Edwards thermodynamics of the jamming transition for frictionless packings: ergodicity test and role of angoricity and compactivity

Understanding slow compression of frictional granular particles by network analysis

We consider frictional granular packings exposed to quasi-static compression rates, with a focus on systems above the jamming transition. For frictionless packings, earlier work (S. Luding et al., Soft Matter, 2022, 18(9), 1868-1884) has uncovered that the system evolution/response involves smooth evolution phases, interrupted by fast transitions (events). The general finding is that the force networks' static quantities correlate closely with the pressure, while their evolution resembles the kinetic energy for both frictionless and frictional packings. The former represents reversible (elastic) particle deformations with affine and non-affine components, whereas the latter also involves much stronger, irreversible (plastic) rearrangements of the packings. Events are associated with jumps in the overall kinetic energy as well as dramatic changes in the force networks describing the particle micro-structure. The frictional nature of particle interactions affects both their frequency and the relevant time scale magnitude. For intermediate friction, events are often followed by an unexpected slow-down during which the kinetic energy drops below its average value. We find that these slow-downs are associated with a significant decrease in the non-affine dynamics of the particles, and are strongly influenced by friction. Friction modifies the structure of the networks, both through the typical number of contacts of a particle, and by influencing topological features of the resulting networks. Furthermore, friction modifies the dynamics of the networks, with larger values of friction leading to smaller changes of the more stable networks.

Is there a granular potential?

Granular materials, such as sand or grain, exhibit many structural and dynamic characteristics similar to those observed in molecular systems, despite temperature playing no role in their properties. This has led to an effort to develop a statistical mechanics for granular materials that has focused on establishing an equivalent to the microcanonical ensemble and a temperature-like thermodynamic variable. Here, we expand on these ideas by introducing a granular potential into the Edwards ensemble, as an analogue to the chemical potential, and explore its properties using a simple model of a granular system. A simple kinetic Monte Carlo simulation of the model shows the effect of mass transport leading to equilibrium and how this is connected to the redistribution of volume in the system. An exact analytical treatment of the model shows that the compactivity and the ratio of the granular potential to the compactivity determine the equilibrium between two open systems that are able to exchange volume and particles, and that mass moves from high to low values of this ratio. Analysis of the granular potential shows that adding a particle to the system increases the entropy at high compactivity, but decreases the entropy at low compactivity. Finally, we demonstrate the use of a small system thermodynamics method for the measurement of granular potential differences.

Equivalence of Fluctuation-Dissipation and Edwards' Temperature in Cyclically Sheared Granular Systems

Using particle trajectory data obtained from x-ray tomography, we determine two kinds of effective temperatures in a cyclically sheared granular system. The first one is obtained from the fluctuation-dissipation theorem which relates the diffusion and mobility of lighter tracer particles immersed in the system. The second is the Edwards compactivity defined via the packing volume fluctuations. We find robust agreement between these two temperatures, independent of the type of the tracers, cyclic shear amplitudes, and particle surface roughness, giving therefore the first experimental evidence that the concept of effective temperature is valid in driven frictional granular systems.

Jammed disks of two sizes and weights in a channel: Alternating sequences

Disks of two sizes and weights in alternating sequence are confined to a long and narrow channel. The axis of the channel is horizontal and its plane vertical. The channel is closed off by pistons that freeze jammed microstates out of loose disk configurations subject to moderate pressure, gravity, and random agitations. Disk sizes and channel width are such that under jamming no disk remains loose and all disks touch one wall. We present exact results for the characterization of jammed macrostates including volume and entropy. The rigorous analysis divides the disk sequences of jammed microstates into overlapping tiles from which we construct a small number of species of statistically interacting particles. Jammed macrostates depend on dimensionless control parameters inferred from ratios between measures of expansion work against the pistons, gravitational potential energy, and intensity of random agitations. These control parameters enter the configurational statistics via the activation energies prior to jamming of the particles. The range of disk weights naturally divides into regimes where qualitatively different features come into play. We sketch a path toward generalizations that include random sequences under a modified jamming protocol.

Experimental Test of the Edwards Volume Ensemble for Tapped Granular Packings

Using x-ray tomography, we experimentally investigate granular packings subject to mechanical tapping for three types of beads with different friction coefficients. We validate the Edwards volume ensemble in these three-dimensional granular systems and establish a granular version of thermodynamic zeroth law. Within the Edwards framework, we also explicitly clarify how friction influences granular statistical mechanics by modifying the density of states, which allows us to determine the entropy as a function of packing fraction and friction. Additionally, we obtain a granular jamming phase diagram based on geometric coordination number and packing fraction.

Structural characterization and statistical properties of jammed soft ellipsoid packing

The jamming transition and jammed packing structures of hydrogel soft ellipsoids are studied using magnetic resonance imaging techniques. As the packing fraction increases, the fluctuation of local free volume decreases and the fluctuation of particle deformation increases. Effective thermodynamic quantities are obtained by characterizing these fluctuations using k-gamma distributions based on an underlying statistical model for granular materials. Surprisingly, the two granular temperatures measuring the relative fluctuations of both free volume and particle deformation remain basically unchanged as the packing fraction increases. The total configurational entropy is also approximately constant for packing with different packing fractions. The significantly different behaviors of these effective thermodynamic quantities compared with hard sphere systems are further attributed to a statistically affine structural transformation of the packing structures along with particle deformations when the packing fraction changes.

Universality of internal structure characteristics in granular media under shear

We examine the signatures of internal structure emerged from quasistatic shear responses of granular materials based on three-dimensional discrete element simulations. Granular assemblies consisting of spheres or nonspherical particles of different polydispersity are sheared from different initial densities and under different loading conditions (drained or undrained) steadily to reach the critical state (a state featured by constant stress and constant volume). The radial distribution function used to measure the packing structure is found to remain almost unchanged during the shearing process, regardless of the initial states or loading conditions of an assembly. Its specific form, however, varies with polydispersities in both grain size and grain shape. Set Voronoi tessellation is employed to examine the characteristics of local volume and anisotropy, and deformation. The local inverse solid fraction and anisotropy index are found following inverse Weibull and log-normal distributions, respectively, which are unique at the critical states. With further normalization, an invariant distribution for local volume and anisotropy is observed whose function can be determined by the polydispersities in both particle size and grain shape but bears no relevance to initial densities or loading conditions (or paths). An invariant Gaussian distribution is found for the local deformation for spherical packings, but no invariant distribution can be found for nonspherical packings where the distribution of normalized local volumetric strain is dependent on initial states.

Structural analysis of high-dimensional basins of attraction

We propose an efficient Monte Carlo method for the computation of the volumes of high-dimensional bodies with arbitrary shape. We start with a region of known volume within the interior of the manifold and then use the multistate Bennett acceptance-ratio method to compute the dimensionless free-energy difference between a series of equilibrium simulations performed within this object. The method produces results that are in excellent agreement with thermodynamic integration, as well as a direct estimate of the associated statistical uncertainties. The histogram method also allows us to directly obtain an estimate of the interior radial probability density profile, thus yielding useful insight into the structural properties of such a high-dimensional body. We illustrate the method by analyzing the effect of structural disorder on the basins of attraction of mechanically stable packings of soft repulsive spheres.

Upper bound on the Edwards entropy in frictional monodisperse hard-sphere packings

We extend the Widom particle insertion method [B. Widom, J. Chem. Phys., 1963, 39, 2808-2812] to determine an upper bound sub on the Edwards entropy in frictional hard-sphere packings. sub corresponds to the logarithm of the number of mechanically stable configurations for a given volume fraction and boundary conditions. To accomplish this, we extend the method for estimating the particle insertion probability through the pore-size distribution in frictionless packings [V. Baranau, et al., Soft Matter, 2013, 9, 3361-3372] to the case of frictional particles. We use computer-generated and experimentally obtained three-dimensional sphere packings with volume fractions φ in the range 0.551-0.65. We find that sub has a maximum in the vicinity of the Random Loose Packing Limit φRLP = 0.55 and decreases then monotonically with increasing φ to reach a minimum at φ = 0.65. Further on, sub does not distinguish between real mechanical stability and packings in close proximity to mechanical stable configurations. The probability to find a given number of contacts for a particle inserted in a large enough pore does not depend on φ, but it decreases strongly with the contact number.

Turning intractable counting into sampling: Computing the configurational entropy of three-dimensional jammed packings

We present a numerical calculation of the total number of disordered jammed configurations Ω of N repulsive, three-dimensional spheres in a fixed volume V. To make these calculations tractable, we increase the computational efficiency of the approach of Xu et al. [Phys. Rev. Lett. 106, 245502 (2011)10.1103/PhysRevLett.106.245502] and Asenjo et al. [Phys. Rev. Lett. 112, 098002 (2014)10.1103/PhysRevLett.112.098002] and we extend the method to allow computation of the configurational entropy as a function of pressure. The approach that we use computes the configurational entropy by sampling the absolute volume of basins of attraction of the stable packings in the potential energy landscape. We find a surprisingly strong correlation between the pressure of a configuration and the volume of its basin of attraction in the potential energy landscape. This relation is well described by a power law. Our methodology to compute the number of minima in the potential energy landscape should be applicable to a wide range of other enumeration problems in statistical physics, string theory, cosmology, and machine learning that aim to find the distribution of the extrema of a scalar cost function that depends on many degrees of freedom.

The structural origin of the hard-sphere glass transition in granular packing

Glass transition is accompanied by a rapid growth of the structural relaxation time and a concomitant decrease of configurational entropy. It remains unclear whether the transition has a thermodynamic origin, and whether the dynamic arrest is associated with the growth of a certain static order. Using granular packing as a model hard-sphere glass, we show the glass transition as a thermodynamic phase transition with a ‘hidden' polytetrahedral order. This polytetrahedral order is spatially correlated with the slow dynamics. It is geometrically frustrated and has a peculiar fractal dimension. Additionally, as the packing fraction increases, its growth follows an entropy-driven nucleation process, similar to that of the random first-order transition theory. Our study essentially identifies a long-sought-after structural glass order in hard-sphere glasses.

Glass transition shows dramatic dynamic slowdown, but its origin remains unclear. Here, Xia et al. observe in granular systems the rapid growth of a geometrically frustrated polytetrahedral order with packing fraction, which is spatially correlated with the slow dynamics.

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.

Devising a protocol-related statistical mechanics framework for granular materials

Devising a statistical mechanics framework for jammed granular materials is a challenging task as those systems do not share some important properties required to characterize them with statistical thermodynamics tools. In a recent paper [Asenjo et al. Phys. Rev. Lett. 112, 098002 (2014)], a new definition of a granular entropy, which puts the protocol used to generate the packings at its roots, has been proposed. Following up these results, it is shown that the protocol used in Asenjo et al. can be recast as a canonical ensemble with a particular value of the temperature. Signature of gaussianity for large system sizes strongly suggests an asymptotic equivalence with a corresponding microcanonical ensemble where jammed states with certain basin volumes are sampled uniformly. We argue that this microcanonical ensemble is not Edwards's microcanonical ensemble and generalize this argument to other protocols.

A dynamic synchrotron X-ray imaging study of effective temperature in a vibrated granular medium

We present a dynamic synchrotron X-ray imaging study of the effective temperature Teff in a vibrated granular medium. By tracking the directed motion and the fluctuation dynamics of the tracers inside, we obtained Teff of the system using the Einstein relationship. We found that as the system unjams with increasing vibration intensities Γ, the structural relaxation time τ increases substantially which can be fitted by an Arrhenius law using Teff. And the characteristic energy scale of structural relaxation yielded by the Arrhenius fitting is E = 0.20 ± 0.02pd(3), where p is the pressure and d is the background particle diameter, which is consistent with those from hard sphere simulations in which the structural relaxation happens via the opening up of free volume against pressure.

Fundamental challenges in packing problems: from spherical to non-spherical particles

Random packings of objects of a particular shape are ubiquitous in science and engineering. However, such jammed matter states have eluded any systematic theoretical treatment due to the strong positional and orientational correlations involved. In recent years progress on a fundamental description of jammed matter could be made by starting from a constant volume ensemble in the spirit of conventional statistical mechanics. Recent work has shown that this approach, first introduced by S. F. Edwards more than two decades ago, can be cast into a predictive framework to calculate the packing fractions of both spherical and non-spherical particles.

Local fluctuations and spatial correlations in granular flows under constant-volume quasistatic shear

We investigate the local fluctuations in dense granular media subjected to athermal, quasistatic shearing, based on three-dimensional discrete element method simulations. By shearing granular assemblies of different polydispersities under constant-volume constraint, we quantify the characteristics of local structures (in terms of local volume and local anisotropy) and local deformation (using local shear strain and nonaffine displacement). The distribution of the local volume in a granular medium is found unchanged during the entire shearing process, which indicates a constant temperaturelike compactivity for the material. The compactivity is not, however, equilibrated among different particle groups in a polydisperse assembly. The local structures of a disordered granular assembly are inherently anisotropic. The fluctuations in local anisotropy can be well captured by a gamma or mixed-gamma distribution function, which is also unchanged during the shear. The local anisotropic orientation evolves towards the coaxial direction of the stress anisotropy with shear. The deformation characteristics of a jammed granular medium have their origins in the structural amorphousness. The local shear strain field depicts clear shear transformation zones which act as plasticity carriers. The spatial correlation of the local shear strains exhibits a fourfold pattern which is stronger in the stress deviatoric planes than in the stress isotropic plane. The fluctuations of nonaffine displacement suggest an isotropic granular temperature and an isotropic spatial correlation independent of the stress state. Both the local strain and the nonaffine displacement exhibit a power-law decayed distribution with a long-range correlation. We further modify the shear-transformation-zone theory to predict the pressure-dependent constitutive behavior of a sheared granular material and compare its prediction with our simulation data.

Numerical calculation of granular entropy

We present numerical simulations that allow us to compute the number of ways in which N particles can pack into a given volume V. Our technique modifies the method of Xu, Frenkel, and Liu [Phys. Rev. Lett. 106, 245502 (2011)] and outperforms existing direct enumeration methods by more than 200 orders of magnitude. We use our approach to study the system size dependence of the number of distinct packings of a system of up to 128 polydisperse soft disks. We show that, even though granular particles are distinguishable, we have to include a factor 1=N! to ensure that the entropy does not change when exchanging particles between systems in the same macroscopic state. Our simulations provide strong evidence that the packing entropy, when properly defined, is extensive. As different packings are created with unequal probabilities, it is natural to express the packing entropy as S = − Σ(i)p(i) ln pi − lnN!, where pi denotes the probability to generate the ith packing. We can compute this quantity reliably and it is also extensive. The granular entropy thus (re)defined, while distinct from the one proposed by Edwards [J. Phys. Condens. Matter 2, SA63 (1990)], does have all the properties Edwards assumed.

Equilibrating temperaturelike variables in jammed granular subsystems

Although jammed granular systems are athermal, several thermodynamiclike descriptions have been proposed which make quantitative predictions about the distribution of volume and stress within a system and provide a corresponding temperaturelike variable. We perform experiments with an apparatus designed to generate a large number of independent, jammed, two-dimensional configurations. Each configuration consists of a single layer of photoelastic disks supported by a gentle layer of air. New configurations are generated by cyclically dilating, mixing, and then recompacting the system through a series of boundary displacements. Within each configuration, a bath of particles surrounds a smaller subsystem of particles with a different interparticle friction coefficient than the bath. The use of photoelastic particles permits us to find all particle positions as well as the vector forces at each interparticle contact. By comparing the temperaturelike quantities in both systems, we find compactivity (conjugate to the volume) does not equilibrate between the systems, while the angoricity (conjugate to the stress) does. Both independent components of the angoricity are linearly dependent on the hydrostatic pressure, in agreement with predictions of the stress ensemble.

Spontaneous crystallization in athermal polymer packings

We review recent results from extensive simulations of the crystallization of athermal polymer packings. It is shown that above a certain packing density, and for sufficiently long simulations, all random assemblies of freely-jointed chains of tangent hard spheres of uniform size show a spontaneous transition into a crystalline phase. These polymer crystals adopt predominantly random hexagonal close packed morphologies. An analysis of the local environment around monomers based on the shape and size of the Voronoi polyhedra clearly shows that Voronoi cells become more spherical and more symmetric as the system transits to the ordered state. The change in the local environment leads to an increase in the monomer translational contribution to the entropy of the system, which acts as the driving force for the phase transition. A comparison of the crystallization of hard-sphere polymers and monomers highlights similarities and differences resulting from the constraints imposed by chain connectivity.