Precise algorithm to generate random sequential addition of hard hyperspheres at saturation

Structural properties of hyperuniform Voronoi networks

Disordered hyperuniform many-particle systems are recently discovered exotic states of matter, characterized by the complete suppression of normalized infinite-wavelength density fluctuations, as in perfect crystals, while lacking conventional long-range order, as in liquids and glasses. In this work, we begin a program to quantify the structural properties of nonhyperuniform and hyperuniform networks. In particular, large two-dimensional (2D) Voronoi networks (graphs) containing approximately 10,000 nodes are created from a variety of different point configurations, including the antihyperuniform hyperplane intersection process (HIP), nonhyperuniform Poisson process, nonhyperuniform random sequential addition (RSA) saturated packing, and both non-stealthy and stealthy hyperuniform point processes. We carry out an extensive study of the Voronoi-cell area distribution of each of the networks by determining multiple metrics that characterize the distribution, including their average areas and corresponding variances as well as higher-order cumulants (i.e., skewness γ_{1} and excess kurtosis γ_{2}). We show that the HIP distribution is far from Gaussian, as evidenced by a high skewness (γ_{1}=3.16) and large positive excess kurtosis (γ_{2}=16.2). The Poisson (with γ_{1}=1.07 and γ_{2}=1.79) and non-stealthy hyperuniform (with γ_{1}=0.257 and γ_{2}=0.0217) distributions are Gaussian-like distributions, since they exhibit a small but positive skewness and excess kurtosis. The RSA (with γ_{1}=0.450 and γ_{2}=-0.0384) and the highest stealthy hyperuniform distributions (with γ_{1}=0.0272 and γ_{2}=-0.0626) are also non-Gaussian because of their low skewness and negative excess kurtosis, which is diametrically opposite of the non-Gaussian behavior of the HIP. The fact that the cell-area distributions of large, finite-sized RSA and stealthy hyperuniform networks (e.g., with N≈10,000 nodes) are narrower, have larger peaks, and smaller tails than a Gaussian distribution implies that in the thermodynamic limit the distributions should exhibit compact support, consistent with previous theoretical considerations. Moreover, we compute the Voronoi-area correlation functions C_{00}(r) for the networks, which describe the correlations between the area of two Voronoi cells separated by a given distance r [M. A. Klatt and S. Torquato, Phys. Rev. E 90, 052120 (2014)1539-375510.1103/PhysRevE.90.052120]. We show that the correlation functions C_{00}(r) qualitatively distinguish the antihyperuniform, nonhyperuniform, and hyperuniform Voronoi networks considered here. Specifically, the antihyperuniform HIP networks possess a slowly decaying C_{00}(r) with large positive values, indicating large fluctuations of Voronoi cell areas across scales. While the nonhyperuniform Poisson and RSA network possess positive and fast decaying C_{00}(r), we find strong anticorrelations in C_{00}(r) (i.e., negative values) for the hyperuniform networks. The latter indicates that the large-scale area fluctuations are suppressed by accompanying large Voronoi cells with small cells (and vice versa) in the systems in order to achieve hyperuniformity. In summary, we have shown that cell-area distributions and pair correlation functions of Voronoi networks enable one to distinguish quantitatively antihyperuniform, standard nonhyperuniform, and hyperuniform networks from one another.

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.

Percolation thresholds of disks with random nonoverlapping patches on four regular two-dimensional lattices

In percolation of patchy disks on lattices, each site is occupied by a disk, and neighboring disks are regarded as connected when their patches contact. Clusters of connected disks become larger as the patchy coverage of each disk χ increases. At the percolation threshold χ_{c}, an incipient cluster begins to span the whole lattice. For systems of disks with n symmetric patches on Archimedean lattices, a recent work [Wang et al., Phys. Rev. E 105, 034118 (2022)2470-004510.1103/PhysRevE.105.034118] found symmetric properties of χ_{c}(n), which are due to the coupling of the patches' symmetry and the lattice geometry. How does χ_{c} behave with increasing n if the patches are randomly distributed on the disks? We consider two typical random distributions of the patches, i.e., the equilibrium distribution and a distribution from random sequential adsorption. Combining Monte Carlo simulations and the critical polynomial method, we numerically determine χ_{c} for 106 models of different n on the square, honeycomb, triangular, and kagome lattices. The rules governing χ_{c}(n) are investigated in detail. They are quite different from those for disks with symmetric patches and could be useful for understanding similar systems.

Properties of Packed Bed Structures Formed during Filtration: A Two and Three-Dimensional Model

Agglomeration is an issue that causes many problems during secondary processing for pharmaceutical companies, causing material to need further processing and costing additional time and resources to ensure a satisfactory outcome. A potential source of agglomeration arises from the particle contacts established during filtration that lead to robust agglomerates forming during drying, so that a necessary first step toward understanding agglomeration is to study the packing properties of filtration beds. Here, we present two and three-dimensional models simulating the formation of packed bed structures during filtration. The models use circular and spherical particles of different sizes, mimicking the bimodal particle size distributions sometimes encountered in industrial practice. The statistics of packing and void formation, along with the distribution of interparticle contacts and percolation structures, are presented and discussed in the context of filtration, drying, and agglomeration. The model paves the way for predictive capabilities that can lead to the rational design of processes to minimize the impact of agglomeration.

Packing spheres in high dimensions with moderate computational effort

We generate nonlattice packings of spheres in up to 22 dimensions using the geometrical constraint satisfaction algorithm RRR. Our aggregated data suggest that it is easy to double the density of Ball's lower bound and, more tentatively, that the exponential decay rate of the density can be improved relative to Minkowski's longstanding 1/2.

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.

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.

Structure of saturated random-sequential-adsorption ellipse packings

Motivated by the recent observation of liquid glass in suspensions of ellipsoidal colloids, we examine the structure of (asymptotically) saturated RSA ellipse packings. We determine the packing fractions ϕ_{s}(α) to high precision, finding an empirical analytic formula that predicts ϕ_{s}(α) to within less than 0.1% for all α≤10. Then we explore how these packings' positional-orientational order varies with α. We find a transition from tip/side- to side/side-contact-dominated structure at α=α_{TS}≃2.4. At this aspect ratio, the peak value g_{max} of packings' positional-orientational pair correlation functions is minimal, and systems can be considered maximally locally disordered. For smaller (larger) α, g_{max} increases exponentially with deceasing (increasing) α. Local nematic order and structures comparable to the precursor domains observed in experiments gradually emerge as α increases beyond three. For α≳5, single-layer lamellae become more prominent and long-wavelength density fluctuations increase with α as packings gradually approach the rodlike limit.

Precise determination of pair interactions from pair statistics of many-body systems in and out of equilibrium

The determination of the pair potential v(r) that accurately yields an equilibrium state at positive temperature T with a prescribed pair correlation function g_{2}(r) or corresponding structure factor S(k) in d-dimensional Euclidean space R^{d} is an outstanding inverse statistical mechanics problem with far-reaching implications. Recently, Zhang and Torquato [Phys. Rev. E 101, 032124 (2020)2470-004510.1103/PhysRevE.101.032124] conjectured that any realizable g_{2}(r) or S(k) corresponding to a translationally invariant nonequilibrium system can be attained by a classical equilibrium ensemble involving only (up to) effective pair interactions. Testing this conjecture for nonequilibrium systems as well as for nontrivial equilibrium states requires improved inverse methodologies. We have devised an optimization algorithm to precisely determine effective pair potentials that correspond to pair statistics of general translationally invariant disordered many-body equilibrium or nonequilibrium systems at positive temperatures. This methodology utilizes a parameterized family of pointwise basis functions for the potential function whose initial form is informed by small-, intermediate- and large-distance behaviors dictated by statistical-mechanical theory. Subsequently, a nonlinear optimization technique is utilized to minimize an objective function that incorporates both the target pair correlation function g_{2}(r) and structure factor S(k) so that the small intermediate- and large-distance correlations are very accurately captured. To illustrate the versatility and power of our methodology, we accurately determine the effective pair interactions of the following four diverse target systems: (1) Lennard-Jones system in the vicinity of its critical point, (2) liquid under the Dzugutov potential, (3) nonequilibrium random sequential addition packing, and (4) a nonequilibrium hyperuniform "cloaked" uniformly randomized lattice. We found that the optimized pair potentials generate corresponding pair statistics that accurately match their corresponding targets with total L_{2}-norm errors that are an order of magnitude smaller than that of previous methods. The results of our investigation lend further support to the Zhang-Torquato conjecture. Furthermore, our algorithm will enable one to probe systems with identical pair statistics but different higher-body statistics, which will shed light on the well-known degeneracy problem of statistical mechanics.

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.

Confinement effects on the random sequential adsorption packings of elongated particles in a slit

The behavior of a system of two-dimensional elongated particles (discorectangles) packed in a slit between the two parallel walls was analyzed using a simulation approach. The packings were produced using the random sequential adsorption model with continuous positional and orientational degrees of freedom. The aspect ratio (length-to-width ratio, ɛ=l/d) of the particles was varied within the range ɛ∈[1;32] while the distance between the walls was varied within the range h/d∈[1;80]. The properties of deposits in jammed state [the coverage, the order parameter, and the long-range (percolation) connectivity between particles] were studied numerically. The values of ɛ and h significantly affected the structure of the packings and the percolation connectivity. Particularly, the observed nontrivial dependencies of the jamming coverage φ(ɛ) or φ(h) were explained by the interplay of the different geometrical factors related to confinement, particle orientation degrees of freedom and excluded volume effects.

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.

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.

All-Nanoparticle Monolayer Broadband Antireflective and Self-Cleaning Transparent Glass Coatings

The vast majority of light-emitting diode and liquid-crystal displays, solar panels, and windows in residential and industrial buildings use glass panels owing to their high mechanical stability, chemical resistance, and optical properties. Glass surfaces reflect about 4-5% of incident light if no antireflective coating is applied. In addition to energy losses in displays, surface reflections diminish picture quality. Engineering of antireflective coatings can be beneficial for all types of glass screens, specifically for large screens and touch-screen devices when scratch-resistance and self-cleaning properties of the glass surface are also desired. A scalable and robust approach to produce antireflective coatings for glass surfaces with desired optical and mechanical properties is introduced in this work. The developed coating mimics the structure of a moth-eye cornea. The coating is a subwavelength-microstructured thin layer on the glass surface made of a monolayer of hemispherical silica nanoparticles obtained by hydrothermal fusion of spherical particles to the glass substrate. The sequence of the particle deposition in the layer-by-layer process is adjusted to balance attractive-repulsive interactions among nanoparticles and between the nanoparticles and the glass surface to generate coatings with a high surface coverage of up to 70%, which exceeds the 54.7% limit of the random sequential addition model. This level of surface coverage allows for a combination of properties beneficial for the described applications: (i) an average reflectance of 0.5 ± 0.2% for a visible and near-infrared optical spectrum, (ii) an improved mechanical stability and scratch resistance, and (iii) non-wetting behavior.

AN IMPROVED DISTINCT ELEMENT METHOD FOR HIGH PACKING FRACTION STOCHASTIC MEDIA MODELING

Due to the generality and flexibility of Monte Carlo method in geometric modeling, Monte Carlo method plays an important role in accurate simulation of random media. At present, rand om sequential addition method (RSA) and distinct element method (DEM) are more accurate and mature explicit modeling methods. The former approach has the problem of upper limit of packing fraction, which is suitable for stochastic geometry with lower filling rate. DEM method can fill random medium model with packing fraction higher than 60%, but DEM is not suitable for non-contact dispersed particles based on the interaction between particles. There fore, an improved DEM method is proposed to solve the problem of modeling non-contact p articles dispersed in the stochastic media with high packing fraction. The virtual surfaces are constructed outside of the outer layer of particles to make them in contact with each other. Thus, the particle system is suitable for DEM method. The construction of virtual surface does not affect the neutron transport process. The correctness of the improved DEM is verified by comparing the total filling particle number and calculation results ofkeffwith RSA method. At the same time, according to the distribution of filling particles, the improved DEM method fills the particles more uniformly.

Nonlinear dynamical modeling of adsorption and desorption processes with power-law kinetics: Application to CO_{2} capture

Modeling of random sequential adsorption (RSA) process is studied in this paper as this kind of process is close to the surface adsorption phenomenon that is, for instance, exploited in gas sensors or for liquid or gas purification. Analysis and simulation of the RSA process is first performed to highlight a power-law kinetic behavior. Such behaviors are often modeled in the literature with fractional models. The paper, however, shows that fractional models are not able to capture some important properties of the RSA process. A nonlinear model and the associated parameters tuning method are, thus, proposed. A discussion on the ability of the proposed model to capture the power-law kinetics without exhibiting some of the drawbacks of fractional models is proposed. This nonlinear model is then modified to take into account the reverse desorption process. The proposed modeling approach is applied to experimental data of CO_{2} capture.

Yuste S. Equation of State of Four- and Five-Dimensional Hard-Hypersphere Mixtures

New proposals for the equation of state of four- and five-dimensional hard-hypersphere mixtures in terms of the equation of state of the corresponding monocomponent hard-hypersphere fluid are introduced. Such proposals (which are constructed in such a way so as to yield the exact third virial coefficient) extend, on the one hand, recent similar formulations for hard-disk and (three-dimensional) hard-sphere mixtures and, on the other hand, two of our previous proposals also linking the mixture equation of state and the one of the monocomponent fluid but unable to reproduce the exact third virial coefficient. The old and new proposals are tested by comparison with published molecular dynamics and Monte Carlo simulation results and their relative merit is evaluated.

Kinetics of random sequential adsorption of two-dimensional shapes on a one-dimensional line

Saturated random sequential adsorption packings built of two-dimensional ellipses, spherocylinders, rectangles, and dimers placed on a one-dimensional line are studied to check analytical prediction concerning packing growth kinetics [A. Baule, Phys. Rev. Lett. 119, 028003 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.028003]. The results show that the kinetics is governed by the power law with the exponent d=1.5 and 2.0 for packings built of ellipses and rectangles, respectively, which is consistent with analytical predictions. However, for spherocylinders and dimers of moderate width-to-height ratio, a transition between these two values is observed. We argue that this transition is a finite-size effect that arises for spherocylinders due to the properties of the contact function. In general, it appears that the kinetics of packing growth can depend on packing size even for very large packings.

Saturated random packing built of arbitrary polygons under random sequential adsorption protocol

Random packings and their properties are a popular and active field of research. Numerical algorithms that can efficiently generate them are useful tools in their study. This paper focuses on random packings produced according to the random sequential adsorption (RSA) protocol. Developing the idea presented by G. Zhang [Phys. Rev. E 97, 043311 (2018)2470-004510.1103/PhysRevE.97.043311], where saturated random packings built of regular polygons were studied, we create an algorithm that generates strictly saturated packings built of any polygons. Then, the algorithm was used to determine the packing fractions for arbitrary triangles. The highest mean packing density, 0.552814±0.000063, was observed for triangles of side lengths 0.63:1:1. Additionally, microstructural properties of such packings, kinetics of their growth, as well as distributions of saturated packing fractions and the number of RSA iterations needed to reach saturation were analyzed.

Random sequential adsorption of particles with tetrahedral symmetry

We study random sequential adsorption (RSA) of a class of solids that can be obtained from a cube by specific cutting of its vertices, in order to find out how the transition from tetrahedral to octahedral symmetry affects the densities of the resulting jammed packings. We find that in general solids of octahedral symmetry form less dense packing; however, the lowest density was obtained for the packing built of tetrahedra. The densest packing is formed by a solid close to a tetrahedron but with vertices and edges slightly cut. Its density is θ_{max}=0.41278±0.00059 and is higher than the mean packing fraction of spheres or cuboids but is lower than that for the densest RSA packings built of ellipsoids or spherocylinders. The density autocorrelation function of the studied packings is typical for random media and vanishes very quickly with distance.

Morphology and kinetics of random sequential adsorption of superballs: From hexapods to cubes

Superballs represent a class of particles whose shapes are defined by the domain |x|^{2p}+|y|^{2p}+|z|^{2p}≤R^{2p}, with p∈(0,∞) being the deformation parameter. 0<p<0.5 represents a family of hexapodlike (concave octahedral-like) particles, 0.5≤p<1 and p>1 represent, respectively, families of convex octahedral-like and cubelike particles, with p=1,0.5, and ∞ representing spheres, octahedra, and cubes. Colloidal zeolite suspensions, catalysis, and adsorption, as well as biomedical magnetic nanoparticles are but a few of the applications of packing of superballs. We introduce a universal method for simulating random sequential adsorption of superballs, which we refer to as the low-entropy algorithm, which is about two orders of magnitude faster than the conventional algorithms that represent high-entropy methods. The two algorithms yield, respectively, precise estimates of the jamming fraction ϕ_{∞}(p) and ν(p), the exponent that characterizes the kinetics of adsorption at long times t, ϕ_{∞}(p)-ϕ(p,t)∼t^{-ν(p)}. Precise estimates of ϕ_{∞}(p) and ν(p) are obtained and shown to be in agreement with the existing analytical and numerical results for certain types of superballs.

Random sequential adsorption of Platonic and Archimedean solids

The aim of this study is the analysis of packings generated according to random sequential adsorption protocol consisting of identical Platonic and Archimedean solids. The computer simulations performed showed that the highest saturated packing fraction θ=0.40210(68) is reached by packings build of truncated tetrahedra and the smallest one θ=0.35635(67) by packings composed of regular tetrahedra. The propagation of translational and orientational order exhibited microstructural properties typically seen in random sequential adsorption packings and the kinetics of three-dimensional packings growth were again observed not to be strictly connected with the dimension of the configuration space. Moreover, a fast overlap criterion for Platonic and Archimedean solids based on separating axis theorem has been described. The criterion, together with other optimizations, allowed us to generate significantly larger packings, which translated directly to a lower statistical error of the results obtained. Additionally, the polyhedral order parameters provided can be utilized in other studies regarding particles of polyhedral symmetry.

Methodology to construct large realizations of perfectly hyperuniform disordered packings

Disordered hyperuniform packings (or dispersions) are unusual amorphous two-phase materials that are endowed with exotic physical properties. Such hyperuniform systems are characterized by an anomalous suppression of volume-fraction fluctuations at infinitely long-wavelengths, compared to ordinary disordered materials. While there has been growing interest in such singular states of amorphous matter, a major obstacle has been an inability to produce large samples that are perfectly hyperuniform due to practical limitations of conventional numerical and experimental methods. To overcome these limitations, we introduce a general theoretical methodology to construct perfectly hyperuniform packings in d-dimensional Euclidean space R^{d}. Specifically, beginning with an initial general tessellation of space by disjoint cells that meets a "bounded-cell" condition, hard particles of general shape are placed inside each cell such that the local-cell particle packing fractions are identical to the global packing fraction. We prove that the constructed packings with a polydispersity in size are perfectly hyperuniform in the infinite-sample-size limit, regardless of particle shapes, positions, and numbers per cell. We use this theoretical formulation to devise an efficient and tunable algorithm to generate extremely large realizations of such packings. We employ two distinct initial tessellations: Voronoi as well as sphere tessellations. Beginning with Voronoi tessellations, we show that our algorithm can remarkably convert extremely large nonhyperuniform packings into hyperuniform ones in R^{2} and R^{3}. Implementing our theoretical methodology on sphere tessellations, we establish the hyperuniformity of the classical Hashin-Shtrikman multiscale coated-spheres structures, which are known to be two-phase media microstructures that possess optimal effective transport and elastic properties. A consequence of our work is a rigorous demonstration that packings that have identical tessellations can either be nonhyperuniform or hyperuniform by simply tuning local characteristics. It is noteworthy that our computationally designed hyperuniform two-phase systems can easily be fabricated via state-of-the-art methods, such as 2D photolithographic and 3D printing technologies. In addition, the tunability of our methodology offers a route for the discovery of novel disordered hyperuniform two-phase materials.

Random sequential adsorption of cuboids

The subject of this study was random sequential adsorption of cuboids of axes length ratio of a : 1 : b for a ∈ [0.3, 1.0] and b ∈ [1.0, 2.0], and the aim of this study was to find a shape that provides the highest packing fraction. The obtained results show that the densest packing fraction is 0.401 87 ± 0.000 97 and is reached for axes ratios near cuboids of 0.75:1:1.30. Kinetics of packing growth was also studied, and it was observed that its power-law character seems not to be governed by the number of cuboid degrees of freedom. The microstructural properties of obtained packings were studied in terms of density correlation function and propagation of orientational ordering.

Computing the 3D Radial Distribution Function from Particle Positions: An Advanced Analytic Approach

The radial distribution function, g( r), is ubiquitously used to analyze the internal structure of particulate systems. However, experimentally derived particle coordinates are always confined to a finite sample volume. This poses a particular challenge on computing g( r): Once the radial distance, r, extends beyond the sample boundaries in at least one dimension, substantial deviations from the true g( r) function can occur. State of the art algorithms for g( r) mitigate this issue for instance by using artificial periodic boundary conditions. However, ignoring the finite nature of the sample volume distorts g( r) significantly. Here, we present a simple, analytic algorithm for the computation of g( r) in finite samples. No additional assumptions about the sample are required. The key idea is to use an analytic solution for the intersection volume between a spherical shell and the sample volume. In addition, we discovered a natural upper bound for the radial distance that only depends on sample size and shape. This analytic approach will prove to be invaluable for the quantitative analysis of the increasing amount of experimentally derived tomography data.

Precise algorithms to compute surface correlation functions of two-phase heterogeneous media and their applications

The quantitative characterization of the microstructure of random heterogeneous media in d-dimensional Euclidean space R^{d} via a variety of n-point correlation functions is of great importance, since the respective infinite set determines the effective physical properties of the media. In particular, surface-surface F_{ss} and surface-void F_{sv} correlation functions (obtainable from radiation scattering experiments) contain crucial interfacial information that enables one to estimate transport properties of the media (e.g., the mean survival time and fluid permeability) and complements the information content of the conventional two-point correlation function. However, the current technical difficulty involved in sampling surface correlation functions has been a stumbling block in their widespread use. We first present a concise derivation of the small-r behaviors of these functions, which are linked to the mean curvature of the system. Then we demonstrate that one can reduce the computational complexity of the problem, without sacrificing accuracy, by extracting the necessary interfacial information from a cut of the d-dimensional statistically homogeneous and isotropic system with an infinitely long line. Accordingly, we devise algorithms based on this idea and test them for two-phase media in continuous and discrete spaces. Specifically for the exact benchmark model of overlapping spheres, we find excellent agreement between numerical and exact results. We compute surface correlation functions and corresponding local surface-area variances for a variety of other model microstructures, including hard spheres in equilibrium, decorated "stealthy" patterns, as well as snapshots of evolving pattern formation processes (e.g., spinodal decomposition). It is demonstrated that the precise determination of surface correlation functions provides a powerful means to characterize a wide class of complex multiphase microstructures.

Kinetics of deposition in the diffusion-controlled limit

The adsorption of particles diffusing in a half space bounded by the substrate and irreversibly sticking to the substrate upon contacts is investigated. We show that when absorbing particles are planar disks diffusing in the three-dimensional half space, the coverage approaches its saturated "jamming" value as t^{-1} in the large time limit (generally as t^{-1/(d-1)} when the substrate is d dimensional and d>1, and as e^{-t/ln(t)} when d=1). We also analyze the asymptotic behavior when particles are spherical and when particles are planar aligned squares.

Nanoparticle adsorption dynamics at fluid interfaces

Understanding the dynamic adsorption of nanoparticles (NPs) at fluid interfaces is important for stabilizing emulsions and for the preparation of 2D NP-based materials. Here we show that the Ward-Tordai equations commonly employed to describe the dynamics of surfactant adsorption at a fluid interface combined with a Frumkin adsorption isotherm can be employed to model the diffusion-limited adsorption of NPs onto a fluid interface. In contrast to surfactants, an additional wetting equation of state (EOS) must be incorporated to characterize the dynamic interfacial tension during the adsorption of NPs to the oil-water interface. Our results show agreement between the model and experiments with NP area fractions <0.3. Slower dynamics are observed at larger area fractions, which are speculated to arise from polydispersity or re-organization at the interface. We show the model can be extended to the competitive adsorption between the NPs and a surface active species.

Precise algorithm to generate random sequential adsorption of hard polygons at saturation

Random sequential adsorption (RSA) is a time-dependent packing process, in which particles of certain shapes are randomly and sequentially placed into an empty space without overlap. In the infinite-time limit, the density approaches a "saturation" limit. Although this limit has attracted particular research interest, the majority of past studies could only probe this limit by extrapolation. We have previously found an algorithm to reach this limit using finite computational time for spherical particles and could thus determine the saturation density of spheres with high accuracy. In this paper, we generalize this algorithm to generate saturated RSA packings of two-dimensional polygons. We also calculate the saturation density for regular polygons of three to ten sides and obtain results that are consistent with previous, extrapolation-based studies.

Rational design of stealthy hyperuniform two-phase media with tunable order

Disordered stealthy hyperuniform materials are exotic amorphous states of matter that have attracted recent attention because of their novel structural characteristics (hidden order at large length scales) and physical properties, including desirable photonic and transport properties. It is therefore useful to devise algorithms that enable one to design a wide class of such amorphous configurations at will. In this paper, we present several algorithms enabling the systematic identification and generation of discrete (digitized) stealthy hyperuniform patterns with a tunable degree of order, paving the way towards the rational design of disordered materials endowed with novel thermodynamic and physical properties. To quantify the degree of order or disorder of the stealthy systems, we utilize the discrete version of the τ order metric, which accounts for the underlying spatial correlations that exist across all relevant length scales in a given digitized two-phase (or, equivalently, a two-spin state) system of interest. Our results impinge on a myriad of fields, ranging from physics, materials science and engineering, visual perception, and information theory to modern data science.

Can exotic disordered "stealthy" particle configurations tolerate arbitrarily large holes?

The probability of finding a spherical cavity or "hole" of arbitrarily large size in typical disordered many-particle systems in the infinite-system-size limit (e.g., equilibrium liquid states) is non-zero. Such "hole" statistics are intimately linked to the thermodynamic and nonequilibrium physical properties of the system. Disordered "stealthy" many-particle configurations in d-dimensional Euclidean space [Doublestruck R]^d are exotic amorphous states of matter that lie between a liquid and crystal that prohibit single-scattering events for a range of wave vectors and possess no Bragg peaks [Torquato et al., Phys. Rev. X, 2015, 5, 021020]. In this paper, we provide strong numerical evidence that disordered stealthy configurations across the first three space dimensions cannot tolerate arbitrarily large holes in the infinite-system-size limit, i.e., the hole probability has compact support. This structural "rigidity" property apparently endows disordered stealthy systems with novel thermodynamic and physical properties, including desirable band-gap, optical and transport characteristics. We also determine the maximum hole size that any stealthy system can possess across the first three space dimensions.

Surface fine structure influence on saturated random packings

Random packings of disks on a mesh are studied numerically using random sequential adsorption algorithm. The mesh is built of straight horizontal and vertical one-dimensional lines of a given distance between them. The packing fraction and structure as well as the kinetics of packing growth dependence on mesh size are analyzed to provide information, whether surface inhomogeneity will affect the properties of random packings. It has been shown that the number of disks in a packing slightly decreases with growing distance between mesh lines while the kinetics may change significantly even for very dense meshes. As packings obtained in random sequential adsorption resemble monolayers produced by irreversible adsorption processes, results of this study show that by measuring properties of a random packing it may be possible to determine fine structure of an underlying surface.

Packing of nonoverlapping cubic particles: Computational algorithms and microstructural characteristics

Packing of cubic particles arises in a variety of problems, ranging from biological materials to colloids and the fabrication of new types of porous materials with controlled morphology. The properties of such packings may also be relevant to problems involving suspensions of cubic zeolites, precipitation of salt crystals during CO_{2} sequestration in rock, and intrusion of fresh water in aquifers by saline water. Not much is known, however, about the structure and statistical descriptors of such packings. We present a detailed simulation and microstructural characterization of packings of nonoverlapping monodisperse cubic particles, following up on our preliminary results [H. Malmir et al., Sci. Rep. 6, 35024 (2016)2045-232210.1038/srep35024]. A modification of the random sequential addition (RSA) algorithm has been developed to generate such packings, and a variety of microstructural descriptors, including the radial distribution function, the face-normal correlation function, two-point probability and cluster functions, the lineal-path function, the pore-size distribution function, and surface-surface and surface-void correlation functions, have been computed, along with the specific surface and mean chord length of the packings. The results indicate the existence of both spatial and orientational long-range order as the the packing density increases. The maximum packing fraction achievable with the RSA method is about 0.57, which represents the limit for a structure similar to liquid crystals.

Microstructural characterization of random packings of cubic particles

Understanding the properties of random packings of solid objects is of critical importance to a wide variety of fundamental scientific and practical problems. The great majority of the previous works focused, however, on packings of spherical and sphere-like particles. We report the first detailed simulation and characterization of packings of non-overlapping cubic particles. Such packings arise in a variety of problems, ranging from biological materials, to colloids and fabrication of porous scaffolds using salt powders. In addition, packing of cubic salt crystals arise in various problems involving preservation of pavements, paintings, and historical monuments, mineral-fluid interactions, CO2 sequestration in rock, and intrusion of groundwater aquifers by saline water. Not much is known, however, about the structure and statistical descriptors of such packings. We have developed a version of the random sequential addition algorithm to generate such packings, and have computed a variety of microstructural descriptors, including the radial distribution function, two-point probability function, orientational correlation function, specific surface, and mean chord length, and have studied the effect of finite system size and porosity on such characteristics. The results indicate the existence of both spatial and orientational long-range order in the packing, which is more distinctive for higher packing densities. The maximum packing fraction is about 0.57.

A Geometric-Structure Theory for Maximally Random Jammed Packings

Maximally random jammed (MRJ) particle packings can be viewed as prototypical glasses in that they are maximally disordered while simultaneously being mechanically rigid. The prediction of the MRJ packing density ϕMRJ, among other packing properties of frictionless particles, still poses many theoretical challenges, even for congruent spheres or disks. Using the geometric-structure approach, we derive for the first time a highly accurate formula for MRJ densities for a very wide class of two-dimensional frictionless packings, namely, binary convex superdisks, with shapes that continuously interpolate between circles and squares. By incorporating specific attributes of MRJ states and a novel organizing principle, our formula yields predictions of ϕMRJ that are in excellent agreement with corresponding computer-simulation estimates in almost the entire α-x plane with semi-axis ratio α and small-particle relative number concentration x. Importantly, in the monodisperse circle limit, the predicted ϕMRJ = 0.834 agrees very well with the very recently numerically discovered MRJ density of 0.827, which distinguishes it from high-density "random-close packing" polycrystalline states and hence provides a stringent test on the theory. Similarly, for non-circular monodisperse superdisks, we predict MRJ states with densities that are appreciably smaller than is conventionally thought to be achievable by standard packing protocols.

Interfacial effect on physical properties of composite media: Interfacial volume fraction with non-spherical hard-core-soft-shell-structured particles

Interfaces are known to be crucial in a variety of fields and the interfacial volume fraction dramatically affects physical properties of composite media. However, it is an open problem with great significance how to determine the interfacial property in composite media with inclusions of complex geometry. By the stereological theory and the nearest-surface distribution functions, we first propose a theoretical framework to symmetrically present the interfacial volume fraction. In order to verify the interesting generalization, we simulate three-phase composite media by employing hard-core-soft-shell structures composed of hard mono-/polydisperse non-spherical particles, soft interfaces, and matrix. We numerically derive the interfacial volume fraction by a Monte Carlo integration scheme. With the theoretical and numerical results, we find that the interfacial volume fraction is strongly dependent on the so-called geometric size factor and sphericity characterizing the geometric shape in spite of anisotropic particle types. As a significant interfacial property, the present theoretical contribution can be further drawn into predicting the effective transport properties of composite materials.

Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations

Systems of particles interacting with "stealthy" pair potentials have been shown to possess infinitely degenerate disordered hyperuniform classical ground states with novel physical properties. Previous attempts to sample the infinitely degenerate ground states used energy minimization techniques, introducing algorithmic dependence that is artificial in nature. Recently, an ensemble theory of stealthy hyperuniform ground states was formulated to predict the structure and thermodynamics that was shown to be in excellent agreement with corresponding computer simulation results in the canonical ensemble (in the zero-temperature limit). In this paper, we provide details and justifications of the simulation procedure, which involves performing molecular dynamics simulations at sufficiently low temperatures and minimizing the energy of the snapshots for both the high-density disordered regime, where the theory applies, as well as lower densities. We also use numerical simulations to extend our study to the lower-density regime. We report results for the pair correlation functions, structure factors, and Voronoi cell statistics. In the high-density regime, we verify the theoretical ansatz that stealthy disordered ground states behave like "pseudo" disordered equilibrium hard-sphere systems in Fourier space. The pair statistics obey certain exact integral conditions with very high accuracy. These results show that as the density decreases from the high-density limit, the disordered ground states in the canonical ensemble are characterized by an increasing degree of short-range order and eventually the system undergoes a phase transition to crystalline ground states. In the crystalline regime (low densities), there exist aperiodic structures that are part of the ground-state manifold but yet are not entropically favored. We also provide numerical evidence suggesting that different forms of stealthy pair potentials produce the same ground-state ensemble in the zero-temperature limit. Our techniques may be applied to sample the zero-temperature limit of the canonical ensemble of other potentials with highly degenerate ground states.

Random sequential adsorption of starlike particles

Random packing of surfaceless starlike particles built of 3 to 50 line segments was studied using random sequential adsorption algorithm. Numerical simulations allow us to determine saturated packing densities as well as the first two virial expansion coefficients for such objects. Measured kinetics of the packing growth supports the power law known to be valid for particles with a finite surface; however, the dependence of the exponent in this law on the number of star arms is unexpected. The density autocorrelation function shows fast superexponential decay as for disks, but the typical distance between closest stars is much smaller than between disks of the similar size, especially for a small number of arms.

Shapes for maximal coverage for two-dimensional random sequential adsorption

Maximal possible saturated random packing fractions and corresponding values of anisotropy level for which they are reached.

Random packing of regular polygons and star polygons on a flat two-dimensional surface

Random packing of unoriented regular polygons and star polygons on a two-dimensional flat continuous surface is studied numerically using random sequential adsorption algorithm. Obtained results are analyzed to determine the saturated random packing ratio as well as its density autocorrelation function. Additionally, the kinetics of packing growth and available surface function are measured. In general, stars give lower packing ratios than polygons, but when the number of vertexes is large enough, both shapes approach disks and, therefore, properties of their packing reproduce already known results for disks.

Properties of random sequential adsorption of generalized dimers

Saturated random packing of particles built of two identical, relatively shifted spheres in two- and three-dimensional flat and homogeneous space was studied numerically using a random sequential adsorption algorithm. The shift between centers of spheres varied from 0.0 to 8.0 sphere diameters. Numerical simulations allowed the determination of random sequential adsorption kinetics, the saturated random coverage ratio, as well as the available surface function and density autocorrelation function.

Kinetics of random sequential adsorption of nearly spherically symmetric particles

Kinetics of random sequential adsorption (RSA) of disks on flat, two-dimensional surfaces is governed by a power law with exponent -1/2. The study has shown that for RSA of nearly spherically symmetric particles this exponent is -1/3, whereas other characteristics typically measured in RSA simulations approach values known for disks with the increase of symmetry of the particles.