Critical properties of frictionless spherical particles below jamming are studied using extensive numerical simulations, paying particular attention to the non-affine part of the displacements during the athermal quasi-static compression. It is shown that the squared norm of the non-affine displacement exhibits a power-law divergence toward the jamming transition point. A possible connection between this critical exponent and that of the shear viscosity is discussed. The participation ratio of the displacements vanishes in the thermodynamic limit at the transition point, meaning that the non-affine displacements are localized marginally with a fractal dimension. Furthermore, the distribution of the displacement is shown to have a power-law tail, the exponent of which is related to the fractal dimension.
We consider the rheology of soft-core frictionless disks in two dimensions in the neighborhood of the athermal jamming transition. From numerical simulations of bidisperse, overdamped, particles, we argue that the divergence of the viscosity below jamming is characteristic of the hard-core limit, independent of the particular soft-core interaction. We develop a mapping from soft-core to hard-core particles that recovers all the critical behavior found in earlier scaling analyses. Using this mapping we derive a duality relation that gives the exponent of the non-linear Herschel-Bulkley rheology…
We simulate numerically the compression driven jamming of athermal, frictionless, soft-core spherocylinders in two dimensions, for a range of particle aspect ratios $α$. We find the critical packing fraction $ϕ_J(α)$ for the jamming transition, and the average number of contacts per particle $z_J(α)$ at jamming. We find that both are nonmonotonic, with a peak at $α\approx 1$. We find that configurations at the compression driven jamming point are always hypostatic for all $α$, with $z_J<z_\mathrm{iso}=2d_f=6$ the isostatic value. We show that, for moderately elongated spherocylinders, there is…
Numerical simulations of soft-core frictionless disks in two dimensions are carried out to study behavior of a simple liquid as a function of thermal temperature $T$, packing fraction $ϕ$, and uniform applied shear strain rate $\dotγ$. Inferring the hard-core limit from our soft-core results, we find that it depends on the two parameters $ϕ$ and $T/\dotγ$. $T/\dotγ\to 0$ defines the athermal limit in which a shear driven jamming transition occurs at a well defined $ϕ_J$. $T/\dotγ\to\infty$ defines the thermalized limit where an equilibrium glass transition may take place at a $ϕ_G$. This concl…
We study numerically a system of athermal, overdamped, frictionless spheres, as in a non-Brownian suspension, in two and three dimensions. Compressing the system isotropically at a fixed rate $\dotε$, we investigate the critical behavior at the jamming transition. The finite compression rate introduces a control timescale, which allows one to probe the critical timescale associated with jamming. As was found previously for steady-state shear-driven jamming, we find for compression-driven jamming that pressure obeys a critical scaling relation as a function of packing fraction $ϕ$ and compressi…
We study athermal jamming as well as the thermal glassy dynamics in systems composed of spheres that interact according to repulsive interactions that exponentially decay as a function of distance. As usual, a cutoff is employed in the simulations. While the athermal jamming transition that is determined by trying to remove overlaps is found to depend on the arbitrary and therefore unphysical choice of the cutoff, we do not find any athermal jamming transition or crossover that only relies on the physical decay length. In contrast, the glassy dynamics mainly depends on the decay length. Our fi…
Recent theoretical advances have led to the creation of a unified phase diagram for the thermal glass and athermal jamming transitions. This diagram makes clear that, while related, the mode-coupling---or dynamic---glass transition is distinct from the jamming transition, occurring at a finite temperature and significantly lower density than the jamming transition. Nonetheless, we demonstrate a pre-jamming transition in athermal frictionless spheres which occurs at the same density as the mode-coupling transition and is marked by percolating clusters of locally rigid particles. At this density…
We use computer simulations to study the microscopic dynamics of an athermal assembly of soft particles near the fluid-to-solid, jamming transition. Borrowing tools developed to study dynamic heterogeneity near glass transitions, we discover a number of original signatures of the jamming transition at the particle scale. We observe superdiffusive, spatially heterogeneous, and collective particle motion over a characteristic scale which displays a surprising non-monotonic behavior across the transition. In the solid phase, the dynamics is an intermittent succession of elastic deformations and p…
A new method is introduced to simulate jamming of polyhedral grains under controlled stress that incorporates global degrees of freedom through the metric tensor of a periodic cell containing grains. Jamming under hydrostatic/isotropic stress and athermal conditions leads to a precise definition of the ideal jamming point at zero shear stress. The structures of tetrahedra jammed hydrostatically exhibit less translational order and lower jamming-point density than previously described `maximally random jammed' hard tetrahedra. Under the same conditions, cubes jam with negligible nematic order. …
While the glass transition at non-zero temperature seems to be hard to access for experimental, theoretical, or simulation studies, jamming at zero temperature has been explored in great detail. It is a widely discussed question whether this athermal jamming transition is related to the glass transition. Motivated by the exploration of the energy landscape that has been successfully used to describe athermal jamming, we introduce a new method to determine whether the configuration space of a soft sphere system can be explored within a reasonable timescale or not, i.e., whether the system is er…
Using confocal microscopy we investigate the effect of attraction on the packing of polydisperse emulsions under gravity. We find that the distributions of neighbors, coordination number, and local packing fraction as a function of attraction are captured by recently proposed geometrical modeling and statistical mechanics approaches to granular matter. This extends the range of applicability of these tools to polydisperse, attractive jammed packings. Furthermore, the dependence of packing density and average coordination number on the strength of attraction provides the first experimental test…
We carry our numerical simulations of athermally sheared, bidisperse, frictionless disks in two dimensions. From an appropriately defined velocity correlation function, we determine that there are two diverging length scales, $ξ$ and $\ell$, as the jamming transition is approached. We analyze our results using a critical scaling ansatz for the correlation function, and argue that the more divergent length $\ell$ is a consequence of a dangerous irrelevant scaling variable, and that it is $ξ$ which is the correlation length that determines the divergence of the system viscosity as jamming is app…
We report recent results of molecular dynamics simulations of frictional athermal particles at constant volume fraction and constant applied shear stress, focusing on a range of control parameters where the system first flows, but then jams after a time tjam. On decreasing the volume fraction, the mean jamming time diverges, while its sample fluctuations become so large that the jamming time probability distribution P(tjam) becomes a power-law. We obtain an insight on the origin of this phenomenology focusing on the flowing regime, which is characterized by the presence of a clear correlation …
We perform numerical simulations of athermal repulsive frictionless disks and spheres in two and three spatial dimensions undergoing cyclic quasi-static simple shear to investigate particle-scale reversible motion. We identify three classes of steady-state dynamics as a function of packing fraction ϕand maximum strain amplitude per cycle γ_{\rm max}. Point-reversible states, where particles do not collide and exactly retrace their intra-cycle trajectories, occur at low ϕand γ_{\rm max}. Particles in loop-reversible states undergo numerous collisions and execute complex trajectories, but return…
We use numerical simulations to study the flow of a bidisperse mixture of athermal, frictionless, soft-core two dimensional spherocylinders driven in uniform steady state shear. Energy dissipation is via a viscous drag with respect to a uniformly sheared host fluid, giving a model for a non-Brownian suspension with a Newtonian rheology. We study pressure $p$ and deviatoric shear stress $σ$ as a function of packing fraction $ϕ$, strain rate $\dotγ$, and a parameter $α$ that measures the asphericity of the particles. We consider the anisotropy of the stress tensor, the macroscopic friction $μ=σ/…
We derive exact results for displacement fields that develop as a response to external pinning forces in two dimensional athermal networks. For a triangular lattice arrangement of particles interacting through soft potentials, we develop a Green's function formalism which we use to derive exact results for displacement fields produced by localized external forces. We show that in the continuum limit the displacement fields decay as $1/r$ at large distances $r$ away from a force dipole. Finally, we extend our formulation to study correlations in the displacement fields produced by the external …
We present the study of the landscape structure of athermal soft spheres both as a function of the packing fraction and of the energy. We find that, on approaching the jamming transition, the number of different configurations available to the system has a steep increase and that a hierarchical organization of the landscape emerges. We use the knowledge of the structure of the landscape to predict the values of thermodynamic observables on the edge of the transition.
By exploring the properties of the energy landscape of a bidisperse system of soft harmonic disks in two dimensions we determine the thermal jamming transition. To be specific, we study whether the ground state of the system where the particle do not overlap can be reached within a reasonable time. Starting with random initial configurations, the energy landscape is probed by energy minimization steps as in case of athermal jamming and in addition steps where an energy barrier can be crossed with a small but non-zero probability. For random initial conditions we find that as a function of pack…
We derive exact results for correlations in the displacement fields $\{ δ\vec{r} \} \equiv \{ δr_{μ= x,y} \}$ in near-crystalline athermal systems in two dimensions. We analyze the displacement correlations produced by different types of microscopic disorder, and show that disorder at the microscopic scale gives rise to long-range correlations with a dependence on the system size $L$ given by $\langle δr_μ δr_ν \rangle \sim c_{μν}(r/L,θ)$. In addition, we show that polydispersity in the constituent particle sizes and random bond disorder give rise to a logarithmic system size scaling, with $c_…
Recent studies of athermal systems such as dry grains and dense, non-Brownian suspensions have shown that shear can lead to solidification through the process of shear jamming in grains and discontinuous shear thickening in suspensions. The similarities observed between these two distinct phenomena suggest that the physical processes leading to shear-induced rigidity in athermal materials are universal. We present a non-equilibrium statistical mechanics model, which exhibits the phenomenology of these shear-driven transitions: shear jamming and discontinuous shear thickening in different regio…
We study relaxation and rheology of dense athermal suspensions of frictionless particles close below the jamming density. Our key quantity, the relaxation time---determined from the exponential decay of the energy after the shearing has suddenly been switched off---is argued to be a determining factor behind the algebraic divergence of various quantities as the jamming density is approached from below. We also define and measure the ``dissipation time'', which is obtained directly in shearing simulations and find that it behaves similarly to the relaxation time. Comparing shear viscosity with …
A mechanically-based structural optimization method is utilized to explore the phenomena of jamming for assemblies of frictionless Platonic solids. Systems of these regular convex polyhedra exhibit mechanically stable phases with density substantially less than optimal for a given shape, revealing that thermal motion is necessary to access high density phases. We confirm that the large system jamming threshold of $0.623 \pm 0.003$ for tetrahedra is consistent with experiments on tetrahedral dice. Also, the extremely short-ranged translational correlations of packed tetrahedra observed in exper…
We introduce a perturbation expansion for athermal systems that allows an exact determination of displacement fields away from the crystalline state as a response to disorder. We show that the displacement fields in energy minimized configurations of particles interacting through central potentials with microscopic disorder, can be obtained as a series expansion in the strength of the disorder. We introduce a hierarchy of force balance equations that allows an order-by-order determination of the displacement fields, with the solutions at lower orders providing sources for the higher order solu…
We study how solidification of model freely rotating polymers under athermal quasistatic compression varies with their bond angle $θ_0$. All systems undergo two discrete, first-order-like transitions: entanglement at $ϕ= ϕ_E(θ_0)$ followed by jamming at $ϕ= ϕ_J(θ_0) \simeq (4/3 \pm 1/10)ϕ_E(θ_0)$. For $ϕ< ϕ_E(θ_0)$, systems are in a "gas" phase wherein all chains remain free to translate and reorient. For $ϕ_E(θ_0) \leq ϕ\leq ϕ_J(θ_0)$, systems are in a liquid-like phase wherein chains are entangled. In this phase, chains' rigid-body-like motion is blocked, yet they can still locally relax via…
We numerically study the local stress distribution within athermal, isotropically stressed, mechanically stable, packings of bidisperse frictionless disks above the jamming transition in two dimensions. Considering the Fourier transform of the local stress, we find evidence for algebraically increasing fluctuations in both isotropic and anisotropic components of the stress tensor at small wavenumbers, contrary to recent theoretical predictions. Such increasing fluctuations imply a lack of self-averaging of the stress on large length scales. The crossover to these increasing fluctuations define…
When an amorphous solid is deformed homogeneously, the response exhibits heterogeneous plastic instabilities with localized cooperative rearrangement of cluster of particles. The heterogeneous behavior plays an important role in deciding the mechanical properties of amorphous solids. In this paper, we employ computer simulation to study the characteristics and the spatial correlations of these clusters characterized by the non-affine displacements in amorphous solids under simple shear deformation in the athermal quasistatic limit. The clusters with large displacements are found to be homogene…
Typical quasistatic compression algorithms for generating jammed packings of athermal, purely repulsive particles begin with dilute configurations and then apply successive compressions with relaxation of the elastic energy allowed between each compression step. It is well-known that during isotropic compression athermal systems with purely repulsive interactions undergo a jamming transition at packing fraction $ϕ_J$ from an unjammed state with zero pressure to a jammed, rigid state with nonzero pressure. Using extensive computer simulations, we show that a novel second-order-like transition, …
We analyze fluctuations of particle displacements and stresses in a sheared athermal suspension of elastic capsules (red blood cells). Upon variation of the volume fraction from the dilute up to the highly concentrated regime, our numerical simulations reveal different characteristic power-law regimes of the fluctuation variances and relaxation times. In the jammed phase and at high shear rates, anomalous scaling exponents are found that deviate from pure dimensional predictions. The observed behavior is rationalized via kinetic arguments and a dissipation balance model that takes into account…
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 (vDOS), which is a key ingredient of linear viscoelastic theory. The vDOS e…
We numerically investigate non-local effects on inhomogeneous flows of soft athermal disks close to but below their jamming transition. We employ molecular dynamics to simulate Kolmogorov flows, in which a sinusoidal flow profile with fixed wave number is externally imposed, resulting in a spatially inhomogeneous shear rate. We find that the resulting rheology is strongly wave number-dependent, and that particle migration, while present, is not sufficient to describe the resulting stress profiles within a conventional local model. We show that, instead, stress profiles can be captured with non…
We apply principal component analysis, a method frequently used in image processing and unsupervised machine learning, to characterize particle displacements observed in the steady shear flow of amorphous solids. PCA produces a low-dimensional representation of the data and clearly reveals the dominant features of elastic (i.e. reversible) and plastic deformation. We show that the principal directions of PCA in the plastic regime correspond to the soft (i.e. zero energy) modes of the elastic propagator that governs the redistribution of shear stress due to localized plastic events. Projections…
We conduct athermal simulations of freely-cooling, viscous soft spheres around the jamming transition density ϕ_{J}, and find evidence for a growing length ξ(t) that governs relaxation to mechanical equilibrium. ξ(t) is manifest in both the velocity correlation function, and the spatial correlations in a scalar measure of local force balance which we define. Data for different densities ϕcan be collapsed onto two master curves by scaling ξ(t) and t by powers of |ϕ-ϕ_{J}|, indicative of critical scaling. Furthermore, particle transport for ϕ>ϕ_{J} exhibits aging and superdiffusion similar to a …
We numerically study a three dimensional system of athermal, overdamped, frictionless spheres, using a simplified model for a non-Brownian suspension. We compute the bulk viscosity under both uniaxial and isotropic compression as a means to address the question of whether stress-anisotropic and stress-isotropic jamming are in the same critical universality class. Carrying out a critical scaling analysis of the system pressure $p$, shear stress $σ$, and macroscopic friction $μ=σ/p$, as functions of particle packing fraction $ϕ$ and compression rate $\dotε$, we find good agreement for all critic…
The current microscopic picture of plasticity in amorphous materials assumes local failure events to produce displacement fields complying with linear elasticity. Indeed, the flow properties of nonaffine systems such as foams, emulsions and granular materials close to jamming, that produce a fluctuating displacement fields when failing, are still controversial. Here we show, via a thorough numerical investigation of jammed materials, that nonaffinity induces a critical scaling of the flow properties dictated by the distance to the jamming point. We rationalize this critical behavior introducin…
In non-equilibrium molecular dynamics simulations of dense athermal shear flows, we observe the transition from shear thinning to shear thickening at a crossover shear rate $\dotγ_c$. Shear thickening occurs when $\frac{{\rm d (ln} T_g)}{{\rm d (ln}\dotγ)}>2$ with $T_g$ the granular temperature. At the transition, the pair distribution function shows the strongest anisotropy. Meanwhile, the dynamics undergo apparent changes, signified by distinct scaling behaviors of the mean squared displacement and relaxation time on both sides of $\dotγ_c$. These features serve as robust signatures of the s…
Dynamics of jammed packings of soft athermal disks under finite-rate shear are studied by means of molecular dynamics simulations. Particularly, we investigate the spatial structures of stress drop events, which are expected to provide information about plasticity. Investigating the displacement fields during stress drop events, we show that there are qualitatively different two types of events in the low rate limit: localized ones and extended ones. We further investigate the time evolution of events and clarify that both types of events are due to oscillatory motion of the stress, which is u…
We study agitated frictional disks in two dimensions with the aim of developing a scaling theory for their diffusion over time. As a function of the area fraction $ϕ$ and mean-square velocity fluctuations $\langle v^2\rangle$ the mean-square displacement of the disks $\langle d^2\rangle$ spans 4-5 orders of magnitude. The motion evolves from a subdiffusive form to a complex diffusive behabvior at long times. The statistics of $\langle d^n\rangle$ at all times are multiscaling, since the probability distribution function (pdf) of displacements has very broad wings. Even where a diffusion consta…
We measure the displacement response of a frictional multicontact interface between identical polymer glasses to a biased shear force oscillation. We evidence the existence, for maximum forces close below the nominal static threshold, of a jamming creep regime governed by an ageing-rejuvenation competition acting within the micrometer-sized contacting asperities. Quantitative analysis of the creep curves suggests that another such mechanism might be at work within the nanometer-thick adhesive junctions.
Critical behavior of soft repulsive particles after quench of temperature near the jamming trasition is numerically investigated. It is found that the plateau of the mean square displacement of tracer particles and the pressure satisfy critical scaling laws. The critical density for the jamming transition depends on the protocol to prepare the system, while the values of the critical exponents which are consistent with the prediction of a phenomenology are independent of the protocol.
The granular jamming transition is experimentally investigated in a two-dimensional system of frictional, bi-dispersed disks subject to quasi-static, uniaxial compression at zero granular temperature. Currently accepted results show the jamming transition occurs at a critical packing fraction $ϕ_c$. In contrast, we observe the first compression cycle exhibits {\it fragility} - metastable configuration with simultaneous jammed and un-jammed clusters - over a small interval in packing fraction ($ϕ_1 < ϕ< ϕ_2$). The fragile state separates the two conditions that define $ϕ_c$ with an exponential …
Geometrical properties of two-dimensional mixtures near the jamming transition point are numerically investigated using harmonic particles under mechanical training. The configurations generated by the quasi-static compression and oscillatory shear deformations exhibit anomalous suppression of the density fluctuations, known as hyperuniformity, below and above the jamming transition. For the jammed system trained by compression above the transition point, the hyperuniformity exponent increases. For the system below the transition point under oscillatory shear, the hyperuniformity exponent also…
Shearing granular materials induces non-affine displacements. Such non-affine displacements have been studied extensively, and are known to correlate with plasticity and other mechanical features of amorphous packings. A well known example is shear transformation zones as captured by the local deviation from affine deformation, $D^2_{min}$, and their relevance to failure and stress fluctuations. We analyze sheared frictional athermal disk packings and show that there exists at least one additional mesoscopic transport mechanism that superimposes itself on top of local diffusive motion. We evid…
Many active materials and biological systems are driven far from equilibrium by embedded agents that spontaneously generate forces and distort the surrounding material. Probing and characterizing these athermal fluctuations is essential for understanding the properties and behaviors of such systems. Here we present a mathematical procedure to estimate the local action of force-generating agents from the observed fluctuating displacement fields. The active agents are modeled as oriented force dipoles or isotropic compression foci, and the matrix on which they act is assumed to be either a compr…
