Soft Sphere Packings Research Articles

Local stability of spheres via the convex hull and the radical Voronoi diagram.

Jamming is an emergent phenomenon wherein the local stability of individual particles percolates to form a globally rigid structure. However, the onset of rigidity does not imply that every particle becomes rigid, and indeed some remain locally unstable. These particles, if they become unmoored from their neighbors, are called rattlers, and their identification is critical to understanding the rigid backbone of a packing, as these particles cannot bear stress. The accurate identification of rattlers, however, can be a time-consuming process, and the currently accepted method lacks a simple geometric interpretation. In this manuscript, we propose two simpler classifications of rattlers in hard sphere systems based on the convex hull of contacting neighbors and the maximum inscribed sphere of the radical Voronoi cell, each of which provides geometric insight into the source of their instability. Furthermore, the convex hull formulation can be generalized to explore stability in hyperstatic soft sphere packings, spring networks, nonspherical packings, and mean-field non-central-force potentials.

Predicting Defects in Soft Sphere Packings near Jamming Using the Force Network Ensemble.

Amorphous systems of soft particles above jamming have more contacts than are needed to achieve mechanical equilibrium. The force network of a granular system with a fixed contact network is thus underdetermined and can be characterized as a random instantiation within the space of the force network ensemble. In this Letter, we show that defect contacts that are not necessary for stability of the system can be uniquely identified by examining the boundaries of this space of allowed force networks. We further show that, for simulations in the near jamming limit, this identification is nearly always correct and that defect contacts are broken under decompression of the system.

Unified quantifier of mechanical disorder in solids.

Mechanical disorder in solids, which is generated by a broad range of physical processes and controls various material properties, appears in a wide variety of forms. Defining unified and measurable dimensionless quantifiers, allowing quantitative comparison of mechanical disorder across widely different physical systems, is therefore an important goal. Two such coarse-grained dimensionless quantifiers (among others) appear in the literature: one is related to the spectral broadening of discrete phononic bands in finite-size systems (accessible through computer simulations) and the other is related to the spatial fluctuations of the shear modulus in macroscopically large systems. The latter has been recently shown to determine the amplitude of wave attenuation rates in the low-frequency limit (accessible through laboratory experiments). Here, using two alternative and complementary theoretical approaches linked to the vibrational spectra of solids, we derive a basic scaling relation between the two dimensionless quantifiers. This scaling relation, which is supported by simulational data, shows that the two apparently distinct quantifiers are in fact intrinsically related, giving rise to a unified quantifier of mechanical disorder in solids. We further discuss the obtained results in the context of the unjamming transition taking place in soft sphere packings at low confining pressures, in addition to their implications for our understanding of the low-frequency vibrational spectra of disordered solids in general, and in particular those of glassy systems.

Dodecagonal Quasicrystals in Mesoporous Silica: A New Route from Hard- to Soft-Sphere Packings

Quasicrystals attract considerable attention due to their unique physical and structural properties. An increasing number of quasicrystals have been reported from self-assembly processes in soft ma...

Unified phase diagram of reversible–irreversible, jamming, and yielding transitions in cyclically sheared soft-sphere packings

Self-organization, and transitions from reversible to irreversible behavior, of interacting particle assemblies driven by externally imposed stresses or deformation is of interest in comprehending diverse phenomena in soft matter. They have been investigated in a wide range of systems, such as colloidal suspensions, glasses, and granular matter. In different density and driving regimes, such behavior is related to yielding of amorphous solids, jamming, memory formation, etc. How these phenomena are related to each other has not, however, been much studied. In order to obtain a unified view of the different regimes of behavior, and transitions between them, we investigate computationally the response of soft-sphere assemblies to athermal cyclic-shear deformation over a wide range of densities and amplitudes of shear deformation. Cyclic-shear deformation induces transitions from reversible to irreversible behavior in both unjammed and jammed soft-sphere packings. Well above the minimum isotropic jamming density ([Formula: see text]), this transition corresponds to yielding. In the vicinity of the jamming point, up to a higher-density limit, we designate [Formula: see text], an unjammed phase emerges between a localized, absorbing phase and a diffusive, irreversible, phase. The emergence of the unjammed phase signals the shifting of the jamming point to higher densities as a result of annealing and opens a window where shear jamming becomes possible for frictionless packings. Below [Formula: see text], two distinct localized states, termed point- and loop-reversible, are observed. We characterize in detail the different regimes and transitions between them and obtain a unified density-shear amplitude phase diagram.

Characterizing nonaffinity upon decompression of soft-sphere packings.

Athermal elastic moduli of soft-sphere packings are known to exhibit universal scaling properties near the unjamming point, most notably the vanishing of the shear-to-bulk moduli ratio G/B upon decompression. Interestingly, the smallness of G/B stems from the large nonaffinity of deformation-induced displacements under shear strains, compared to insignificant nonaffinity of displacements under compressive strains. In this work, we show using numerical simulations that the relative weights of the affine and nonaffine contributions to the bulk modulus, and their dependence on the proximity to the unjamming point, can differ qualitatively between different models that feature the same generic unjamming phenomenology. In canonical models of unjamming, we observe that the ratio of the nonaffine to total bulk moduli B_{na}/B approaches a constant upon decompression, while in other, less well-studied models, it vanishes. We show that the vanishing of B_{na}/B in noncanonical models stems from the emergence of an invariance of net (zero) forces on the constituent particles to compressive strains at the onset of unjamming. We provide a theoretical scaling analysis that fully explains our numerical observations, and allows us to predict the scaling behavior of B_{na}/B upon unjamming, given the functional form of the pairwise interaction potential.

Effect of instantaneous and continuous quenches on the density of vibrational modes in model glasses

Computational studies of supercooled liquids often focus on various analyses of their "underlying inherent states"-the glassy configurations at zero temperature obtained by an infinitely fast (instantaneous) quench from equilibrium supercooled states. Similar protocols are also regularly employed in investigations of the unjamming transition at which the rigidity of decompressed soft-sphere packings is lost. Here we investigate the statistics and localization properties of low-frequency vibrational modes of glassy configurations obtained by such instantaneous quenches. We show that the density of vibrational modes grows as ω^{β} with β depending on the parent temperature T_{0} from which the glassy configurations were instantaneously quenched. For quenches from high temperature liquid states we find β≈3, whereas β appears to approach the previously observed value β=4 as T_{0} approaches the glass transition temperature. We discuss the consistency of our findings with the theoretical framework of the soft potential model, and contrast them with similar measurements performed on configurations obtained by continuous quenches at finite cooling rates. Our results suggest that any physical quench at rates sufficiently slower than the inverse vibrational time scale-including all physically realistic quenching rates of molecular or atomistic glasses-would result in a glass whose density of vibrational modes is universally characterized by β=4.

Nonlocal Elasticity near Jamming in Frictionless Soft Spheres.

We use simulations of frictionless soft sphere packings to identify novel constitutive relations for linear elasticity near the jamming transition. By forcing packings at varying wavelengths, we directly access their transverse and longitudinal compliances. These are found to be wavelength dependent, in violation of conventional (local) linear elasticity. Crossovers in the compliances select characteristic length scales, which signify the appearance of nonlocal effects. Two of these length scales diverge as the pressure vanishes, indicating that critical effects near jamming control the breakdown of local elasticity. We expect these nonlocal constitutive relations to be applicable to a wide range of weakly jammed solids, including emulsions, foams, and granulates.

Beyond linear elasticity: jammed solids at finite shear strain and rate.

The shear response of soft solids can be modeled with linear elasticity, provided the forcing is slow and weak. Both of these approximations must break down when the material loses rigidity, such as in foams and emulsions at their (un)jamming point - suggesting that the window of linear elastic response near jamming is exceedingly narrow. Yet precisely when and how this breakdown occurs remains unclear. To answer these questions, we perform computer simulations of stress relaxation and shear start-up tests in athermal soft sphere packings, the canonical model for jamming. By systematically varying the strain amplitude, strain rate, distance to jamming, and system size, we identify characteristic strain and time scales that quantify how and when the window of linear elasticity closes, and relate these scales to changes in the microscopic contact network.

Jamming graphs: A local approach to global mechanical rigidity

We revisit the concept of minimal rigidity as applied to frictionless, repulsive soft sphere packings in two dimensions with the introduction of the jamming graph. Minimal rigidity is a purely combinatorial property encoded via Laman's theorem in two dimensions. It constrains the global, average coordination number of the graph, for example. However, minimal rigidity does not address the geometry of local mechanical stability. The jamming graph contains both properties of global mechanical stability at the onset of jamming and local mechanical stability. We demonstrate how jamming graphs can be constructed using local moves via the Henneberg construction such that these graphs fall under the jurisdiction of correlated percolation. We then probe how jamming graphs destabilize, or become unjammed, by deleting a bond and computing the resulting rigid cluster distribution. We also study how the system restabilizes with the addition of new contacts and how a jamming graph with extra (redundant) contacts destabilizes. The latter endeavor allows us to probe a disk packing in the rigid phase and uncover a potentially new diverging length scale associated with the random deletion of contacts as compared to the study of cut-out (or frozen-in) subsystems.

Shear dilatancy in marginal solids

Shearing stresses can change the volume of a material via a nonlinear effect known as shear dilatancy. We calculate the elastic dilatancy coefficient of soft sphere packings and random spring networks, two canonical models of marginal solids close to their rigidity transition. We predict a dramatic enhancement of dilatancy near rigidity loss in both materials, with a surprising distinction: while packings expand under shear, networks contract. We show that contraction in networks is due to the destabilizing influence of increasing hydrostatic or uniaxial loads, which is counteracted in packings by the formation of new contacts.

Soft-Sphere Packings at Finite Pressure but Unstable to Shear

When are athermal soft-sphere packings jammed? Any experimentally relevant definition must, at the very least, require a jammed packing to resist shear. We demonstrate that widely used (numerical) protocols, in which particles are compressed together, can and do produce packings that are unstable to shear-and that the probability of generating such packings reaches one near jamming. We introduce a new protocol which, by allowing the system to explore different box shapes as it equilibrates, generates truly jammed packings with strictly positive shear moduli G. For these packings, the scaling of the average of G is consistent with earlier results, while the probability distribution P(G) exhibits novel and rich scalings.

Relaxations and Rheology near Jamming

We determine the form of the complex shear modulus G* in soft sphere packings near jamming. Viscoelastic response at finite frequency is closely tied to a packing's intrinsic relaxational modes, which are distinct from the vibrational modes of undamped packings. We demonstrate and explain the appearance of an anomalous excess of slowly relaxing modes near jamming, reflected in a diverging relaxational density of states. From the density of states, we derive the dependence of G* on the frequency and distance to the jamming transition, which is confirmed by numerics.

Jamming of frictional spheres and random loose packing

The role of the friction coefficient, μ, on the jamming properties of disordered, particle packings is studied using computer simulations. Compressed, soft-sphere packings are brought towards the jamming transition—the point where a packing loses mechanical stability—by decreasing the packing fraction. The values of the packing fraction at the jamming transition, ϕμc, gradually decrease from the random close packing point for zero friction, to a value coincident with random loose packing as the friction coefficient is increased over several orders of magnitude. This is accompanied by a decrease in the coordination number at the jamming transition, zμc, which varies from approximately six to four with increasing friction. Universal power law scaling is observed in the pressure and coordination number as a function of distance from the generalised, friction-dependent jamming point. Various power laws are also reported between the ϕμc, zμc, and μ. Dependence on preparation history of the packings is also investigated.

Structural signatures of the unjamming transition at zero temperature

We study the pair correlation function g(r) for zero-temperature, disordered, soft-sphere packings just above the onset of jamming. We find distinct signatures of the transition in both the first and split second peaks of this function. As the transition is approached from the jammed side (at higher packing fraction) the first peak diverges and narrows on the small-r side to a delta function. On the high-r side of this peak, g(r) decays as a power law. In the split second peak, the two subpeaks are both singular at the transition, with power-law behavior on their low-r sides and step-function drop-offs on their high-r sides. These singularities at the transition are reminiscent of empirical criteria that have previously been used to distinguish glassy structures from liquid ones.

Fine structure in randomly packed, dense clusters of hard spheres

Very dense, randomly-packed clusters of 500 equal spheres produced by computer using the algorithm described show significant differences from extended dense packings. The free boundary makes volume available to allow collective rearrangements which result in packing densities 4.5% higher than otherwise possible. The radial distribution functions of the dense clusters show a reversal in intensity of the split second peak as density is increased, and an additional small peak at about 1.4 diameters in the very high density packings can be correlated with the local arrangements found in the clusters. The topological statistics of the Voronoi polyhedra show differences from the extended random packing which can also be interpreted in terms of the extra freedom conferred by the free boundary. In contrast with soft sphere packings, the icosahedral arrangements are squeezed out at high densities in favour of more complex local regions. This supports previous suggestions that the detailed structure of a random packing depends upon both the softness and nature of the intermolecular potential function, the nature of the boundary, and the conditions of preparation, and warns that packing models of real amorphous solids must be used with care.