We numerically study the relaxation dynamics and associated criticality of non-Brownian frictionless spheres below jamming in spatial dimensions $d=2$, $3$, $4$, and $8$, and in the mean-field Mari-Kurchan model. We discover non-trivial finite-size and volume fraction dependences of the relaxation time associated to the relaxation of unjammed packings. In particular, the relaxation time is shown to diverge logarithmically with system size at any density below jamming, and no critical exponent can characterise its behaviour approaching jamming. In mean-field, the relaxation time is instead well-defined: it diverges at jamming with a critical exponent that we determine numerically and differs from an earlier mean-field prediction. We rationalise the finite $d$ logarithmic divergence using an extreme-value statistics argument in which the relaxation time is dominated by the most connected region of the system. The same argument shows that the earlier proposition that relaxation dynamics and shear viscosity are directly related breaks down in large systems. The shear viscosity of non-Brownian packings is well-defined in all $d$ in the thermodynamic limit, but large finite-size effects plague its measurement close to jamming.
We determine the form of the complex shear modulus $G^*$ in soft sphere packings near jamming. Viscoelastic response at finite frequency is closely tied to a packing's intrinsic relaxational modes, which are distinct from the vibrational modes of undamped packings. We demonstrate and explain the appearance of an anomalous excess of slowly relaxing modes near jamming, reflected in a diverging relaxational density of states. From the density of states, we derive the dependence of $G^*$ on frequency and distance to the jamming transition, which is confirmed by numerics.
Like other critical phenomena, the jamming transition accompanies the divergence of the relaxation time $τ$. A recent numerical study of frictionless spherical particles proves that $τ$ is inversely proportional to the lowest non-zero eigenvalue $λ_1$ of the dynamical matrix. In this note, we derive the scaling of $λ_1$ below the jamming transition point $φ_J$ by solving the linearized dynamical equation. The resultant critical exponent agrees with a previous theoretical result for sheared suspension obtained by applying the virtual work theorem to a simple shear, highlighting the universality…
We show that non-Brownian suspensions of repulsive spheres below jamming display a slow relaxational dynamics with a characteristic time scale that diverges at jamming. This slow time scale is fully encoded in the structure of the unjammed packing and can be readily measured via the vibrational density of states. We show that the corresponding dynamic critical exponent is the same for randomly generated and sheared packings. Our results show that a wide variety of physical situations, from suspension rheology to algorithmic studies of the jamming transition are controlled by a unique diverging…
The dynamical mechanisms controlling the rheology of dense suspensions close to jamming are investigated numerically, using simplified models for the relevant dissipative forces. We show that the velocity fluctuations control the dissipation rate and therefore the effective viscosity of the suspension. These fluctuations are similar in quasi-static simulations and for finite strain rate calculations with various damping schemes. We conclude that the statistical properties of grain trajectories -- in particular the critical exponent of velocity fluctuations with respect to volume fraction ϕ-- o…
We introduce a stochastic microscopic model to investigate the jamming and reorganization of grains induced by an object moving through a granular medium. The model reproduces the experimentally observed periodic sawtooth fluctuations in the jamming force and predicts the period and the power spectrum in terms of the controllable physical parameters. It also predicts that the avalanche sizes, defined as the number of displaced grains during a single advance of the object, follow a power-law, $P(s)\sim s^{-τ}$, where the exponent is independent of the physical parameters.
We add relaxation mechanisms that mimic the effect of temperature and non-equilibrium driving to the recently-proposed spiral model which jams at a critical density rho_c < 1. This enables us to explore unjamming by temperature or driving at rho_c < rho < 1. We numerically calculate the relaxation time of the persistence function and its spatial heterogeneity. We disentangle the three different relaxation mechanisms responsible for unjamming when varying density, temperature, and driving strength, respectively. We show that the spatial scale of dynamic heterogeneity depends on density much mor…
The reversible aggregates formation in a shear thickening, concentrated colloidal suspension is investigated through speckle visibility spectroscopy, a dynamic light scattering technique recently introduced [P.K. Dixon and D.J. Durian, Phys. Rev. Lett. 90, 184302 (2003)]. Formation of particles aggregates is observed in the jamming regime, and their relaxation after shear cessation is monitored as a function of the applied shear stress. The aggregates relaxation time increases when a larger stress is applied. Several phenomena have been proposed to interpret this behavior: an increase of the a…
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 investigate the statistical mechanics of glass-forming materials and jamming matter by means of a geometrically driven approach based on a revised cell theory. By considering the system as constituted of jammed blocks of increasing sizes, we obtain a unified picture that describes accurately the whole process from low densities to limit densities at the glass/jamming transition. The approach retrieves many of the aspects of existing theories unifying them into a coherent framework. In particular, at low densities we find a free volume regime, based on local relaxation process, at intermedia…
A systematic microscopic theory for the rheology of dense non-Brownian suspensions characterized by the volume fraction $φ$ is developed. The theory successfully derives the critical behavior in the vicinity of the jamming point (volume fraction $φ_{J}$), for both the pressure $P$ and the shear stress $σ_{xy}$, i.e. $P \sim σ_{xy} \sim \dotγη_0 δφ^{-2}$, where $\dotγ$ is the shear rate, $η_0$ is the shear viscosity of the solvent, and $δφ= φ_J - φ> 0$ is the distance from the jamming point. It also successfully describes the behavior of the stress ratio $μ= σ_{xy}/P$ with respect to the viscou…
In marginally jammed solids confined by walls, we calculate the particle and ensemble averaged value of an order parameter, $\left<Ψ(r)\right>$, as a function of the distance to the wall, $r$. Being a microscopic indicator of structural disorder and particle mobility in solids, $Ψ$ is by definition the response of the mean square particle displacement to the increase of temperature in the harmonic approximation and can be directly calculated from the normal modes of vibration of the zero-temperature solids. We find that, in confined jammed solids, $\left<Ψ(r)\right>$ curves at different pressu…
We have explored isotropically jammed states of semi-2D granular materials through cyclic compression. In each compression cycle, systems of either identical ellipses or bi-disperse disks, transition between jammed and unjammed states. We determine the evolution of the average pressure, P, and structure through consecutive jammed states. We observe a transition point, ϕm, above which P persists over many cycles; below ϕm, P relaxes slowly. The relaxation time scale associated with P increases with packing fraction, while the relaxation time scale for collective particle motion remains constant…
We investigate the dynamics of soft sphere liquids through computer simulations for spatial dimensions from $d =3$ to $8$, over a wide range of temperatures and densities. Employing a scaling of density-temperature dependent relaxation times, we precisely identify the density $ϕ_0$ which marks the ideal glass transition in the hard sphere limit, and a crossover from sub- to super-Arrhenius temperature dependence. The difference between $ϕ_0$ and the athermal jamming density $ϕ_J$, small in 3 and 4 dimensions, increases with dimension, with $ϕ_0 > ϕ_J$ for $d > 4$. We compare our results with r…
Discontinuous shear-thickening in dense suspensions naturally emerges from the activation of frictional forces by shear flow in non-Brownian systems close to jamming. Yet, this physical picture is incomplete as most experiments study soft colloidal particles subject to thermal fluctuations. To characterise discontinuous shear-thickening in colloidal suspensions we use computer simulations to provide a complete description of the competition between athermal jamming, frictional forces, thermal motion, particle softness, and shear flow. We intentionally neglect hydrodynamics, electrostatics, lub…
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…
The dynamics of the jamming transition in a three-dimensional granular system under vertical vibration is studied using diffusing-wave spectroscopy. When the maximum acceleration of the external vibration is large, the granular system behaves like a fluid, with the dynamic correlation function G(t) relaxing rapidly. As the acceleration of vibration approaches the gravitational acceleration g, the relaxation of G(t) slows down dramatically, and eventually stops. Thus the system undergoes a phase transition and behaves like a solid. Near the transition point, we find that the structural relaxati…
We study experimentally the relaxation towards mechanical equilibrium of a granular pile which has just experienced an avalanche and discuss it in the more general context of the granular jamming transition. Two coexisting dynamics are observed in the surface layer: a short time exponential decay consisting in rapid and independent moves of grains and intermittent bursts consisting in spatially correlated moves lasting for longer time. The competition of both dynamics results in long-lived intermittent transients, the total duration of which can late more than a thousand of seconds. We measure…
Rheological properties of dense flows of hard particles are singular as one approaches the jamming threshold where flow ceases, both for aerial granular flows dominated by inertia, and for over-damped suspensions. Concomitantly, the lengthscale characterizing velocity correlations appears to diverge at jamming. Here we introduce a theoretical framework that proposes a tentative, but potentially complete scaling description of stationary flows. Our analysis, which focuses on frictionless particles, applies {\it both} to suspensions and inertial flows of hard particles. We compare our prediction…
We use large scale computer simulations and finite size scaling analysis to study the shear rheology of dense three-dimensional suspensions of frictionless non-Brownian particles in the vicinity of the jamming transition. We perform simulations of soft repulsive particles at constant shear rate, constant pressure, and finite system size, and study carefully the asymptotic limits of large system sizes and infinitely hard particle repulsion. Extending earlier analysis by about two orders of magnitude, we first study the asymptotic behavior of the shear viscosity in the hard particle limit. We co…
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 …
Static and dynamic properties of two-dimensional bidisperse dissipative particles are numerically studied near the jamming transition. We investigate the dependency of the critical scaling on the ratio of the different diameters and find a new scaling of the maximum overlap (not consistent with the scaling of the mean overlap). The ratio of kinetic and potential energies drastically slows down near the jamming transition, i.e. the relaxation time diverges at the jamming point.
We simulate a relaxation process of non-brownian particles in a sheared viscous medium; the small shear strain is initially applied to a system, which then undergoes relaxation. The relaxation time and the correlation length are estimated as functions of density, which algebraically diverge at the jamming density. This implies that the relaxation time can be scaled by the correlation length using the dynamic critical exponent, which is estimated as 4.6(2). It is also found that shear stress undergoes power-law decay at the jamming density, which is reminiscent of critical slowing down.
We present experiments on several distinct effective temperatures in a granular system at a sequence of increasing packing densities and at a sequence of decreasing driving rates. This includes single-grain measurements based on the mechanical energies of both the grains and an embedded oscillator, as well as a collective measurement based on the Einstein relation between diffusivity and mobility, which all probe different time scales. Remarkably, all effective temperatures agree. Furthermore, mobility data along the two trajectories collapse when plotted vs effective temperature and exhibit 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 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 startup experiments in athermal soft sphere packings, the canonical model for jamming. By systematical…
Particles suspended in a Newtonian fluid raise the viscosity and also generally give rise to a shear-rate dependent rheology. In particular, pronounced shear thickening may be observed at large solid volume fractions. In a recent article (R. Seto, R. Mari, J. F. Morris, and M. M. Denn., Phys. Rev. Lett., 111:218301, 2013) we have considered the minimum set of components to reproduce the experimentally observed shear thickening behavior, including Discontinuous Shear Thickening (DST). We have found frictional contact forces to be essential, and were able to reproduce the experimental behavior b…
Using Brownian Dynamics computer simulations we show that the relaxation of a supercooled Brownian system is qualitatively the same as that of a Newtonian system. In particular, near the so-called mode-coupling transition temperature, dynamic properties of the Brownian system exhibit the same deviations from power-law behavior as those of the Newtonian one. Thus, similar dynamical events cut off the idealized mode-coupling transition in Brownian and Newtonian systems. We discuss implications of this finding for extended mode-coupling theory. In addition, we point out and discuss the difference…
Amorphous solids, such as glasses, have complex responses to deformations, with significant consequences in material design and applications. In this respect two intertwined aspects are important: stability and reversibility. It is crucial to understand on the one hand how a glass may become unstable due to increased plasticity under shear deformations; on the other hand, to what extent the response is reversible, meaning how much a system is able to recover the original configuration once the perturbation is released. Here we focus on assemblies of hard spheres as the simplest model of amorph…
When weakly jammed packings of soft, viscous, non-Brownian spheres are probed mechanically, they respond with a complex admixture of elastic and viscous effects. While many of these effects are understood for specific, approximate models of the particles' interactions, there are a number of proposed force laws in the literature, especially for viscous interactions. We numerically measure the complex shear modulus $G^*$ of jammed packings for various viscous force laws that damp relative velocities between pairs of contacting particles or between a particle and the continuous fluid phase. We fi…
We generalize the force-level Nonlinear Langevin Equation theory of single particle hopping to include collective effects associated with long range elastic distortion of the liquid. The activated alpha relaxation event is of a mixed spatial character, involving two distinct, but inter-related, local and collective barriers. There are no divergences at volume fractions below jamming or temperatures above zero Kelvin. The ideas are first developed and implemented analytically and numerically in the context of hard sphere fluids. In an intermediate volume fraction crossover regime, the local cag…
We show the linear response theory of spatial-scale-dependent relaxation moduli for overdamped Brownian particle systems. We employ the Irving-Kirkwood stress tensor field as the microscopic stress tensor field. We show that the scale-dependent relaxation modulus tensor, which characterizes the response of the stress tensor field to the applied velocity gradient field, can be expressed by using the correlation function of the Irving-Kirkwood stress tensor field. The spatial Fourier transform of the relaxation modulus tensor gives the wavenumber-dependent relaxation modulus. For isotropic and h…
We solved the mode-coupling equations for the Kob-Andersen binary mixture using the structure factors calculated from Brownian dynamics simulations of the same system. We found, as was previously observed, that the mode-coupling temperature, Tc, inferred from simulations is about two times greater than that predicted by the theory. However, we find that many time dependent quantities agree reasonably well with the predictions of the mode-coupling theory if they are compared at the same reduced temperature epsilon = (T-Tc)/Tc, and if epsilon is not too small. Specifically, the simulation result…
We investigate the dynamics of ice mélange by analyzing rapid motion recorded by a time-lapse camera and terrestrial radar during several calving events that occurred at Jakobshavn Isbræ, Greenland. During calving events (1) the kinetic energy of the ice mélange is two orders of magnitude smaller than the total energy released during the events, (2) a jamming front propagates through the ice mélange at a rate that is an order of magnitude faster than the motion of individual icebergs, (3) the ice mélange undergoes initial compaction followed by slow relaxation and extension, and (4) motion of …
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…
We present a new formulation of the jamming phase diagram for a class of glass-forming fluids consisting of spheres interacting via finite-ranged repulsions at temperature $T$, packing fraction $ϕ$ or pressure $p$, and applied shear stress $Σ$. We argue that the natural choice of axes for the phase diagram are the dimensionless quantities $T/pσ^3$, $pσ^3/ε$, and $Σ/p$, where $T$ is the temperature, $p$ is the pressure, $Σ$ is the stress, $σ$ is the sphere diameter, $ε$ is the interaction energy scale, and $m$ is the sphere mass. We demonstrate that the phase diagram is universal at low $pσ^3/ε…
We investigate the persistence probability of a Brownian particle in a harmonic potential, which decays to zero at long times -- leading to an unbounded motion of the Brownian particle. We consider two functional forms for the decay of the confinement, an exponential and an algebraic decay. Analytical calculations and numerical simulations show, that for the case of the exponential relaxation, the dynamics of Brownian particle at short and long times are independent of the parameters of the relaxation. On the contrary, for the algebraic decay of the confinement, the dynamics at long times is d…
Many disordered systems experience a transition from a fluid-like state to a solid-like state following a sudden arrest in dynamics called jamming. In contrast to jamming in spatially homogeneous systems, jamming in hoppers occurs under extremely inhomogeneous conditions as the gravity-driven flow of grains enclosed by rigid walls converges towards a small opening. In this work, we study velocity fluctuations in a collisional flow near jamming using event-driven simulations. The average flow in a hopper geometry is known to have strong gradients, especially near the walls and the orifice. We f…
A consensus is emerging that discontinuous shear thickening (DST) in dense suspensions marks a transition from a flow state where particles remain well separated by lubrication layers, to one dominated by frictional contacts. We show here that reasonable assumptions about contact proliferation predict two distinct types of DST in the absence of inertia. The first occurs at densities above the jamming point of frictional particles; here the thickened state is completely jammed and (unless particles deform) cannot flow without inhomogeneity or fracture. The second regime shows strain- rate hyste…
Granular materials jam when developing a network of contact forces able to resist the applied stresses. Through numerical simulations of the dynamics of the jamming process, we show that the jamming transition does not occur when the kinetic energy vanishes. Rather, as the system jams, the kinetic energy becomes dominated by rattlers particles, that scatter withing their cages. The relaxation of the kinetic energy in the jammed configuration exhibits a double power-law decay, which we interpret in terms of the interplay between backbone and rattlers particles.
We show that a system of hard discs confined to a narrow channel exhibits a fragile-strong fluid crossover located at the maximum of the isobaric heat capacity and that the relaxation times for different channel widths fall onto a single master curve when rescaled by the relaxation times and temperatures of the crossover. Calculations of the configurational entropy and the inherent structure equation of state find that the crossover is related to properties of the jamming landscape for the model but that the Adams-Gibbs relation does not predict the relaxation behavior. We also show that a fac…
While the rheology of non-Brownian suspensions in the dilute regime is well-understood, their behavior in the dense limit remains mystifying. As the packing fraction of particles increases, particle motion becomes more collective, leading to a growing length scale and scaling properties in the rheology as the material approaches the jamming transition. There is no accepted microscopic description of this phenomenon. However, in recent years it has been understood that the elasticity of simple amorphous solids is governed by a critical point, the unjamming transition where the pressure vanishes…
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…
Unified Microscopic Theory of Stress Relaxation, Structural Evolution, and Memory Effects in Dense Glass Forming Brownian Suspensions After Flow Cessation
University of Illinois（伊利诺伊大学）；Department of Materials Science and Engineering, University of Illinois（伊利诺伊大学材料科学与工程系）；Materials Research Laboratory, University of Illinois（伊利诺伊大学材料研究实验室）；Department of Chemistry, University of Illinois（伊利诺伊大学化学系）；Department of Chemical & Biomolecular Engineering, University of Illinois（伊利诺伊大学化学与生物分子工程系）
We carry out overdamped simulations in a simple model of jamming - a collection of bi-disperse soft core frictionless disks in two dimensions - with the aim to explore the finite size dependence of different quantities, both the relaxation time obtained from the relaxation of the energy and the pressure-equivalent of the shear viscosity. The motivation for the paper is the observation [Nishikawa et al., J. Stat. Phys, 182, 37 (2021)] that there are finite size effects in the relaxation time, $τ$, that give problems in the determination of the critical divergence, and the claim that this is due…
The yet virtually unexplored class of soft colloidal rods with small aspect ratio is investigated and shown to exhibit a very rich phase and dynamic behavior, spanning from liquid to nearly melt state. Instead of nematic order, these short and soft nanocylinders alter their organization with increasing concentration from isotropic liquid with random orientation to one with preferred local orientation and eventually a multi-domain arrangement with local orientational order. The latter gives rise to a kinetically suppressed state akin to structural glass with detectable terminal relaxation, whic…
