Perspective: Basic understanding of condensed phases of matter via packing models

Generating Ultradense Jammed Ellipse Packings Using Biased SWAP

Using a Lubachevsky-Stillinger-like growth algorithm combined with biased SWAP Monte Carlo and transient degrees of freedom, we generate ultradense disordered jammed ellipse packings. For all aspect ratios α, these packings exhibit significantly smaller intermediate-wavelength density fluctuations and greater local nematic order than their less-dense counterparts. The densest packings are disordered despite having packing fractions ϕJ(α) that are within less than 0.5% of that of the monodisperse-ellipse crystal [ϕxtal = π/(2√3) ≃ 0.9069] over the range 1.2 ≲ α ≲ 1.45 and coordination numbers ZJ(α) that are within less than 0.5% of isostaticity [Ziso = 6] over the range 1.2 ≲ α ≲ 2.6. Lower-α packings are strongly fractionated and consist of polycrystals of intermediate-size particles, with the largest and smallest particles isolated at the grain boundaries. Higher-α packings are also fractionated, but in a qualitatively different fashion; they are composed─of increasingly large locally nematic domains─reminiscent of liquid glasses.

From Hard to Soft Dense Sphere Packings: The Cohesive Energy of Barlow Structures Using Exact Lattice Summations for a General Lennard-Jones Potential

There are infinitely many distinguishable dense sphere packings in dimension three (Barlow structures) with maximum packing density of ρ = π/√18. While Hales [Hales, T. C. Ann. Math. 2005, 162, 1065-1185.] proved that the packing density of the most dense cubic packing cannot be surpassed, it is currently not known how the infinite possibilities of densely packed Barlow structures with different sequences of hexagonal close packed layers are energetically related compared to the well-known face centered cubic (fcc) and hexagonal close packed (hcp) structures. We demonstrate that, for a general Lennard-Jones potential, which includes the hard-sphere model as a limiting case, Barlow packings lie energetically between fcc and hcp. Exceptions to this energy sequence only occur when fcc and hcp are energetically quasi-degenerate.

Optimal three-dimensional particle shapes for maximally dense saturated packing

Saturated packing is a random packing state of particles widely applied in investigating the physicochemical properties of granular materials. Optimizing particle shape to maximize packing density is a crucial challenge in saturated packing research. The known optimal three-dimensional shape is an ellipsoid with a saturated packing density of 0.437 72(51). In this work, we generate saturated packings of three-dimensional asymmetric shapes, including spherocylinders, cones, and tetrahedra, via the random sequential adsorption algorithm and investigate their packing properties. Results show that the optimal shape of asymmetric spherocylinders gives the maximum density of 0.4338(9), while cones achieve a higher value of 0.4398(10). Interestingly, tetrahedra exhibit two distinct optimal shapes with significantly high densities of 0.4789(19) and 0.4769(18), which surpass all previous results in saturated packing. The study of adsorption kinetics reveals that the two optimal shapes of tetrahedra demonstrate notably higher degrees of freedom and faster growth rates of the particle number. The analysis of packing structures via the density pair-correlation function shows that the two optimal shapes of tetrahedra possess faster transitions from local to global packing densities.

Evident structural anisotropies arising from near-zero particle asphericity in granular spherocylinder packings

With magnetic resonance imaging experiments, we study packings of granular spherocylinders with merely 2% asphericity. Evident structural anisotropies across all length scales are identified. Most interestingly, the global nematic order decreases with increasing packing fraction, while the local contact anisotropy shows an opposing trend. We attribute this counterintuitive phenomenon to a competition between gravity-driven ordering aided by frictional contacts and a geometric frustration effect at the marginally jammed state. It is also surprising to notice that such slight particle asphericity can trigger non-negligible correlations between contact-level and mesoscale structures, manifested in drastically different nonaffine structural rearrangements upon compaction from that of granular spheres. These observations can help improve statistical mechanical models for the orientational order transformation of nonspherical granular particle packings, which involves complex interplays between particle shape, frictional contacts, and external force field.

Estimating random close packing density from circle radius distributions

Circles of a single size can pack together densely in a hexagonal lattice, but adding in size variety disrupts the order of those packings. We conduct simulations which generate dense random packings of circles with specified size distributions and measure the area fraction in each case. While the size distributions can be arbitrary, we find that for a wide range of size distributions the random close-packing area fraction ϕ_{rcp} under this general protocol is determined to high accuracy by the polydispersity and skewness of the size distribution. At low skewness, all packings tend to a minimum packing fraction ϕ_{0}≈0.840 independent of polydispersity. In the limit of high skewness, ϕ_{rcp} becomes independent of skewness, asymptoting to a polydispersity-dependent limit. We show how these results can be predicted from the behavior of bidisperse or bi-Gaussian circle size distributions.

Spheroid models to elaborate the broken symmetry and equivalent volume of molecules in crystalline phase

Dense packing of particles has provided powerful models to elaborate the important structural features of matter in various systems such as liquid, glassy, and crystalline phases. The simplest sphere packing models can represent and capture salient properties of the building blocks for covalent, metallic, and ionic crystals; it, however, becomes insufficient to reflect the broken symmetry of the commonly anisotropic molecules in molecular crystals. Here, we develop spheroid models with a minimal degree of anisotropy, which serve as a simple geometrical representation for a rich spectrum of molecules-including both isotropic and anisotropic, convex and concave ones-in crystalline phases. Our models are determined via an inverse packing approach: Given a molecular crystal, an optimal spheroid model is constructed using a contact diagram, which depicts the packing relationship between neighboring molecules within the crystal. The spheroid models are capable of accurately capturing the broken symmetry and characterizing the equivalent volume of molecules in the crystalline phases. Moreover, our model retrieves such molecular information from low-quality x-ray diffraction data with poorly resolved structures, and by using soft spheroids, it can also describe the packing behavior in cocrystals.

Order-disorder transition during shear thickening in bidisperse dense suspensions

Shear thickening of multimodal suspensions has proven difficult to understand because the rheology depends largely on the microscopic details of stress-induced frictional contacts at different particle size distributions (PSDs). Our discrete particle simulations below a critical volume fraction ϕc over a broad range of shear rates and PSDs elucidate the basic mechanism of order-disorder transition. Around the theoretical optimal PSD (relative content of small particles ζ1= 0.26), particles order into a layered structure in the Newtonian regime. At the onset of shear thickening, this layered structure transforms to a disordered one, accompanied by an abrupt viscosity jump. Minor increase in large-large particle contacts after the order-disorder transition causes apparent increase in radial force along the compressional axis. Bidisperse suspensions with less regular but stable layered structure at ζ1= 0.50 show good fluidity in the shear thickening regime. This work shows that in inertial flows where particle collisions dominate, order-disorder transition could play an essential role in shear thickening for bidisperse suspensions.

The rule of four: anomalous distributions in the stoichiometries of inorganic compounds

Why are materials with specific characteristics more abundant than others? This is a fundamental question in materials science and one that is traditionally difficult to tackle, given the vastness of compositional and configurational space. We highlight here the anomalous abundance of inorganic compounds whose primitive unit cell contains a number of atoms that is a multiple of four. This occurrence-named here the rule of four-has to our knowledge not previously been reported or studied. Here, we first highlight the rule's existence, especially notable when restricting oneself to experimentally known compounds, and explore its possible relationship with established descriptors of crystal structures, from symmetries to energies. We then investigate this relative abundance by looking at structural descriptors, both of global (packing configurations) and local (the smooth overlap of atomic positions) nature. Contrary to intuition, the overabundance does not correlate with low-energy or high-symmetry structures; in fact, structures which obey the rule of four are characterized by low symmetries and loosely packed arrangements maximizing the free volume. We are able to correlate this abundance with local structural symmetries, and visualize the results using a hybrid supervised-unsupervised machine learning method.

Versatile Deposition of Complex Colloidal Assemblies from the Evaporation of Hanging Drops

Existing strategies designed to produce ordered arrangements of colloidal particles on solid supports are of great interest for their wide range of applications, from colloidal lithography, plasmonic and biomimetic surfaces to tags for anti-counterfeiting, but they all share various degrees of complexity hampering their facile implementation. Here, a drastically simplified methodology is presented to achieve ordered particle deposition, consisting in adding micromolar amounts of cationic surfactant to a colloidal suspension drop and let it evaporate in an upside-down configuration. Confinement at the air/water interface enables particle assembly into monolayers, which are then transferred on the substrate producing highly ordered structures displaying vivid, orientation-dependent structural colors. The method is compatible with many particle types and substrates, while controlling system parameters allows tuning the deposit size and morphology, from monocrystals to polycrystalline disks and "irises", from single-component to crystal alloys with Moiré patterns, demonstrating its practicality for a variety of processes.

Dense random packing of disks with a power-law size distribution in thermodynamic limit

The correlation properties of a random system of densely packed disks, obeying a power-law size distribution, are analyzed in reciprocal space in the thermodynamic limit. This limit assumes that the total number of disks increases infinitely, while the mean density of the disk centers and the range of the size distribution are kept constant. We investigate the structure factor dependence on momentum transfer across various number of disks and extrapolate these findings to the thermodynamic limit. The fractal power-law decay of the structure factor is recovered in reciprocal space within the fractal range, which corresponds to the range of the size distribution in real space. The fractal exponent coincides with the exponent of the power-law size distribution as was shown previously by the authors of the work of Cherny et al. [J. Chem. Phys. 158(4), 044114 (2023)]. The dependence of the structure factor on density is examined. As is found, the power-law exponent remains unchanged but the fractal range shrinks when the packing fraction decreases. Additionally, the finite-size effects are studied at extremely low momenta of the order of the inverse system size. We show that the structure factor is parabolic in this region and calculate the prefactor analytically. The obtained results reveal fractal-like properties of the packing and can be used to analyze small-angle scattering from such systems.

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.

Identification and analysis of 3D pores in packed particulate materials

We took the classic 'guess the number of beans in a jar game' and amplified the research question. Rather than estimate the quantity of particles in the jar, we sought to characterize the spaces between them. Here we present an approach for delineating the pockets of empty space (three-dimensional pores) between packed particles, which are hotspots for activity in applications and natural phenomena that deal with particulate materials. We utilize techniques from graph theory to exploit information about particle configuration that allows us to locate important spatial landmarks within the void space. These landmarks are the basis for our pore segmentation, where we consider both interior pores as well as entrance and exit pores into and out of the structure. Our method is robust for particles of varying size, form, stiffness and configuration, which allows us to study and compare three-dimensional pores across a range of packed particle types. We report striking relationships between particles and pores that are described mathematically, and we offer a visual library of pore types. With a meaningful discretization of void space, we demonstrate that packed particles can be understood not by their solid space, but by their empty space.

Correlations in randomly stacked solids

Packing of spheres is a problem with a long history dating back to Kepler's conjecture in 1611. The highest density is realized in face-centered-cubic (FCC) and hexagonal-close-packed (HCP) arrangements. These are only limiting examples of an infinite family of maximal-density structures called Barlow stackings. They are constructed by stacking triangular layers, with each layer shifted with respect to the one below. At the other extreme, Torquato-Stillinger stackings are believed to yield the lowest possible density while preserving mechanical stability. They form an infinite family of structures composed of stacked honeycomb layers. In this article, we characterize layer-correlations in both families when the stacking is random. To do so, we take advantage of the Hägg code-a mapping between a Barlow stacking and a one-dimensional Ising magnet. The layer correlation is related to a moment-generating function of the Ising model. We first determine the layer correlation for random Barlow stacking, finding exponential decay. We next introduce a bias favoring one of two stacking chiralities-equivalent to a magnetic field in the Ising model. Although this bias favors FCC ordering, there is no long-ranged order as correlations still decay exponentially. Finally, we consider Torquato-Stillinger stackings, which map to a combination of an Ising magnet and a three-state Potts model. With random stacking, the correlations decay exponentially with a form that is similar to the Barlow problem. We discuss relevance to ordering in clusters of stacked solids and for layer-deposition-based synthesis methods.

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 ϕMRJ hsc attains a maximum. This work confirms the form of the graph ϕMRJ hsc versus L/D: here, comparably to certain previous investigations, it is found that the maximal ϕMRJ hsc = 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.

Approach to hyperuniformity in a metallic glass-forming material exhibiting a fragile to strong glass transition

We investigate a metallic glass-forming (GF) material (Al₉₀Sm₁₀) exhibiting a fragile-strong (FS) glass-formation by molecular dynamics simulation to better understand this highly distinctive pattern of glass-formation in which many of the usual phenomenological relations describing relaxation times and diffusion of ordinary GF liquids no longer apply, and where instead genuine thermodynamic features are observed in response functions and little thermodynamic signature is exhibited at the glass transition temperature, T_g. Given the many unexpected similarities between the thermodynamics and dynamics of this metallic GF material with water, we first focus on the anomalous static scattering in this liquid, following recent studies on water, silicon and other FS GF liquids. We quantify the "hyperuniformity index" H of our liquid, which provides a quantitative measure of molecular "jamming". To gain insight into the T-dependence and magnitude of H, we also estimate another more familiar measure of particle localization, the Debye-Waller parameter 〈u²〉 describing the mean-square particle displacement on a timescale on the order of the fast relaxation time, and we also calculate H and 〈u²〉 for heated crystalline Cu. This comparative analysis between H and 〈u²〉 for crystalline and metallic glass materials allows us to understand the critical value of H on the order of 10^-3 as being analogous to the Lindemann criterion for both the melting of crystals and the "softening" of glasses. We further interpret the emergence of FS GF and liquid-liquid phase separation in this class of liquids to arise from a cooperative self-assembly process in the GF liquid.

Optimal shapes of disk assembly in saturated random packings

Particle morphology is one of the most significant factors influencing the packing structures of granular materials. With certain targeted properties or optimization criteria, inverse packing problems have drawn extensive attention in terms of their adaptability to many material design tasks. An important question hard to answer is which particle shape, especially within given shape families, forms the densest (loosest) random packing? In this paper, we address this issue for the disk assembly model in two dimensions with an infinite variety of shapes, which are simulated in the random sequential adsorption process to suppress crystallization. Via a unique shape representation method, particle shapes are transformed into genotype sequences in the continuous shape space where we utilize the genetic algorithm as an efficient shape optimizer. Specifically, we consider three representative species of disk assembly, i.e., congruent tangent disks, incongruent tangent disks, and congruent overlapping disks, and carry out shape optimization on their packing densities in the saturated random state. We numerically search optimal shapes in the three species with a variable number of constituent disks which yield the maximal and minimal packing densities. We obtain an isosceles circulo-triangle and an unclosed ring for the maximal and minimal packing density in saturated random packings, respectively. The perfect sno-cone and isosceles circulo-triangle are also specifically investigated which give remarkably high packing densities of around 0.6, much denser than those of ellipses. This study is beneficial for guiding the design of particle shapes as well as the inverse design of granular materials.

Ligand Effects in Assembly of Cubic and Spherical Nanocrystals: Applications to Packing of Perovskite Nanocubes

We establish the formula representing cubic nanocrystals (NCs) as hard cubes taking into account the role of the ligands and describe how these results generalize to any other NC shapes. We derive the conditions under which the hard cube representation breaks down and provide explicit expressions for the effective size. We verify the results from the detailed potential of mean force calculations for two nanocubes in different orientations as well as with spherical nanocrystals. Our results explicitly demonstrate the relevance of certain ligand conformations, i.e., "vortices", and show that edges and corners provide natural sites for their emergence. We also provide both simulations and experimental results with single component cubic perovskite nanocrystals assembled into simple cubic superlattices, which further corroborate theoretical predictions. In this way, we extend the Orbifold Topological Model (OTM) accounting for the role of ligands beyond spherical nanocrystals and discuss its extension to arbitrary nanocrystal shapes. Our results provide detailed predictions for recent superlattices of perovskite nanocubes and spherical nanocrystals. Problems with existing united atom force fields are discussed.

Virial coefficients of hard, homonuclear dumbbells in two- to four-dimensional Euclidean spaces

We calculated virial coefficients up to the eighth order for hard dumbbells in two-, three-, and four-dimensional Euclidean spaces employing Mayer-sampling Monte Carlo simulations. We improved and extended available data in two dimensions, provide virial coefficients in R^{4} in dependence on their aspect ratio, and recalculated virial coefficients for three-dimensional dumbbells. Highly accurate, semianalytical values for the second virial coefficient of homonuclear, four-dimensional dumbbells are provided. We compare the influence of the aspect ratio and the dimensionality to the virial series for this concave geometry. Lower-order reduced virial coefficients B[over ̃]_{i}=B_{i}/B_{2}^{i-1} depend in first approximation linearly from the inverse excess part of their mutual excluded volume.

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.

Dense random packing with a power-law size distribution: The structure factor, mass-radius relation, and pair distribution function

We consider a dense random packing of disks with a power-law distribution of radii and investigate their correlation properties. We study the corresponding structure factor, mass-radius relation, and pair distribution function of the disk centers. A toy model of dense segments in one dimension (1D) is solved exactly. It is shown theoretically in 1D and numerically in 1D and 2D that such a packing exhibits fractal properties. It is found that the exponent of the power-law distribution and the fractal dimension coincide. An approximate relation for the structure factor in arbitrary dimensions is derived, which can be used as a fitting formula in small-angle scattering. These findings can be useful for understanding the microstructural properties of various systems such as ultra-high performance concrete, high-internal-phase-ratio emulsions, or biological systems.

Realizability of iso-g2 processes via effective pair interactions

An outstanding problem in statistical mechanics is the determination of whether prescribed functional forms of the pair correlation function g2( r) [or equivalently, structure factor S( k)] at some number density ρ can be achieved by many-body systems in d-dimensional Euclidean space. The Zhang–Torquato conjecture states that any realizable set of pair statistics, whether from a nonequilibrium or equilibrium system, can be achieved by equilibrium systems involving up to two-body interactions. To further test this conjecture, we study the realizability problem of the nonequilibrium iso- g2 process, i.e., the determination of density-dependent effective potentials that yield equilibrium states in which g2 remains invariant for a positive range of densities. Using a precise inverse algorithm that determines effective potentials that match hypothesized functional forms of g2( r) for all r and S( k) for all k, we show that the unit-step function g2, which is the zero-density limit of the hard-sphere potential, is remarkably realizable up to the packing fraction ϕ = 0.49 for d = 1. For d = 2 and 3, it is realizable up to the maximum “terminal” packing fraction ϕ c = 1/2 d, at which the systems are hyperuniform, implying that the explicitly known necessary conditions for realizability are sufficient up through ϕ c. For ϕ near but below ϕ c, the large- r behaviors of the effective potentials are given exactly by the functional forms exp[ − κ( ϕ) r] for d = 1, r−1/2 exp[ − κ( ϕ) r] for d = 2, and r−1 exp[ − κ( ϕ) r] (Yukawa form) for d = 3, where κ−1( ϕ) is a screening length, and for ϕ = ϕ c, the potentials at large r are given by the pure Coulomb forms in the respective dimensions as predicted by Torquato and Stillinger [Phys. Rev. E 68, 041113 (2003)]. We also find that the effective potential for the pair statistics of the 3D “ghost” random sequential addition at the maximum packing fraction ϕ c = 1/8 is much shorter ranged than that for the 3D unit-step function g2 at ϕ c; thus, it does not constrain the realizability of the unit-step function g2. Our inverse methodology yields effective potentials for realizable targets, and, as expected, it does not reach convergence for a target that is known to be non-realizable, despite the fact that it satisfies all known explicit necessary conditions. Our findings demonstrate that exploring the iso- g2 process via our inverse methodology is an effective and robust means to tackle the realizability problem and is expected to facilitate the design of novel nanoparticle systems with density-dependent effective potentials, including exotic hyperuniform states of matter.

Densest plane group packings of regular polygons

Packings of regular convex polygons (n-gons) that are sufficiently dense have been studied extensively in the context of modeling physical and biological systems as well as discrete and computational geometry. Former results were mainly regarding densest lattice or double-lattice configurations. Here we consider all two-dimensional crystallographic symmetry groups (plane groups) by restricting the configuration space of the general packing problem of congruent copies of a compact subset of the two-dimensional Euclidean space to particular isomorphism classes of the discrete group of isometries. We formulate the plane group packing problem as a nonlinear constrained optimization problem. By means of the Entropic Trust Region Packing Algorithm that approximately solves this problem, we examine some known and unknown densest packings of various n-gons in all 17 plane groups and state conjectures about common symmetries of the densest plane group packings for every n-gon.

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 ϕ_MRJ hr 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 → +∞, ϕ_MRJ hr L/D = (10 + 1)/2, is compared to the available experimental data.

Asymptotics and Summation of the Effective Properties of Suspensions, Simple Liquids and Composites

We review the problem of summation for a very short truncation of a power series by means of special resummation techniques inspired by the field-theoretical renormalization group. Effective viscosity (EV) of active and passive suspensions is studied by means of a special algebraic renormalization approach applied to the first and second-order expansions in volume fractions of particles. EV of the 2D and 3D passive suspensions is analysed by means of various self-similar approximants such as iterated roots, exponential approximants, super-exponential approximants and root approximants. General formulae for all concentrations are derived. A brief introduction to the rheology of micro-swimmers is given. Microscopic expressions for the intrinsic viscosity of the active system of puller-like microswimmers are obtained. Special attention is given to the problem of the calculation of the critical indices and amplitudes of the EV and to the sedimentation rate in the vicinity of known critical points. Critical indices are calculated from the short truncation by means of minimal difference and minimal derivative conditions on the fixed points imposed directly on the critical properties. Accurate expressions are presented for the non-local diffusion coefficient of a simple liquid in the vicinity of a critical point. Extensions and corrections to the celebrated Kawasaki formula are discussed. We also discuss the effective conductivity for the classical analog of graphene and calculate the effective critical index for superconductivity dependent on the concentration of vacancies. Finally, we discuss the effective conductivity of a random 3D composite and calculate the superconductivity critical index of a random 3D composite.

Revised scattering exponents for a power-law distribution of surface and mass fractals

We consider scattering exponents arising in small-angle scattering from power-law polydisperse surface and mass fractals. It is shown that a set of fractals, whose sizes are distributed according to a power law, can change its fractal dimension when the power-law exponent is sufficiently big. As a result, the scattering exponent corresponding to this dimension appears due to the spatial correlations between positions of different fractals. For large values of the momentum transfer, the correlations do not play any role, and the resulting scattering intensity is given by a sum of intensities of all composing fractals. The restrictions imposed on the power-law exponents are found. The obtained results generalize Martin's formulas for the scattering exponents of the polydisperse fractals.

Treating random sequential addition via the replica method

While many physical processes are non-equilibrium in nature, the theory and modeling of such phenomena lag behind theoretical treatments of equilibrium systems. The diversity of powerful theoretical tools available to describe equilibrium systems has inspired strategies that map non-equilibrium systems onto equivalent equilibrium analogs so that interrogation with standard statistical mechanical approaches is possible. In this work, we revisit the mapping from the non-equilibrium random sequential addition process onto an equilibrium multi-component mixture via the replica method, allowing for theoretical predictions of non-equilibrium structural quantities. We validate the above approach by comparing the theoretical predictions to numerical simulations of random sequential addition.

Virial coefficients of hard hyperspherocylinders in R^{4}: Influence of the aspect ratio

We provide second- to sixth-order virial coefficients of hard hyperspherocylinders in dependence on their aspect ratio ν. Virial coefficients of an anisotropic geometry in four dimensions are calculated employing an optimized Mayer-sampling algorithm. As the second virial coefficient of a hard particle is identical to its excluded hypervolume, the numerically obtained second virial coefficients can be compared to analytical relations for the excluded hypervolume based on geometric measures of the respective, convex geometry in dependence on its aspect ratio ν.

Structural evolution of granular cubes packing during shear-induced ordering

Packings of granular particles may transform into ordered structures under external agitation, which is a special type of out-of-equilibrium self-assembly. Here, evolution of the internal packing structures of granular cubes under cyclic rotating shearing has been analyzed using magnetic resonance imaging techniques. Various order parameters, different types of contacts and clusters composed of face-contacting cubes, as well as the free volume regions in which each cube can move freely have been analyzed systematically to quantify the ordering process and the underlying mechanism of this granular self-assembly. The compaction process is featured by a first rapid formation of orientationally ordered local structures with faceted contacts, followed by further densification driven by free-volume maximization with an almost saturated degree of order. The ordered structures are strongly anisotropic with contacting ordered layers in the vertical direction while remaining liquid-like in the horizontal directions. Therefore, the constraint of mechanical stability for granular packings and the thermodynamic principle of entropy maximization are both effective in this system, which we propose can be reconciled by considering different depths of supercooling associated with various degrees of freedom.

Molecular dynamics investigation of a one-component model for the stacking motif in complex alloy structures

Developing realistic three-dimensional growth models for quasicrystals is a fundamental requirement. The present work employs classical molecular dynamics simulations to investigate the adsorption of Al on a close-packed Al layer containing atomic vacancies. Simulation results show that the adsorbed Al atoms are located preferentially above and below the atomic vacancies in the close-packed layer, and the results obtained from a one-component system of atoms interacting via an interatomic pair potential for Al–Al appropriately reproduce the stacking motif seen in complex alloys such as the μ-Al4Mn phase. The simulations also reveal the formation of a deformed icosahedron. These results provide new insights into the growth mechanism and origin of complex alloys and quasicrystals.

Diffusion spreadability as a probe of the microstructure of complex media across length scales

Understanding time-dependent diffusion processes in multiphase media is of great importance in physics, chemistry, materials science, petroleum engineering, and biology. Consider the time-dependent problem of mass transfer of a solute between two phases and assume that the solute is initially distributed in one phase (phase 2) and absent from the other (phase 1). We desire the fraction of total solute present in phase 1 as a function of time, S(t), which we call the spreadability, since it is a measure of the spreadability of diffusion information as a function of time. We derive exact direct-space formulas for S(t) in any Euclidean space dimension d in terms of the autocovariance function as well as corresponding Fourier representations of S(t) in terms of the spectral density, which are especially useful when scattering information is available experimentally or theoretically. These are singular results because they are rare examples of mass transport problems where exact solutions are possible. We derive closed-form general formulas for the short- and long-time behaviors of the spreadability in terms of crucial small- and large-scale microstructural information, respectively. The long-time behavior of S(t) enables one to distinguish the entire spectrum of microstructures that span from hyperuniform to nonhyperuniform media. For hyperuniform media, disordered or not, we show that the "excess" spreadability, S(∞)-S(t), decays to its long-time behavior exponentially faster than that of any nonhyperuniform two-phase medium, the "slowest" being antihyperuniform media. The stealthy hyperuniform class is characterized by an excess spreadability with the fastest decay rate among all translationally invariant microstructures. We obtain exact results for S(t) for a variety of specific ordered and disordered model microstructures across dimensions that span from hyperuniform to antihyperuniform media. Moreover, we establish a remarkable connection between the spreadability and an outstanding problem in discrete geometry, namely, microstructures with "fast" spreadabilities are also those that can be derived from efficient "coverings" of space. We also identify heretofore unnoticed, to our best knowledge, remarkable links between the spreadability S(t) and NMR pulsed field gradient spin-echo amplitude as well as diffusion MRI measurements. This investigation reveals that the time-dependent spreadability is a powerful, dynamic-based figure of merit to probe and classify the spectrum of possible microstructures of two-phase media across length scales.

Phase behavior of hard circular arcs

By using Monte Carlo numerical simulation, this work investigates the phase behavior of systems of hard infinitesimally thin circular arcs, from an aperture angle θ→0 to an aperture angle θ→2π, in the two-dimensional Euclidean space. Except in the isotropic phase at lower density and in the (quasi)nematic phase, in the other phases that form, including the isotropic phase at higher density, hard infinitesimally thin circular arcs autoassemble to form clusters. These clusters are either filamentous, for smaller values of θ, or roundish, for larger values of θ. Provided the density is sufficiently high, the filaments lengthen, merge, and straighten to finally produce a filamentary phase while the roundels compact and dispose themselves with their centers of mass at the sites of a triangular lattice to finally produce a cluster hexagonal phase.

Inherent structure landscape of hard spheres confined to narrow cylindrical channels

The inherent structure landscape for a system of hard spheres confined to a hard cylindrical channel, such that spheres can only contact their first and second neighbors, is studied using an analytical model that extends previous results [Phys. Rev. Lett. 115, 025702 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.025702] to provide a comprehensive picture of jammed packings over a range of packing densities. In the model, a packing is described as an arrangement of k helical sections, separated by defects, that have alternating helical twist directions and where all spheres satisfy local jamming constraints. The structure of each helical section is determined by a single helical twist angle, and a jammed packing is obtained by minimizing the length of the channel per particle with respect to the k helical section angles. An analysis of a small system of N=20 spheres shows that the basins on the inherent structure landscape associated with these helical arrangements split into a number of distinct jammed states separated by low barriers giving rise to a degree of hierarchical organization. The model accurately predicts the geometric properties of packings generated using the Lubachevsky and Stillinger compression scheme (N=10^{4}) and provides insight into the nature of the probability distribution of helical section lengths.

Random sequential adsorption of oriented rectangles with random aspect ratio

We studied random sequential adsorption (RSA) of parallel rectangles with random aspect ratio but fixed area using a newly developed algorithm that allows to generate strictly saturated packing of this kind. We determined saturated packing fraction for several different distributions of a random variable used for selecting side length ratio of deposited rectangles. It was also shown that the anisotropy of deposited rectangles changes during packing generation. Additionally, we examined the kinetics of packing growth, which near saturation obeys the power law with the exponent 1/d≈1/3, typical for the RSA of unoriented anisotropic shapes on a two-dimensional surface. Kinetics in the low coverage limit is determined using the concept of the available surface function. The microstructural properties of obtained random packings are evaluated in terms of two-point density correlation function.

Molecular dynamics study of six-dimensional hard hypersphere crystals

Six-dimensional hard hypersphere systems in the A₆, D₆, and E₆ crystalline phases have been studied using event-driven molecular dynamics simulations in periodic, skew cells that reflect the underlying lattices. In all the simulations, the systems had sufficient numbers of hyperspheres to capture the first coordination shells, and the larger simulations also included the complete second coordination shell. The equations of state, for densities spanning the fluid, metastable fluid, and solid regimes, were determined. Using molecular dynamics simulations with the hyperspheres tethered to lattice sites allowed the computation of the free energy for each of the crystal lattices relative to the fluid phase. From these free energies, the fluid-crystal coexistence region was determined for the E₆, D₆, and A₆ lattices. Pair correlation functions for all the examined states were computed. Interestingly, for all the states examined, the pair correlation functions displayed neither a split second peak nor a shoulder in the second peak. These behaviors have been previously used as a signature of the freezing of the fluid phase for hard hyperspheres in two to five dimensions.

Non-hyperuniform metastable states around a disordered hyperuniform state of densely packed spheres: stochastic density functional theory at strong coupling

The disordered and hyperuniform structures of densely packed spheres near and at jamming are characterized by vanishing of long-wavelength density fluctuations, or equivalently by long-range power-law decay of the direct correlation function (DCF). We focus on previous simulation results that exhibit the degradation of hyperuniformity in jammed structures while maintaining the long-range nature of the DCF to a certain length scale. Here we demonstrate that the field-theoretic formulation of stochastic density functional theory is relevant to explore the degradation mechanism. The strong-coupling expansion method of stochastic density functional theory is developed to obtain the metastable chemical potential considering the intermittent fluctuations in dense packings. The metastable chemical potential yields the analytical form of the metastable DCF that has a short-range cutoff inside the sphere while retaining the long-range power-law behavior. It is confirmed that the metastable DCF provides the zero-wavevector limit of the structure factor in quantitative agreement with the previous simulation results of degraded hyperuniformity. We can also predict the emergence of soft modes localized at the particle scale by plugging this metastable DCF into the linearized Dean-Kawasaki equation, a stochastic density functional equation.

Critical pore radius and transport properties of disordered hard- and overlapping-sphere models

Transport properties of porous media are intimately linked to their pore-space microstructures. We quantify geometrical and topological descriptors of the pore space of certain disordered and ordered distributions of spheres, including pore-size functions and the critical pore radius δ_{c}. We focus on models of porous media derived from maximally random jammed sphere packings, overlapping spheres, equilibrium hard spheres, quantizer sphere packings, and crystalline sphere packings. For precise estimates of the percolation thresholds, we use a strict relation of the void percolation around sphere configurations to weighted bond percolation on the corresponding Voronoi networks. We use the Newman-Ziff algorithm to determine the percolation threshold using universal properties of the cluster size distribution. The critical pore radius δ_{c} is often used as the key characteristic length scale that determines the fluid permeability k. A recent study [Torquato, Adv. Wat. Resour. 140, 103565 (2020)10.1016/j.advwatres.2020.103565] suggested for porous media with a well-connected pore space an alternative estimate of k based on the second moment of the pore size 〈δ^{2}〉, which is easier to determine than δ_{c}. Here, we compare δ_{c} to the second moment of the pore size 〈δ^{2}〉, and indeed confirm that, for all porosities and all models considered, δ_{c}^{2} is to a good approximation proportional to 〈δ^{2}〉. However, unlike 〈δ^{2}〉, the permeability estimate based on δ_{c}^{2} does not predict the correct ranking of k for our models. Thus, we confirm 〈δ^{2}〉 to be a promising candidate for convenient and reliable estimates of the fluid permeability for porous media with a well-connected pore space. Moreover, we compare the fluid permeability of our models with varying degrees of order, as measured by the τ order metric. We find that (effectively) hyperuniform models tend to have lower values of k than their nonhyperuniform counterparts. Our findings could facilitate the design of porous media with desirable transport properties via targeted pore statistics.

УЩІЛЬНЕННЯ (КОМПАКТИЗАЦІЯ) ВПАКУВАННЯ У БІ-КОМПОНЕНТНІЙ МІКРОМЕХАНІЧНІЙ (ГРАНУЛЬОВАНІЙ) СУМІШІ

Вступ. Одна з традиційно актуальних проблем теоретичного базису виробництва і технологій - це опис, параметризація та прогнозування властивостей суміші залежно від параметрів компонентів. Однією із найсуттєвіших проблем,які заважають ефективному використанню гранульованих матеріалів, наприклад у будівельній промисловості, є складність забезпечення їх максимального ущільнення для підвищення ефективності практичного застосування.Проблематика. Розуміння принципів, завдяки яким формуються основні параметри багатокомпонентних систем спирається на базові моделі, які дозволяють параметризувати дані вимірів в термінах величин, що характеризують окремі чисті компоненти (reference data). Побудова таких моделей є складною задачею та вимагає феноменологічної інформації із декількох альтернативних джерел.Мета. Спираючись на апарат теорії Кірквуда-Баффа, модельні рівняння стану та данні аналізу експериментальних даних з вивчення макроскопічних параметрів бідисперсної мікромеханічної суміші побудувати теоретичний алгоритм опису та параметризації їх фізико-механічних характеристик в термінах зв’язків макроскопічних та парціальних властивостей залежно від об’ємної (або молярної) фракції одного з компонентів.Матеріали й методи. Моделі гранульованих бікомпонентних сумішей; теорія Кірквуда-Баффа; модельні рівняння стану для модельних сумішей твердих кульок типу Карнахана-Старлінга; феноменологічна інформація про динаміку ущільнення простих гранульованих сумішей.Результати. За допомогою теорії Кірквуда-Баффа, модельних співвідношень для сумішей твердих кульок, із використанням феноменологічних даних про характер ущільнення гранульованих матеріалів, розроблено алгоритм для опису макроскопічних властивостей бінарних гранульованих систем зокрема компактизації.Висновки. Отримані дані підтверджують наявність впливу мультидисперсності на динаміку ущільнення тобто, на можливість суміші під дією зовнішніх впливів прогнозовано змінювати локальну структуру впакування та її параметри. 

Algorithms to generate saturated random sequential adsorption packings built of rounded polygons

We present the algorithm for generating strictly saturated random sequential adsorption packings built of rounded polygons. It can be used in studying various properties of such packings built of a wide variety of different shapes, and in modeling monolayers obtained during irreversible adsorption processes of complex molecules. Here, we apply the algorithm to study the densities of packings built of rounded regular polygons. Contrary to packings built of regular polygons, where the packing fraction grows with an increasing number of polygon sides, here the packing fraction reaches its maximum for packings built of rounded regular triangles. With a growing number of polygon sides and increasing rounding radius, the packing fractions tend to the limit given by a packing built of disks. However, they are still slightly higher, even for the rounded 25-gon, which is the highest-sided regular polygon studied here.

Optimized Factor Approximants and Critical Index

Based on expansions with only two coefficients and known critical points, we consider a minimal model of critical phenomena. The method of analysis is both based on and inspired with the symmetry properties of functional self-similarity relation between the consecutive functional approximations. Factor approximants are applied together with various natural optimization conditions of non-perturbative nature. The role of control parameter is played by the critical index by itself. The minimal derivative condition imposed on critical amplitude appears to bring the most reasonable, uniquely defined results. The minimal difference condition also imposed on amplitudes produces upper and lower bound on the critical index. While one of the bounds is close to the result from the minimal difference condition, the second bound is determined by the non-optimized factor approximant. One would expect that for the minimal derivative condition to work well, the bounds determined by the minimal difference condition should be not too wide. In this sense the technique of optimization presented above is self-consistent, since it automatically supplies the solution and the bounds. In the case of effective viscosity of passive suspensions the bounds could be found that are too wide to make any sense from either of the solutions. Other optimization conditions imposed on the factor approximants, lead to better estimates for the critical index for the effective viscosity. The optimization is based on equating two explicit expressions following from two different definitions of the critical index, while optimization parameter is introduced as the trial third-order coefficient in the expansion.

Elongation and percolation of defect motifs in anisotropic packing problems

We examine the regime between crystalline and amorphous packings of anisotropic objects on surfaces of different genus by continuously varying their size distribution or shape from monodispersed spheres to bidispersed mixtures or monodispersed ellipsoidal particles; we also consider an anisotropic variant of the Thomson problem with a mixture of charges. With increasing anisotropy, we first observe the disruption of translational order with an intermediate orientationally ordered hexatic phase as proposed by Nelson, Rubinstein and Spaepen, and then a transition to amorphous state. By analyzing the structure of the disclination motifs induced, we show that the hexatic-amorphous transition is caused by the growth and connection of disclination grain boundaries, suggesting this transition lies in the percolation universality class in the scenarios considered.

Hidden Order Beyond Hyperuniformity in Critical Absorbing States

Disordered hyperuniformity is a description of hidden correlations in point distributions revealed by an anomalous suppression in fluctuations of local density at various coarse-graining length scales. In the absorbing phase of models exhibiting an active-absorbing state transition, this suppression extends up to a hyperuniform length scale that diverges at the critical point. Here, we demonstrate the existence of additional many-body correlations beyond hyperuniformity. These correlations are hidden in the higher moments of the probability distribution of the local density and extend up to a longer length scale with a faster divergence than the hyperuniform length on approaching the critical point. Our results suggest that a hidden order beyond hyperuniformity may generically be present in complex disordered systems.

Kinetic Frustration Effects on Dense Two-Dimensional Packings of Convex Particles and Their Structural Characteristics

The study of hard-particle packings is of fundamental importance in physics, chemistry, cell biology, and discrete geometry. Much of the previous work on hard-particle packings concerns their densest possible arrangements. By contrast, we examine kinetic effects inevitably present in both numerical and experimental packing protocols. Specifically, we determine how changing the compression/shear rate of a two-dimensional packing of noncircular particles causes it to deviate from its densest possible configuration, which is always periodic. The adaptive shrinking cell (ASC) optimization scheme maximizes the packing fraction of a hard-particle packing by first applying random translations and rotations to the particles and then isotropically compressing and shearing the simulation box repeatedly until a possibly jammed state is reached. We use a stochastic implementation of the ASC optimization scheme to mimic different effective time scales by varying the number of particle moves between compressions/shears. We generate dense, effectively jammed, monodisperse, two-dimensional packings of obtuse scalene triangle, rhombus, curved triangle, lens, and "ice cream cone" (a semicircle grafted onto an isosceles triangle) shaped particles, with a wide range of packing fractions and degrees of order. To quantify these kinetic effects, we introduce the kinetic frustration index K, which measures the deviation of a packing from its maximum possible packing fraction. To investigate how kinetics affect short- and long-range ordering in these packings, we compute their spectral densities χ̃_V(k) and characterize their contact networks. We find that kinetic effects are most significant when the particles have greater asphericity, less curvature, and less rotational symmetry. This work may be relevant to the design of laboratory packing protocols.

A sphere packing model for shear bands in dense soils

The rhombic sphere packing can be used to model the biaxial test on granular soils in a very simple way. According to the angle of assemblage, the packing is dilatant or contractive. Correspondingly, overall stresses are transmitted as chains of forces or oblique forces of contact. The connection of the soil stress-strain behaviour and the packing void ratio is achieved by mapping both of the plots. The mapping shows that dense soils are dilatant and loose soils are contractive, separated by the critical state. It also shows that the bifurcation point and the peak strength are features only of dense soils. The band of strain localization is analysed in the elastic regime, and its inclination is found maximizing the intensity of the mobilized stress ratio. The stresses within the shear band are obtained by assuming a partially coaxial packing rotated to reach the full plastic state. The equilibrium of the overall stress at the line of discontinuity reveals a relationship between the peak friction angle and the coefficient of lateral pressure at rest. As long as these parameters are obtained independently of each other, they allow the validation of the theory.

Percus–Yevick structure factors made simple

Measuring the structure factor,S(q), of a dispersion of particles by small-angle X-ray scattering provides a unique method to investigate the spatial arrangement of colloidal particles. However, it is impossible to find the exact location of the particles fromS(q) because some information is inherently lacking in the measured signal. The two standard ways to analyse an experimentalS(q) are then to compare it either with structure factors computed from simulated systems or with analytical ones calculated from approximated systems. However, such approaches may prove inadequate for dispersions of variously polydisperse particles. While Vrij, Bloom and Stell established a mean-field approach that could yield fairly accurate approximations for experimentalS(q), this solution has remained underused because of its mathematical complexity. In the present work, the complete Percus–Yevick solution for general polydisperse hard-sphere systems is derived in a concise form that is straightforward to use. The form of the solution has been simplified enough to provide experimentalists with ready solutions of several commonly encountered particle-radius distributions in real systems (Schulz, truncated normal and inverse Gaussian). The approach is also illustrated with a case study of the exponential radius distribution. Finally, the application of the proposed solution to the power-law radius distribution is discussed in detail by comparing the calculations with experimentally measuredS(q) for an Apollonian packing of spherical droplets recently reported in high-internal-phase-ratio emulsions.

Dense packings of hard circular arcs

This work investigates dense packings of congruent hard infinitesimally thin circular arcs in the two-dimensional Euclidean space. It focuses on those denotable as major whose subtended angle θ∈(π,2π]. Differently than those denotable as minor whose subtended angle θ∈[0,π], it is impossible for two hard infinitesimally thin circular arcs with θ∈(π,2π] to arbitrarily closely approach once they are arranged in a configuration, e.g., on top of one another, replicable ad infinitum without introducing any overlap. This makes these hard concave particles, in spite of being infinitesimally thin, most densely pack with a finite number density. This raises the question as to what are these densest packings and what is the number density that they achieve. Supported by Monte Carlo numerical simulations, this work shows that one can analytically construct compact closed circular groups of hard major circular arcs in which a specific, θ-dependent, number of them (counter) clockwise intertwine. These compact closed circular groups then arrange on a triangular lattice. These analytically constructed densest-known packings are compared to corresponding results of Monte Carlo numerical simulations to assess whether they can spontaneously turn up.

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.