Prediction of polydisperse hard-sphere mixture behavior using tridisperse systems

Influence of friction on the packing efficiency and short-to-intermediate range structure of hard-sphere systems

Using particle-resolved computer simulations, we investigate the effect of friction on the packing structure of hard-sphere mixtures with two kinds of particles under external compression. We first show that increasing friction between the particles results in a more disordered and less efficient packing of the local structure on the nearest neighbor scale. It is also found that standard two-point correlation functions, i.e., radial distribution function and static structure factor, show basically no detectable changes beyond short-range distances upon varying inter-particle friction. Further analysis of the structure using a four-point correlation method reveals that these systems have on the intermediate-range scale a three-dimensional structure with an icosahedral/dodecahedral symmetry that exhibits a pronounced dependence on friction: small friction gives rise to an orientational order that extends to larger distances. Our results also demonstrate that composition plays a role in that the degree of structural order and the structural correlation length are mainly affected by the friction coefficients associated with the more abundant species.

Even Strong Energy Polydispersity Does Not Affect the Average Structure and Dynamics of Simple Liquids

Size-polydisperse liquids have become standard models for avoiding crystallization, thereby enabling studies of supercooled liquids and glasses formed, e.g., by colloidal systems. Purely energy-polydisperse liquids have been studied much less, but provide an interesting alternative. We here study numerically the difference in structure and dynamics obtained by introducing these two kinds of polydispersity into systems of particles interacting via the Lennard-Jones and EXP pair potentials. To a very good approximation, the average pair structure and dynamics are unchanged even for strong energy polydispersity, which is not the case for size-polydisperse systems. When the system at extreme energy polydispersity undergoes a continuous phase separation into lower and higher particle-energy regions whose structure and dynamics are different from the average, the average structure and dynamics are still virtually the same as for the monodisperse system. Our findings are consistent with the fact that the distribution of forces on the individual particles do not change when energy polydispersity is introduced, while they do change in the case of size polydispersity. A theoretical explanation remains to be found, however.

Estimating random close packing in polydisperse and bidisperse hard spheres via an equilibrium model of crowding

We show that an analogy between crowding in fluid and jammed phases of hard spheres captures the density dependence of the kissing number for a family of numerically generated jammed states. We extend this analogy to jams of mixtures of hard spheres in d = 3 dimensions and, thus, obtain an estimate of the random close packing volume fraction, ϕ_RCP, as a function of size polydispersity. We first consider mixtures of particle sizes with discrete distributions. For binary systems, we show agreement between our predictions and simulations using both our own results and results reported in previous studies, as well as agreement with recent experiments from the literature. We then apply our approach to systems with continuous polydispersity using three different particle size distributions, namely, the log-normal, Gamma, and truncated power-law distributions. In all cases, we observe agreement between our theoretical findings and numerical results up to rather large polydispersities for all particle size distributions when using as reference our own simulations and results from the literature. In particular, we find ϕ_RCP to increase monotonically with the relative standard deviation, s_σ, of the distribution and to saturate at a value that always remains below 1. A perturbative expansion yields a closed-form expression for ϕ_RCP that quantitatively captures a distribution-independent regime for s_σ < 0.5. Beyond that regime, we show that the gradual loss in agreement is tied to the growth of the skewness of size distributions.

Bulk modulus along jamming transition lines of bidisperse granular packings

We present three-dimensional discrete element method simulations of bidisperse granular packings to investigate their jamming densities ϕ_{J} and dimensionless bulk moduli K as functions of the size ratio δ and the concentration of small particles X_{S}. We determine the partial and total bulk moduli for packings near their jamming densities, including a second transition that occurs for sufficiently small δ and X_{S} when the system is compressed beyond its first jamming transition. While the first transition is sharp, exclusively with large-large contacts, the second is rather smooth, carried by small-large interactions at densities much higher than the monodisperse random packing baseline, ϕ_{J}^{mono}≈0.64. When only nonrattlers are considered, all the effective transition densities are reduced, and the density of the second transition emerges rather close to the reduced baseline, ϕ[over ̃]_{J}^{mono}≈0.61, due to its smooth nature. At size ratios δ≤0.22 a concentration X_{S}^{*} divides the diagram-either with most small particles nonjammed or jammed jointly with large ones. For X_{S}<X_{S}^{*}, the modulus K displays different behaviors at first and second jamming transitions. Along the second transition, K rises relative to the values found at the first transition; however, is still small compared to K at X_{S}^{*}. Explicitly, for our smallest δ=0.15, the discontinuous jump in K as a function of X_{S} is obtained at X_{S}^{*}≈0.21 and coincides with the maximum jamming density where both particle species mix most efficiently. Our results will allow tuning or switching the bulk modulus K or other properties, such as the wave speed, by choosing specific sizes and concentrations based on a better understanding of whether small particles contribute to the jammed structure or not, and how the micromechanical structure behaves at either transition.

Connecting Packing Efficiency of Binary Hard Sphere Systems to Their Intermediate Range Structure

Using computed x-ray tomography we determine the three dimensional (3D) structure of binary hard sphere mixtures as a function of composition and size ratio of the particles q. Using a recently introduced four-point correlation function we reveal that this 3D structure has on intermediate and large length scales a surprisingly regular order, the symmetry of which depends on q. The related structural correlation length has a minimum at the composition at which the packing fraction is highest. At this composition also the number of different local particle arrangements has a maximum, indicating that efficient packing of particles is associated with a structure that is locally maximally disordered.

Dense shearing flows of soft, frictional cylinders

We perform discrete numerical simulations at a constant volume of dense, steady, homogeneous flows of true cylinders interacting via Hertzian contacts, with and without friction, in the absence of preferential alignment. We determine the critical values of the solid volume fraction and the average number of contacts per particle above which rate-independent components of the stresses develop, along with a sharp increase in the fluctuations of angular velocity. We show that kinetic theory, extended to account for a velocity correlation at solid volume fractions larger than 0.49, can quantitatively predict the measured fluctuations of translational velocity, at least for sufficiently rigid cylinders, for any value of the cylinder aspect ratio and friction investigated here. The measured pressure above and below the critical solid volume fraction is in agreement with a recent theory originally intended for spheres that conjugates extended kinetic theory, the finite duration of collisions between soft particles and the development of an elastic network of long-lasting contacts responsible for the rate-independency of the flows in the supercritical regime. Finally, we find that, for sufficiently rigid cylinders, the ratio of shear stress to pressure in the subcritical regime is a linear function of the ratio of the shear rate to a suitable measure of the fluctuations of translational velocity, in qualitative accordance with kinetic theory, with an intercept that increases with friction. A decrease in the particle stiffness gives rise to nonlinear effects that greatly diminishes the stress ratio.

Inverse design of equilibrium cluster fluids applied to a physically informed model

Inverse design strategies have proven highly useful for the discovery of interaction potentials that prompt self-assembly of a variety of interesting structures. However, often the optimized particle interactions do not have a direct relationship to experimental systems. In this work, we show that Relative Entropy minimization is able to discover physically meaningful parameter sets for a model interaction built from depletion attraction and electrostatic repulsion that yield self-assembly of size-specific clusters. We then explore the sensitivity of the optimized interaction potentials with respect to deviations in the underlying physical quantities, showing that clustering behavior is largely preserved even as the optimized parameters are perturbed.

Hidden Scale Invariance in Polydisperse Mixtures of Exponential Repulsive Particles

Polydisperse systems of particles interacting by the purely repulsive exponential (EXP) pair potential are studied in regard to how structure and dynamics vary along isotherms, isochores, and isomorphs. The sizable size polydispersities of 23%, 29%, 35%, and 40%, as well as energy polydispersity 35%, were considered. For each system an isomorph was traced out covering about one decade in density. For all systems studied, the structure and dynamics vary significantly along the isotherms and isochores but are invariant to a good approximation along the isomorphs. We conclude that the single-component EXP system's hidden scale invariance (implying isomorph invariance of structure and dynamics) is maintained even when a sizable polydispersity is introduced into the system.

Structural and thermodynamic properties of hard-sphere fluids

This Perspective article provides an overview of some of our analytical approaches to the computation of the structural and thermodynamic properties of single-component and multicomponent hard-sphere fluids. For the structural properties, they yield a thermodynamically consistent formulation, thus improving and extending the known analytical results of the Percus-Yevick theory. Approximate expressions linking the equation of state of the single-component fluid to the one of the multicomponent mixtures are also discussed.

Structural universality in disordered packings with size and shape polydispersity

We numerically investigate disordered jammed packings with both size and shape polydispersity, using frictionless superellipsoidal particles. We implement the set Voronoi tessellation technique to evaluate the local specific volume, i.e., the ratio of cell volume over particle volume, for each individual particle. We focus on the average structural properties for different types of particles binned by their sizes and shapes. We generalize the basic observation that the larger particles are locally packed more densely than the smaller ones in a polydisperse-sized packing into systems with coupled particle shape dispersity. For this purpose, we define the normalized free volume vf to measure the local compactness of a particle and study its dependency on the normalized particle size A. The definition of vf relies on the calibrated monodisperse specific volume for a certain particle shape. For packings with shape dispersity, we apply the previously introduced concept of equivalent diameter for a non-spherical particle to define A properly. We consider three systems: (A) linear superposition states of mixed-shape packings, (B) merely polydisperse-sized packings, and (C) packings with coupled size and shape polydispersity. For (A), the packing is simply considered as a mixture of different subsystems corresponding to monodisperse packings for different shape components, leading to A = 1, and vf = 1 by definition. We propose a concise model to estimate the shape-dependent factor αc, which defines the equivalent diameter for a certain particle. For (B), vf collapses as a function of A, independent of specific particle shape and size polydispersity. Such structural universality is further validated by a mean-field approximation. For (C), we find that the master curve vf(A) is preserved when particles possess similar αc in a packing. Otherwise, the dispersity of αc among different particles causes the deviation from vf(A). These findings show that a polydisperse packing can be estimated as the combination of various building blocks, i.e., bin components, with a universal relation vf(A).

Additive rheology of complex granular flows

Granular flows are omnipresent in nature and industrial processes, but their rheological properties such as apparent friction and packing fraction are still elusive when inertial, cohesive and viscous interactions occur between particles in addition to frictional and elastic forces. Here we report on extensive particle dynamics simulations of such complex flows for a model granular system composed of perfectly rigid particles. We show that, when the apparent friction and packing fraction are normalized by their cohesion-dependent quasistatic values, they are governed by a single dimensionless number that, by virtue of stress additivity, accounts for all interactions. We also find that this dimensionless parameter, as a generalized inertial number, describes the texture variables such as the bond network connectivity and anisotropy. Encompassing various stress sources, this unified framework considerably simplifies and extends the modeling scope for granular dynamics, with potential applications to powder technology and natural flows.

Granular materials are abundant in nature, but we haven’t fully understood their rheological properties as complex interactions between particles are involved. Here, Vo et al. show that granular flows can be described by a generalized dimensionless number based on stress additivity.

Effects of particle shape mixture on strength and structure of sheared granular materials

Using bi-dimensional discrete element simulations, the shear strength and microstructure of granular mixtures composed of particles of different shapes are systematically analyzed as a function of the proportion of grains of a given number of sides and the combination of different shapes (species) in one sample. We varied the angularity of the particles by varying the number of sides of the polygons from 3 (triangles) up to 20 (icosagons) and disks. The samples analyzed were built keeping in mind the following cases: (1) increase of angularity and species starting from disks; (2) decrease of angularity and increase of species starting from triangles; (3) random angularity and increase of species starting from disks and from polygons. The results show that the shear strength vary monotonically with increasing numbers of species (it may increase or decrease), even in the random mixtures (case 3). At the micro-scale, the variation in shear strength as a function of the number of species is due to different mechanisms depending on the cases analyzed. It may result from the increase of both the geometrical and force anisotropies, from only a decrease of frictional anisotropy, or from compensation mechanisms involving geometrical and force anisotropies.

Equation of state of polydisperse hard-disk mixtures in the high-density regime

A proposal to link the equation of state of a monocomponent hard-disk fluid to the equation of state of a polydisperse hard-disk mixture is presented. Event-driven molecular dynamics simulations are performed to obtain data for the compressibility factor of the monocomponent fluid and of 26 polydisperse mixtures with different size distributions. Those data are used to assess the proposal and to infer the values of the compressibility factor of the monocomponent hard-disk fluid in the metastable region from those of mixtures in the high-density region. The collapse of the curves for the different mixtures is excellent in the stable region. In the metastable regime, except for two mixtures in which crystallization is present, the outcome of the approach exhibits a rather good performance. The simulation results indicate that a (reduced) variance of the size distribution larger than about 0.01 is sufficient to avoid crystallization and explore the metastable fluid branch.

Merging fluid and solid granular behavior

Simple homogeneous shear flows of frictionless, deformable particles are studied by particle simulations at large shear rates and for differently soft, deformable particles. Particle stiffness sets a time-scale that can be used to scale the physical quantities; thus the dimensionless shear rate, i.e. the inertial number I (inversely proportional to pressure), can alternatively be expressed as inversely proportional to the square root of particle stiffness. Asymptotic scaling relations for the field variables pressure, shear stress and granular temperature are inferred from simulations in both fluid and solid regimes, corresponding to unjammed and jammed conditions. Then the limit cases are merged to unique constitutive relations that cover also the transition zone in the proximity of jamming. By exploiting the diverging behavior of the scaling laws at the jamming density, we arrive at continuous and differentiable phenomenological constitutive relations for stresses and granular temperature as functions of the volume fraction, shear rate, particle stiffness and distance from jamming. In contrast to steady shear flows of hard particles the (shear) stress ratio μ does not collapse as a function of the inertial number, indicating the need for an additional control parameter. In the range of particle stiffnesses investigated, in the solid regime, only pressure is rate independent, whereas shear stress exhibits a slight shear rate- and stiffness-dependency.

Effect of Size Polydispersity on the Nature of Lennard-Jones Liquids

Polydisperse fluids are encountered everywhere in biological and industrial processes. These fluids naturally show a rich phenomenology exhibiting fractionation and shifts in critical point and freezing temperatures. We study here the effect of size polydispersity on the basic nature of Lennard-Jones (LJ) liquids, which represent most molecular liquids without hydrogen bonds, via two- and three-dimensional molecular dynamics computer simulations. A single-component liquid constituting spherical particles and interacting via the LJ potential is known to exhibit strong correlations between virial and potential energy equilibrium fluctuations at constant volume. This correlation significantly simplifies the physical description of the liquid, and these liquids are now known as Roskilde-simple (RS) liquids. We show that this simple nature of the single-component LJ liquid is preserved even for very high polydispersities (above 40% polydispersity for the studied uniform distribution). We also investigate isomorphs of moderately polydisperse LJ liquids. Isomorphs are curves in the phase diagram of RS liquids along which structure, dynamics, and some thermodynamic quantities are invariant in dimensionless units. We find that isomorphs are a good approximation even for polydisperse LJ liquids. The theory of isomorphs thus extends readily to size polydisperse fluids and can be used to improve even further the understanding of these intriguing systems.

Effects of shape and size polydispersity on strength properties of granular materials

By means of extensive contact dynamics simulations, we analyze the combined effects of polydispersity both in particle size and in particle shape, defined as the degree of shape irregularity, on the shear strength and microstructure of sheared granular materials composed of pentagonal particles. We find that the shear strength is independent of the size span, but unexpectedly, it declines with increasing shape polydispersity. At the same time, the solid fraction is an increasing function of both the size span and the shape polydispersity. Hence, the densest and loosest packings have the same shear strength. At the scale of the particles and their contacts, we analyze the connectivity of particles, force transmission, and friction mobilization as well as their anisotropies. We show that stronger forces are carried by larger particles and propped by an increasing number of small particles. The independence of shear strength with regard to size span is shown to be a consequence of contact network self-organization, with the falloff of contact anisotropy compensated by increasing force anisotropy.

Simple effective rule to estimate the jamming packing fraction of polydisperse hard spheres

A recent proposal in which the equation of state of a polydisperse hard-sphere mixture is mapped onto that of the one-component fluid is extrapolated beyond the freezing point to estimate the jamming packing fraction ϕJ of the polydisperse system as a simple function of M1M3/M22, where Mk is the kth moment of the size distribution. An analysis of experimental and simulation data of ϕJ for a large number of different mixtures shows a remarkable general agreement with the theoretical estimate. To give extra support to the procedure, simulation data for seventeen mixtures in the high-density region are used to infer the equation of state of the pure hard-sphere system in the metastable region. An excellent collapse of the inferred curves up to the glass transition and a significant narrowing of the different out-of-equilibrium glass branches all the way to jamming are observed. Thus, the present approach provides an extremely simple criterion to unify in a common framework and to give coherence to data coming from very different polydisperse hard-sphere mixtures.