Can a Large Packing be Assembled from Smaller Ones?

Jamming is a first-order transition with quenched disorder in amorphous materials sheared by cyclic quasistatic deformations

Jamming is an athermal transition between flowing and rigid states in amorphous systems such as granular matter, colloidal suspensions, complex fluids and cells. The jamming transition seems to display mixed aspects of a first-order transition, evidenced by a discontinuity in the coordination number, and a second-order transition, indicated by power-law scalings and diverging lengths. Here we demonstrate that jamming is a first-order transition with quenched disorder in cyclically sheared systems with quasistatic deformations, in two and three dimensions. Based on scaling analyses, we show that fluctuations of the jamming density in finite-sized systems have important consequences on the finite-size effects of various quantities, resulting in a square relationship between disconnected and connected susceptibilities, a key signature of the first-order transition with quenched disorder. This study puts the jamming transition into the category of a broad class of transitions in disordered systems where sample-to-sample fluctuations dominate over thermal fluctuations, suggesting that the nature and behavior of the jamming transition might be better understood within the developed theoretical framework of the athermally driven random-field Ising model.

Scaling regimes and fluctuations of observables in computer glasses approaching the unjamming transition

Under decompression, disordered solids undergo an unjamming transition where they become under-coordinated and lose their structural rigidity. The mechanical and vibrational properties of these materials have been an object of theoretical, numerical, and experimental research for decades. In the study of low-coordination solids, understanding the behavior and physical interpretation of observables that diverge near the transition is of particular importance. Several such quantities are length scales (ξ or l) that characterize the size of excitations, the decay of spatial correlations, the response to perturbations, or the effect of physical constraints in the boundary or bulk of the material. Additionally, the spatial and sample-to-sample fluctuations of macroscopic observables such as contact statistics or elastic moduli diverge approaching unjamming. Here, we discuss important connections between all of these quantities and present numerical results that characterize the scaling properties of sample-to-sample contact and shear modulus fluctuations in ensembles of low-coordination disordered sphere packings and spring networks. Overall, we highlight three distinct scaling regimes and two crossovers in the disorder quantifiers χz and χμ as functions of system size N and proximity to unjamming δz. As we discuss, χX relates to the standard deviation σX of the sample-to-sample distribution of the quantity X (e.g., excess coordination δz or shear modulus μ) for an ensemble of systems. Importantly, χμ has been linked to experimentally accessible quantities that pertain to sound attenuation and the density of vibrational states in glasses. We investigate similarities and differences in the behaviors of χz and χμ near the transition and discuss the implications of our findings on current literature, unifying findings in previous studies.

Dynamical Approach to the Jamming Problem

A simple dynamical model, biased random organization (BRO), appears to produce configurations known as random close packing (RCP) as BRO's densest critical point in dimension d=3. We conjecture that BRO likewise produces RCP in any dimension; if so, then RCP does not exist in d=1-2 (where BRO dynamics lead to crystalline order). In d=3-5, BRO produces isostatic configurations and previously estimated RCP volume fractions 0.64, 0.46, and 0.30, respectively. For all investigated dimensions (d=2-5), we find that BRO belongs to the Manna universality class of dynamical phase transitions by measuring critical exponents associated with the steady-state activity and the long-range density fluctuations. Additionally, BRO's distribution of near contacts (gaps) displays behavior consistent with the infinite-dimensional theoretical treatment of RCP when d≥4. The association of BRO's densest critical configurations with random close packing implies that RCP's upper-critical dimension is consistent with the Manna class d_{uc}=4.

Hard-sphere jamming through the lens of linear optimization

The jamming transition is ubiquitous. It is present in granular matter, foams, colloids, structural glasses, and many other systems. Yet, it defines a critical point whose properties still need to be fully understood. Recently, a major breakthrough came about when the replica formalism was extended to build a mean-field theory that provides an exact description of the jamming transition of spherical particles in the infinite-dimensional limit. While such theory explains the jamming critical behavior of both soft and hard spheres, investigating the transition in finite-dimensional systems poses very difficult and different problems, in particular from the numerical point of view. Soft particles are modeled by continuous potentials; thus, their jamming point can be reached through efficient energy minimization algorithms. In contrast, the latter methods are inapplicable to hard-sphere (HS) systems since the interaction energy among the particles is always zero by construction. To overcome these difficulties, here we recast the jamming of hard spheres as a constrained optimization problem and introduce the CALiPPSO algorithm, capable of readily producing jammed HS packings without including any effective potential. This algorithm brings a HS configuration of arbitrary dimensions to its jamming point by solving a chain of linear optimization problems. We show that there is a strict correspondence between the force balance conditions of jammed packings and the properties of the optimal solutions of CALiPPSO, whence we prove analytically that our packings are always isostatic and in mechanical equilibrium. Furthermore, using extensive numerical simulations, we show that our algorithm is able to probe the complex structure of the free-energy landscape, finding qualitative agreement with mean-field predictions. We also characterize the algorithmic complexity of CALiPPSO and provide an open-source implementation of it.

Experimental observations of marginal criticality in granular materials

SignificanceAmorphous materials, such as grains, foams, colloids, and glasses, are ubiquitous in nature and our daily life. They can undergo glass transitions or jamming transitions to obtain rigidity either by fast quench or compression, but show subtle changes in the structures compared to the liquid states or liquid-like states. Recent progress on the first-principle replica theory unifies the glass transition and the jamming transition and points out the marginal phase with fractal free-energy landscape within the stable glass phase. Independently, marginal stability analysis predicts the relations between the exponents of the marginal phase. Here, we perform experiments with photoelastic disks and provide direct evidence of these theories in real-world amorphous materials.

Geometrical properties of mechanically annealed systems near the jamming transition

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 increases until the shear amplitude reaches the threshold value. The threshold value matches with the transition point from the point-reversible phase where the particles experience no collision to the loop-reversible phase where the particles' displacements are non-affine during a shear cycle before coming back to an original position. The results demonstrated in this paper are explained in terms of neither of universal criticality of the jamming transition nor the nonequilibrium phase transitions.

Finite-size effects in the microscopic critical properties of jammed configurations: A comprehensive study of the effects of different types of disorder

Jamming criticality defines a universality class that includes systems as diverse as glasses, colloids, foams, amorphous solids, constraint satisfaction problems, neural networks, etc. A particularly interesting feature of this class is that small interparticle forces (f) and gaps (h) are distributed according to nontrivial power laws. A recently developed mean-field (MF) theory predicts the characteristic exponents of these distributions in the limit of very high spatial dimension, d→∞ and, remarkably, their values seemingly agree with numerical estimates in physically relevant dimensions, d=2 and 3. These exponents are further connected through a pair of inequalities derived from stability conditions, and both theoretical predictions and previous numerical investigations suggest that these inequalities are saturated. Systems at the jamming point are thus only marginally stable. Despite the key physical role played by these exponents, their systematic evaluation has yet to be attempted. Here, we carefully test their value by analyzing the finite-size scaling of the distributions of f and h for various particle-based models for jamming. Both dimension and the direction of approach to the jamming point are also considered. We show that, in all models, finite-size effects are much more pronounced in the distribution of h than in that of f. We thus conclude that gaps are correlated over considerably longer scales than forces. Additionally, remarkable agreement with MF predictions is obtained in all but one model, namely near-crystalline packings. Our results thus help to better delineate the domain of the jamming universality class. We furthermore uncover a secondary linear regime in the distribution tails of both f and h. This surprisingly robust feature is understood to follow from the (near) isostaticity of our configurations.

Memory Formation in Jammed Hard Spheres

Liquids equilibrated below an onset condition share similar inherent states, while those above that onset have inherent states that markedly differ. Although this type of materials memory was first reported in simulations over 20 years ago, its physical origin remains controversial. Its absence from mean-field descriptions, in particular, has long cast doubt on its thermodynamic relevance. Motivated by a recent theoretical proposal, we reassess the onset phenomenology in simulations using a fast hard sphere jamming algorithm and find it to be both thermodynamically and dimensionally robust. Remarkably, we also uncover a second type of memory associated with a Gardner-like regime of the jamming algorithm.

Marginally jammed states of hard disks in a one-dimensional channel

We have studied a class of marginally jammed states in a system of hard disks confined in a narrow channel-a quasi-one-dimensional system-whose exponents are not those predicted by theories valid in the infinite dimensional limit. The exponent γ which describes the distribution of small gaps takes the value 1 rather than the infinite dimensional value 0.41269⋯. Our work shows that there exist jammed states not found within the tiling approach of Ashwin and Bowles. The most dense of these marginal states is an unusual state of matter that is asymptotically crystalline.

Jamming Below Upper Critical Dimension

Extensive numerical simulations in the past decades proved that the critical exponents of the jamming of frictionless spherical particles are the same in two and three dimensions. This implies that the upper critical dimension is d_{u}=2 or lower. In this Letter, 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 scaling of the excess contact number shows the linear scaling. Furthermore, the gap distribution remains finite even at the jamming transition point. These results are qualitatively different from those of the bulk systems in two and three dimensions.