Jammed Packings Research Articles

Shear thickening and hysteresis in dense suspensions: The effect of particle shape

The flow of dense suspension of non-Brownian particles has been considered by various studies affected by their significance in a variety of industries and natural phenomena. In this study, we investigate the effect of polyhedron morphology on shear thickening, shear jamming, and hysteresis characteristics of non-Brownian suspension of acrylate particles. Particles with the same chemical nature and three different shapes of spherical (aspect ratio Γ = 1), elliptical paraboloid (Γ ≈ 1), and boat-shaped (Γ ≈ 3) are fabricated via photopolymerization-based methods. Studied suspensions show the shear-thinning behavior at low shear stresses and shear thickening behavior at the higher range of shear stress. Also, the strength of observed shear thickening is enhanced for the suspensions of polyhedron particles, which can be attributed to the heightened degree of interparticle frictional contacts. Furthermore, it is found that angularity not only shifts the predicted frictionless and frictional jamming packing fractions to lower values but also expands the shear jamming packing fraction range. Finally, a history-dependent hysteresis is observed in all samples due to the different particle spatial structures forming in ascending and descending flow modes. The observed hysteresis loops strongly depend on the volume fraction and diminish near the jamming packing fraction due to the restricted mobility space of particles. In addition, the tumbling of elongated particles also can decrease the hysteresis loop by enhancing viscosity in the ascending flow mode, where the structures are not fully developed.

Exceptionally dense and resilient critically jammed polydisperse disk packings.

Understanding the way disordered particle packings transition between jammed (rigid) and unjammed (fluid) states is of both great practical importance and strong fundamental interest. The values of critical packing fraction (and other state variables) at the jamming transition are protocol dependent. Here, we demonstrate that this variability can be systematically traced to structural measures of packing, as well as to energy measures inside the jammed regime. A novel generalized simultaneous particle swap algorithm constructs overjammed states of desired energy, which upon decompression lead to predictable critical packing fractions. Thus, for a given set of particle sizes, states with extraordinarily high critical packing fractions can be found efficiently, which sustain substantial shear strain and preserve their special structure over the entire jammed domain. The close relation revealed here between the energy landscape of overjammed soft-particle packings and the behavior near the jamming transition points towards new ways of understanding and constructing disordered materials with exceptional properties.

Hyperuniformity of maximally random jammed packings of hyperspheres across spatial dimensions.

The maximally random jammed (MRJ) state is the most random (i.e., disordered) configuration of strictly jammed (mechanically rigid) nonoverlapping objects. MRJ packings are hyperuniform, meaning their long-wavelength density fluctuations are anomalously suppressed compared to typical disordered systems, i.e., their structure factors S(k) tend to zero as the wave number |k| tends to zero. Here we show that generating high-quality strictly jammed states for Euclidean space dimensions d=3,4, and 5 is of paramount importance in ensuring hyperuniformity and extracting precise values of the hyperuniformity exponent α>0 for MRJ states, defined by the power-law behavior of S(k)∼|k|^{α} in the limit |k|→0. Moreover, we show that for fixed d it is more difficult to ensure jamming as the particle number N increases, which results in packings that are nonhyperuniform. Free-volume theory arguments suggest that the ideal MRJ state does not contain rattlers, which act as defects in numerically generated packings. As d increases, we find that the fraction of rattlers decreases substantially. Our analysis of the largest truly jammed packings suggests that the ideal MRJ packings for all dimensions d≥3 are hyperuniform with α=d-2, implying the packings become more hyperuniform as d increases. The differences in α between MRJ packings and the recently proposed Manna-class random close packed (RCP) states, which were reported to have α=0.25 in d=3 and be nonhyperuniform (α=0) for d=4 and d=5, demonstrate the vivid distinctions between the large-scale structure of RCP and MRJ states in these dimensions. Our paper clarifies the importance of the link between true jamming and hyperuniformity and motivates the development of an algorithm to produce rattler-free three-dimensional MRJ packings.

Dense Disordered Jammed Packings of Hard Spherocylinders with a Low Aspect Ratio: A Characterization of Their Structure.

This work numerically investigates dense disordered (maximally random) jammed packings of hard spherocylinders of cylinder length L and diameter D by focusing on L/D ∈ [0,2]. It is within this interval that one expects that the packing fraction of these dense disordered jammed packings ϕMRJhsc attains a maximum. This work confirms the form of the graph ϕMRJhsc versus L/D: here, comparably to certain previous investigations, it is found that the maximal ϕMRJhsc = 0.721 ± 0.001 occurs at L/D = 0.45 ± 0.05. Furthermore, this work meticulously characterizes the structure of these dense disordered jammed packings via the special pair-correlation function of the interparticle distance scaled by the contact distance and the ensuing analysis of the statistics of the hard spherocylinders in contact: here, distinctly from all previous investigations, it is found that the dense disordered jammed packings of hard spherocylinders with 0.45 ≲ L/D ≤ 2 are isostatic.

Local and global measures of the shear moduli of jammed disk packings.

Strain-controlled isotropic compression gives rise to jammed packings of repulsive, frictionless disks with either positive or negative global shear moduli. We carry out computational studies to understand the contributions of the negative shear moduli to the mechanical response of jammed disk packings. We first decompose the ensemble-averaged, global shear modulus as 〈G〉=(1-F_{-})〈G_{+}〉+F_{-}〈G_{-}〉, where F_{-} is the fraction of jammed packings with negative shear moduli and 〈G_{+}〉 and 〈G_{-}〉 are the average values from packings with positive and negative moduli, respectively. We show that 〈G_{+}〉 and 〈|G_{-}|〉 obey different power-law scaling relations above and below pN^{2}∼1. For pN^{2}>1, both 〈G_{+}〉N and 〈|G_{-}|〉N∼(pN^{2})^{β}, where β∼0.5 for repulsive linear spring interactions. Despite this, 〈G〉N∼(pN^{2})^{β^{'}} with β^{'}≳0.5 due to the contributions from packings with negative shear moduli. We show further that the probability distribution of global shear moduli P(G) collapses at fixed pN^{2} and different values of p and N. We calculate analytically that P(G) is a Γ distribution in the pN^{2}≪1 limit. As pN^{2} increases, the skewness of P(G) decreases and P(G) becomes a skew-normal distribution with negative skewness in the pN^{2}≫1 limit. We also partition jammed disk packings into subsystems using Delaunay triangulation of the disk centers to calculate local shear moduli. We show that the local shear moduli defined from groups of adjacent triangles can be negative even when G>0. The spatial correlation function of local shear moduli C(r[over ⃗]) displays weak correlations for pn_{sub}^{2}<10^{-2}, where n_{sub} is the number of particles within each subsystem. However, C(r[over ⃗]) begins to develop long-ranged spatial correlations with fourfold angular symmetry for pn_{sub}^{2}≳10^{-2}.

A heuristic approach for the densest packing fraction of hard-sphere mixtures

In a previous work, a simple approach to derive the jamming packing fraction of a hard-sphere mixture from the knowledge of the random close-packing fraction of the monocomponent system was proposed. Now, an extension of that approach is applied to provide an approximate formula for the densest packing fraction of a given hard-sphere mixture in terms of the fcc close-packing fraction of a monocomponent hard-sphere system and of a single parameter encapsulating the dependence on the size ratios and the number of spheres in the unit cell. Comparison with recent results for such densest packing fraction of binary and ternary systems is performed and reasonable agreement is obtained.

Comparison of compression versus shearing near jamming, for a simple model of athermal frictionless disks in suspension.

Using a simplified model for a non-Brownian suspension, we numerically study the response of athermal, overdamped, frictionless disks in two dimensions to isotropic and uniaxial compression, as well as to pure and simple shearing, all at finite constant strain rates ε[over ̇]. We show that isotropic and uniaxial compression result in the same jamming packing fraction ϕ_{J}, while pure-shear- and simple-shear-induced jamming occurs at a slightly higher ϕ_{J}^{*}, consistent with that found previously for simple shearing. A critical scaling analysis of pure shearing gives critical exponents consistent with those previously found for both isotropic compression and simple shearing. Using orientational order parameters for contact bond directions, we compare the anisotropy of the force and contact networks at both lowest nematic order, as well as higher 2n-fold order.

Structure of jammed ellipse packings with a wide range of aspect ratios.

Motivated in part by the recent observation of liquid glass in suspensions of ellipsoidal colloids, we examine the structure of jammed two-dimensional ellipse packings over a much wider range of particle aspect ratios (α, the ratio of the major and minor axis lengths) than has been previously attempted. We determine the jamming densities ϕJ(α) to high precision, and find empirical analytic formulae that predict ϕJ(α) to within less than 0.1% for all 1≤α≤10, for three different particle dispersities. Then we explore how these packings' local structural order varies with α. We find that the densest packings possess unusually-well-defined nearest-neighbor shells, including both a higher fraction fZ=6 of particles with exactly six contacts and a previously-unreported short-range order marked by "kinetically suppressed" regions in their positional-orientational pair correlation function g(r,Δθ). We also show that the previously-reported approach to isostaticity (coordination number ZJ → Ziso ≡ 6) with increasing α is interrupted and then reversed as local nematic order increases: ZJ(α) drops towards 4 as ellipses are more often trapped by contacts with a parallel-oriented neighbor on either side and a perpendicularly-oriented neighbor on either end. Finally we show that ϕJ/ϕs (where ϕs is the saturated RSA packing density) is nearly α-independent for systems that do not develop substantial local hexatic or nematic order during compression.

Dense disordered jammed packings of hard very elongate particles: A new derivation of the random contact equation.

This work describes a derivation of the random contact equation that predicts the packing fraction ϕMRJhr of a dense disordered (maximally random) jammed state of hard, very elongate particles. This derivation is based on (i) the compressibility equation connecting the compressibility of a uniform system with its pair-correlation function: it is assumed equal to zero at jamming; (ii) the pair-correlation function of the interparticle distance scaled with respect to the orientationally dependent contact distance: it is assumed equal to the sum of a delta function and a unit-step function at jamming, where the former function accounts for the interparticle contacts, while the latter function accounts for the background. On assuming that the hard, very elongate particles are cylindrically symmetric with a length L and a diameter D and isostaticity occurs at jamming, the prediction, in particular that, in the limit of L/D → +∞, ϕMRJhr L/D = (10 + 1)/2, is compared to the available experimental data.

Hyperuniform jammed sphere packings have anomalous material properties.

A spatial distribution is hyperuniform if it has local density fluctuations that vanish in the limit of long length scales. Hyperuniformity is a well known property of both crystals and quasicrystals. Of recent interest, however, is disordered hyperuniformity: the presence of hyperuniform scaling without long-range configurational order. Jammed granular packings have been proposed as an example of disordered hyperuniformity, but recent numerical investigation has revealed that many jammed systems instead exhibit a complex set of distinct behaviors at long, emergent length scales. We use the Voronoi tessellation as a tool to define a set of rescaling transformations that can impose hyperuniformity on an arbitrary weighted point process, and show that these transformations can be used in simulations to iteratively generate hyperuniform, mechanically stable packings of athermal soft spheres. These hyperuniform jammed packings display atypical mechanical properties, particularly in the low-frequency phononic excitations, which exhibit an isolated band of highly collective modes and a band gap around zero frequency.

Stress transmission in entangled granular structures

We study the transmission of compressive and tensile stresses, and the development of stress - induced anisotropy in entangled granular structures composed of nonconvex S-shaped hooks and staples. Utilizing discrete element simulations, we find that these systems exhibit fundamentally different behavior compared to standard convex particle systems, including the ability to entangle which contributes to a lower jamming packing fraction and facilitates the transmission of tensile stresses. We present direct evidence of tensile stress chains, and show that these chains are generally sparser, shorter and shorter-lived than the compressive chains found in convex particle packings. We finally study the probability distribution, angular density and anisotropic spatial correlation of the minor (compressive) and major (tensile) particle stresses. The insight gained for these systems can help the design of reconfigurable and recyclable granular structures capable of bearing considerable loads, without any need for reinforcement.Graphical abstract

Finding the grain size distribution that produces the densest arrangement in frictional sphere packings: Revisiting and rediscovering the century-old Fuller and Thompson distribution

By means of discrete-element methods, we investigate the joint effects of the grain size distribution (GSD) and contact friction on the structure of three-dimensional samples composed of spherical grains. Specifically, we compress these systems isotropically until jamming and then analyze their structure in terms of density, connectivity, coefficients of uniformity and curvature, and parameters of grading entropy. Our study focuses on power-law GSDs and particularly on the Fuller and Thompson distribution, proposed over a century ago. First, we show that, among the set of GSDs investigated, this particular distribution produces the densest and best-connected systems, falsifying a conjecture recently posed in the literature. Second, we find that the jamming packing fraction can be accurately predicted as a function of simple descriptors of the GSD, but among these descriptors the granular entropy concept proves to be the most useful. This allows for an alternative interpretation of both jamming and grading entropy concepts. Finally, we compare the Fuller and Thompson distribution with two well-known GSDs: that of the Apollonian sphere packing and that towards which granular systems evolve after intensive grain fracturing. Surprisingly, we find that these three GSDs are practically coincident in the limit of large size spans, despite having been introduced or discovered in different scientific contexts (i.e., engineering, mathematics, and earth sciences, respectively).

Effect of adhesive interaction on strain stiffening and dissipation in granular gels undergoing yielding

Shear induced yielding in disordered solids, characterized by irreversibility and enhanced dissipation, is important for a wide range of industrial and geological processes. Although such phenomena in thermal systems have been extensively studied, they remain poorly understood for granular solids. Here, using oscillatory shear rheology we study energy dissipation in a disordered solid formed by dense granular suspensions of adhesive frictional particles. We find non-linear flow regimes showing intra-cycle strain stiffening and plasticity that strongly depend on both the applied strain amplitude and particle volume fraction, which can be captured by the normalized energy dissipation. Furthermore, in-situ optical imaging reveals irreversible particle rearrangements correlating with the spatio-temporal fluctuations in local velocity across the yielding transition. By directly measuring the critical jamming packing fraction using particle settling experiments, we propose a phase diagram that unravels the effect of inter-particle interactions on flow properties of the system for a large parameter space.

Determining the equivalent packing diameter of two-dimensional shapes

How to quantify the effective size of particles in a mixture, with different shapes and size dispersity, remains ambiguous. Recently, this issue has been extensively studied for disordered jammed packings in three dimensions, with particular emphasis on the concept of equivalent packing diameter (De). However, this approach cannot be simply generalized to two dimensions (2D). In this paper, we address this issue by calibrating De via quaternary mixtures in 2D, which effectively suppress the ordering in packings. The quaternary mixture consists of two bi-disperse systems of certain shapes with invariant size ratio 1.4. We verify the linear superposition of the specific volume of a shape-mixed disordered packing under identical De together with the transferability of the equivalent packing size ratio between different 2D shapes. We further devise a concise predictive model to estimate De for ellipses and superdisks. These results should constitute the foundation of 2D polydisperse packings.

Orientational arrest in dense suspensions of elliptical particles under oscillatory shear flows

We study the rheological response of dense suspensions of elliptical particles, with an aspect ratio equal to 3, under oscillatory shear flows and imposed pressure by numerical simulations. Like for the isotropic particles, we find that the oscillatory shear flows respect the Cox-Merz rule at large oscillatory strains but differ at low strains, with a lower viscosity than the steady shear and higher shear jamming packing fractions. However, unlike the isotropic cases (i.e., discs and spheres), frictionless ellipses get dynamically arrested in their initial orientational configuration at small oscillatory strains. We illustrate this by starting at two different configurations with different nematic order parameters and the average orientation of the particles. Surprisingly, the overall orientation in the frictionless case is uncoupled to the rheological response close to jamming, and the rheology is only controlled by the average number of contacts and the oscillatory strain. Having larger oscillatory strains or adding friction does, however, help the system escape these orientational arrested states, which are evolving to a disordered state independent of the initial configuration at low strains and ordered ones at large strains.

Long-Range Anomalous Decay of the Correlation in Jammed Packings.

We numerically study the structure of the interactions occurring in three-dimensional systems of hard spheres at jamming, focusing on the large-scale behavior. Given the fundamental role in the configuration of jammed packings, we analyze the propagation through the system of the weak forces and of the variation of the coordination number with respect to the isostaticity condition, ΔZ. We show that these correlations can be successfully probed by introducing a correlation function weighted on the density-density fluctuations. The results of this analysis can be further improved by introducing a representation of the system based on the contact points between particles. In particular, we find evidence that the weak forces and the ΔZ fluctuations support the hypothesis of randomly jammed packings of spherical particles being hyperuniform by exhibiting an anomalous long-range decay. Moreover, we find that the large-scale structure of the density-density correlation exhibits a complex behavior due to the superimposition of two exponentially damped oscillating signals propagating with linearly depending frequencies.

Shape effects on mechanical properties of maximally random jammed packings of intersecting spherocylinders

The packing state of granular material has significant effects on its mechanical properties. It is essential to compare the mechanical properties in the same physical state. Spherocylinders and intersecting spherocylinders are widely used particle shapes both in scientific researches and engineering applications. In this work, we generate the maximally random jammed (jamming onset) packings of spherocylinders and intersecting spherocylinders via the adaptive shrinking cell algorithm. Then we obtain the bulk modulus and measure the structural stability of these packings by the discrete element method. For all the shapes concerned, the bulk modulus first increases and then decreases with the increase of the aspect ratio and is positively correlated with the packing density. We also observe that the structural stability increases as the aspect ratio grows, and the friction improves the bulk modulus and structural stability simultaneously. These results are beneficial for guiding granular material design in engineering applications.

Mean-Field Predictions of Scaling Prefactors Match Low-Dimensional Jammed Packings

No known analytic framework precisely explains all the phenomena observed in jamming. The replica theory for glasses and jamming is a mean-field theory which attempts to do so by working in the limit of infinite dimensions, such that correlations between neighbors are negligible. As such, results from this mean-field theory are not guaranteed to be observed in finite dimensions. However, many results in mean field for jamming have been shown to be exact or nearly exact in low dimensions. This suggests that the infinite dimensional limit is not necessary to obtain these results. In this Letter, we perform precision measurements of jamming scaling relationships between pressure, excess packing fraction, and number of excess contacts from dimensions 2-10 in order to extract the prefactors to these scalings. While these prefactors should be highly sensitive to finite dimensional corrections, we find the mean-field predictions for these prefactors to be exact in low dimensions. Thus the mean-field approximation is not necessary for deriving these prefactors. We present an exact, first-principles derivation for one, leaving the other as an open question. Our results suggest that mean-field theories of critical phenomena may compute more for d≥d_{u} than has been previously appreciated.

Effects of mechanical constraint on thermally induced reverse martensitic transformation in granular shape memory ceramic packings

Zirconia-based ceramics exhibit shape memory and superelastic effects based on the reversible martensitic transformation between tetragonal and monoclinic crystal structures. In the form of granular packings, these shape memory ceramics can be scaled up for bulk applications despite their intrinsic brittleness, while displaying drastically different transformation characteristics than the monolithic counterparts. Here, we present a comparative study to understand the thermally induced reverse martensitic transformation in granular packings and the influence of mechanical constraints. This study employs ZrO2–CeO2 shape memory ceramics of the same composition but with different degrees of mechanical constraints. The explored material forms include loose and jammed granular packings, themselves consisting of polycrystalline or single crystal particles, as well as sintered bulk polycrystals. Except for the latter, no endothermic peak is observed in the heat flow measurement of the reverse transformation process. This unusual behavior is shown to stem from the weak inter-particle mechanical constraint and the transformation heterogeneity among individual particles, rather than stress relaxation or particle rearrangement. To compare, conspicuous endothermic peaks only appear in bursting-type transformations under a strong mechanical constraint. For granular packings, the intra-particle mechanical constraint does not affect the presence of any endothermic peaks in thermal reversion but can influence the austenite start temperature.

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.