We perform numerical simulations of the jamming transition of non-spherical particles in two dimensions. In particular, we systematically investigate how the physical quantities at the jamming transition point behave when the shapes of the particle deviate slightly from the perfect disks. For efficient numerical simulation, we first derive an analytical expression of the gap function, using the perturbation theory around the reference disks. Starting from disks, we observe the effects of the deformation of the shapes of particles by the $n$-th order term of the Fourier series $\sin(nθ)$. We show that the several physical quantities, such as the number of contacts, gap distribution, and characteristic frequencies of the vibrational density of states, show the power-law behaviors with respect to the linear deviation from the reference disks. The power-law behaviors do not depend on $n$ and are fully consistent with the mean-field theory of the jamming of non-spherical particles. This result suggests that the mean-field theory holds very generally for nearly spherical particles whose shape can be expressed by the Fourier series.
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…
The unconventional thermal properties of jammed amorphous solids are directly related to their density of vibrational states. While the vibrational spectrum of jammed soft sphere solids has been fully described, the vibrational spectrum of hard spheres, a model for colloidal glasses, is still unknown due to the difficulty of treating the non-analytic interaction potential. We bypass this difficulty using the recently described effective interaction potential for the free energy of thermal hard spheres. By minimizing this effective free energy we mimic a quench and produce typical configuration…
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…
Amorphous solids tend to present an abundance of soft elastic modes, which diminish their transport properties, generate heterogeneities in their elastic response, and affect non-linear processes like thermal activation of plasticity. This is especially true in packings of particles near their jamming transition, for which effective medium theory and variational arguments can both predict the density of vibrational modes. However, recent numerics support that one hypothesis of the variational argument does not hold. We provide a novel variational argument which overcomes this problem, and corr…
Studying the jamming transition of granular and colloidal systems, has lead to a proliferation of theoretical and numerical results formulated in the language of the eigenspectrum of the dynamical matrix for these disordered system. Only recently however, these modes have been accessed experimentally in colloidal and granular media, computing the eigenmodes of the covariance matrix of the particle positions. At the same time new conceptual and methodological questions have appeared, regarding the interpretation of these results. In the present paper, we first give an overview of the theoretica…
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.
We review the physics of jamming from the theoretical, experimental and numerical perspectives. We summarize the mean-field theory of jamming and the marginally stable solid phase, with particular emphasis on the connection with the Replica Symmetry Breaking theory of glasses. We report validations of the mean-field theory of jamming from experimental and numerical studies of critical behaviors near the transition. In particular, we describe the physics of jamming of frictionless, spherical particles, as well as more recent work on jamming of frictionless, non-spherical particles and frictiona…
Jammed particulate systems composed of various shapes of particles undergo the jamming transition as they are compressed or decompressed. To date, sphere packings have been extensively studied in many previous works, where isostaticity at the transition and scaling laws with the pressure of various quantities, including the contact number and the vibrational density of states, have been established. Additionally, much attention has been paid to nonspherical packings, and particularly recent work has made progress in understanding ellipsoidal packings. In the present work, we study the dimer pa…
Amorphous packings of non-spherical particles such as ellipsoids and spherocylinders are known to be hypostatic: the number of mechanical contacts between particles is smaller than the number of degrees of freedom, thus violating Maxwell's mechanical stability criterion. In this work, we propose a general theory of hypostatic amorphous packings and the associated jamming transition. First, we show that many systems fall into a same universality class. As an example, we explicitly map ellipsoids into a system of `breathing' particles. We show by using a marginal stability argument that in both …
We study the localization of vibrational modes of frictionless granular media. We introduce a new method, motivated by earlier work on non-Hermitian quantum problems, which works well both in the localized regime where the localization length $ξ$ is much less than the linear size $L$ and in the regime $ξ$ grater or of order $L$ when modes are extended throughout our finite system. Our very lowest frequency modes show "quasi-localized" resonances away from the jamming point; the spatial extent of these regions increases as the jamming point is approached, as expected theoretically. Throughout t…
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 report an analytical study of the vibrational spectrum of the simplest model of jamming, the soft perceptron. We identify two distinct classes of soft modes. The first kind of modes are related to isostaticity and appear only in the close vicinity of the jamming transition. The second kind of modes instead are present everywhere in the glass phase and are related to the hierarchical structure of the potential energy landscape. Our results highlight the universality of the spectrum of normal modes in disordered systems, and open the way towards a detailed analytical understanding of the vibr…
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 demonstrate the existence of high--order jamming crossovers in systems of particles with repulsive contact interactions, which originate from the collapse of successive coordination shells. At zero temperature, these crossovers induce an anomalous behavior of the bulk modulus, which varies non--monotonically with the density, while at finite temperature they induce density anomalies consisting in an increased diffusivity upon isothermal compression and in a negative thermal expansion coefficient. We rationalize the dependence of these crossovers on the softness of the interaction potential,…
A theory for the microscopic structure and the vibrational properties of soft sphere glass at finite temperature is presented. With an effective potential, derived here, the phase diagram and vibrational properties are worked out around the Maxwell critical point at zero temperature $T$ and pressure $p$. Variational arguments and effective medium theory identically predict a non-trivial temperature scale $T^*\sim p^{(2-a)/(1-a)}$ with $a \approx 0.17$ such that low-energy vibrational properties are hard-sphere like for $T \gtrsim T^*$, and zero-temperature soft-sphere like otherwise. However, …
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…
While frictionless spheres at jamming are isostatic, frictional spheres at jamming are not. As a result, frictional spheres near jamming do not necessarily exhibit an excess of soft modes. However, a generalized form of isostaticity can be introduced if fully mobilized contacts at the Coulomb friction threshold are considered as slipping contacts. We show here that, in this framework, the vibrational density of states (DOS) of frictional discs exhibits a plateau when the generalized isostaticity line is approached. The crossover frequency to elastic behavior scales linearly with the distance f…
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…
Jamming is a fundamental transition that governs the mechanical behavior of particulate media, including sand, foam and dense suspensions but also biological tissues: Upon compression, particulate media can change from freely flowing to a disordered solid. Jamming has previously been conceived as a bulk phenomenon involving particle motions in fixed geometries. In a diverse class of soft materials, however, solidification can take place in a deformable geometry, such as on the surface of a fluid droplet or in the formation of a bijel. In these systems the nature and dynamics of jamming remains…
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…
We use molecular simulations to study jamming of a crumpled bead-spring model polymer in a finite container and compare to jamming of repulsive spheres. After proper constraint counting, the onset of rigidity is seen to occur isostatically as in the case of repulsive spheres. Despite this commonality, the presence of the curved container wall and polymer backbone bonds introduce new mechanical properties. Notably, these include additional bands in the vibrational density of states that reflect the material structure as well as oscillations in local contact number and density near the wall but …
As a function of packing fraction at zero temperature and applied stress, an amorphous packing of spheres exhibits a jamming transition where the system is sensitive to boundary conditions even in the thermodynamic limit. Upon further compression, the system should become insensitive to boundary conditions provided it is sufficiently large. Here we explore the linear response to a large class of boundary perturbations in 2 and 3 dimensions. We consider each finite packing with periodic-boundary conditions as the basis of an infinite square or cubic lattice and study properties of vibrational m…
Amorphous packings of frictionless, spherical particles are isostatic at jamming onset, with the number of constraints (contacts) equal to the number of degrees of freedom. Their structural and mechanical properties are controlled by the interparticle contact network. In contrast, amorphous packings of frictional particles are typically hyperstatic at jamming onset. We perform extensive numerical simulations in two dimensions of the geometrical asperity (GA) model for static friction, to further investigate the role of isostaticity. In the GA model, interparticle forces are obtained by summing…
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…
The structural properties of static, jammed packings of monodisperse spheres in the vicinity of the jamming transition are investigated using large-scale computer simulations. At small wavenumber $k$, we argue that the anomalous behavior in the static structure factor, $S(k) \sim k$, is consequential of an excess of low-frequency, collective excitations seen in the vibrational spectrum. This anomalous feature becomes more pronounced closest to the jamming transition, such that $S(0) \to 0$ at the transition point. We introduce an appropriate dispersion relation that accounts for these phenomen…
We report on the critical properties of minimaly-polydisperse crystals, hexagonal in 2d and face-centered cubic in 3 dimensions, at the isostatic jamming point. The force and gap distributions display power-law tails for small values. The vibrational density of states (VDOS) is flat. The scaling behavior of forces of extended floppy modes and the VDOS are universal and in agreement with an infinite-dimensional mean-field theory and maximally amorphous packings down to 2 dimensions. The distributions of gaps and forces of localized floppy modes of near-crystals appear non-universal. A small fra…
Several mean-field theories predict that Hessian matrices of amorphous solids can be written by using the random matrix in the limit of the large spatial dimensions $d\to\infty$. Motivated by these results, we here propose a way to map a Hessian of the amorphous solid to a random matrix. This is possible by determining the coefficients of a random matrix so that the trace of the random matrix coincides with the Hessian of the original system. We compare our result with that of previous numerical simulations of harmonic spheres in several spatial dimensions $d=3$, $5$, and $9$. For small pressu…
We consider a system of hard spheres close to jamming, where translation invariance is broken by pinning a randomly chosen set of particles. Using two different protocols, we generate two kinds of packings at the jamming point, isostatic and hyperstatic packings. In the case where the packings are isostatic, jamming transition is not affected by random pinning: the jamming density is only slightly reduced, a generalized isostaticity condition holds, the system is marginally rigid over the entire glass phase, and the typical structural signatures of jamming are unchanged. Besides, random pinnin…
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…
We investigate the jamming transition in a quasi-2D granular material composed of regular pentagons or disks subjected to quasistatic uniaxial compression. We report six major findings based on experiments with monodisperse photoelastic particles with static friction coefficient $μ\approx 1$. (1) For both pentagons and disks, the onset of rigidity occurs when the average coordination number of non-rattlers, $Z_{nr}$ , reaches 3, and the dependence of $Z_{nr}$ on the packing fraction $ϕ$ changes again when $Z_{nr}$ reaches 4. (2) Though the packing fractions $ϕ_{c1}$ and $ϕ_{c2}$ at these trans…
The spherical model is a popular solvable model and has been applied to describe several critical phenomena such as the ferromagnetic transition, Bose-Einstein condensation, spin-glass transition, glass transition, jamming transition, and so on. Motivated by recent developments of active matter, here we consider the spherical model driven by the Ornstein-Uhlenbeck type self-propulsion force with persistent time $τ_p$. We show that the model exhibits the Ising universality for finite $τ_p$. On the contrary, the model exhibits the random field Ising universality in the limit $τ_p\to\infty$.
We propose a `phase diagram' for particulate systems that interact via purely repulsive contact forces, such as granular media and colloidal suspensions. We identify and characterize two distinct classes of behavior as a function of the input kinetic energy per degree of freedom $T_0$ and packing fraction deviation above and below jamming onset $Δϕ=ϕ- ϕ_J$ using numerical simulations of purely repulsive frictionless disks. Iso-coordinated solids (ICS) only occur above jamming for $Δϕ> Δϕ_c(T_0)$; they possess average coordination number equal to the isostatic value ($< z> = z_{\rm iso}$) requi…
We characterize vibrational motion occurring at low temperatures in dense suspensions of soft repulsive spheres over a broad range of volume fractions encompassing the jamming transition at (T = 0, phi = phi_J). We find that characteristic time and length scales of thermal vibrations obey critical scaling in the vicinity of the jamming transition. We show in particular that the amplitude and the time scale of dynamic fluctuations diverge symmetrically on both sides of the transition, and directly reveal a diverging correlation length. The critical region near phi_J is divided in three differen…
This paper describes methods to measure flow rate and jamming onset of granules discharged through a flat-bottom cylindrical hopper with a circular orifice. The intrinsic jamming onset for ideal particles (spherical, monodisperse, smooth) is experimentally measured by two independent methods, with good agreement. For non-ideal particles, the normalized jamming onset increases with elongated granule shape, broadened size distribution and increased friction as measured by the drained angle of repose. An empirical model of the jamming onset is introduced to quantify these effects over the range o…
We conduct experiments on two-dimensional packings of colloidal thermosensitive hydrogel particles whose packing fraction can be tuned above the jamming transition by varying the temperature. By measuring displacement correlations between particles, we extract the vibrational properties of a corresponding "shadow" system with the same configuration and interactions, but for which the dynamics of the particles are undamped. The vibrational spectrum and the nature of the modes are very similar to those predicted for zero-temperature idealized sphere models and found in atomic and molecular glass…
Swap algorithms can shift the glass transition to lower temperatures, a recent unexplained observation constraining the nature of this phenomenon. Here we show that swap dynamic is governed by an effective potential describing both particle interactions as well as their ability to change size. Requiring its stability is more demanding than for the potential energy alone. This result implies that stable configurations appear at lower energies with swap dynamics, and thus at lower temperatures when the liquid is cooled. \maa{ The magnitude of this effect is proportional to the width of the radii…
We study the structural and mechanical properties of jammed ellipse packings, and find that the nature of the jamming transition in these systems is fundamentally different from that for spherical particles. Ellipse packings are generically hypostatic with more degrees of freedom than constraints. The spectra of low energy excitations possess two gaps and three distinct branches over a range of aspect ratios. In the zero compression limit, the energy of the modes in the lowest branch increases {\it quartically} with deformation amplitude, and the density of states possesses a $δ$-function at z…
Random sequential addition (RSA) models are used in a large variety of contexts to model particle aggregation and jamming. A key feature of these models is the algebraic time dependence of the asymptotic jamming coverage as $t\to\infty$. For the RSA of monodisperse non-spherical particles the scaling is generally believed to be $~t^{-ν}$, where $ν=1/d_{\rm f}$ for a particle with $d_{\rm f}$ degrees of freedom. While the $d_{\rm f}=1$ result of spheres (Renyi's classical car parking problem) can be derived analytically, evidence for the $1/d_{\rm f}$ scaling for arbitrary particle shapes has s…
We propose a theoretical framework for predicting the protocol dependence of the jamming transition for frictionless spherical particles that interact via purely repulsive contact forces. We study isostatic jammed disk packings obtained via two protocols: isotropic compression and simple shear. We show that for frictionless systems, all jammed packings can be obtained via either protocol. However, the probability to obtain a particular jammed packing depends on the packing-generation protocol. We predict the average shear strain required to induce jamming in initially unjammed packings from th…
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 study shear-driven jamming of ellipsoidal particles at zero temperature with a focus on the microscopic dynamics. We find that a change from spherical particles to ellipsoids with aspect ratio $α=1.02$ gives dramatic changes of the microscopic dynamics with much lower translational velocities and a new role for the rotations. Whereas the velocity difference at contacts---and thereby the dissipation---in collections of spheres is dominated by the translational velocities and reduced by the rotations, the same quantity is in collections of ellipsoids instead totally dominated by the rotationa…
We report experimental and computational observations of dynamic contact networks for colloidal suspensions undergoing shear thickening. The dense suspensions are comprised of sterically stabilized poly(methyl methacrylate) hard sphere colloids that are spherically symmetric and have varied surface roughness. Confocal rheometry and dissipative particle dynamics simulations show that the shear thickening strength scales exponentially with the scaled deficit contact number and the scaled jamming distance. Rough colloids, which experience additional tangential and rolling constraints, require an …
Contact breaking and Hertzian interactions between grains can both give rise to nonlinear vibrational response of static granular packings. We perform molecular dynamics simulations at constant energy in 2D of frictionless bidisperse disks that interact via Hertzian spring potentials as a function of energy and measure directly the vibrational response from the Fourier transform of the velocity autocorrelation function. We compare the measured vibrational response of static packings near jamming onset to that obtained from the eigenvalues of the dynamical matrix to determine the temperature ab…
At the jamming transition, amorphous packings are known to display anomalous vibrational modes with a density of states (DOS) that remains constant at low frequency. The scaling of the DOS at higher densities remains, however, unclear. One might expect to find simple Debye scaling, but recent results from effective medium theory and the exact solution of mean-field models both predict an anomalous, non-Debye scaling. Being mean-field solutions, however, these solutions are only strictly applicable to the limit of infinite spatial dimension, and it is unclear what value they have for finite-dim…
We numerically investigate the mechanical properties of static packings of ellipsoidal particles in 2D and 3D over a range of aspect ratio and compression $Δϕ$. While amorphous packings of spherical particles at jamming onset ($Δϕ=0$) are isostatic and possess the minimum contact number $z_{\rm iso}$ required for them to be collectively jammed, amorphous packings of ellipsoidal particles generally possess fewer contacts than expected for collective jamming ($z < z_{\rm iso}$) from naive counting arguments, which assume that all contacts give rise to linearly independent constraints on interpar…
Dense suspensions of hard particles in a liquid can exhibit strikingly counter-intuitive behavior, such as discontinuous shear thickening (DST) [1-8] and reversible shear jamming (SJ) into a state with finite yield stress [9-13]. Recent studies identified a stress-activated crossover from hydrodynamic interactions to frictional particle contacts to be key for these behaviors [2-4, 6-8, 10, 14]. However, many suspensions exhibit only DST and not SJ. Here we show that particle surface chemistry can play a central role in creating conditions that allow for SJ. We find the system's ability to form…
Effect of high and low Reynolds number is studied on low frequency vibrational modes of a spherical virus embedded in the aqueous medium. We have used an analytical approach based on fluid dynamic and classical Lamb theory to calculate the vibrational modes of a virus with material parameters of lysozyme crystal in water. The obvious size effect on the vibrational modes is observed. The estimated damping time which is of the order of picosecond varies with Reynolds number and shows a high value for a critical Reynolds number. The stationary eigenfrequency regions are observed for every quantum…
Amorphous packings prepared in the vicinity of the jamming transition play a central role in theoretical studies of the vibrational spectrum of glasses. Two mean-field theories predict that the vibrational density of states $g(ω)$ obeys a characteristic power law, $g(ω)\simω^2$, called the non-Debye scaling in the low-frequency region. Numerical studies have however reported that this scaling breaks down at low frequencies, due to finite dimensional effects. In this study, we prepare amorphous packings of up to $128000$ particles in spatial dimensions from $d=3$ to $d=9$ to characterise the ra…
We numerically simulate the uniform athermal shearing of bidisperse, frictionless, two dimensional spherocylinders and three dimensional prolate ellipsoids. We focus on the orientational ordering of particles as an asphericity parameter $α\to 0$ and particles approach spherical. We find that the nematic order parameter $S_2$ is non-monotonic in the packing fraction $ϕ$, and that as $α\to 0$ $S_{2}$ stays finite at jamming and above. The approach to spherical particles thus appears to be singular. We also find that sheared particles continue to rotate above jamming, and that particle contacts p…
