Jamming transition and inherent structures of hard spheres and disks

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.

Anoplophora graafi longhorn beetle coloration is due to disordered diamond-like packed spheres

While artificial photonic materials are typically highly ordered, photonic structures in many species of birds and insects do not possess a long-range order. Studying their order-disorder interplay sheds light on the origin of the photonic band gap. Here, we investigated the scale morphology of the Anoplophora graafi longhorn beetle. Combining small-angle X-ray scattering and slice-and-view FIB-SEM tomography with molecular dynamics and optical simulations, we characterised the chitin sphere assemblies within blue and green A. graafi scales. The low volume fraction of spheres and the number of their nearest neighbours are incompatible with any known close-packed sphere morphology. A short-range diamond lattice with long-range disorder best describes the sphere assembly, which will inspire the development of new colloid-based photonic materials.

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.

Dependence of the Glass Transition and Jamming Densities on Spatial Dimension

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 recent theoretical calculations.

Comparison of compression versus shearing near jamming, for a simple model of athermal frictionless disks in suspension

Using a simplified model for a non-Brownian suspension, we numerically study the response of athermal, overdamped, frictionless disks in two dimensions to isotropic and uniaxial compression, as well as to pure and simple shearing, all at finite constant strain rates ε[over ̇]. We show that isotropic and uniaxial compression result in the same jamming packing fraction ϕ_{J}, while pure-shear- and simple-shear-induced jamming occurs at a slightly higher ϕ_{J}^{*}, consistent with that found previously for simple shearing. A critical scaling analysis of pure shearing gives critical exponents consistent with those previously found for both isotropic compression and simple shearing. Using orientational order parameters for contact bond directions, we compare the anisotropy of the force and contact networks at both lowest nematic order, as well as higher 2n-fold order.

Revealing the characteristic length of random close packing via critical-like random pinning

By randomly pinning particles in fluidized states and finding the local energy minima, we form static packings of mono-disperse disks that resemble random close packing, when only n_c = 2.6% of the particles are pinned. The packings are isostatic and exhibit typical critical scalings of the jamming transition. The non-triviality of n_c is manifested mainly in two aspects. First, n_c acts as a critical point, leading to bifurcated critical scalings in its vicinity. The criticality of n_c is also demonstrated in the packings of weakly polydisperse disks. Second, n_c sets a length scale in agreement with the characteristic length of random close packing. With robust evidence, we show that this agreement is generally true for both mono- and poly-disperse particles and in both two and three dimensions. The randomness inherited from fluidized states by random pinning thus interprets the randomness of random close packing from a unique perspective.

Inherent structure landscape of hard spheres confined to narrow cylindrical channels

The inherent structure landscape for a system of hard spheres confined to a hard cylindrical channel, such that spheres can only contact their first and second neighbors, is studied using an analytical model that extends previous results [Phys. Rev. Lett. 115, 025702 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.025702] to provide a comprehensive picture of jammed packings over a range of packing densities. In the model, a packing is described as an arrangement of k helical sections, separated by defects, that have alternating helical twist directions and where all spheres satisfy local jamming constraints. The structure of each helical section is determined by a single helical twist angle, and a jammed packing is obtained by minimizing the length of the channel per particle with respect to the k helical section angles. An analysis of a small system of N=20 spheres shows that the basins on the inherent structure landscape associated with these helical arrangements split into a number of distinct jammed states separated by low barriers giving rise to a degree of hierarchical organization. The model accurately predicts the geometric properties of packings generated using the Lubachevsky and Stillinger compression scheme (N=10^{4}) and provides insight into the nature of the probability distribution of helical section lengths.

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.

Critical pore radius and transport properties of disordered hard- and overlapping-sphere models

Transport properties of porous media are intimately linked to their pore-space microstructures. We quantify geometrical and topological descriptors of the pore space of certain disordered and ordered distributions of spheres, including pore-size functions and the critical pore radius δ_{c}. We focus on models of porous media derived from maximally random jammed sphere packings, overlapping spheres, equilibrium hard spheres, quantizer sphere packings, and crystalline sphere packings. For precise estimates of the percolation thresholds, we use a strict relation of the void percolation around sphere configurations to weighted bond percolation on the corresponding Voronoi networks. We use the Newman-Ziff algorithm to determine the percolation threshold using universal properties of the cluster size distribution. The critical pore radius δ_{c} is often used as the key characteristic length scale that determines the fluid permeability k. A recent study [Torquato, Adv. Wat. Resour. 140, 103565 (2020)10.1016/j.advwatres.2020.103565] suggested for porous media with a well-connected pore space an alternative estimate of k based on the second moment of the pore size 〈δ^{2}〉, which is easier to determine than δ_{c}. Here, we compare δ_{c} to the second moment of the pore size 〈δ^{2}〉, and indeed confirm that, for all porosities and all models considered, δ_{c}^{2} is to a good approximation proportional to 〈δ^{2}〉. However, unlike 〈δ^{2}〉, the permeability estimate based on δ_{c}^{2} does not predict the correct ranking of k for our models. Thus, we confirm 〈δ^{2}〉 to be a promising candidate for convenient and reliable estimates of the fluid permeability for porous media with a well-connected pore space. Moreover, we compare the fluid permeability of our models with varying degrees of order, as measured by the τ order metric. We find that (effectively) hyperuniform models tend to have lower values of k than their nonhyperuniform counterparts. Our findings could facilitate the design of porous media with desirable transport properties via targeted pore statistics.

A jamming plane of sphere packings

The concept of jamming has attracted great research interest due to its broad relevance in soft-matter, such as liquids, glasses, colloids, foams, and granular materials, and its deep connection to sphere packing and optimization problems. Here, we show that the domain of amorphous jammed states of frictionless spheres can be significantly extended, from the well-known jamming-point at a fixed density, to a jamming-plane that spans the density and shear strain axes. We explore the jamming-plane, via athermal and thermal simulations of compression and shear jamming, with initial equilibrium configurations prepared by an efficient swap algorithm. The jamming-plane can be divided into reversible-jamming and irreversible-jamming regimes, based on the reversibility of the route from the initial configuration to jamming. Our results suggest that the irreversible-jamming behavior reflects an escape from the metastable glass basin to which the initial configuration belongs to or the absence of such basins. All jammed states, either compression- or shear-jammed, are isostatic and exhibit jamming criticality of the same universality class. However, the anisotropy of contact networks nontrivially depends on the jamming density and strain. Among all state points on the jamming-plane, the jamming-point is a unique one with the minimum jamming density and the maximum randomness. For crystalline packings, the jamming-plane shrinks into a single shear jamming-line that is independent of initial configurations. Our study paves the way for solving the long-standing random close-packing problem and provides a more complete framework to understand jamming.

Dilatancy, shear jamming, and a generalized jamming phase diagram of frictionless sphere packings

Granular packings display the remarkable phenomenon of dilatancy, wherein their volume increases upon shear deformation. Conventional wisdom and previous results suggest that dilatancy, also being the related phenomenon of shear-induced jamming, requires frictional interactions. Here, we show that the occurrence of isotropic jamming densities φ_j above the minimal density (or the J-point density) φ_J leads both to the emergence of shear-induced jamming and dilatancy in frictionless packings. Under constant pressure shear, the system evolves into a steady-state at sufficiently large strains, whose density only depends on the pressure and is insensitive to the initial jamming density φ_j. In the limit of vanishing pressure, the steady-state exhibits critical behavior at φ_J. While packings with different φ_j values display equivalent scaling properties under compression, they exhibit striking differences in rheological behaviour under shear. The yield stress under constant volume shear increases discontinuously with density when φ_j > φ_J, contrary to the continuous behaviour in generic packings that jam at φ_J. Our results thus lead to a more coherent, generalised picture of jamming in frictionless packings, which also have important implications on how dilatancy is understood in the context of frictional granular matter.

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.

Interfacial viscoelasticity and jamming of colloidal particles at fluid-fluid interfaces: a review

Colloidal particles can be adsorbed at fluid-fluid interfaces, a phenomenon frequently observed in particle-stabilized foams, Pickering emulsions, and bijels. Particles adsorbed at interfaces exhibit unique physical and chemical behaviors, which affect the mechanical properties of the interface. Therefore, interfacial colloidal particles are of interest in terms of both fundamental and applied research. In this paper, we review studies on the adsorption of colloidal particles at fluid-fluid interfaces, from both thermodynamic and mechanical points of view, and discuss the differences as compared with surfactants and polymers. The unique particle interactions induced by the interfaces as well as the particle dynamics including lateral diffusion and contact line relaxation will be presented. We focus on the rearrangement of the particles and the resultant interfacial viscoelasticity. Particular emphasis will be given to the effects of particle shape, size, and surface hydrophobicity on the interfacial particle assembly and the mechanical properties of the obtained particle layer. We will also summarize recent advances in interfacial jamming behavior caused by adsorption of particles at interfaces. The buckling and cracking behavior of particle layers will be discussed from a mechanical perspective. Finally, we suggest several potential directions for future research in this area.

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.

Unified phase diagram of reversible-irreversible, jamming, and yielding transitions in cyclically sheared soft-sphere packings

Self-organization, and transitions from reversible to irreversible behavior, of interacting particle assemblies driven by externally imposed stresses or deformation is of interest in comprehending diverse phenomena in soft matter. They have been investigated in a wide range of systems, such as colloidal suspensions, glasses, and granular matter. In different density and driving regimes, such behavior is related to yielding of amorphous solids, jamming, memory formation, etc. How these phenomena are related to each other has not, however, been much studied. In order to obtain a unified view of the different regimes of behavior, and transitions between them, we investigate computationally the response of soft-sphere assemblies to athermal cyclic-shear deformation over a wide range of densities and amplitudes of shear deformation. Cyclic-shear deformation induces transitions from reversible to irreversible behavior in both unjammed and jammed soft-sphere packings. Well above the minimum isotropic jamming density ([Formula: see text]), this transition corresponds to yielding. In the vicinity of the jamming point, up to a higher-density limit, we designate [Formula: see text], an unjammed phase emerges between a localized, absorbing phase and a diffusive, irreversible, phase. The emergence of the unjammed phase signals the shifting of the jamming point to higher densities as a result of annealing and opens a window where shear jamming becomes possible for frictionless packings. Below [Formula: see text], two distinct localized states, termed point- and loop-reversible, are observed. We characterize in detail the different regimes and transitions between them and obtain a unified density-shear amplitude phase diagram.

Building Reconfigurable Devices Using Complex Liquid-Fluid Interfaces

Liquid-fluid interfaces provide a platform both for structuring liquids into complex shapes and assembling dimensionally confined, functional nanomaterials. Historically, attention in this area has focused on simple emulsions and foams, in which surface-active materials such as surfactants or colloids stabilize structures against coalescence and alter the mechanical properties of the interface. In recent decades, however, a growing body of work has begun to demonstrate the full potential of the assembly of nanomaterials at liquid-fluid interfaces to generate functionally advanced, biomimetic systems. Here, a broad overview is given, from fundamentals to applications, of the use of liquid-fluid interfaces to generate complex, all-liquid devices with a myriad of potential applications.

Temperature dependence of the transition packing fraction of thermal jamming in a harmonic soft sphere system

The glassy dynamics of soft harmonic spheres are often mapped onto the dynamics of hard spheres by considering an effective diameter for the soft particles and therefore an effective packing fraction. While in this approach the thermal fluctuations within valleys of the energy landscape are covered, the crossing of energy barriers from one valley into another usually is neglected. Here we argue-motivated by studies of the glass transition based on explorations of the energy landscape-that the crossing of energy barriers can be attributed by an effective decrease of the glass transition packing fraction with increasing temperature T according to T ^0.2. Furthermore, we reanalyzing data of soft sphere simulations. Since fitting scaling laws to simulation data always allows for some arbitrariness, we cannot prove based on the simulation data that our idea of a shift of the glass transition packing fraction due to barrier crossings is the only possible way to explain the discrepancies that have been observed previously. However, we show that a possible explanation of the simulation data is given by our approach to characterize the dynamics of soft spheres by both, the previously-considered temperature-dependent effective packing fraction due to the increase of the mean overlap between neighboring particles with stronger thermal fluctuations and the newly introduced increase of the glass transition packing with an increasing probability of barrier crossings.

Marginally stable phases in mean-field structural glasses

A novel form of amorphous matter characterized by marginal stability was recently discovered in the mean-field theory of structural glasses. Using this approach, we provide complete phase diagrams delimiting the location of the marginally stable glass phase for a large variety of pair interactions and physical conditions, extensively exploring physical regimes relevant to granular matter, foams, emulsions, hard and soft colloids, and molecular glasses. We find that all types of glasses may become marginally stable, but the extent of the marginally stable phase highly depends on the preparation protocol. Our results suggest that marginal phases should be observable for colloidal and non-Brownian particles near jamming and for poorly annealed glasses. For well-annealed glasses, two distinct marginal phases are predicted. Our study unifies previous results on marginal stability in mean-field models and will be useful to guide numerical simulations and experiments aimed at detecting marginal stability in finite-dimensional amorphous materials.

Spin-glass-like aging in colloidal and granular glasses

Motivated by the mean field prediction of a Gardner phase transition between a "normal glass" and a "marginally stable glass", we investigate the off-equilibrium dynamics of three-dimensional polydisperse hard spheres, used as a model for colloidal or granular glasses. Deep inside the glass phase, we find that a sharp crossover pressure PG separates two distinct dynamical regimes. For pressure P < PG, the glass behaves as a normal solid, displaying fast dynamics that quickly equilibrate within the glass free energy basin. For P > PG, instead, the dynamics become strongly anomalous, displaying very large equilibration timescales, aging, and a constantly increasing dynamical susceptibility. The crossover at PG is strongly reminiscent of the one observed in three-dimensional spin-glasses in an external field, suggesting that the two systems could be in the same universality class, consistent with theoretical expectations.

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.

Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical approximation formulas. These property formulas include interfacial bounds as well as universal scaling laws for the mean survival time and the fluid permeability. We also estimate the principal relaxation time associated with Brownian motion among perfectly absorbing traps. For the propagation of electromagnetic waves in the long-wavelength limit, we show that a dispersion of dielectric MRJ spheres within a matrix of another dielectric material forms, to a very good approximation, a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio. We compare the effective properties of the MRJ sphere packings to those of overlapping spheres, equilibrium hard-sphere packings, and lattices of hard spheres. Moreover, we generalize results to micro- and macroscopically anisotropic packings of spheroids with tensorial effective properties. The analytic bounds predict the qualitative trend in the physical properties associated with these structures, which provides guidance to more time-consuming simulations and experiments. They especially provide impetus for experiments to design materials with unique bulk properties resulting from hyperuniformity, including structural-color and color-sensing applications.

Shear-stress fluctuations and relaxation in polymer glasses

We investigate by means of molecular dynamics simulation a coarse-grained polymer glass model focusing on (quasistatic and dynamical) shear-stress fluctuations as a function of temperature T and sampling time Δt. The linear response is characterized using (ensemble-averaged) expectation values of the contributions (time averaged for each shear plane) to the stress-fluctuation relation μ_{sf} for the shear modulus and the shear-stress relaxation modulus G(t). Using 100 independent configurations, we pay attention to the respective standard deviations. While the ensemble-averaged modulus μ_{sf}(T) decreases continuously with increasing T for all Δt sampled, its standard deviation δμ_{sf}(T) is nonmonotonic with a striking peak at the glass transition. The question of whether the shear modulus is continuous or has a jump singularity at the glass transition is thus ill posed. Confirming the effective time-translational invariance of our systems, the Δt dependence of μ_{sf} and related quantities can be understood using a weighted integral over G(t).

Response of jammed packings to thermal fluctuations

We focus on the response of mechanically stable (MS) packings of frictionless, bidisperse disks to thermal fluctuations, with the aim of quantifying how nonlinearities affect system properties at finite temperature. In contrast, numerous prior studies characterized the structural and mechanical properties of MS packings of frictionless spherical particles at zero temperature. Packings of disks with purely repulsive contact interactions possess two main types of nonlinearities, one from the form of the interaction potential (e.g., either linear or Hertzian spring interactions) and one from the breaking (or forming) of interparticle contacts. To identify the temperature regime at which the contact-breaking nonlinearities begin to contribute, we first calculated the minimum temperatures T_{cb} required to break a single contact in the MS packing for both single- and multiple-eigenmode perturbations of the T=0 MS packing. We find that the temperature required to break a single contact for equal velocity-amplitude perturbations involving all eigenmodes approaches the minimum value obtained for a perturbation in the direction connecting disk pairs with the smallest overlap. We then studied deviations in the constant volume specific heat C[over ¯]_{V} and deviations of the average disk positions Δr from their T=0 values in the temperature regime T_{C[over ¯]_{V}}<T<T_{r}, where T_{r} is the temperature beyond which the system samples the basin of a new MS packing. We find that the deviation in the specific heat per particle ΔC[over ¯]_{V}^{0}/C[over ¯]_{V}^{0} relative to the zero-temperature value C[over ¯]_{V}^{0} can grow rapidly above T_{cb}; however, the deviation ΔC[over ¯]_{V}^{0}/C[over ¯]_{V}^{0} decreases as N^{-1} with increasing system size. To characterize the relative strength of contact-breaking versus form nonlinearities, we measured the ratio of the average position deviations Δr^{ss}/Δr^{ds} for single- and double-sided linear and nonlinear spring interactions. We find that Δr^{ss}/Δr^{ds}>100 for linear spring interactions is independent of system size. This result emphasizes that contact-breaking nonlinearities are dominant over form nonlinearities in the low-temperature range T_{cb}<T<T_{r} for model jammed systems.

Shear Modulus and Shear-Stress Fluctuations in Polymer Glasses

Using molecular dynamics simulation of a standard coarse-grained polymer glass model, we investigate by means of the stress-fluctuation formalism the shear modulus μ as a function of temperature T and sampling time Δt. While the ensemble-averaged modulus μ(T) is found to decrease continuously for all Δt sampled, its standard deviation δμ(T) is nonmonotonic, with a striking peak at the glass transition. Confirming the effective time-translational invariance of our systems, μ(Δt) can be understood using a weighted integral over the shear-stress relaxation modulus G(t). While the crossover of μ(T) gets sharper with an increasing Δt, the peak of δμ(T) becomes more singular. It is thus elusive to predict the modulus of a single configuration at the glass transition.

Shear-induced solidification of athermal systems with weak attraction

We find that unjammed packings of frictionless particles with rather weak attraction can always be driven into solidlike states by shear. The structure of shear-driven solids evolves continuously with packing fraction from gel-like to jamminglike, but is almost independent of the shear stress. In contrast, both the density of vibrational states (DOVS) and force network evolve progressively with the shear stress. There exists a packing fraction independent shear stress σ_{c}, at which the shear-driven solids are isostatic and have a flattened DOVS. Solidlike states induced by a shear stress greater than σ_{c} possess properties of marginally jammed solids and are thus strictly defined shear jammed states. Below σ_{c}, shear-driven solids with rather different structures are all under isostaticity and share common features in the DOVS and force network. Our study leads to a jamming phase diagram for weakly attractive particles, which reveals the significance of the shear stress in determining properties of shear-driven solids and the connections and distinctions between jamminglike and gel-like states.

Characterization of maximally random jammed sphere packings. II. Correlation functions and density fluctuations

In the first paper of this series, we introduced Voronoi correlation functions to characterize the structure of maximally random jammed (MRJ) sphere packings across length scales. In the present paper, we determine a variety of different correlation functions that arise in rigorous expressions for the effective physical properties of MRJ sphere packings and compare them to the corresponding statistical descriptors for overlapping spheres and equilibrium hard-sphere systems. Such structural descriptors arise in rigorous bounds and formulas for effective transport properties, diffusion and reactions constants, elastic moduli, and electromagnetic characteristics. First, we calculate the two-point, surface-void, and surface-surface correlation functions, for which we derive explicit analytical formulas for finite hard-sphere packings. We show analytically how the contact Dirac delta function contribution to the pair correlation function g_{2}(r) for MRJ packings translates into distinct functional behaviors of these two-point correlation functions that do not arise in the other two models examined here. Then we show how the spectral density distinguishes the MRJ packings from the other disordered systems in that the spectral density vanishes in the limit of infinite wavelengths; i.e., these packings are hyperuniform, which means that density fluctuations on large length scales are anomalously suppressed. Moreover, for all model systems, we study and compute exclusion probabilities and pore size distributions, as well as local density fluctuations. We conjecture that for general disordered hard-sphere packings, a central limit theorem holds for the number of points within an spherical observation window. Our analysis links problems of interest in material science, chemistry, physics, and mathematics. In the third paper of this series, we will evaluate bounds and estimates of a host of different physical properties of the MRJ sphere packings that are based on the structural characteristics analyzed in this paper.

Growing timescales and lengthscales characterizing vibrations of amorphous solids

Low-temperature properties of crystalline solids can be understood using harmonic perturbations around a perfect lattice, as in Debye's theory. Low-temperature properties of amorphous solids, however, strongly depart from such descriptions, displaying enhanced transport, activated slow dynamics across energy barriers, excess vibrational modes with respect to Debye's theory (i.e., a boson peak), and complex irreversible responses to small mechanical deformations. These experimental observations indirectly suggest that the dynamics of amorphous solids becomes anomalous at low temperatures. Here, we present direct numerical evidence that vibrations change nature at a well-defined location deep inside the glass phase of a simple glass former. We provide a real-space description of this transition and of the rapidly growing time- and lengthscales that accompany it. Our results provide the seed for a universal understanding of low-temperature glass anomalies within the theoretical framework of the recently discovered Gardner phase transition.

Equilibrium Sampling of Hard Spheres up to the Jamming Density and Beyond

We implement and optimize a particle-swap Monte Carlo algorithm that allows us to thermalize a polydisperse system of hard spheres up to unprecedentedly large volume fractions, where previous algorithms and experiments fail to equilibrate. We show that no glass singularity intervenes before the jamming density, which we independently determine through two distinct nonequilibrium protocols. We demonstrate that equilibrium fluid and nonequilibrium jammed states can have the same density, showing that the jamming transition cannot be the end point of the fluid branch.

The random packing density of nearly spherical particles

Obtaining general relations between macroscopic properties of random assemblies, such as density, and the microscopic properties of their constituent particles, such as shape, is a foundational challenge in the study of amorphous materials. By leveraging existing understanding of the random packing of spherical particles, we estimate the random packing density for all sufficiently spherical shapes. Our method uses the ensemble of random packing configurations of spheres as a reference point for a perturbative calculation, which we carry to linear order in the deformation. A fully analytic calculation shows that all sufficiently spherical shapes pack more densely than spheres. Additionally, we use simulation data for spheres to calculate numerical estimates for nonspherical particles and compare these estimates to simulations.

Glass transition of two-dimensional 80-20 Kob-Andersen model at constant pressure

We reconsider numerically the two-dimensional version of the Kob-Andersen model (KA2d) with a fraction of 80% of large spheres. A constant moderate pressure is imposed while the temperature T is systematically quenched from the liquid limit through the glass transition at [Formula: see text] down to very low temperatures. Monodisperse Lennard-Jones (mdLJ) bead systems, forming a crystal phase at low temperatures, are used to highlight several features of the KA2d model. As can be seen, e.g. from the elastic shear modulus G(T), determined using the stress-fluctuation formalism, our KA2d model is a good glass-former. A continuous cusp-singularity, [Formula: see text] with [Formula: see text], is observed in qualitative agreement with other recent numerical and theoretical work, however in striking conflict with the additive jump discontinuity predicted by mode-coupling theory.

Inherent structures, fragility, and jamming: insights from quasi-one-dimensional hard disks

We study a quasi-one-dimensional system of hard disks confined between hard lines to explore the relationship between the inherent structure landscape, the thermodynamics, and the dynamics of the fluid. The transfer matrix method is used to obtain an exact description of the landscape, equation of state, and provide a mapping of configurations of the equilibrium fluid to their local jammed structures. This allows us to follow how the system samples the landscape as a function of occupied volume fraction ϕ. Configurations of the ideal gas map to the maximum in the distribution of inherent structures, with a jamming volume fraction ϕ(J)(*), and sample more dense basins with increasing ϕ. This suggests jammed states with a density below ϕ(J)(*) are inaccessible from the equilibrium fluid. The configurational entropy of the fluid decreases rapidly at intermediate ϕ before plateauing at a low value and going to zero as the most dense packing is approached. This leads to the appearance of a maximum in both the isobaric heat capacity and the inherent structure pressure. We also show that the system exhibits a crossover from fragile to strong fluid behavior, located at the heat capacity maximum. Structural relaxation in the fragile fluid is shown to be controlled by the presence of high order saddle points caused by neighboring defects that are unstable with respect to jamming and spontaneously rearrange to form a stable local environment. In the strong fluid, the defect concentration is low so that defects do not interact and relaxation occurs through the hopping of isolated defects between stable local packing environments.

Shear-stress relaxation and ensemble transformation of shear-stress autocorrelation functions

We revisit the relation between the shear-stress relaxation modulus G(t), computed at finite shear strain 0<γ≪1, and the shear-stress autocorrelation functions C(t)|(γ) and C(t)|(τ) computed, respectively, at imposed strain γ and mean stress τ. Focusing on permanent isotropic spring networks it is shown theoretically and computationally that in general G(t)=C(t)|(τ)=C(t)|(γ)+G(eq) for t>0 with G(eq) being the static equilibrium shear modulus. G(t) and C(t)|(γ) thus must become different for solids and it is impossible to obtain G(eq) alone from C(t)|(γ) as often assumed. We comment briefly on self-assembled transient networks where G(eq)(f) must vanish for a finite scission-recombination frequency f. We argue that G(t)=C(t)|(τ)=C(t)|(γ) should reveal an intermediate plateau set by the shear modulus G(eq)(f=0) of the quenched network.

Potential energy landscape and inherent dynamics of a hard-sphere fluid

Hard-sphere models exhibit many of the same kinds of supercooled-liquid behavior as more realistic models of liquids, but the highly nonanalytic character of their potentials makes it a challenge to think of that behavior in potential energy landscape terms. We show here that it is possible to calculate an important topological property of hard-sphere landscapes, the geodesic pathways through those landscapes, and to do so without artificially coarse-graining or softening the potential. We show, moreover, that the rapid growth of the lengths of those pathways with increasing packing fraction quantitatively predicts the precipitous decline in diffusion constants in a glass-forming hard-sphere mixture model. The geodesic paths themselves can be considered as defining the intrinsic dynamics of hard spheres, so it is also revealing to find that they (and therefore the features of the underlying potential energy landscape) correctly predict the occurrence of dynamic heterogeneity and nonzero values of the non-Gaussian parameter. The success of these landscape predictions for the dynamics of such a singular model emphasizes that there is more to potential energy landscapes than is revealed by looking at the minima and saddle points.

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.

Detailed characterization of rattlers in exactly isostatic, strictly jammed sphere packings

We generate jammed disordered packings of 100≤N≤2000 monodisperse hard spheres in three dimensions whose strictly jammed backbones are demonstrated to be exactly isostatic with unprecedented numerical accuracy. This is accomplished by using the Torquato-Jiao (TJ) packing algorithm as a means of studying the maximally random jammed (MRJ) state. The rattler fraction of these packings converges towards 0.015 in the infinite-system limit, which is markedly lower than previous estimates for the MRJ state using the Lubachevsky-Stillinger protocol. This is because the packings that the TJ algorithm creates are closer to the true MRJ state, as shown using bond-orientational and translational order metrics. The rattler pair correlation statistics exhibit strongly correlated behavior contrary to the conventional understanding that they be randomly (Poisson) distributed. Dynamically interacting "polyrattlers" may be found imprisoned in shared cages as well as interacting through "bottlenecks" in the backbone and these clusters are mainly responsible for the sharp increase in the rattler pair correlation function near contact. We discover the surprising existence of polyrattlers with cluster sizes of up to five rattlers (which is expected to increase with system size) and present a distribution of polyrattler occurrence as a function of cluster size and system size. We also enumerate all of the rattler interaction topologies we observe and present images of several examples, showing that MRJ packings of monodisperse spheres can contain large rattler cages while still obeying the strict jamming criterion. The backbone spheres that encage the rattlers are significantly hypostatic, implying that correspondingly hyperstatic regions must exist elsewhere in these isostatic packings. We also observe that rattlers in hard-sphere packings share an apparent connection with the low-temperature two-level system anomalies that appear in real amorphous insulators and semiconductors.

Transition state theory and the dynamics of hard disks

The dynamics of two- and five-disk systems confined in a square has been studied using molecular dynamics simulations and compared with the predictions of transition state theory. We determine the partition functions Z and Z(‡) of transition state theory using a procedure first used by Salsburg and Wood for the pressure. Our simulations show this procedure and transition state theory are in excellent agreement with the simulations. A generalization of the transition state theory to the case of a large number of disks N is made and shown to be in full agreement with simulations of disks moving in a narrow channel. The same procedure for hard spheres in three dimensions leads to the Vogel-Fulcher-Tammann formula for their alpha relaxation time.

Disordered strictly jammed binary sphere packings attain an anomalously large range of densities

Previous attempts to simulate disordered binary sphere packings have been limited in producing mechanically stable, isostatic packings across a broad spectrum of packing fractions. Here we report that disordered strictly jammed binary packings (packings that remain mechanically stable under general shear deformations and compressions) can be produced with an anomalously large range of average packing fractions 0.634≤φ≤0.829 for small to large sphere radius ratios α restricted to α≥0.100. Surprisingly, this range of average packing fractions is obtained for packings containing a subset of spheres (called the backbone) that are exactly strictly jammed, exactly isostatic, and also generated from random initial conditions. Additionally, the average packing fractions of these packings at certain α and small sphere relative number concentrations x approach those of the corresponding densest known ordered packings. These findings suggest for entropic reasons that these high-density disordered packings should be good glass formers and that they may be easy to prepare experimentally. We also identify an unusual feature of the packing fraction of jammed backbones (packings with rattlers excluded). The backbone packing fraction is about 0.624 over the majority of the α-x plane, even when large numbers of small spheres are present in the backbone. Over the (relatively small) area of the α-x plane where the backbone is not roughly constant, we find that backbone packing fractions range from about 0.606 to 0.829, with the volume of rattler spheres comprising between 1.6% and 26.9% of total sphere volume. To generate isostatic strictly jammed packings, we use an implementation of the Torquato-Jiao sequential linear programming algorithm [Phys. Rev. E 82, 061302 (2010)], which is an efficient producer of inherent structures (mechanically stable configurations at the local maxima in the density landscape). The identification and explicit construction of binary packings with such high packing fractions could have important practical implications for granular composites where density is critical both to material properties and fabrication cost, including for solid propellants, concrete, and ceramics. The densities and structures of jammed binary packings at various α and x are also relevant to the formation of a glass phase in multicomponent metallic systems.

Decorrelation of the static and dynamic length scales in hard-sphere glass formers

We show that, in the equilibrium phase of glass-forming hard-sphere fluids in three dimensions, the static length scales tentatively associated with the dynamical slowdown and the dynamical length characterizing spatial heterogeneities in the dynamics unambiguously decorrelate. The former grow at a much slower rate than the latter when density increases. This observation is valid for the dynamical range that is accessible to computer simulations, which roughly corresponds to that accessible in colloidal experiments. We also find that, in this same range, no one-to-one correspondence between relaxation time and point-to-set correlation length exists. These results point to the coexistence of several relaxation mechanisms in the dynamically accessible regime of three-dimensional hard-sphere glass formers.