We carry out a finite size scaling analysis of the jamming transition in frictionless bi-disperse soft core disks in two dimensions. We consider two different jamming protocols: (i) quench from random initial positions, and (ii) quasistatic shearing. By considering the fraction of jammed states as a function of packing fraction for systems with different numbers of particles, we determine the spatial correlation length critical exponent $ν\approx 1$, and show that corrections to scaling are crucial for analyzing the data. We show that earlier numerical results yielding $ν<1$ are due to the improper neglect of these corrections.
The jamming transition of particles with finite-range interactions is characterized by a variety of critical phenomena, including power law distributions of marginal contacts. We numerically study a recently proposed simple model of jamming, which is conjectured to lie in the same universality class as the jamming of spheres in all dimensions. We extract numerical estimates of the critical exponents, θ = 0.451 $\pm$ 0.006 and γ = 0.404 $\pm$ 0.004, that match the exponents observed in sphere packing systems. We analyze finite-size scaling effects that manifest in a subcritical cutoff regime an…
The critical rheology of sheared frictional granular materials near jamming transition is numer- ically investigated. It is confirmed that there exist a true critical density which characterizes the onset of the yield stress, and two fictitious critical densities which characterize the scaling laws of rheological properties. We find the existence of a hysteresis loop between two of the critical densities for each friction coefficient. It is noteworthy that the critical scaling law for frictionless jamming transition seems to be still valid even for frictional jamming despite using fictitious c…
The existence of universal scaling in the vicinity of the jamming transition of sheared granular materials is predicted by a phenomenology. The critical exponents are explicitly determined, which are independent of the spatial dimension. The validity of the theory is verified by the molecular dynamics simulation.
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 the origin of the scaling behavior in frictionless granular media above the jamming transition by analyzing their linear response. The response to local forcing is non-self-averaging and fluctuates over a length scale that diverges at the jamming transition. The response to global forcing becomes increasingly non-affine near the jamming transition. This is due to the proximity of floppy modes, the influence of which we characterize by the local linear response. We show that the local response also governs the anomalous scaling of elastic constants and contact number.
We discuss the physics of crystals with small polydispersity near the jamming transition point. For this purpose, we introduce an effective single-particle model taking into account the nearest neighbor structure of crystals. The model can be solved analytically by using the replica method in the limit of large dimensions. In the absence of polydispersity, the replica symmetric solution is stable until the jamming transition point, which leads to the standard scaling of perfect crystals. On the contrary, for finite polydispersity, the model undergoes the full replica symmetry breaking (RSB) tr…
We carry out numerical simulations to study transport behavior about the jamming transition of a model granular material in two dimensions at zero temperature. Shear viscosity ηis computed as a function of particle volume density ρand applied shear stress σ, for diffusively moving particles with a soft core interaction. We find an excellent scaling collapse of our data as a function of the scaling variable σ/|ρ_c-ρ|^Δ, where ρ_c is the critical density at σ=0 ("point J"), and Δis the crossover scaling critical exponent. Our results show that jamming is a true critical phenomenon, extending to …
We present an analysis of finite-size effects in jammed packings of N soft, frictionless spheres at zero temperature. There is a 1/N correction to the discrete jump in the contact number at the transition so that jammed packings exist only above isostaticity. As a result, the canonical power-law scalings of the contact number and elastic moduli break down at low pressure. These quantities exhibit scaling collapse with a non-trivial scaling function, demonstrating that the jamming transition can be considered a phase transition. Scaling is achieved as a function of N in both 2 and 3 dimensions,…
Extensive numerical simulations in the past decades proved that the critical exponents of the jamming of frictionless spherical particles remain unchanged in two and three dimensions. This implies that the upper critical dimension is $d_u=2$ or lower. In this work, we study the jamming transition below the upper critical dimension. We investigate a quasi-one-dimensional system: disks confined in a narrow channel. We show that the system is isostatic at the jamming transition point as in the case of standard jamming transition of the bulk systems in two and three dimensions. Nevertheless, the s…
Athermal packings of soft repulsive spheres exhibit a sharp jamming transition in the thermodynamic limit. Upon further compression, various structural and mechanical properties display clean power-law behavior over many decades in pressure. As with any phase transition, the rounding of such behavior in finite systems close to the transition plays an important role in understanding the nature of the transition itself. The situation for jamming is surprisingly rich: the assumption that jammed packings are isotropic is only strictly true in the large-size limit, and finite-size has a profound ef…
We determine the dimensional dependence of the percolative exponents of the jamming transition via numerical simulations in four and five spatial dimensions. These novel results complement literature ones, and establish jamming as a mixed first-order percolation transition, with critical exponents $β=0$, $γ= 2$, $α= 0$ and the finite size scaling exponent $ν^* = 2/d$ for values of the spatial dimension $d \geq 2$. We argue that the upper critical dimension is $d_u=2$ and the connectedness length exponent is $ν=1$.
We propose a scaling ansatz for the elastic energy of a system near the critical jamming transition in terms of three relevant fields: the compressive strain $Δϕ$ relative to the critical jammed state, the shear strain $ε$, and the inverse system size $1/N$. We also use $ΔZ$, the number of contacts relative to the minimum required at jamming, as an underlying control parameter. Our scaling theory predicts new exponents, exponent equalities and scaling collapses for energy, pressure and shear stress that we verify with numerical simulations of jammed packings of soft spheres. It also yields new…
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…
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…
We probe flows of soft, viscous spheres near the jamming point, which acts as a critical point for static soft spheres. Starting from energy considerations, we find nontrivial scaling of velocity fluctuations with strain rate. Combining this scaling with insights from jamming, we arrive at an analytical model that predicts four distinct regimes of flow, each characterized by rational-valued scaling exponents. Both the number of regimes and values of the exponents depart from prior results. We validate predictions of the model with simulations.
In thermal amorphous systems, the first peak of the pair correlation function $g(r)$ shows a maximum height $g_1^{\rm max}$ at a volume fraction $ϕ=ϕ_v$ that increases with the temperature. $g_1^{\rm max}$ diverges at the T=0 jamming transition at $ϕ=ϕ_c$. Molecular dynamics simulations show that some typical quantities, such as the pressure, bulk modulus, shear modulus, and boson peak frequency that behave power law scalings with $ϕ-ϕ_c$ in marginally jammed solids at T=0, all show scalings with $ϕ-ϕ_c$ when $ϕ> ϕ_v$, while the scalings break down when $ϕ< ϕ_v$. The presence of $g_1^{\rm max}…
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-a…
Rheological properties of a dense granular material consisting of frictionless spheres are investigated. It is found that the shear stress, the pressure, and the kinetic temperature obey critical scaling near the jamming transition point, which is considered as a critical point. These scaling laws have some peculiar properties in view of conventional critical phenomena because the exponents depend on the interparticle force models so that they are not universal. It is also found that these scaling laws imply the relation between the exponents that describe the growing correlation length.
We perform numerical simulations to determine the shear stress and pressure of steady-state shear flow in a soft-disk model in two dimensions at zero temperature in the vicinity of the jamming transition ϕ_J. We use critical point scaling analyses to determine the critical behavior at jamming, and we find that it is crucial to include corrections to scaling for a reliable analysis. We find that the relative size of these corrections are much smaller for pressure than for shear stress. We furthermore find a superlinear behavior for pressure and shear stress above ϕ_J, both from the scaling anal…
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 investigate the rheology of sheared granular materials near the jamming transition point. We numerically determine the values of the critical fraction and the exponents for the jamming transition using a finite size scaling and the nonlinear minimization method known as the Levenberg-Marquardt algorithm. The exponents are close to our previous theoretical prediction, but there is a small discrepancy, if the critical point is independently determined.
We carry out constant volume simulations of steady-state, shear driven, rheology in a simple model of bidisperse, soft-core, frictionless disks in two dimensions, using a dissipation law that gives rise to Bagnoldian rheology. We carry out a detailed critical scaling analysis of our resulting data for pressure $p$ and shear stress $σ$, in order to determine the critical exponent $β$ that describes the algebraic divergence of the Bagnold transport coefficients, as the jamming transition is approached from below. We show that it is necessary, for the strain rates considered in this work, to cons…
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.
Collections of bidisperse frictionless particles at zero temperature in three dimensions are simulated with a shear-driven dynamics with the aim to compare with behavior in two dimensions. Contrary to the prevailing picture, and in contrast to results from isotropic jamming from compression or quench, we find that the critical exponents in three dimensions are different from those in two dimensions and conclude that shear-driven jamming in two and three dimensions belong to different universality classes.
During the jamming of thermal colloids, the first peak of the pair distribution function shows a maximum height $g_1^{\rm max}$. We find that $g_1^{\rm max}$ is accompanied by significant change of material properties and thus signifies the transition from unjammed to jammed glasses. The scaling laws at $g_1^{\rm max}$ lead to scaling collapse of structural and thermodynamic quantities, indicating the criticality of the T=0 jamming transition. The physical significance of $g_1^{\rm max}$ is highlighted by its coincidence with the equality of the kinetic and potential energy and the maximum flu…
Recent simulations have predicted that near jamming for collections of spherical particles, there will be a discontinuous increase in the mean contact number, Z, at a critical volume fraction, phi_c. Above phi_c, Z and the pressure, P are predicted to increase as power laws in phi-phi_c. In experiments using photoelastic disks we corroborate a rapid increase in Z at phi_c and power-law behavior above phi_c for Z and P. Specifically we find power-law increase as a function of phi-phi_c for Z-Z_c with an exponent beta around 0.5, and for P with an exponent psi around 1.1. These exponents are in …
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…
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…
Recent computer simulations have uncovered the striking difference between the jamming transition of spherical and non-spherical particles. While systems of spherical particles are isostatic at the jamming point, systems of nonspherical particles are not: the contact number and shear modulus of the former exhibit a square root singularity near jamming, while those of the latter are linearly proportional to the distance from jamming. Furthermore, while our theoretical understanding of jamming of spherical particles is well developed, the same is not true for nonspherical particles. To understan…
We investigate the glass and the jamming transitions of hard spheres in finite dimensions $d$, through a revised cell theory, that combines the free volume and the Random First Order Theory (RFOT). Recent results show that in infinite dimension the ideal glass transition and jamming transitions are distinct, while based on our theory we argue that they indeed coincide for finite $d$. As a consequence, jamming results into a percolation transition described by RFOT, with a static length diverging with exponent $ν=2/d$, which we verify through finite size scaling, and standard critical exponents…
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…
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 …
Amorphous materials as diverse as foams, emulsions, colloidal suspensions and granular media can jam into a rigid, disordered state where they withstand finite shear stresses before yielding. Here we review the current understanding of the transition to jamming and the nature of the jammed state for disordered packings of particles that act through repulsive contact interactions and are at zero temperature and zero shear stress. We first discuss the breakdown of affine assumptions that underlies the rich mechanics near jamming. We then extensively discuss jamming of frictionless soft spheres. …
We comprehensively study mechanical and vibrational properties of dimer packings in three-dimensional space with particular attention on critical scaling behaviors near the jamming transition. First, we confirm the dependence of the packing fraction at the transition on the aspect ratio, the isostatic contact number at the transition, and the scaling dependence of the excess contact number on the excess density. Second, we study the elastic moduli, bulk and shear moduli, and establish power-law scaling of them. Finally, we study the vibrational density of states and its characteristic frequenc…
We study properties of jammed packings of frictionless spheres over a wide range of volume fractions. There exists a crossover volume fraction which separates deeply jammed solids from marginally jammed solids. In deeply jammed solids, all the scalings presented in marginally jammed solids are replaced with remarkably different ones with potential independent exponents. Correspondingly, there are structural changes in the pair distribution function associated with the crossover. The normal modes of vibration of deeply jammed solids also exhibit some anomalies, e.g. strengthened quasi-localizat…
We study the effect of dilute pinning on the jamming transition. Pinning reduces the average contact number needed to jam unpinned particles and shifts the jamming threshold to lower densities, leading to a pinning susceptibility, $χ_p$. Our main results are that this susceptibility obeys scaling form and diverges in the thermodynamic limit as $χ_p \propto |ϕ- ϕ_c^\infty|^{-γ_p}$ where $ϕ_c^\infty$ is the jamming threshold in the absence of pins. Finite-size scaling arguments yield these values with associated statistical (systematic) errors $γ_p = 1.018 \pm 0.026 (0.291) $ in $d=2$ and $γ_p =…
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…
Critical behaviors of sheared dense and frictionless granular materials in the vicinity of the jamming transition are numerically investigated. From the extensive molecular dynamics simulation, we verify the validity of the scaling theory near the jamming transition proposed by Otsuki and Hayakawa (Prog. Theor. Phys., 121, 647 (2009)). We also clarify the critical behaviors of the shear viscosity and the pair correlation function based on both a phenomenology and the simulation.
We study numerically the critical behavior and marginal stability of the shear jamming transition for frictionless soft spheres, observed to occur over a finite range of densities, associated with isotropic jamming for densities above the minimum jamming (J-point) density. Several quantities are shown to scale near the shear jamming point in the same way as the isotropic jamming point. We compute the exponents associated with the small force distribution and the interparticle gap distribution,and show that the corresponding exponents are consistent with the marginal stability condition observe…
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…
In this thesis I present most of the results obtained during my PhD, where I worked on different subjects regarding jamming in systems of frictionless spheres. In particular, I focused on microscopic properties of jammed packings, such as the distribution of contact forces and interparticle gaps, as well as the single particle dynamics that occur near the jamming point. Several of these results have already been presented in Refs. [1,2], but here I include a more detailed analysis of some of them. Besides, all the results of the second chapter are new, even if related to such works.
Jamming and avalanche statistics are studied in a simulation of the discharge of a polydisperse ensemble of disks from a 2-D silo. Exponential distributions are found for the avalanche sizes for all sizes of the exit opening, in agreement with reported experiments. The average avalanche size grows quite fast with the size of the exit opening. Data for this growth agree better with a critical divergence with a large critical exponent, as reported for 3-D experiments, than with the exponential growth reported for 2-D experiments.
The dynamical properties of a dense horizontally vibrated bidisperse granular monolayer are experimentally investigated. The quench protocol produces states with a frozen structure of the assembly, but the remaining degrees of freedom associated with contact dynamics control the appearance of macroscopic rigidity. We provide decisive experimental evidence that this transition is a critical phenomenon, with increasingly collective and heterogeneous rearrangements occurring at length scales much smaller than the grains' diameter, presumably reflecting the contact force network fluctuations. Dyna…
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 rheology near jamming of a suspension of soft colloidal spheres is studied using a custom microfluidic rheometer that provides stress versus strain rate over many decades. We find non-Newtonian behavior below the jamming concentration and yield stress behavior above it. The data may be collapsed onto two branches with critical scaling exponents that agree with expectations based on Hertzian contacts and viscous drag. These results support the conclusion that jamming is similar to a critical phase transition, but with interaction-dependent exponents.
We investigate a three-dimensional kinetically-constrained model that exhibits two types of phase transitions at different densities. At the jamming density $ ρ_J $ there is a mixed-order phase transition in which a finite fraction of the particles become frozen, but the other particles may still diffuse throughout the system. At the caging density $ ρ_C > ρ_J $, the mobile particles are trapped in finite cages and no longer diffuse. The caging transition occurs due to a percolation transition of the unfrozen sites, and we numerically find that it is a continuous transition with the same criti…
The jamming transition of non-spherical particles is fundamentally different from the spherical case. Non-spherical particles are hypostatic at their jamming points, while isostaticity is ensured in the case of the jamming of spherical particles. This structural difference implies that the presence of asphericity affects the critical exponents related to the contact number and the vibrational density of states. Moreover, while the force and gap distributions of isostatic jamming present power-law behaviors, even an infinitesimal asphericity is enough to smooth out these singularities. In a rec…
We study the rheology of amorphous packings of soft, frictionless particles close to jamming. Implementing a quasistatic simulation method we generate a well defined ensemble of states that directly samples the system at its yield-stress. A continuous jamming transition from a freely-flowing state to a yield stress situation takes place at a well defined packing fraction, where the scaling laws characteristic of isostatic solids are observed. We propose that long-range correlations observed below the transition are dominated by this isostatic point, while those that are observed above the tran…
Recently, we proposed a universal scaling framework that shows shear thickening in dense suspensions is governed by the crossover between two critical points: one associated with frictionless isotropic jamming and a second corresponding to frictional shear jamming. Here, we show that orthogonal perturbations to the flows, an effective method for tuning shear thickening, can also be folded into this universal scaling framework. Specifically, we show that the effect of adding in orthogonal shear perturbations (OSP) can be incorporated by simply altering the scaling variable to include a multipli…
By finding local minima of an enthalpy-like energy, we can generate jammed packings of frictionless spheres under constant shear stress $σ$ and obtain the yield stress $σ_y$ by sampling the potential energy landscape. For three-dimensional systems with harmonic repulsion, $σ_y$ satisfies the finite size scaling with the limiting scaling relation $σ_y\simϕ- ϕ_{_{c,\infty}}$, where $ϕ_{_{c,\infty}}$ is the critical volume fraction of the jamming transition at $σ=0$ in the thermodynamic limit. The width or uncertainty of the yield stress decreases with decreasing $ϕ$ and decays to zero in the the…
