Jamming III: Characterizing randomness via the entropy of jammed matter

The nature of randomness in disordered packings of frictional and frictionless spheres is investigated using theory and simulations of identical spherical grains. The entropy of the packings is defined through the force and volume ensemble of jammed matter and this is shown to be difficult to calculate analytically. A mesoscopic ensemble of isostatic states is then utilized in an effort to predict the entropy through the definition of a volume function that is dependent on the coordination number. Equations of state are obtained relating entropy, volume fraction and compactivity characterizing the different states of jammed matter, and elucidating the phase diagram for jammed granular matter. Analytical calculations are compared to numerical simulations using volume fluctuation analysis and graph theoretical methods, with reasonable agreement. The entropy of the jammed system reveals that random loose packings are more disordered than random close packings, allowing for an unambiguous interpretation of both limits. Ensemble calculations show that the entropy vanishes at random close packing (RCP), while numerical simulations show that a finite entropy remains in the microscopic states at RCP. The notion of a negative compactivity, which explores states with volume fractions below those achievable by existing simulation protocols, is also explored, expanding the equations of state. The mesoscopic theory reproduces the simulations results in shape well, though a difference in magnitude implies that the entire entropy of the packing may not be captured by the methods presented herein. We discuss possible extensions to the present mesoscopic approach describing packings from random loose packing (RLP) to RCP to the ordered branch of the equation of state in an effort to understand the entropy of jammed matter in the full range of densities from RLP to face-centered cubic (FCC) packing.     

Structural transitions in granular packs: statistical mechanics and statistical geometry investigations

We investigate equal spheres packings generated from several experiments and from a large number of different numerical simulations. The structural organization of these disordered packings is studied in terms of the network of common neighbours. This geometrical analysis reveals sharp changes in the network’s clustering occurring at the packing fractions (fraction of volume occupied by the spheres respect to the total volume, ρ) corresponding to the so called Random Loose Packing limit (RLP, ρ ～ 0.555) and Random Close Packing limit (RCP, ρ ～ 0.645). At these packing fractions we also observe abrupt changes in the fluctuations of the portion of free volume around each sphere. We analyze such fluctuations by means of a statistical mechanics approach and we show that these anomalies are associated to sharp variations in a generalized thermodynamical variable which is the analogous for these a-thermal systems to the specific heat in thermal systems.     

Jamming II: Edwards’ statistical mechanics of random packings of hard spheres

The problem of finding the most efficient way to pack spheres has an illustrious history, dating back to the crystalline arrays conjectured by Kepler and the random geometries explored by Bernal in the 1960s. This problem finds applications spanning from the mathematician’s pencil, the processing of granular materials, the jamming and glass transitions, all the way to fruit packing in every grocery. There are presently numerous experiments showing that the loosest way to pack spheres gives a density of ∼55% (named random loose packing, RLP) while filling all the loose voids results in a maximum density of ∼63%-64% (named random close packing, RCP). While those values seem robustly true, to this date there is no well-accepted physical explanation or theoretical prediction for them. Here we develop a common framework for understanding the random packings of monodisperse hard spheres whose limits can be interpreted as the experimentally observed RLP and RCP. The reason for these limits arises from a statistical picture of jammed states in which the RCP can be interpreted as the ground state of the ensemble of jammed matter with zero compactivity, while the RLP arises in the infinite compactivity limit. We combine an extended statistical mechanics approach ‘a la Edwards’ (where the role traditionally played by the energy and temperature in thermal systems is substituted by the volume and compactivity) with a constraint on mechanical stability imposed by the isostatic condition. We show how such approaches can bring results that can be compared to experiments and allow for an exploitation of the statistical mechanics framework. The key result is the use of a relation between the local Voronoi volumes of the constituent grains (denoted the volume function) and the number of neighbors in contact that permits us to simply combine the two approaches to develop a theory of volume fluctuations in jammed matter. Ultimately, our results lead to a phase diagram that provides a unifying view of the disordered hard sphere packing problem and further sheds light on a diverse spectrum of data, including the RLP state. Theoretical results are well reproduced by numerical simulations that confirm the essential role played by friction in determining both the RLP and RCP limits. The RLP values depend on friction, explaining why varied experimental results can be obtained.     

Random packings of frictionless particles

We conduct numerical simulations of random packings of frictionless particles at T = 0. The packing fraction where the pressure becomes nonzero is the same as the jamming threshold, where the static shear modulus becomes nonzero. The distribution of threshold packing fractions narrows, and its peak approaches random close packing as the system size increases. For packing fractions within the peak, there is no self-averaging, leading to exponential decay of the interparticle force distribution.     

A first-order phase transition defines the random close packing of hard spheres

Randomly packing spheres of equal size into a container consistently results in a static configuration with a density of ∼64%. The ubiquity of random close packing (RCP) rather than the optimal crystalline array at 74% begs the question of the physical law behind this empirically deduced state. Indeed, there is no signature of any macroscopic quantity with a discontinuity associated with the observed packing limit. Here we show that RCP can be interpreted as a manifestation of a thermodynamic singularity, which defines it as the “freezing point” in a first-order phase transition between ordered and disordered packing phases. Despite the athermal nature of granular matter, we show the thermodynamic character of the transition in that it is accompanied by sharp discontinuities in volume and entropy. This occurs at a critical compactivity, which is the intensive variable that plays the role of temperature in granular matter. Our results predict the experimental conditions necessary for the formation of a jammed crystal by calculating an analogue of the “entropy of fusion”. This approach is useful since it maps out-of-equilibrium problems in complex systems onto simpler established frameworks in statistical mechanics.     

From crystal to amorphous: A novel route to unjamming in soft disk packings

Numerical studies on the unjamming packing fraction of bi- and polydisperse disk packings, which are generated through compression of a monodisperse crystal, are presented. In bidisperse systems, a fraction f ₊ = 0.400 up to 0.800 of the total number of particles has their radii increased by D \Delta R , while the rest has their radii decreased by the same amount. Polydisperse packings are prepared by changing all particle radii according to a uniform distribution in the range [- D \Delta R,D \Delta R] . The results indicate that the critical packing fraction is never larger than the value for the initial monodisperse crystal, f₀ \phi_{0}^{} = p \pi/?{12} \sqrt{{12}} , and that the lowest value achieved is approximately the one for random close packing. These results are seen as a consequence of the interplay between the increase in small-small particle contacts and the local crystalline order provided by the large-large particle contacts.     

Diffusion limited cluster aggregation with irreversible slippery bonds

Irreversible diffusion limited cluster aggregation (DLCA) of hard spheres was simulated using Brownian cluster dynamics. Bound spheres were allowed to move freely within a specified range, but no bond breaking was allowed. The structure and size distribution of the clusters was investigated before gelation. The pair correlation function and the static structure factor of the gels were determined as a function of the volume fraction and time. Slippery bonds led to local densification of the clusters and the gels, with a certain degree of order. At low volume fractions densification of the clusters occurred during their growth, but at higher volume fractions it occurred mainly after gelation. At very low volume fractions, the large-scale structure (fractal dimension), size distribution and growth kinetics of the clusters was found to be close to that known for DLCA with rigid bonds. Restructuring of the gels continued for long times, indicating that aging processes in systems with strong attraction do not necessarily involve bond breaking. The mean-square displacement of particles in the gels was determined. It is shown to be highly heterogeneous and to increase with decreasing volume fraction.     

Dense packing of binary and polydisperse hard spheres

The packing of binary and polydisperse unimodal and bimodal ensembles of hard spheres in the limit of high pressure is studied using a sequential addition algorithm. Upon fixing the number of particles, and their size distribution, the average (maximum) packing fraction is determined for systems of up to 20 000 particles. The structures obtained correspond to amorphous states close to the dense random close packing density. Binary distributions obtained are denser than the equivalent monodisperse distribution and agree with the theoretical prediction for an infinite size ratio limit. Unimodal normal and lognormal polydisperse distributions obtained compare favourably with available simulation and experimental data. Results for bimodal lognormal distributions are presented. In all cases it is seen how an increase in polydispersity increases the packing fraction of the system. The results can be employed to gain insight into optimal formulations for dense emulsions.     

Jamming in two-dimensional packings

We investigate the existence of random close and random loose packing limits in two-dimensional packings of monodisperse hard disks. A statistical mechanics approach-based on several approximations to predict the probability distribution of volumes-suggests the existence of the limiting densities of the jammed packings according to their coordination number and compactivity. This result has implications for the understanding of disordered states in the disk packing problem as well as the existence of a putative glass transition in two-dimensional systems.     

Velocity profiles in repulsive athermal systems under shear

We conduct molecular dynamics simulations of athermal systems undergoing boundary-driven planar shear flow in two and three spatial dimensions. We find that these systems possess nonlinear mean velocity profiles when the velocity u of the shearing wall exceeds a critical value u(c). Above u(c), we also show that the packing fraction and mean-square velocity profiles become spatially dependent with dilation and enhanced velocity fluctuations near the moving boundary. In systems with overdamped dynamics, u(c) is only weakly dependent on packing fraction phi. However, in systems with underdamped dynamics, u(c) is set by the speed of shear waves in the material and tends to zero as phi approaches phi(c), which is near random close packing at small damping. For underdamped systems with phi相似文献   

Packing spheres tightly: influence of mechanical stability on close-packed sphere structures

Many experiments and simulations of packings of monodisperse hard spheres report a dominance of the face-centered cubic structure in the hexagonally close-packed limit, even though it has no significant energetic or entropic gain over other close-packed configurations. Combining simulations and experiments, we demonstrate that a simple mechanical instability which occurs during the packing process may play an important role in selecting the face-centered cubic structure over other close-packed alternatives. Our argument is supported by detailed quantitative analyses of key configurations in sphere packings and highlights the importance of the packing dynamics. The proposed mechanism is elementary and should therefore play a role in a wide range of sphere systems.     

Modelling local voids using an irregular polyhedron based on natural neighbourhood and application to characterize near-dense random packing (DRP)

A model has been developed for finding local voids in randomly packed monodisperse spheres. The voids are polyhedral in shape and are based on the natural neighbourhood concept. The natural neighbourhood is defined in the same spirit of Sibson, who introduced the concept as a refinement of Voronoi tessellation. The model is basically the construction of a Delaunay star, where the centre of the Delaunay star is an arbitrary point in the void and the vertices of the star are the sphere centres. The method is best suited for sampling study. Since the model does not use the radius of the spheres, it can even be used for point distribution in three-dimensional (3-D) space. The model can be improved by using Voronoi vertices as seed points (instead of the arbitrary points) and can be used for crystallochemical studies, where only the electron density distribution is known. It is applicable to non-spherical atoms/particles also. The method is used to analyze near-dense random packing (DRP) and the statistical properties of void structures, e.g. number of vertices per void, cell volume, void volume and void fraction, which do not change from packing to packing in the limit of DRP. The overall local void properties are insensitive to sampling; repeatedly taking 500 void samples in an ensemble did not show considerable change. Most of the voids have 9–12 vertices.     

Shear- and vibration-induced order-disorder transitions in granular media

By molecular dynamics simulations we investigate the order-disorder transitions induced in granular media by an applied drive combining vibrations and shear. As the steady state is attained, the pack is found in disordered configurations for comparatively high intensities of the drive; conversely, ordering and packing fractions exceeding the random close packing are found when vibrations and shear are weak. As forcing amplitudes get smaller, we find diverging time scales in the dynamics, as the system enters a jamming region. Under this perspective, our picture supports the intuition that externally applied forcing has, in driven granular media, a role similar to temperature in thermal systems.     

Shear-induced configurations of confined colloidal suspensions

We show that geometric confinement dramatically affects the shear-induced configurations of dense monodisperse colloidal suspensions; a new structure emerges, where layers of particles buckle to stack in a more efficient packing. The volume fraction in the shear zone is controlled by a balance between the viscous stresses and the osmotic pressure of a contacting reservoir of unsheared particles. We present a model that accounts for our observations and helps elucidate the complex interplay between particle packing and shear stress for confined suspensions.     

Degenerate quasicrystal of hard triangular bipyramids

We report a degenerate quasicrystal in Monte Carlo simulations of hard triangular bipyramids each composed of two regular tetrahedra sharing a single face. The dodecagonal quasicrystal is similar to that recently reported for hard tetrahedra [Haji-Akbari et al., Nature (London) 462, 773 (2009)] but degenerate in the pairing of tetrahedra, and self-assembles at packing fractions above 54%. This notion of degeneracy differs from the degeneracy of a quasiperiodic random tiling arising through phason flips. Free energy calculations show that a triclinic crystal is preferred at high packing fractions.     

Dynamic Simulation of Random Packing of Polydispersive Fine Particles

In this paper, we perform molecular dynamic (MD) simulations to study the two-dimensional packing process of both monosized and random size particles with radii ranging from 1.0 to 7.0 μm. The initial positions as well as the radii of five thousand fine particles were defined inside a rectangular box by using a random number generator. Both the translational and rotational movements of each particle were considered in the simulations. In order to deal with interacting fine particles, we take into account both the contact forces and the long-range dispersive forces. We account for normal and static/sliding tangential friction forces between particles and between particle and wall by means of a linear model approach, while the long-range dispersive forces are computed by using a Lennard-Jones-like potential. The packing processes were studied assuming different long-range interaction strengths. We carry out statistical calculations of the different quantities studied such as packing density, mean coordination number, kinetic energy, and radial distribution function as the system evolves over time. We find that the long-range dispersive forces can strongly influence the packing process dynamics as they might form large particle clusters, depending on the intensity of the long-range interaction strength.     

Mixtures of hard spherocylinders and hard spheres

Monte Carlo simulations have been performed for equimolar mixtures of hard prolate spherocylinders of length: breadth ratio 2:1 and hard spheres, in the fluid region. Two systems have been studied. In the first the breadth of the spherocylinder was equal to the hard sphere diameter, and in the second system both components were of equal molecular volume.

The compressibility factor, PV/NkT, has been obtained for both mixtures at four reduced densities (packing fractions) from 0·20 to 0·45. The results have been compared with the predictions of several analytical equations appropriate to mixtures of hard convex molecules, and an equation due to Pavlicek et al. was found to be very accurate. The results have been used to calculate the excess volumes of mixing at constant pressure, in an attempt to establish the relative importance of the effects of differences in molecular volume and shape on the thermodynamic properties.

The structural properties of the mixtures have also been investigated by calculating pair distribution functions for the three types of pair interactions present in these mixtures.     

A study on the magnetic properties of two-phase particulate magnetic composites

The magnetic properties of two-phase particulate magnetic composites with a hard ferromagnetic component are studied theoretically and experimentally. The magnetic properties considered here are phase-distribution sensitive properties, including remanence, coercivity and the shape of hysteresis loop. These properties depend mainly on the properties of its constituents, volume fractions, phase distribution, packing fraction and orientation distribution for anisotropic particles. With fixed packing fraction and orientation distribution, the magnetic properties of the two phase mixture can be calculated in terms of its component properties, volume fractions and phase distribution. Here, the component properties include not only remanence B_r and coercivity H_c but also a variable m which is the rate of change of magnetic induction B with respect to field intensity H. For two-phase systems satisfying B - H relation of the type B = B_r + mH where m is a constant, the equations for calculating the magnetic properties B_r, H_c, etc., in terms of m are derived. The method for calculating m is also developed for the cases of parallel and series distributions. Bounds for m-values were also established. A modified Landauer's type equation is developed to calculate m-values in terms of the component properties of the mixture. Experiments were conducted to verify the theoretical calculations. Good agreements between the theoretical calculations and experimental results were obtained.     

Statistical Analysis of Simulated Random Packings of Spheres

This paper describes two algorithms for the generation of random packings of spheres with arbitrary diameter distribution. The first algorithm is the force‐biased algorithm of Mościński and Bargieł. It produces isotropic packings of very high density. The second algorithm is the Jodrey‐Tory sedimentation algorithm, which simulates successive packing of a container with spheres following gravitation. It yields packings of a lower density and of weak anisotropy. The results obtained with these algorithms for the cases of log‐normal and two‐point sphere diameter distributions are analysed statistically, i. e. standard characteristics of spatial statistics such as porosity (or volume fraction), pair correlation function of the system of sphere centres and spherical contact distribution function of the set‐theoretical union of all spheres are determined. Furthermore, the mean coordination numbers are analysed. These results are compared for both algorithms and with data from the literature based on other numerical simulations or from experiments with real spheres.     

Dense random packings of hard and compressible spheres

The sublattices of the tetrahedrally co-ordinated random network of Connell and Temkin are related to the dense random packing of equal spheres. Thus, by relaxation of the atomic co-ordinates of the former, the dependence of the pair distribution function and packing fraction of the latter on sphere compressibility can be investigated. The results are compared with experimental data on NiP alloys.