Jammed Configurations Research Articles

A nonsmooth program for jamming hard spheres

We study packings of $$n$$ hard spheres of equal radius in the $$d$$ -dimensional unit cube. We present a nonsmooth function whose local extrema are the radii of jammed packings (where no subset of spheres can be moved keeping all others fixed) and show that for a fixed number of spheres there are only finitely many radii of such jammed configurations. We propose an algorithm for the maximization of this maximal radius function and present examples for five to eight disks in the unit square and four to six spheres in the unit cube. The method allows straightforward generalization to packings of spheres in other compact containers.

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

This paper illustrates how the tools of equilibrium statistical mechanics can help to describe a far-from-equilibrium problem: the jamming transition in frictionless granular materials. Edwards ideas consist of proposing a statistical ensemble of volume and stress fluctuations through the thermodynamic notion of entropy, compactivity, X, and angoricity, A (two temperature-like variables). We find that Edwards thermodynamics is able to describe the jamming transition (J point) in frictionless packings. Using the ensemble formalism we elucidate the following: (i) We test the combined volume-stress ensemble by comparing the statistical properties of jammed configurations obtained by dynamics with those averaged over the ensemble of minima in the potential energy landscape as a test of ergodicity. Agreement between both methods supports the idea of ergodicity and "thermalization" at a given angoricity and compactivity. (ii) A microcanonical ensemble analysis supports the maximum entropy principle for grains. (iii) The intensive variables A and X describe the approach to jamming through a series of scaling relations as A → 0+ and X → 0-. Due to the force-strain coupling in the interparticle forces, the jamming transition is probed thermodynamically by a "jamming temperature" T(J) composed of contributions from A and X. (iv) The thermodynamic framework reveals the order of the jamming phase transition by showing the absence of critical fluctuations at jamming in static observables like pressure and volume, and we discuss other critical scenarios for the jamming transition. (v) Finally, we elaborate on a comparison with relevant studies by Gao, Blawzdziewicz, and O'Hern [Phys. Rev. E 74, 061304 (2006)], showing a breakdown of equiprobability of microstates obtained via fast quenches. A network analysis of the energy landscape reveals the origin of the inhomogeneities in the uneven distribution of the areas of the basins. Such inhomogeneities are also found in other out-of-equilibrium systems like Lennard-Jones glasses and their existence does not preclude the use of statistical mechanics for jammed systems.

Sample size effects on the large strain bursts in submicron aluminum pillars

In situ transmission electron microscope compression testing of submicron Al pillars shows two sample size regimes with contrasting behavior underlying the large strain bursts. For small pillars, the bursts originate from explosive and highly correlated dislocation generation, characterized by very high collapse stresses and nearly dislocation-free post-collapse microstructure. For larger pillars, the bursts result from the reconstruction of jammed dislocation configurations, featuring relative low stress levels and retention of dislocation network after bursts.

Random sequential adsorption of partially oriented lineark-mers on a square lattice

Jamming phenomena on a square lattice are investigated for two different models of anisotropic random sequential adsorption (RSA) of linear k-mers (particles occupying k adjacent adsorption sites along a line). The length of a k-mer varies from 2 to 256. The effect of k-mer alignment on the jamming threshold is examined. For completely ordered systems where all the k-mers are aligned along one direction (e.g., vertical), the obtained simulation data are very close to the known analytical results for one-dimensional systems. In particular, the jamming threshold tends to the Rényi's parking constant for large k. In the other extreme case, when k-mers are fully disordered, our results correspond to the published results for short k-mers. It was observed that for partially oriented systems the jamming configurations consist of the blocks of vertically and horizontally oriented k-mers (v and h blocks, respectively) and large voids between them. The relative areas of different blocks and voids depend on the order parameter s, k-mer length, and type of the model. For small k-mers (k⩽4), denser configurations are observed in disordered systems as compared to those of completely ordered systems. However, longer k-mers exhibit the opposite behavior.

Slow dynamics of stress and strain relaxation in randomly crumpled elasto-plastic sheets

Stress and strain relaxation in randomly folded paper sheets under axial compression is studied both experimentally and theoretically. Equations providing the best fit to the experimental data are found. Our findings suggest that, in an axially compressed ball folded from an elastic or elasto-plastic material, the relaxation dynamics is ruled by activated processes of an energy foci rearrangement in the crumpling network. The dynamics of relaxation is discussed within a framework of Edwards's statistical mechanics. The functional forms of the activation barrier between admissible jammed folding configurations of the crumpling network under axial compression are derived. It is shown that relaxation kinetics can be mapped to activated dynamics of depinning and creep of elastic interface in a disordered medium.

Wireless Secrecy Regions With Friendly Jamming

Inspired by recent results on information-theoretic security, we consider the transmission of confidential messages over wireless networks, in which the legitimate communication partners are aided by friendly jammers. We characterize the security level of a confined region in a quasi-static fading environment by computing the probability of secrecy outage in connection with two new measures of physical-layer security: the jamming coverage and the jamming efficiency. Our analysis for various jamming strategies based on different levels of channel state information provides insight into the design of optimal jamming configurations and shows that a single jammer is not sufficient to maximize both figures of merit simultaneously. Moreover, a single jammer requires full channel state information to provide security gains in the vicinity of the legitimate receiver.

A statistical mechanics framework captures the packing of monodisperse particles

We present a generalization of the granocentric model proposed in [Clusel et al., Nature, 2009, 460, 611–615] that is capable of describing the local fluctuations inside not only polydisperse but also monodisperse packings of spheres. This minimal model does not take into account the relative particle positions, yet it captures positional disorder through local stochastic processes sampled by efficient Monte Carlo methods. The disorder is characterized by the distributions of local parameters, such as the number of neighbors and contacts, filled solid angle around a central particle and the cell volumes. The model predictions are in good agreement with our experimental data on monodisperse random close packings of PMMA particles. Moreover, the model can be used to predict the distributions of local fluctuations in any packing, as long as the average number of neighbors, contacts and the packing fraction are known. These distributions give a microscopic foundation to the statistical mechanics framework for jammed matter and allow us to calculate thermodynamic quantities such as the compactivity in the phase space of possible jammed configurations.

Jammed hard-particle packings: From Kepler to Bernal and beyond

This review describes the diversity of jammed configurations attainable by frictionless convex nonoverlapping (hard) particles in Euclidean spaces and for that purpose it stresses individual-packing geometric analysis. A fundamental feature of that diversity is the necessity to classify individual jammed configurations according to whether they are locally, collectively, or strictly jammed. Each of these categories contains a multitude of jammed configurations spanning a wide and (in the large system limit) continuous range of intensive properties, including packing fraction $\phi$, mean contact number $Z$, and several scalar order metrics. Application of these analytical tools to spheres in three dimensions (an analog to the venerable Ising model) covers a myriad of jammed states, including maximally dense packings (as Kepler conjectured), low-density strictly-jammed tunneled crystals, and a substantial family of amorphous packings. With respect to the last of these, the current approach displaces the historically prominent but ambiguous idea of ``random close packing" (RCP) with the precise concept of ``maximally random jamming" (MRJ). This review also covers recent advances in understanding jammed packings of polydisperse sphere mixtures, as well as convex nonspherical particles, e.g., ellipsoids, ``superballs", and polyhedra. Because of their relevance to error-correcting codes and information theory, sphere packings in high-dimensional Euclidean spaces have been included as well. We also make some remarks about packings in (curved) non-Euclidean spaces. In closing this review, several basic open questions for future research to consider have been identified.

On the Concept of Jammed Configurations from a Structural Mechanics Perspective

Applying a method that is widely known in nonlinear structural mechanics, the paper offers an alternative approach for the jamming analysis of granular assemblies. The main advantage of the proposed approach is that deformable particles with general shapes can be handled with it, in contrast to the previous methods that are restricted to rigid grains with spherical shape. The paper first gives an overview on the existing concepts of jammed states. Then an alternative set of definitions is proposed; the definitions are based on the stability analysis of the considered assemblies. After that, a calculation method (restricted to elastic contacts) is introduced for the jamming analysis. The method is based on determining the eigenvalues of the stiffness matrix that contains the effect of the particle properties as well as the already existing contact forces that are present in the system.

Statistical dynamics of dislocations in simple models of plastic deformation: Phase transitions and related phenomena

Dislocation assemblies in crystalline substrates are a beautiful example of the broad class of systems that are governed by the presence of kinematical constraints induced by interactions, geometry, and/or disorder. The interactions between dislocation lines of different type together with the dynamic constraints which tie the motion of the dislocations to their slip planes lead to the possibility of forming metastable jammed configurations even in the absence of any disorder in the material. However, general dislocation assemblies will also be affected by the presence of disorder-induced pinning forces. In this paper we study several aspects concerning the dynamics of dislocation assemblies in simple models of plastic deformation that we believe we can successfully address with the conceptual and technical tools provided by Statistical Mechanics. In particular, we discuss the yielding (or jamming) transition between stationary and moving states in these models, the intermittent and globally slow relaxations observed around this transition, and the stress–strain relationships measured in the steady regime of deformation. We also briefly describe the main implications of sample geometry and of quenched disorder in dislocation assemblies present in the vortex lattice of type II superconductors, where we obtain some of the statistical dynamic properties of experimental interest.

Lattice-based random jammed configurations for hard particles.

A nontrivial subset of the jammed packings for rigid disks and spheres are those that can be obtained by sequential removal of particles from periodic crystalline arrays. This paper considers the enumeration problems presented by such packings that are based on the close-packed triangular disk lattice, and the face-centered and body-centered cubic sphere lattices. Three distinct categories of packings have been distinguished, depending on their behavior with respect to nonoverlap geometric constraints and/or externally imposed virtual displacements: locally jammed, collectively jammed, and strictly jammed. Each of these possesses an upper limiting vacancy concentration beyond which no packings of the types considered can exist. For each of the three lattices, specific vacancy clusters have been identified whose presence would destroy local jamming, and some of the corresponding patterns that would destroy collective jamming in the triangular disk lattice have also been found. Within the allowable range of vacancy concentration for each case, the number of distinct jammed packings is expected to rise exponentially with system size. By using the concept of local attrition factors, approximate enumerations have been constructed for the three lattice classes of locally jammed packings. In the interests of later extension of this work, we stress that at least some aspects of these enumeration problems might benefit from the formal transcription to a lattice-gas/Ising-model representation with vacancy interactions chosen to enforce the packing category of interest.

Granular materials: Towards the statistical mechanics of jammed configurations

The properties of granular materials can be well defined, that is be a branch of physics, but conventional statistical mechanics is inadequate to handle what amounts to the physics of disordered packings of hard-core particles, either static or driven by external forces. A new approach that employs statistical-mechanical concepts is offered for the description of such systems. An analysis of the stress field in static granular packings is given within the framework of this approach. There are more conventional systems such as polymer glasses which have a rather similar statistical physics to granular media, and some speculative ideas are offered which are a real departure from conventional glass theories.

Diversity of order and densities in jammed hard-particle packings.

Recently the conventional notion of random close packing has been supplanted by the more appropriate concept of the maximally random jammed (MRJ) state. This inevitably leads to the necessity of distinguishing the MRJ state among the entire collection of jammed packings. While the ideal method of addressing this question would be to enumerate and classify all possible jammed hard-sphere configurations, practical limitations prevent such a method from being employed. Instead, we generate numerically a large number of representative jammed hard-sphere configurations (primarily relying on a slight modification of the Lubachevsky-Stillinger algorithm to do so) and evaluate several commonly employed order metrics for each of these packings. Our investigation shows that, even in the large-system limit, jammed systems of hard spheres can be generated with a wide range of packing fractions from phi approximately 0.52 to the fcc limit (phi approximately 0.74). Moreover, at a fixed packing fraction, the variation in the order can be substantial, indicating that the density alone does not uniquely characterize a packing. Interestingly, each order metric evaluated yielded a relatively consistent estimate for the packing fraction of the maximally random jammed state (phi(MRJ) approximately 0.63). This estimate, however, is compromised by the weaknesses in the order metrics available, and we propose several guiding principles for future efforts to define more broadly applicable metrics.

Multiplicity of Generation, Selection, and Classification Procedures for Jammed Hard-Particle Packings

Hard-particle packings have served as useful starting points to study the structure of diverse systems such as liquids, living cells, granular media, glasses, and amorphous solids. Howard Reiss has played a major role in helping to illuminate our understanding of hard-particle systems, which still offer scientists many interesting conundrums. Jammed configurations of hard particles are of great fundamental and practical interest. What one precisely means by a "jammed" configuration is quite subtle and considerable ambiguity remains in the literature on this question. We will show that there is a multiplicity of generation, selection, and classification procedures for jammed configurations of identical d-dimensional spheres. We categorize common ordered lattices according to our definitions and discuss implications for random disk and sphere packings. We also show how the concept of rigidity percolation (which has been used to understand the mechanical properties of network glasses) can be generalized to further characterize hard-sphere packings.

Self-organization of traffic jams in cities: Effects of stochastic dynamics and signal periods

We propose a cellular automata model for vehicular traffic in cities by combining (and appropriately modifying) ideas borrowed from the Biham-Middleton-Levine (BML) model of city traffic and the Nagel-Schreckenberg (NS) model of highway traffic. We demonstrate a phase transition from the "free-flowing" dynamical phase to the completely "jammed" phase at a vehicle density which depends on the time periods of the synchronized signals and the separation between them. The intrinsic stochasticity of the dynamics, which triggers the onset of jamming, is similar to that in the NS model, while the phenomenon of complete jamming through self-organization as well as the final jammed configurations are similar to those in the BML model. Using our new model, we have made an investigation of the time-dependence of the average speeds of the cars in the "free-flowing" phase as well as the dependence of flux and jamming on the time period of the signals.

Self-Organization and Phase Transition in Two-Dimensional Traffic-Flow Model

An analytical investigation is made on two-dimensional traffic-flow model with alternative movement and exclude-volume effect between right and up arrows. Several exact results are obtained, including the upper critical density above which there are only jamming configurations, and the lower critical density below which there are only moving configurations. The observed jamming transition takes place at another critical density , which is in the intermediate region between the lower and upper critical densities. This transition is suggested to be a second-order phase transition, the order parameter is found. The nature of self-organization, ergodicity breaking and synchronization are discussed. Comparison with the sandpile model is made.

Deposition-evaporation stochastic systems in two and higher dimensions

We study the stochastic dynamics of deposition-evaporation cooperative processes of dimers, trimers, etc., in two- and higher-dimensional lattices. The dimer system in bipartite lattices allows for an exact solution of dynamic correlations and scaling functions by means of a quantum spin equivalence. Autocorrelations exhibit a diffusive asymptotic kinetics and crossovers of different dynamic regimes in highly anisotropic lattices. Monte Carlo simulations combined with finite-size scaling arguments support the validity of the diffusive picture in more general situations. Steady-state coverages and diffusion constants are obtained using mean-field approaches, spin wave calculations, and random walk analyses in nearly jammed configurations.

Properties of jamming configurations built up by the adsorption of Brownian particles onto solid surfaces.

We study a realistic model for the absorption of colloid particles or latex spheres from solution on a solid surface. The adsorbing particles are represented by hard spheres that undergo Brownian motion in the vicinity of the surface. Once a sphere touches the surface, it becomes permanently fixed in place. We show that the mean coverage and structure of the jammed configurations that result from this irreversible adsorption process are indistinguishable from those of their counterparts generated by a random sequential adsorption (RSA) algorithm. This result demonstrates, posteriori, the importance of the RSA model in the study of static properties of systems of absorbed particles.