Hypoconstrained Jammed Packings of Nonspherical Hard Particles: Ellipses and Ellipsoids

Continuing on recent computational and experimental work on jammed packings of hard ellipsoids [Donev et al., Science, vol. 303, 990-993] we consider jamming in packings of smooth strictly convex nonspherical hard particles. We explain why the isocounting conjecture, which states that for large disordered jammed packings the average contact number per particle is twice the number of degrees of freedom per particle (\bar{Z}=2d_{f}), does not apply to nonspherical particles. We develop first- and second-order conditions for jamming, and demonstrate that packings of nonspherical particles can be jammed even though they are hypoconstrained (\bar{Z}<2d_{f}). We apply an algorithm using these conditions to computer-generated hypoconstrained ellipsoid and ellipse packings and demonstrate that our algorithm does produce jammed packings, even close to the sphere point. We also consider packings that are nearly jammed and draw connections to packings of deformable (but stiff) particles. Finally, we consider the jamming conditions for nearly spherical particles and explain quantitatively the behavior we observe in the vicinity of the sphere point.Comment: 33 pages, third revisio

Understanding the Frequency Distribution of Mechanically Stable Disk Packings

Relative frequencies of mechanically stable (MS) packings of frictionless bidisperse disks are studied numerically in small systems. The packings are created by successively compressing or decompressing a system of soft purely repulsive disks, followed by energy minimization, until only infinitesimal particle overlaps remain. For systems of up to 14 particles most of the MS packings were generated. We find that the packings are not equally probable as has been assumed in recent thermodynamic descriptions of granular systems. Instead, the frequency distribution, averaged over each packing-fraction interval $\Delta \phi$, grows exponentially with increasing $\phi$. Moreover, within each packing-fraction interval MS packings occur with frequencies $f_k$ that differ by many orders of magnitude. Also, key features of the frequency distribution do not change when we significantly alter the packing-generation algorithm--for example frequent packings remain frequent and rare ones remain rare. These results indicate that the frequency distribution of MS packings is strongly influenced by geometrical properties of the multidimensional configuration space. By adding thermal fluctuations to a set of the MS packings, we were able to examine a number of local features of configuration space near each packing including the time required for a given packing to break to a distinct one, which enabled us to estimate the energy barriers that separate one packing from another. We found a positive correlation between the packing frequencies and the heights of the lowest energy barriers $\epsilon_0$. We also examined displacement fluctuations away from the MS packings to correlate the size and shape of the local basins near each packing to the packing frequencies.Comment: 21 pages, 20 figures, 1 tabl

Rods are less fragile than spheres: Structural relaxation in dense liquids composed of anisotropic particles

We perform extensive molecular dynamics simulations of dense liquids composed of bidisperse dimer- and ellipse-shaped particles in 2D that interact via repulsive contact forces. We measure the structural relaxation times obtained from the long-time decay of the self-part of the intermediate scattering function for the translational and rotational degrees of freedom (DOF) as a function of packing fraction \phi, temperature T, and aspect ratio \alpha. We are able to collapse the \phi and T-dependent structural relaxation times for disks, and dimers and ellipses over a wide range of \alpha, onto a universal scaling function {\cal F}_{\pm}(|\phi-\phi_0|,T,\alpha), which is similar to that employed in previous studies of dense liquids composed of purely repulsive spherical particles in 3D. {\cal F_{\pm}} for both the translational and rotational DOF are characterized by the \alpha-dependent scaling exponents \mu and \delta and packing fraction \phi_0(\alpha) that signals the crossover in the scaling form {\cal F}_{\pm} from hard-particle dynamics to super-Arrhenius behavior for each aspect ratio. We find that the fragility at \phi_0, m(\phi_0), decreases monotonically with increasing aspect ratio for both ellipses and dimers. Moreover, the results for the slow dynamics of dense liquids composed of dimer- and ellipse-shaped particles are qualitatively the same, despite the fact that zero-temperature static packings of dimers are isostatic, while static packings of ellipses are hypostatic.Comment: 10 pages, 17 figures, and 1 tabl

Phase Behavior of Colloidal Superballs: Shape Interpolation from Spheres to Cubes

The phase behavior of hard superballs is examined using molecular dynamics within a deformable periodic simulation box. A superball's interior is defined by the inequality $|x|^{2q} + |y|^{2q} + |z|^{2q} \leq 1$, which provides a versatile family of convex particles ($q \geq 0.5$) with cube-like and octahedron-like shapes as well as concave particles ($q < 0.5$) with octahedron-like shapes. Here, we consider the convex case with a deformation parameter q between the sphere point (q = 1) and the cube (q = 1). We find that the asphericity plays a significant role in the extent of cubatic ordering of both the liquid and crystal phases. Calculation of the first few virial coefficients shows that superballs that are visually similar to cubes can have low-density equations of state closer to spheres than to cubes. Dense liquids of superballs display cubatic orientational order that extends over several particle lengths only for large q. Along the ordered, high-density equation of state, superballs with 1 < q < 3 exhibit clear evidence of a phase transition from a crystal state to a state with reduced long-ranged orientational order upon the reduction of density. For $q \geq 3$, long-ranged orientational order persists until the melting transition. The width of coexistence region between the liquid and ordered, high-density phase decreases with q up to q = 4.0. The structures of the high-density phases are examined using certain order parameters, distribution functions, and orientational correlation functions. We also find that a fixed simulation cell induces artificial phase transitions that are out of equilibrium. Current fabrication techniques allow for the synthesis of colloidal superballs, and thus the phase behavior of such systems can be investigated experimentally.Comment: 33 pages, 14 figure

A first-order phase transition at 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.Comment: 33 pages, 10 figure

Jammed frictionless discs: connecting local and global response

By calculating the linear response of packings of soft frictionless discs to quasistatic external perturbations, we investigate the critical scaling behavior of their elastic properties and non-affine deformations as a function of the distance to jamming. Averaged over an ensemble of similar packings, these systems are well described by elasticity, while in single packings we determine a diverging length scale $\ell^*$ up to which the response of the system is dominated by the local packing disorder. This length scale, which we observe directly, diverges as $1/\Delta z$, where $\Delta z$ is the difference between contact number and its isostatic value, and appears to scale identically to the length scale which had been introduced earlier in the interpretation of the spectrum of vibrational modes. It governs the crossover from isostatic behavior at the small scale to continuum behavior at the large scale; indeed we identify this length scale with the coarse graining length needed to obtain a smooth stress field. We characterize the non-affine displacements of the particles using the \emph{displacement angle distribution}, a local measure for the amount of relative sliding, and analyze the connection between local relative displacements and the elastic moduli.Comment: 19 pages, 15 figures, submitted to Phys. Rev.

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 explain 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). 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 "thermalization" at a given angoricity and compactivity. (ii) A microcanonical ensemble analysis supports the idea of maximum entropy principle for grains. (iii) The intensive variables describe the approach to jamming through a series of scaling relations as A {\to} 0+ and X {\to} 0-. Due to the force-volume coupling, the jamming transition can be probed thermodynamically by a "jamming temperature" TJ comprised 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 observables like pressure and volume. (v) Finally, we elaborate on a comparison with relevant studies showing a breakdown of equiprobability of microstates.Comment: 22pages, 24 figure