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Can liquid-like and gas-like states be distinguished beyond the critical point, where the liquid-gas phase transition no longer exists and conventionally only a single supercritical fluid phase is defined? Recent experiments and simulations report strong evidence of dynamical crossovers above the critical temperature and pressure. Despite using different criteria, many existing theoretical explanations consider a single crossover line separating liquid-like and gas-like states in the supercritical fluid phase. We argue that such a single-line scenario is inconsistent with the supercritical behavior of the Ising model, which has two crossover lines due to its symmetry, violating the universality principle of critical phenomena. To reconcile the inconsistency, we define two thermodynamic crossover lines in supercritical fluids as boundaries of liquid-like, indistinguishable, and gas-like states. Near the critical point, the two crossover lines follow critical scalings with exponents of the Ising universality class, supported by calculations of theoretical models and analyses of experimental data from the standard database. The upper line agrees with crossovers independently estimated from the inelastic X-ray scattering data of supercritical argon, and from the small-angle neutron scattering data of supercritical carbon dioxide. The lower line is verified by the equation of states for the compressibility factor. This work provides a fundamental framework for understanding supercritical physics in general phase transitions.
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Amorphous materials undergo a transition from liquid-like to solid-like states through processes like rapid quenching or densification. Under external loads, they exhibit yielding, with minimal structural changes compared to crystals. However, these universal characteristics are rarely explored comprehensively in a single granular experiment due to the added complexity of inherent friction. The discernible differences between static configurations before and after yielding are largely unaddressed, and a comprehensive examination from both statistical physics and mechanical perspectives is lacking. To address these gaps, we conducted experiments using photoelastic disks, simultaneously tracking particles and measuring forces. Our findings reveal that the yielding transition demonstrates critical behavior from a statistical physics standpoint and marginal stability from a mechanical perspective, akin to the isotropic jamming transition. This criticality differs significantly from spinodal criticality in frictionless amorphous solids, highlighting unique characteristics of granular yielding. Furthermore, our analysis confirms the marginal stability of granular yielding by assessing the contact number and evaluating the balance between weak forces and small gaps. These factors serve as structural indicators for configurations before and after yielding. Our results not only contribute to advancing our understanding of the fundamental physics of granular materials but also bear significant implications for practical applications in various fields.
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SignificanceAmorphous materials, such as grains, foams, colloids, and glasses, are ubiquitous in nature and our daily life. They can undergo glass transitions or jamming transitions to obtain rigidity either by fast quench or compression, but show subtle changes in the structures compared to the liquid states or liquid-like states. Recent progress on the first-principle replica theory unifies the glass transition and the jamming transition and points out the marginal phase with fractal free-energy landscape within the stable glass phase. Independently, marginal stability analysis predicts the relations between the exponents of the marginal phase. Here, we perform experiments with photoelastic disks and provide direct evidence of these theories in real-world amorphous materials.
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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.
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Apparent critical phenomena, typically indicated by growing correlation lengths and dynamical slowing down, are ubiquitous in nonequilibrium systems such as supercooled liquids, amorphous solids, active matter, and spin glasses. It is often challenging to determine if such observations are related to a true second-order phase transition as in the equilibrium case or simply a crossover and even more so to measure the associated critical exponents. Here we show that the simulation results of a hard-sphere glass in three dimensions are consistent with the recent theoretical prediction of a Gardner transition, a continuous nonequilibrium phase transition. Using a hybrid molecular simulation-machine learning approach, we obtain scaling laws for both finite-size and aging effects and determine the critical exponents that traditional methods fail to estimate. Our study provides an approach that is useful to understand the nature of glass transitions and can be generalized to analyze other nonequilibrium phase transitions.
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Around a glass transition, the dynamics of a supercooled liquid dramatically slow down, exhibited by caging of particles, while the structural changes remain subtle. Alternative to recent machine learning studies searching for structural predictors of glassy dynamics, here we propose to learn directly particle caging features defined purely according to dynamics. We focus on three transitions in a simulated hard sphere glass model, the melting of ultra-stable glasses, the Gardner transition and the liquid to ordinary glass transition. Implementing the machine learning algorithm based on a two-level nested neural network, we attain not only appropriate caging order parameters for all three transitions, but also a phase classification for input samples. A finite-size scaling analysis of the phase classification results identifies the order of melting (first) and Gardner (second) transitions. A false positive is avoided, as the liquid to glass transition is indicated as a crossover, rather than a phase transition with a well-defined transition point. This study paves the way to a generic approach for learning dynamical features in glassy systems, with a minimum requirement of system-specific knowledge.
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The two-state stochastic phenomenon is observed in various systems and is attracting more interest, and it can be described by the two-state random walk (TSRW) model. The TSRW model is a typical two-state renewal process alternating between the continuous-time random walk state and the Lévy walk state, in both of which the sojourn time distributions follow a power law. In this paper, by discussing the statistical properties and calculating the ensemble averaged and time averaged mean squared displacement, the ergodic property and the ultimate diffusive behavior of the aging TSRW is studied. Results reveal that because of the two-state intermittent feature, ergodicity and nonergodicity can coexist in the aging TSRW, which behave as the time scalings of the time averages and ensemble averages not being identically equal. In addition, we find that the unique state occupation mechanism caused by the diverging mean of the sojourn times of one state, determines the aging TSRW's ultimate diffusive behavior at extremely large timescales, i.e., instead of the term with the larger diffusion exponent, the diffusion is surprisingly characterized by the term with the smaller one, which is distinctly different from previous conclusions and known results. At last, we note that the Lévy walk with rests model which also displays aging and ergodicity breaking, can be generalized by the TSRW model.
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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 φj above the minimal density (or the J-point density) φJ 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 φj. In the limit of vanishing pressure, the steady-state exhibits critical behavior at φJ. While packings with different φj 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 φj > φJ, contrary to the continuous behaviour in generic packings that jam at φJ. 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.
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Measuring and optimizing the influence of nodes in big-data online social networks are important for many practical applications, such as the viral marketing and the adoption of new products. As the viral spreading on a social network is a global process, it is commonly believed that measuring the influence of nodes inevitably requires the knowledge of the entire network. Using percolation theory, we show that the spreading process displays a nucleation behavior: Once a piece of information spreads from the seeds to more than a small characteristic number of nodes, it reaches a point of no return and will quickly reach the percolation cluster, regardless of the entire network structure; otherwise the spreading will be contained locally. Thus, we find that, without the knowledge of the entire network, any node's global influence can be accurately measured using this characteristic number, which is independent of the network size. This motivates an efficient algorithm with constant time complexity on the long-standing problem of best seed spreaders selection, with performance remarkably close to the true optimum.
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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.
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We systematically generate a large set of random micro-particle packings over a wide range of adhesion and friction by means of adhesive contact dynamics simulation. The ensemble of generated packings covers a range of volume fractions Ï from 0.135 ± 0.007 to 0.639 ± 0.004, and of coordination numbers Z from 2.11 ± 0.03 to 6.40 ± 0.06. We determine Ï and Z at four limits (random close packing, random loose packing, adhesive close packing, and adhesive loose packing), and find a universal equation of state Ï(Z) to describe packings with arbitrary adhesion and friction. From a mechanical equilibrium analysis, we determine the critical friction coefficient µf,c: when the friction coefficient µf is below µf,c, particles' rearrangements are dominated by sliding, otherwise they are dominated by rolling. Because of this reason, both Ï(µf) and Z(µf) change sharply across µf,c. Finally, we generalize the Maxwell counting argument to micro-particle packings, and show that the loosest packing, i.e., adhesive loose packing, satisfies the isostatic condition at Z = 2.
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One of the most actively debated issues in the study of the glass transition is whether a mean-field description is a reasonable starting point for understanding experimental glass formers. Although the mean-field theory of the glass transition--like that of other statistical systems--is exact when the spatial dimension d â ∞, the evolution of systems properties with d may not be smooth. Finite-dimensional effects could dramatically change what happens in physical dimensions,d = 2, 3. For standard phase transitions finite-dimensional effects are typically captured by renormalization group methods, but for glasses the corrections are much more subtle and only partially understood. Here, we investigate hopping between localized cages formed by neighboring particles in a model that allows to cleanly isolate that effect. By bringing together results from replica theory, cavity reconstruction, void percolation, and molecular dynamics, we obtain insights into how hopping induces a breakdown of the Stokes-Einstein relation and modifies the mean-field scenario in experimental systems. Although hopping is found to supersede the dynamical glass transition, it nonetheless leaves a sizable part of the critical regime untouched. By providing a constructive framework for identifying and quantifying the role of hopping, we thus take an important step toward describing dynamic facilitation in the framework of the mean-field theory of glasses.
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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.
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A significant amount of attention was dedicated in recent years to the phenomenon of jamming of athermal amorphous solids by increasing the volume fraction of the microscopic constituents. At a critical value of the volume fraction, pressure shoots up from zero to finite values with a host of critical exponents discovered and discussed. In this paper, we advance evidence for the existence of a second transition, within the jammed state of two-dimensional granular systems, that separates two regimes of different mechanical responses. Explicitly, highly packed systems are quasielastic with quadrupole screening, and more loosely jammed systems exhibit anomalous mechanics with dipole screening. Evidence is given for a clear transition between these two regimes, reminiscent of the intermediate hexatic phase of crystal melting in two-dimensional crystals. Theoretical estimates of the screening parameters and the pressure where transition takes place are provided.
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We generalize to higher spatial dimensions the Stokes-Einstein relation (SER) as well as the leading correction to diffusivity in finite systems with periodic boundary conditions, and validate these results with numerical simulations. We then investigate the evolution of the high-density SER violation with dimension in simple hard sphere glass formers. The analysis suggests that this SER violation disappears around dimension d(u) = 8, above which it is not observed. The critical exponent associated with the violation appears to evolve linearly in 8 - d, below d = 8, as predicted by Biroli and Bouchaud [J. Phys.: Condens. Matter 19, 205101 (2007)], but the linear coefficient is not consistent with the prediction. The SER violation with d establishes a new benchmark for theory, and its complete description remains an open problem.
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The jamming transition, generally manifested by a rapid increase of rigidity under compression (i.e. compression hardening), is ubiquitous in amorphous materials. Here we study shear hardening in deeply annealed frictionless packings generated by numerical simulations, reporting critical scalings absent in compression hardening. We demonstrate that hardening is a natural consequence of shear-induced memory destruction. Based on an elasticity theory, we reveal two independent microscopic origins of shear hardening: (i) the increase of the interaction bond number and (ii) the emergence of anisotropy and long-range correlations in the orientations of bonds-the latter highlights the essential difference between compression and shear hardening. Through the establishment of physical laws specific to anisotropy, our work completes the criticality and universality of jamming transition, and the elasticity theory of amorphous solids.
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The holographic duality has proven successful in linking seemingly unrelated problems in physics. Recently, intriguing correspondences between the physics of soft matter and gravity are emerging, including strong similarities between the rheology of amorphous solids, effective field theories for elasticity, and the physics of black holes. However, direct comparisons between theoretical predictions and experimental/simulation observations remain limited. Here, we study the effects of nonlinear elasticity on the mechanical and thermodynamic properties of amorphous materials responding to shear, using effective field and gravitational theories. The predicted correlations among the nonlinear elastic exponent, the yielding strain/stress, and the entropy change due to shear are supported qualitatively by simulations of granular matter models. Our approach opens a path toward understanding the complex mechanical responses of amorphous solids, such as mixed effects of shear softening and shear hardening, and offers the possibility to study the rheology of solid states and black holes in a unified framework.
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Amorphous solids, such as glasses, have complex responses to deformations, with substantial consequences in material design and applications. In this respect, two intertwined aspects are important: stability and reversibility. It is crucial to understand, on the one hand, how a glass may become unstable due to increased plasticity under shear deformations, and, on the other hand, to what extent the response is reversible, meaning how much a system is able to recover the original configuration once the perturbation is released. Here, we focus on assemblies of hard spheres as the simplest model of amorphous solids such as colloidal glasses and granular matter. We prepare glass states quenched from equilibrium supercooled liquid states, which are obtained by using the swap Monte Carlo algorithm and correspond to a wide range of structural relaxation time scales. We exhaustively map out their stability and reversibility under volume and shear strains using extensive numerical simulations. The region on the volume-shear strain phase diagram where the original glass state remains solid is bounded by the shear yielding and the shear jamming lines that meet at a yielding-jamming crossover point. This solid phase can be further divided into two subphases: the stable glass phase, where the system deforms purely elastically and is totally reversible, and the marginal glass phase, where it experiences stochastic plastic deformations at mesoscopic scales and is partially irreversible. The details of the stability-reversibility map depend strongly on the quality of annealing of the glass. This study provides a unified framework for understanding elasticity, plasticity, yielding, and jamming in amorphous solids.
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For amorphous solids, it has been intensely debated whether the traditional view on solids, in terms of the ground state and harmonic low energy excitations on top of it, such as phonons, is still valid. Recent theoretical developments of amorphous solids revealed the possibility of unexpectedly complex free-energy landscapes where the simple harmonic picture breaks down. Here we demonstrate that standard rheological techniques can be used as powerful tools to examine nontrivial consequences of such complex free-energy landscapes. By extensive numerical simulations on a hard sphere glass under quasistatic shear at finite temperatures, we show that above the so-called Gardner transition density, the elasticity breaks down, the stress relaxation exhibits slow, and ageing dynamics and the apparent shear modulus becomes protocol-dependent. Being designed to be reproducible in laboratories, our approach may trigger explorations of the complex free-energy landscapes of a large variety of amorphous materials.