intervortex spacing in the reversible regime, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau (d)$$\end{document}τ(d) shows a significant drop, indicating a transition or crossover from a loop-reversible state with vortex-vortex collisions to a collisionless point-reversible state. In either regime, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau (d)$$\end{document}τ(d) exhibits a power-law divergence at the same \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{\mathrm {c}}$$\end{document}dc with nearly the same exponent.

Multiperiodic orbits from interacting soft spots in cyclically sheared amorphous solids

When an amorphous solid is deformed cyclically, it may reach a steady state in which the paths of constituent particles trace out closed loops that repeat in each driving cycle. A remarkable variant has been noticed in simulations where the period of particle motions is a multiple of the period of driving, but the reasons for this behavior have remained unclear. Motivated by mesoscopic features of displacement fields in experiments on jammed solids, we propose and analyze a simple model of interacting soft spots-locations where particles rearrange under stress and that resemble two-level systems with hysteresis. We show that multiperiodic behavior can arise among just three or more soft spots that interact with each other, but in all cases it requires frustrated interactions, illuminating this otherwise elusive type of interaction. We suggest directions for seeking this signature of frustration in experiments and for achieving it in designed systems.

Yielding in an amorphous solid subject to constant stress at finite temperatures

Understanding the nature of the yield transition is a long-standing problem in the physics of amorphous solids. Here we use molecular dynamics simulations to study the response of amorphous solids to constant stresses at finite temperatures. We compare amorphous solids that are prepared using fast and slow quenches and show that for thermal systems, the steady-state velocity exhibits a continuous transition from very slow creep to a finite strain rate as a function of the stress. This behavior is observed for both well-annealed and poorly annealed systems. However, the transient dynamics is different in the latter and involves overcoming an energy barrier. Due to the different simulation protocol, the strain rate as a function of stress and temperature follows a scaling relation that is different from the ones that are shown for systems where the strain is controlled. Collapsing the data using this scaling relation allows us to calculate critical exponents for the dynamics close to yield, including an exponent for thermal rounding. We also demonstrate that strain slips due to avalanche events above yield follow standard scaling relations and we extract critical exponents that are comparable to the ones obtained in previous studies that performed simulations of both molecular dynamics and elastoplastic models using strain-rate control.

Topology of the energy landscape of sheared amorphous solids and the irreversibility transition

Recent experiments and simulations of amorphous solids plastically deformed by an oscillatory drive have found a surprising behavior-for small strain amplitudes the dynamics can be reversible, which is contrary to the usual notion of plasticity as an irreversible form of deformation. This reversibility allows the system to reach limit cycles in which plastic events repeat indefinitely under the oscillatory drive. It was also found that reaching reversible limit cycles can take a large number of driving cycles and it was surmised that the plastic events encountered during the transient period are not encountered again and are thus irreversible. Using a graph representation of the stable configurations of the system and the plastic events connecting them, we show that the notion of reversibility in these systems is more subtle. We find that reversible plastic events are abundant and that a large portion of the plastic events encountered during the transient period are actually reversible in the sense that they can be part of a reversible deformation path. More specifically, we observe that the transition graph can be decomposed into clusters of configurations that are connected by reversible transitions. These clusters are the strongly connected components of the transition graph and their sizes turn out to be power-law distributed. The largest of these are grouped in regions of reversibility, which in turn are confined by regions of irreversibility whose number proliferates at larger strains. Our results provide an explanation for the irreversibility transition-the divergence of the transient period at a critical forcing amplitude. The long transients result from transition between clusters of reversibility in a search for a cluster large enough to contain a limit cycle of a specific amplitude. For large enough amplitudes, the search time becomes very large, since the sizes of the limit cycles become incompatible with the sizes of the regions of reversibility.

Shear modulus and reversible particle trajectories of frictional granular materials under oscillatory shear

In this study, we numerically investigated the mechanical responses and trajectories of frictional granular particles under oscillatory shear in the reversible phase where particle trajectories form closed loops below the yielding point. When the friction coefficient is small, the storage modulus exhibits softening, and the loss modulus remains finite in the quasi-static limit. As the friction coefficient increases, the softening and residual loss modulus are suppressed. The storage and loss moduli satisfy scaling laws if they are plotted as functions of the areas of the loop trajectories divided by the strain amplitude and diameter of grains, at least for small values of the areas.

Acoustic resonance in periodically sheared glass: damping due to plastic events

Using molecular dynamics simulation, we study acoustic resonance in a low-temperature model glass by applying a small periodic shear at a boundary wall. Shear wave resonance occurs as the frequency ω approaches ωl = πc⊥l/L (l = 1, 2…). Here, c⊥ is the transverse sound speed and L is the cell width. At resonance, large-amplitude sound waves appear after many cycles even if the applied strain γ0 is very small. They then induce plastic events, which are heterogeneous on the mesoscopic scale and intermittent on timescales longer than the oscillation period tp = 2π/ω. We visualize them together with the extended elastic strains around them. These plastic events serve to damp sounds. We obtain the nonlinear damping Q-1 = tan δ due to the plastic events near the first resonance at ω ≅ ω1, which is linear in γ0 and independent of ω. After many resonant cycles, we observe an increase in the shear modulus (measured after switching-off the oscillation). We also observe catastrophic plastic events after a very long time (∼103tp), which induce system-size elastic strains and cause a transition from resonant to off-resonant states. At resonance, stroboscopic diffusion becomes detectable.

Quantification of plasticity via particle dynamics above and below yield in a 2D jammed suspension

The yield transition of amorphous materials is characterized by a swift increase of energy dissipation. The connection between particle dynamics, dissipation, and overall material rheology, however, has still not been elucidated. Here, we take a new approach relating trajectories to yielding, using a custom built interfacial stress rheometer, which allows for measurement of shear moduli (G',G'') of a dense athermal suspension's microstructure while simultaneously tracking particle trajectories undergoing cyclic shear. We find an increase in total area traced by particle trajectories as the system is stressed well below to well above yield. Trajectories may be placed into three categories: reversibly elastic paths; reversibly plastic paths, associated with smooth limit cycles; and irreversibly plastic paths, in which particles do not return to their original position. We find that above yield, reversibly plastic trajectories are predominantly found near to the shearing surface, whereas reversibly elastic paths are more prominent near the stationary wall. This spatial transition between particles acting as liquids to those acting as solids is characteristic of a 'melting front', which is observed to shift closer to the wall with increasing strain. We introduce a non-dimensional measure of plastic dissipation based on particle trajectories that scales linearly with strain amplitude both above and below yield, and that is unity at the rheological yield point. Surprisingly, this relation collapses for three systems of varying degrees of disorder.

Forced deterministic dynamics on a random energy landscape: Implications for the physics of amorphous solids

The dynamics of supercooled liquids and plastically deformed amorphous solids is known to be dominated by the structure of their rough energy landscapes. Recent experiments and simulations on amorphous solids subjected to oscillatory shear at athermal conditions have shown that for small strain amplitudes these systems reach limit cycles of different periodicities after a transient. However, for larger strain amplitudes the transients become longer and for strain amplitudes exceeding a critical value the system reaches a diffusive steady state. This behavior cannot be explained using the current mean-field models of amorphous plasticity. Here we show that this phenomenology can be described and explained using a simple model of forced dynamics on a multidimensional random energy landscape. In this model, the existence of limit cycles can be ascribed to confinement of the dynamics to a small part of the energy landscape which leads to self-intersection of state-space trajectories and the transition to the diffusive regime for larger forcing amplitudes occurs when the forcing overcomes this confinement.

Embedding orthogonal memories in a colloidal gel through oscillatory shear

It has recently been shown that in a broad class of disordered systems oscillatory shear training can embed memories of specific shear protocols in relevant physical parameters such as the yield strain. These shear protocols can be used to change the physical properties of the system and memories of the protocol can later be "read" out. Here we investigate shear training memories in colloidal gels, which include an attractive interaction and network structure, and discover that such systems can support memories both along and orthogonal to the training flow direction. We use oscillatory shear protocols to set and read out the yield strain memories and confocal microscopy to analyze the rearranging gel structure throughout the shear training. We find that the gel bonds remain largely isotropic in the shear-vorticity plane throughout the training process suggesting that structures formed to support shear along the training shear plane are also able to support shear along the orthogonal plane. Orthogonal memory extends the usefulness of shear memories to more applications and should apply to many other disordered systems as well.

Absorbing-State Transitions in Granular Materials Close to Jamming

We consider a model for driven particulate matter in which absorbing states can be reached both by particle isolation and by particle caging. The model predicts a nonequilibrium phase diagram in which analogs of hydrodynamic and elastic reversibility emerge at low and high volume fractions respectively, partially separated by a diffusive, nonabsorbing region. We thus find a single phase boundary that spans the onset of chaos in sheared suspensions to the onset of yielding in jammed packings. This boundary has the properties of a nonequilibrium second order phase transition, leading us to write a Manna-like mean field description that captures the model predictions. Dependent on contact details, jamming marks either a direct transition between the two absorbing states, or occurs within the diffusive region.

Nucleation Theory for Yielding of Nearly Defect-Free Crystals: Understanding Rate Dependent Yield Points

Experiments and simulations show that when an initially defect-free rigid crystal is subjected to deformation at a constant rate, irreversible plastic flow commences at the so-called yield point. The yield point is a weak function of the deformation rate, which is usually expressed as a power law with an extremely small nonuniversal exponent. We reanalyze a representative set of published data on nanometer sized, mostly defect-free Cu, Ni, and Au crystals in light of a recently proposed theory of yielding based on nucleation of stable stress-free regions inside the metastable rigid solid. The single relation derived here, which is not a power law, explains data covering 15 orders of magnitude in timescales.

Annealing and rejuvenation in a two-dimensional model amorphous solid under oscillatory shear

We study the annealing and rejuvenation behavior of a two-dimensional amorphous solid model under oscillatory shear. We show that, depending on the cooling protocol used to create the initial configuration, the mean potential energy can either decrease or increase under subyield oscillatory shear. For post-yield oscillatory shear, the mean potential energy increases and is independent on the initial conditions. We explain this behavior by modeling the dynamics using a simple model of forced dynamics on a random energy landscape and show that the model reproduces the qualitative behavior of the mean potential energy and mean-square displacement observed in the particle based simulations. This suggests that some important aspects of the dynamics of amorphous solids can be understood by studying the properties of random energy landscapes and without explicitly taking into account the complex real-space interactions which are involved in plastic deformation.

Experimental signatures of a nonequilibrium phase transition near the crossover point of a Langmuir monolayer

We investigate the response of the two-dimensional (2D) continuous non-particulate film of surfactant sorbitan tristearate confined at the air-water interface under oscillatory shear deformation. The time dependence of various rheological parameters show critical-like behavior at a value of strain amplitude close to the crossover point of elastic ([Formula: see text]) and viscous ([Formula: see text]) shear moduli. Imposing oscillatory shear of different strain amplitudes ([Formula: see text]) above and below the crossover strain amplitude ([Formula: see text]) over a large number of cycles, we quantify the temporal dependence of interfacial viscous modulus, phase angle ([Formula: see text]) as well as higher harmonic components of stress. The number of shear cycles ([Formula: see text]) required for these quantities to reach the steady state value diverges near [Formula: see text]. The steady state values of the third harmonic ([Formula: see text]) show order parameter like behavior indicating the importance of higher order harmonics near the nonequilibrium transition. We further show that the energy dissipation per cycle per unit volume has a marked change near [Formula: see text], consistent with continuum level nonequilibrium shear-transformation-zone model of amorphous viscoplasticity.

Directed aging, memory, and nature's greed

Disordered materials are often out of equilibrium and evolve very slowly in a rugged and tortuous energy landscape. This slow evolution, referred to as aging, is deemed undesirable as it often leads to material degradation. However, we show that aging also encodes a memory of the stresses imposed during preparation. Because of inhomogeneous local stresses, the material itself decides how to evolve by modifying stressed regions differently from those under less stress. Because material evolution occurs in response to stresses, aging can be "directed" to produce sought-after responses and unusual functionalities that do not inherently exist. Aging obeys a natural "greedy algorithm" as, at each instant, the material simply follows the path of most rapid and accessible relaxation. Our experiments and simulations illustrate directed aging in examples in which the material's elasticity transforms as desired because of an imposed deformation.

Critical behavior near the reversible-irreversible transition in periodically driven vortices under random local shear

When many-particle (vortex) assemblies with disordered distribution are subjected to a periodic shear with a small amplitude \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{d}}$$\end{document}d, the particles gradually self-organize to avoid next collisions and transform into an organized configuration. We can detect it from the time-dependent voltage \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{V}}{\boldsymbol{(}}{\boldsymbol{t}}{\boldsymbol{)}}$$\end{document}V(t) (average velocity) that increases towards a steady-state value. For small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{d}}$$\end{document}d, the particles settle into a reversible state where all the particles return to their initial position after each shear cycle, while they reach an irreversible state for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{d}}$$\end{document}d above a threshold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{d}}}_{{\boldsymbol{c}}}$$\end{document}dc. Here, we investigate the general phenomenon of a reversible-irreversible transition (RIT) using periodically driven vortices in a strip-shaped amorphous film with random pinning that causes local shear, as a function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{d}}$$\end{document}d. By measuring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{V}}{\boldsymbol{(}}{\boldsymbol{t}}{\boldsymbol{)}}$$\end{document}V(t), we observe a critical behavior of RIT, not only on the irreversible side, but also on the reversible side of the transition, which is the first under random local shear. The relaxation time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\tau }}{\boldsymbol{(}}{\boldsymbol{d}}{\boldsymbol{)}}$$\end{document}τ(d) to reach either the reversible or irreversible state shows a power-law divergence at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{d}}}_{{\boldsymbol{c}}}$$\end{document}dc. The critical exponent is determined with higher accuracy and is, within errors, in agreement with the value expected for an absorbing phase transition in the two-dimensional directed-percolation universality class. As \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{d}}$$\end{document}d is decreased down to the intervortex spacing in the reversible regime, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\tau }}{\boldsymbol{(}}{\boldsymbol{d}}{\boldsymbol{)}}$$\end{document}τ(d) deviates downward from the power-law relation, reflecting the suppression of intervortex collisions. We also suggest the possibility of a narrow smectic-flow regime, which is predicted to intervene between fully reversible and irreversible flow.

Networks and Hierarchies: How Amorphous Materials Learn to Remember

We consider the slow and athermal deformations of amorphous solids and show how the ensuing sequence of discrete plastic rearrangements can be mapped onto a directed network. The network topology reveals a set of highly connected regions joined by occasional one-way transitions. The highly connected regions include hierarchically organized hysteresis cycles and subcycles. At small to moderate strains this organization leads to near-perfect return point memory. The transitions in the network can be traced back to localized particle rearrangements (soft spots) that interact via Eshelby-type deformation fields. By linking topology to dynamics, the network representations provide new insight into the mechanisms that lead to reversible and irreversible behavior in amorphous solids.

Classification of the reversible-irreversible transitions in particle trajectories across the jamming transition point

The reversible-irreversible (RI) transition of particle trajectories in athermal colloidal suspensions under cyclic shear deformation is an archetypal nonequilibrium phase transition which has attracted much attention recently. Most studies of the RI transitions have focused on either dilute limit or very high densities well above the jamming transition point. The transition between the two limiting cases is largely unexplored. In this paper, we study the RI transition of athermal frictionless colloidal particles over a wide range of densities, with emphasis on the region below φ_J, by using oscillatory sheared molecular dynamics simulation. We reveal that the nature of the RI transitions in the intermediate densities is very rich. As demonstrated by the previous work by Schreck et al. [Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2013, 88, 052205], there exist the point-reversible and the loop-reversible phases depending on the density and the shear strain amplitude. We find that, between the two reversible phases, a quasi-irreversible phase where the particles' trajectories are highly non-affine and diffusive. The averaged number of contacts of particles is found to characterize the phase boundaries. We also find that the system undergoes the yielding transition below but in the vicinity of φ_J when the strain with a small but finite strain rate is applied. This yielding transition line matches with the RI transition line separating the loop-reversible from the irreversible phases. Surprisingly, the nonlinear rheological response called "softening" has been observed even below φ_J. These findings imply that geometrical properties encoded in the sheared configurations control the dynamical transitions.

Yield Precursor Dislocation Avalanches in Small Crystals: The Irreversibility Transition

The transition from elastic to plastic deformation in crystalline metals shares history dependence and scale-invariant avalanche signature with other nonequilibrium systems under external loading such as colloidal suspensions. These other systems exhibit transitions with clear analogies to work hardening and yield stress, with many typically undergoing purely elastic behavior only after "training" through repeated cyclic loading; studies in these other systems show a power-law scaling of the hysteresis loop extent and of the training time as the peak load approaches a so-called reversible-to-irreversible transition (RIT). We discover here that deformation of small crystals shares these key characteristics: yielding and hysteresis in uniaxial compression experiments of single-crystalline Cu nano- and micropillars decay under repeated cyclic loading. The amplitude and decay time of the yield precursor avalanches diverge as the peak stress approaches failure stress for each pillar, with a power-law scaling virtually equivalent to RITs in other nonequilibrium systems.

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.

Strength of Mechanical Memories is Maximal at the Yield Point of a Soft Glass

We show experimentally that both single and multiple mechanical memories can be encoded in an amorphous bubble raft, a prototypical soft glass, subject to an oscillatory strain. In line with recent numerical results, we find that multiple memories can be formed sans external noise. By systematically investigating memory formation for a range of training strain amplitudes spanning yield, we find clear signatures of memory even beyond yielding. Most strikingly, the extent to which the system recollects memory is largest for training amplitudes near the yield strain and is a direct consequence of the spatial extent over which the system reorganizes during the encoding process. Our study further suggests that the evolution of force networks on training plays a decisive role in memory formation in jammed packings.

Critical diffusivity in the reversibility-irreversibility transition of amorphous solids under oscillatory shear

Recently it was shown that under oscillatory shear at zero temperature an amorphous solid transitions from asymptotically periodic to asymptotically diffusive steady-state at a critical maximal strain amplitude. Current understanding of the physics behind this transition is lacking. Here we show, using computer simulations, evidence that the diffusivity of the vector of coordinates of the particles comprising an amorphous solid, when subject to oscillatory shear, undergoes a second order phase transition at the reversibility-irreversibility transition point. We explain how such a transition is consistent with dissipative forced dynamics on a complex energy landscape, such as is known to exist in amorphous solids. We demonstrate that as the forcing increases, more and more state-space volume becomes accessible to the system, making it less probable for the state-space trajectory of the system to self-intersect and form a limit-cycle, which explains the slowing-down observed at the transition.

On the existence of thermodynamically stable rigid solids

Customarily, crystalline solids are defined to be rigid since they resist changes of shape determined by their boundaries. However, rigid solids cannot exist in the thermodynamic limit where boundaries become irrelevant. Particles in the solid may rearrange to adjust to shape changes eliminating stress without destroying crystalline order. Rigidity is therefore valid only in the metastable state that emerges because these particle rearrangements in response to a deformation, or strain, are associated with slow collective processes. Here, we show that a thermodynamic collective variable may be used to quantify particle rearrangements that occur as a solid is deformed at zero strain rate. Advanced Monte Carlo simulation techniques are then used to obtain the equilibrium free energy as a function of this variable. Our results lead to a unique view on rigidity: While at zero strain a rigid crystal coexists with one that responds to infinitesimal strain by rearranging particles and expelling stress, at finite strain the rigid crystal is metastable, associated with a free energy barrier that decreases with increasing strain. The rigid phase becomes thermodynamically stable when an external field, which penalizes particle rearrangements, is switched on. This produces a line of first-order phase transitions in the field-strain plane that intersects the origin. Failure of a solid once strained beyond its elastic limit is associated with kinetic decay processes of the metastable rigid crystal deformed with a finite strain rate. These processes can be understood in quantitative detail using our computed phase diagram as reference.

Shaken and stirred: Random organization reduces viscosity and dissipation in granular suspensions

The viscosity of suspensions of large (≥10 μm) particles diverges at high solid fractions due to proliferation of frictional particle contacts. Reducing friction, to allow or improve flowability, is usually achieved by tuning the composition, either by changing particle sizes and shapes or by adding lubricating molecules. We present numerical simulations that demonstrate a complementary approach whereby the viscosity divergence is shifted by driven flow tuning, using superimposed shear oscillations in various configurations to facilitate a primary flow. The oscillations drive the suspension toward an out-of-equilibrium, absorbing state phase transition, where frictional particle contacts that dominate the viscosity are reduced in a self-organizing manner. The method can allow otherwise jammed states to flow; even for unjammed states, it can substantially decrease the energy dissipated per unit strain. This creates a practicable route to flow enhancement across a broad range of suspensions where compositional tuning is undesirable or problematic.

Reconsideration of the Rheology of Silica Filled Natural Rubber Compounds

There is substantial progress along with giant debate in reinforcement mechanisms in relation to structured filler network and heterogeneously retarded polymer dynamics, while the dissipation behaviors have never been clarified for nanoparticle filled polymers. Herein dynamic rheological behaviors of silica filled natural rubber were investigated. Master curves of linear rheology in the hydrodynamic regime and those of the nonlinear Payne effect at a predetermined frequency were created, disclosing a leading role of dynamically retarded bulk rubbery phase to the hydrodynamic regime and a leading role of molecular disentanglement in the bulk phase to the Payne effect. The methodology is able to account for both reinforcement and dissipation of the compounds as a function of filler content. Furthermore, a frequency-dependent hydrodynamic to non-hydrodynamic transition is revealed, revealing the importance of the relaxation of chains in the bulk phase to both reinforcement and dissipation of the compounds. It is suggested that the dynamics of the bulk phase play a critical role for the rheology in the hydrodynamic regime while the fractal filler aggregates become dominative only in the terminal non-hydrodynamic regime where the bulk phase relaxes sufficiently.

Direct Observation of Percolation in the Yielding Transition of Colloidal Glasses

When strained beyond the linear regime, soft colloidal glasses yield to steady-state plastic flow in a way that is similar to the deformation of conventional amorphous solids. Because of the much larger size of the colloidal particles with respect to the atoms comprising an amorphous solid, colloidal glasses allow us to obtain microscopic insight into the nature of the yielding transition, as we illustrate here combining experiments, atomistic simulations, and mesoscopic modeling. Our results unanimously show growing clusters of nonaffine deformation percolating at yielding. In agreement with percolation theory, the spanning cluster is fractal with a fractal dimension d_{f}≃2, and the correlation length diverges upon approaching the critical yield strain. These results indicate that percolation of highly nonaffine particles is the hallmark of the yielding transition in disordered glassy systems.

The yielding transition in amorphous solids under oscillatory shear deformation

Amorphous solids are ubiquitous among natural and man-made materials. Often used as structural materials for their attractive mechanical properties, their utility depends critically on their response to applied stresses. Processes underlying such mechanical response, and in particular the yielding behaviour of amorphous solids, are not satisfactorily understood. Although studied extensively, observed yielding behaviour can be gradual and depend significantly on conditions of study, making it difficult to convincingly validate existing theoretical descriptions of a sharp yielding transition. Here we employ oscillatory deformation as a reliable probe of the yielding transition. Through extensive computer simulations for a wide range of system sizes, we demonstrate that cyclically deformed model glasses exhibit a sharply defined yielding transition with characteristics that are independent of preparation history. In contrast to prevailing expectations, the statistics of avalanches reveals no signature of the impending transition, but exhibit dramatic, qualitative, changes in character across the transition.

The onset of yielding can be difficult to define unambiguously for amorphous materials. Here the authors undertake computer simulations of model glasses of varying system sizes and show that, under oscillatory shear, they exhibit a sharp transition independent of preparation history.

Depinning and nonequilibrium dynamic phases of particle assemblies driven over random and ordered substrates: a review

We review the depinning and nonequilibrium phases of collectively interacting particle systems driven over random or periodic substrates. This type of system is relevant to vortices in type-II superconductors, sliding charge density waves, electron crystals, colloids, stripe and pattern forming systems, and skyrmions, and could also have connections to jamming, glassy behaviors, and active matter. These systems are also ideal for exploring the broader issues of characterizing transient and steady state nonequilibrium flow phases as well as nonequilibrium phase transitions between distinct dynamical phases, analogous to phase transitions between different equilibrium states. We discuss the differences between elastic and plastic depinning on random substrates and the different types of nonequilibrium phases which are associated with specific features in the velocity-force curves, fluctuation spectra, scaling relations, and local or global particle ordering. We describe how these quantities can change depending on the dimension, anisotropy, disorder strength, and the presence of hysteresis. Within the moving phase we discuss how there can be a transition from a liquid-like state to dynamically ordered moving crystal, smectic, or nematic states. Systems with periodic or quasiperiodic substrates can have multiple nonequilibrium second or first order transitions in the moving state between chaotic and coherent phases, and can exhibit hysteresis. We also discuss systems with competing repulsive and attractive interactions, which undergo dynamical transitions into stripes and other complex morphologies when driven over random substrates. Throughout this work we highlight open issues and future directions such as absorbing phase transitions, nonequilibrium work relations, inertia, the role of non-dissipative dynamics such as Magnus effects, and how these results could be extended to the broader issues of plasticity in crystals, amorphous solids, and jamming phenomena.

Collective nonaffine displacements in amorphous materials during large-amplitude oscillatory shear

Using molecular dynamics simulations, we study the transient response of a binary Lennard-Jones glass subjected to periodic shear deformation. The amorphous solid is modeled as a three-dimensional Kob-Andersen binary mixture at a low temperature. The cyclic loading is applied to slowly annealed, quiescent samples, which induces irreversible particle rearrangements at large strain amplitudes, leading to stress-strain hysteresis and a drift of the potential energy towards higher values. We find that the initial response to cyclic shear near the critical strain amplitude involves disconnected clusters of atoms with large nonaffine displacements. In contrast, the amplitude of shear stress oscillations decreases after a certain number of cycles, which is accompanied by the initiation and subsequent growth of a shear band.

Colloidal polycrystalline monolayers under oscillatory shear

In this paper we probe the structural response to oscillatory shear deformations of polycrystalline monolayers of soft repulsive colloids with varying area fraction over a broad range of frequencies and amplitudes. The particles are confined at a fluid interface, sheared using a magnetic microdisk, and imaged through optical microscopy. The structural and mechanical response of soft materials is highly dependent on their microstructure. If crystals are well understood and deform through the creation and mobilization of specific defects, the situation is much more complex for disordered jammed materials, where identifying structural motifs defining plastically rearranging regions remains an elusive task. Our materials fall between these two classes and allow the identification of clear pathways for structural evolution. In particular, we demonstrate that large enough strains are able to fluidize the system, identifying critical strains that fulfill a local Lindemann criterion. Conversely, smaller strains lead to localized and erratic irreversible particle rearrangements due to the motion of structural defects. In this regime, oscillatory shear promotes defect annealing and leads to the growth of large crystalline domains. Numerical simulations help identify the population of rearranging particles with those exhibiting the largest deviatoric stresses and indicate that structural evolution proceeds towards the minimization of the stress stored in the system. The particles showing high deviatoric stresses are localized around grain boundaries and defects, providing a simple criterion to spot regions likely to rearrange plastically under oscillatory shear.

Macroscopic yielding in jammed solids is accompanied by a nonequilibrium first-order transition in particle trajectories

We use computer simulations to analyze the yielding transition during large-amplitude oscillatory shear of a simple model for soft jammed solids. Simultaneous analysis of global mechanical response and particle-scale motion demonstrates that macroscopic yielding, revealed by a smooth crossover in mechanical properties, is accompanied by a sudden change in the particle dynamics, which evolves from nondiffusive motion to irreversible diffusion as the amplitude of the shear is increased. We provide numerical evidence that this sharp change corresponds to a nonequilibrium first-order dynamic phase transition, thus establishing the existence of a well-defined microscopic dynamic signature of the yielding transition in amorphous materials in oscillatory shear.

Nonaffine rearrangements of atoms in deformed and quiescent binary glasses

The influence of periodic shear deformation on nonaffine atomic displacements in an amorphous solid is examined via molecular dynamics simulations. We study the three-dimensional Kob-Andersen binary mixture model at a finite temperature. It is found that when the material is periodically strained, most of the atoms undergo repetitive nonaffine displacements with amplitudes that are broadly distributed. We show that particles with large amplitudes of nonaffine displacements are organized into compact clusters. With increasing strain amplitude, spatial correlations of nonaffine displacements become increasingly long-ranged, although they remain present even in a quiescent system due to thermal fluctuations.