Finite-size scaling at the jamming transition: corrections to scaling and the correlation-length critical exponent

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.

Unified Understanding of Nonlinear Rheology near the Jamming Transition Point

When slowly sheared, jammed packings respond elastically before yielding. This linear elastic regime becomes progressively narrower as the jamming transition point is approached, and rich nonlinear rheologies such as shear softening and hardening or melting emerge. However, the physical mechanism of these nonlinear rheologies remains elusive. To clarify this, we numerically study jammed packings of athermal frictionless soft particles under quasistatic shear γ. We find the universal scaling behavior for the ratio of the shear stress σ and the pressure P, independent of the preparation protocol of the initial configurations. In particular, we reveal shear softening σ/P∼γ^{1/2} over an unprecedentedly wide range of strain up to the yielding point, which a simple scaling argument can rationalize.

Jamming, relaxation, and memory in a minimally structured glass former

Structural glasses form through various out-of-equilibrium processes, including temperature quenches, rapid compression (crunches), and shear. Although each of these processes should be formally understandable within the recently formulated dynamical mean-field theory (DMFT) of glasses, the numerical tools needed to solve the DMFT equations up to the relevant physical regime do not yet exist. In this context, numerical simulations of minimally structured (and therefore mean-field-like) model glass formers can aid the search for and understanding of such solutions, thanks to their ability to disentangle structural from dimensional effects. We study here the infinite-range Mari-Kurchan model under simple out-of-equilibrium processes, and we compare results with the random Lorentz gas [J. Phys. A 55, 334001 (2022)10.1088/1751-8121/ac7f06]. Because both models are mean-field-like and formally equivalent in the limit of infinite spatial dimensions, robust features are expected to appear in the DMFT as well. The comparison provides insight into temperature and density onsets, memory, as well as anomalous relaxation. This work also further enriches the algorithmic understanding of the jamming density.

Relaxation times, rheology, and finite size effects for non-Brownian disks in two dimensions

We carry out overdamped simulations in a simple model of jamming-a collection of bidisperse 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)0022-471510.1007/s10955-021-02710-8] 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 to a finite size dependence, τ∼lnN, which makes τ an ill-defined quantity. Beside analyses to determine the relaxation time for the whole system we determine particle relaxation times which allow us to determine both histograms of particle relaxation times and the average particle relaxation times-two quantities that are very useful for the analyses. The starting configurations for the relaxation simulations are of two different kinds-completely random or taken from steady shearing simulations-and we find that the difference between these two cases are bigger than previously noted and that the observed problems in the determination of the critical divergence obtained when starting from random configurations are not present when instead starting the relaxations from shearing configurations. We also argue that the the effect that causes the lnN dependence is not as problematic as asserted. When it comes to the finite size dependence of the pressure equivalent of the shear viscosity we find that our data don't give support for the claimed strong finite size dependence, but also that the finite size dependence is at odds with what one would normally expect for a system with a diverging correlation length, and that this calls for an alternative understanding of the phenomenon of shear-driven jamming.

Universality of stress-anisotropic and stress-isotropic jamming of frictionless spheres in three dimensions: Uniaxial versus isotropic compression

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 ε[over ̇], we find good agreement for all critical parameters comparing the isotropic and anisotropic cases. In particular, we determine that the bulk viscosity diverges as p/ε[over ̇]∼(ϕ_{J}-ϕ)^{-β}, with β=3.36±0.09, as jamming is approached from below. We further demonstrate that the average contact number per particle Z can also be written in a scaling form as a function of ϕ and ε[over ̇]. Once again, we find good agreement between the uniaxial and isotropic cases. We compare our results to prior simulations and theoretical predictions.

Testing mean-field theory for jamming of non-spherical particles: contact number, gap distribution, and vibrational density of states

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 [Formula: see text]. 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.

Nonlocal Effects Reflect the Jamming Criticality in Frictionless Granular Flows Down Inclines

The jamming transition is accompanied by a rich phenomenology such as hysteresis or nonlocal effects that is still not well understood. Here, we experimentally investigate a model frictionless granular layer flowing down an inclined plane as a way to disentangle generic collective effects from those arising from frictional interactions. We find that thin frictionless granular layers are devoid of hysteresis of the avalanche angle, yet the layer stability increases as it gets thinner. Steady rheological laws obtained for different layer thicknesses can be collapsed into a unique master curve, supporting the idea that nonlocal effects are the consequence of the usual finite-size effects associated with the presence of a critical point. This collapse indicates that the so-called isostatic length l^{*}, the scale on which pinning a boundary freezes all remaining floppy modes, governs the effect of boundaries on flow and rules out other propositions made in the past.

Critical scaling of compression-driven jamming of athermal frictionless spheres in suspension

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 ε[over ̇], 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 compression rate ε[over ̇], and that the bulk viscosity p/ε[over ̇] diverges upon jamming. A scaling analysis determines the critical exponents associated with the compression-driven jamming transition. Our results suggest that stress-isotropic, compression-driven jamming may be in the same universality class as stress-anisotropic, shear-driven jamming.

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.

Dynamic length scales in athermal, shear-driven jamming of frictionless disks in two dimensions

We carry out numerical simulations of athermally sheared, bidisperse, frictionless disks in two dimensions. From an appropriately defined velocity correlation function, we determine that there are two diverging length scales, ξ and ℓ, as the jamming transition is approached. We analyze our results using a critical scaling ansatz for the correlation function and argue that the more divergent length ℓ is a consequence of a dangerous irrelevant scaling variable and that it is ξ, which is the correlation length that determines the divergence of the system viscosity as jamming is approached from below in the liquid phase. We find that ξ∼(ϕ_{J}-ϕ)^{-ν} diverges with the critical exponent ν=1. We provide evidence that ξ measures the length scale of fluctuations in the rotation of the particle velocity field, while ℓ measures the length scale of fluctuations in the divergence of the velocity field.

Machine-learning iterative calculation of entropy for physical systems

Characterizing the entropy of a system is a crucial, and often computationally costly, step in understanding its thermodynamics. It plays a key role in the study of phase transitions, pattern formation, protein folding, and more. Current methods for entropy estimation suffer from a high computational cost, lack of generality, or inaccuracy and inability to treat complex, strongly interacting systems. In this paper, we present a method, termed machine-learning iterative calculation of entropy (MICE), for calculating the entropy by iteratively dividing the system into smaller subsystems and estimating the mutual information between each pair of halves. The estimation is performed with a recently proposed machine-learning algorithm which works with arbitrary network architectures that can be chosen to fit the structure and symmetries of the system at hand. We show that our method can calculate the entropy of various systems, both thermal and athermal, with state-of-the-art accuracy. Specifically, we study various classical spin systems and identify the jamming point of a bidisperse mixture of soft disks. Finally, we suggest that besides its role in estimating the entropy, the mutual information itself can provide an insightful diagnostic tool in the study of physical systems.

Translational and rotational velocities in shear-driven jamming of ellipsoidal particles

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 rotational velocities. By also examining the effect of different aspect ratios we find that the examined quantities show either a peak or a change in slope at α≈1.2, which thus gives evidence for a crossover between different regions of low and high aspect ratio.

Depletion forces in athermally sheared mixtures of frictionless disks and rods in two dimensions

We carry out numerical simulations to study the behavior of an athermal mixture of frictionless circular disks and elongated rods in two dimensions, under three different types of global linear deformation at a finite strain rate: (i) simple shearing, (ii) pure shearing, and (iii) isotropic compression. We find that the fluctuations induced by such deformations lead to depletion forces that cause rods to group in parallel oriented clusters for the cases of simple and pure shear, but not for isotropic compression. For simple shearing, we find that as the fraction of rods increases, this clustering increases, leading to an increase in the average rate of rotation of the rods, and a decrease in the magnitude of their nematic ordering.

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.

Structured randomness: jamming of soft discs and pins

Simulations are used to find the zero temperature jamming threshold, φj, for soft, bidisperse disks in the presence of small fixed particles, or "pins", arranged in a lattice. The presence of pins leads, as one expects, to a decrease in φj. Structural properties of the system near the jamming threshold are calculated as a function of the pin density. While the correlation length exponent remains ν = 1/2 at low pin densities, the system is mechanically stable with more bonds, yet fewer contacts than the Maxwell criterion implies in the absence of pins. In addition, as pin density increases, novel bond orientational order and long-range spatial order appear, which are correlated with the square symmetry of the pin lattice.

Tuning and jamming reduced to their minima

Inspired by protein folding, we smooth out the complex cost function landscapes of two processes: the tuning of networks and the jamming of ideal spheres. In both processes, geometrical frustration plays a role-tuning pressure differences between pairs of target nodes far from the source in a flow network impedes tuning of nearby pairs more than the reverse process, while unjamming the system in one region can make it more difficult to unjam elsewhere. By modifying the cost functions to control the order in which functions are tuned or regions unjam, we smooth out local minima while leaving global minima unaffected, increasing the success rate for reaching global minima.

Orientational Ordering in Athermally Sheared, Aspherical, Frictionless Particles

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→0 and particles approach spherical. We find that the nematic order parameter S_{2} is nonmonotonic in the packing fraction ϕ and that, as α→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 preferentially lie along the narrowest width of the particles, even as α→0.

Criticality of the zero-temperature jamming transition probed by self-propelled particles

We perform simulations of athermal systems of self-propelled particles (SPPs) interacting via harmonic repulsion in the vicinity of the zero-temperature jamming transition at point J. Every particle is propelled by a constant force f pointing to a randomly assigned and fixed direction. When f is smaller than the yield force f_y, the system is statically jammed. We find that f_y increases with packing fraction and exhibits finite size scaling, implying the criticality of point J. When f > f_y, SPPs flow forever and their velocities satisfy the k-Gamma distribution. Velocity distributions at various packing fractions and f collapse when the particle velocity is scaled by the average velocity v[combining macron], suggesting that v[combining macron] is a reasonable quantity to characterize the response to f. We thus define a response function R(ϕ,f) = v[combining macron](ϕ,f)/f. The function exhibits critical scaling nicely, implying again the criticality of point J. Our analysis and results indicate that systems of SPPs behave analogically to sheared systems, although their driving mechanisms are apparently distinct.

Compression-driven jamming of athermal frictionless spherocylinders in two dimensions

We simulate numerically the compression-driven jamming of athermal, frictionless, soft-core spherocylinders in two dimensions, for a range of particle aspect ratios α. We find the critical packing fraction ϕ_{J}(α) for the jamming transition and the average number of contacts per particle z_{J}(α) at jamming. We find that both are nonmonotonic, with a peak at α≈1. We find that configurations at the compression-driven jamming point are always hypostatic for all α, with z_{J}<z_{iso}=2d_{f}=6 the isostatic value. We show that, for moderately elongated spherocylinders, there is no orientational ordering upon athermal compression through jamming. We analyze in detail the eigenmodes of the dynamical matrix close to the jamming point for a few different values of the aspect ratio, from nearly circular to moderately elongated. We find that there are low frequency bands containing N(z_{iso}-z_{J})/2 modes, such that the frequencies of these modes vanish as ϕ→ϕ_{J}. We consider the extended versus localized nature of these low frequency modes, and the extent to which they involve translational or rotational motion, and find many low frequency sliding modes where particles can move with little rotation. We highlight the importance of treating side-to-side contacts, along flat sides of the spherocylinder, properly for the correct determination of z_{J}. We note the singular nature of taking the α→0 limit. We discuss the similarities and differences with previous work on jammed ellipses and ellipsoids, to illustrate the effects that different particle shapes have on configurations at jamming.

Scaling ansatz for the jamming transition

We propose a Widom-like scaling ansatz for the critical jamming transition. Our ansatz for the elastic energy shows that the scaling of the energy, compressive strain, shear strain, system size, pressure, shear stress, bulk modulus, and shear modulus are all related to each other via scaling relations, with only three independent scaling exponents. We extract the values of these exponents from already known numerical or theoretical results, and we numerically verify the resulting predictions of the scaling theory for the energy and residual shear stress. We also derive a scaling relation between pressure and residual shear stress that yields insight into why the shear and bulk moduli scale differently. Our theory shows that the jamming transition exhibits an emergent scale invariance, setting the stage for the potential development of a renormalization group theory for jamming.

Pinning Susceptibility: The Effect of Dilute, Quenched Disorder on Jamming

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}∝|ϕ-ϕ_{c}^{∞}|^{-γ_{p}} where ϕ_{c}^{∞} 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±0.026(0.291) in d=2 and γ_{p}=1.534±0.120(0.822) in d=3. Logarithmic corrections raise the exponent in d=2 to close to the d=3 value, although the systematic errors are very large.

Critical scaling of Bagnold rheology at the jamming transition of frictionless two-dimensional disks

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 discuss in detail the critical scaling ansatz for the shear-driven jamming transition and carry out a detailed scaling analysis of our resulting data for pressure p and shear stress σ. Our analysis determines the critical exponent β that describes the algebraic divergence of the Bagnold transport coefficients lim_{γ[over ̇]→0}p/γ[over ̇]^{2},σ/γ[over ̇]^{2}∼(ϕ_{J}-ϕ)^{-β} as the jamming transition ϕ_{J} is approached from below. For the low strain rates considered in this work, we show that it is still necessary to consider the leading correction-to-scaling term in order to achieve a self-consistent analysis of our data, in which the critical parameters become independent of the size of the window of data used in the analysis. We compare our resulting value β≈5.0±0.4 against previous numerical results and competing theoretical models. Our results confirm that the shear-driven jamming transition in Bagnoldian systems is well described by a critical scaling theory and we relate this scaling theory to the phenomenological constituent laws for dilatancy and friction.

Search for hyperuniformity in mechanically stable packings of frictionless disks above jamming

We numerically simulate mechanically stable packings of soft-core, frictionless, bidisperse disks in two dimensions, above the jamming packing fraction ϕ(J). For configurations with a fixed isotropic global stress tensor, we investigate the fluctuations of the local packing fraction ϕ(r) to test whether such configurations display the hyperuniformity that has been claimed to exist exactly at ϕ(J). For our configurations, generated by a rapid quench protocol, we find that hyperuniformity persists only out to a finite length scale and that this length scale appears to remain finite as the system stress decreases towards zero, i.e., towards the jamming transition. Our result suggests that the presence of hyperuniformity at jamming may be sensitive to the specific protocol used to construct the jammed configurations.

Maximum entropy and the stress distribution in soft disk packings above jamming

We show that the maximum entropy hypothesis can successfully explain the distribution of stresses on compact clusters of particles within disordered mechanically stable packings of soft, isotropically stressed, frictionless disks above the jamming transition. We show that, in our two-dimensional case, it becomes necessary to consider not only the stress but also the Maxwell-Cremona force-tile area as a constraining variable that determines the stress distribution. The importance of the force-tile area had been suggested by earlier computations on an idealized force-network ensemble.

Membrane Protein Structure, Function, and Dynamics: a Perspective from Experiments and Theory

Membrane proteins mediate processes that are fundamental for the flourishing of biological cells. Membrane-embedded transporters move ions and larger solutes across membranes; receptors mediate communication between the cell and its environment and membrane-embedded enzymes catalyze chemical reactions. Understanding these mechanisms of action requires knowledge of how the proteins couple to their fluid, hydrated lipid membrane environment. We present here current studies in computational and experimental membrane protein biophysics, and show how they address outstanding challenges in understanding the complex environmental effects on the structure, function, and dynamics of membrane proteins.

Relaxation times and rheology in dense athermal suspensions

We study the jamming transition in a model of elastic particles under shear at zero temperature. The key quantity is the relaxation time τ which is obtained by stopping the shearing and letting energy and pressure decay to zero. At many different densities and initial shear rates we do several such relaxations to determine the average τ. We establish that τ diverges with the same exponent as the viscosity and determine another exponent from the relation between τ and the coordination number. Though most of the simulations are done for the model with dissipation due to the motion of particles relative to an affinely shearing substrate, we also examine a model, where the dissipation is instead due to velocity differences of disks in contact, and confirm that the above-mentioned exponent is the same for these two models. We also consider finite size effects on both τ and the coordination number.

Statistics of conserved quantities in mechanically stable packings of frictionless disks above jamming

We numerically simulate mechanically stable packings of soft-core, frictionless, bidisperse disks in two dimensions, above the jamming packing fraction ϕ(J). For configurations with a fixed isotropic global stress tensor, we compute the averages, variances, and correlations of conserved quantities (stress Γ(C), force-tile area A(C), Voronoi volume V(C), number of particles N(C), and number of small particles N(sC)) on compact subclusters of particles C, as a function of the cluster size and the global system stress. We find several significant differences depending on whether the cluster C is defined by a fixed radius R or a fixed number of particles M. We comment on the implications of our findings for maximum entropy models of jammed packings.

Diverging viscosity and soft granular rheology in non-Brownian suspensions

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 carefully study the asymptotic limits of large system sizes and infinitely hard particle repulsion. We first focus on the asymptotic behavior of the shear viscosity in the hard particle limit. By measuring the viscosity increase over about 5 orders of magnitude, we are able to confirm its asymptotic power law divergence close to the jamming transition. However, a precise determination of the critical density and critical exponent is difficult due to the "multiscaling" behavior of the viscosity. Additionally, finite-size scaling analysis suggests that this divergence is accompanied by a growing correlation length scale, which also diverges algebraically. Finally, we study the effect of particle softness and propose a natural extension of the standard granular rheology, which we test against our simulation data. Close to the jamming transition, this "soft granular rheology" offers a detailed description of the nonlinear rheology of soft particles, which differs from earlier empirical scaling forms.

Universality of jamming criticality in overdamped shear-driven frictionless disks

We investigate the criticality of the jamming transition for overdamped shear-driven frictionless disks in two dimensions for two different models of energy dissipation: (i) Durian's bubble model with dissipation proportional to the velocity difference of particles in contact, and (ii) Durian's "mean-field" approximation to (i), with dissipation due to the velocity difference between the particle and the average uniform shear flow velocity. By considering the finite-size behavior of pressure, the pressure analog of viscosity, and the macroscopic friction σ/p, we argue that these two models share the same critical behavior.

Jamming in finite systems: stability, anisotropy, fluctuations, and scaling

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 effect on the very meaning of jamming. Here, we provide a comprehensive numerical study of finite-size effects in sphere packings above the jamming transition, focusing on stability as well as the scaling of the contact number and the elastic response.

Aspects of jamming in two-dimensional athermal frictionless systems

In this work we provide an overview of jamming transitions in two dimensional systems focusing on the limit of frictionless particle interactions in the absence of thermal fluctuations. We first discuss jamming in systems with short range repulsive interactions, where the onset of jamming occurs at a critical packing density and where certain quantities show a divergence indicative of critical behavior. We describe how aspects of the dynamics change as the jamming density is approached and how these dynamics can be explored using externally driven probes. Different particle shapes can produce jamming densities much lower than those observed for disk-shaped particles, and we show how jamming exhibits fragility for some shapes while for other shapes this is absent. Next we describe the effects of long range interactions and jamming behavior in systems such as charged colloids, vortices in type-II superconductors, and dislocations. We consider the effect of adding obstacles to frictionless jamming systems and discuss connections between this type of jamming and systems that exhibit depinning transitions. Finally, we discuss open questions such as whether the jamming transition in all these different systems can be described by the same or a small subset of universal behaviors, as well as future directions for studies of jamming transitions in two dimensional systems, such as jamming in self-driven or active matter systems.

Size and density avalanche scaling near jamming

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 field 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 by introducing a new universal jamming exponent and hyperscaling relationships, and we use these results to describe the volume fraction dependence of the friction coefficient.

Finite size analysis of zero-temperature jamming transition under applied shear stress by minimizing a thermodynamic-like potential

By finding local minima of a thermodynamic-like potential, we generate jammed packings of frictionless spheres under constant shear stress σ and obtain the yield stress σy by sampling the potential energy landscape. For three-dimensional systems with harmonic repulsion, σy satisfies the finite size scaling with the limiting scaling relation σy∼ϕ-ϕc,∞, where ϕc,∞ is the critical volume fraction of the jamming transition at σ=0 in the thermodynamic limit. The finite size scaling implies a length ξ∼(ϕ-ϕc,∞)-ν with ν=0.81±0.05, which turns out to be a robust and universal length scale exhibited as well in the finite size scaling of multiple quantities measured without shear and independent of particle interaction. Moreover, comparison between our new approach and quasistatic shear reveals that quasistatic shear tends to explore low-energy states.

Pressure distribution and critical exponent in statically jammed and shear-driven frictionless disks

We numerically study the distributions of global pressure that are found in ensembles of statically jammed and quasistatically sheared systems of bidisperse, frictionless disks at fixed packing fraction ϕ in two dimensions. We use these distributions to address the question of how pressure increases as ϕ increases above the jamming point ϕ(J), p ∼ |ϕ-ϕ(J)(y). For statically jammed ensembles, our results are consistent with the exponent y being simply related to the power law of the interparticle soft-core interaction. For sheared systems, however, the value of y is consistent with a nontrivial value, as found previously in rheological simulations.

Jamming graphs: a local approach to global mechanical rigidity

We revisit the concept of minimal rigidity as applied to frictionless, repulsive soft sphere packings in two dimensions with the introduction of the jamming graph. Minimal rigidity is a purely combinatorial property encoded via Laman's theorem in two dimensions. It constrains the global, average coordination number of the graph, for example. However, minimal rigidity does not address the geometry of local mechanical stability. The jamming graph contains both properties of global mechanical stability at the onset of jamming and local mechanical stability. We demonstrate how jamming graphs can be constructed using local moves via the Henneberg construction such that these graphs fall under the jurisdiction of correlated percolation. We then probe how jamming graphs destabilize, or become unjammed, by deleting a bond and computing the resulting rigid cluster distribution. We also study how the system restabilizes with the addition of new contacts and how a jamming graph with extra (redundant) contacts destabilizes. The latter endeavor allows us to probe a disk packing in the rigid phase and uncover a potentially new diverging length scale associated with the random deletion of contacts as compared to the study of cut-out (or frozen-in) subsystems.

Jamming in systems with quenched disorder

We numerically study the effect of adding quenched disorder in the form of randomly placed pinning sites on jamming transitions in a disk packing that jams at a well-defined point J in the clean limit. Quenched disorder decreases the jamming density and introduces a depinning threshold. The onset of a finite threshold coincides with point J at the lowest pinning densities, but for higher pinning densities there is always a finite depinning threshold even well below jamming. We find that proximity to point J strongly affects the transport curves and noise fluctuations, and we observe a change from plastic behavior below jamming, where the system is highly heterogeneous, to elastic depinning above jamming. Many of the general features we find are related to other systems containing quenched disorder, including the peak effect observed in vortex systems.

Shear flow of non-Brownian suspensions close to jamming

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-only weakly depend on the dissipation mechanism. Rather they are determined by steric effects, which are the main driving forces in the quasistatic simulations. The critical exponent of the suspension viscosity with respect to φ can then be deduced, and is consistent with experimental data.

The physics of the colloidal glass transition

As one increases the concentration of a colloidal suspension, the system exhibits a dramatic increase in viscosity. Beyond a certain concentration, the system is said to be a colloidal glass; structurally, the system resembles a liquid, yet motions within the suspension are slow enough that it can be considered essentially frozen. For several decades, colloids have served as a valuable model system for understanding the glass transition in molecular systems. The spatial and temporal scales involved allow these systems to be studied by a wide variety of experimental techniques. The focus of this review is the current state of understanding of the colloidal glass transition, with an emphasis on experimental observations. A brief introduction is given to important experimental techniques used to study the glass transition in colloids. We describe features of colloidal systems near and in glassy states, including increases in viscosity and relaxation times, dynamical heterogeneity and ageing, among others. We also compare and contrast the glass transition in colloids to that in molecular liquids. Other glassy systems are briefly discussed, as well as recently developed synthesis techniques that will keep these systems rich with interesting physics for years to come.

A unified framework for non-brownian suspension flows and soft amorphous solids

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, and where elastic properties display scaling and a diverging length scale. The correspondence between these two transitions is at present unclear. Here we show that for a simple model of dense flow, which we argue captures the essential physics near the jamming threshold, a formal analogy can be made between the rheology of the flow and the elasticity of simple networks. This analogy leads to a new conceptual framework to relate microscopic structure to rheology. It enables us to define and compute numerically normal modes and a density of states. We find striking similarities between the density of states in flow, and that of amorphous solids near unjamming: both display a plateau above some frequency scale ω(∗) ∼ |z(c) - z|, where z is the coordination of the network of particle in contact, z(c) = 2D where D is the spatial dimension. However, a spectacular difference appears: the density of states in flow displays a single mode at another frequency scale ω(min) ≪ ω(∗) governing the divergence of the viscosity.

Contact percolation transition in athermal particulate systems

Typical quasistatic compression algorithms for generating jammed packings of purely repulsive, frictionless particles begin with dilute configurations and then apply successive compressions with the relaxation of the elastic energy allowed between each compression step. It is well known that during isotropic compression these systems undergo a first-order-like jamming transition at packing fraction φ(J) from an unjammed state with zero pressure and no force-bearing contacts to a jammed, rigid state with nonzero pressure, a percolating network of force-bearing contacts, and contact number z=2d, where d is the spatial dimension. Using computer simulations of two-dimensional systems with monodisperse and bidisperse particle size distributions, we investigate the second-order-like contact percolation transition, which precedes the jamming transition with φ(P)<φ(J) and signals the formation of a system-spanning cluster of non-force-bearing contacts between particles. By measuring the number of nonfloppy modes of the dynamical matrix, the displacement field between successive compression steps, and the overlap between the adjacency matrix, which represents the network of contacting grains, at φ and φ(J), we find that the contact percolation transition also signals the onset of a nontrivial mechanical response to applied stress. Our results show that cooperative particle motion occurs in unjammed systems significantly below the jamming transition for φ(P)<φ<φ(J), not only for jammed systems with φ>φ(J).

Similarities between protein folding and granular jamming

Grains and glasses, widely different materials, arrest their motions upon decreasing temperature and external load, respectively, in common ways, leading to a universal jamming phase diagram conjecture. However, unified theories are lacking, mainly because of the disparate nature of the particle interactions. Here we demonstrate that folded proteins exhibit signatures common to both glassiness and jamming by using temperature- and force-unfolding molecular dynamics simulations. Upon folding, proteins develop a peak in the interatomic force distributions that falls on a universal curve with experimentally measured forces on jammed grains and droplets. Dynamical signatures are found as a dramatic slowdown of stress relaxation upon folding. Together with granular similarities, folding is tied not just to the jamming transition, but a more nuanced picture of anisotropy, preparation protocol and internal interactions emerges. Results have implications for designing stable polymers and can open avenues to link protein folding to jamming theory.