Experimental Study of Ordering of Hard Cubes by Shearing

Breakdown of Magic Numbers in Spherical Confinement

Magic numbers in finite particle systems correspond to specific system sizes that allow configurations with low free energy, often exhibiting closed surface shells to maximize the number of nearest neighbors. Since their discovery in atomic nuclei, magic numbers have been essential for understanding the number-structure-property relationships in finite clusters across different scales. However, as the system size increases, the significance of magic numbers diminishes, and the precise system size at which magic number phenomena disappear remains uncertain. In this study, we investigate colloidal clusters formed through confined self-assembly. Small magic number clusters display icosahedral symmetry with closed surface shells, corresponding to pronounced free energy minima. Our findings reveal that beyond a critical system size, closed surface shells disappear, and free energy minima become less pronounced. Instead, we observe a distinct type of colloidal cluster, termed a football cluster, which retains icosahedral symmetry but features lower-coordinated facets disconnected by terraces. A sphere packing model demonstrates that forming closed surface shells becomes impossible beyond a critical system size, explaining the breakdown of magic numbers in large confined systems.

Use of the radial distribution function with the Chebyshev distance to characterize orientation of two-dimensional square crystals

In the present work, the orientation of two-dimensional square crystals is studied using the radial distribution function (RDF) defined with the Chebyshev distance. Traditional RDF methods, based on the Euclidean metric, fail to detect changes in crystal orientation due to their isotropic nature. To overcome this limitation, we introduce a modification of the RDF employing the Chebyshev metric, which adds sensitivity to directional changes within square lattices. This approach expands the information conventionally obtained from the RDF by incorporating not only translational properties but also orientational information. The study provides a mathematical analysis of the peak distribution in this enhanced RDF approach when applied to square crystals rotated through various orientations. We found that the orientation of square crystals can be efficiently determined by calculating the abscissa of the first peak in the Chebyshev RDF, which reduces computational times, even for fine numerical resolutions of differential annular widths as small as 10^{-3} or less. This approach can be applied to any two-dimensional (2D) Bravais lattice structure, with potential modifications based on lattice parameters and is robust for analyzing imperfect lattices with noise and vacancies. In this study, we demonstrate its application on rotated 2D Bravais square lattices, laying the groundwork for future studies on other 2D lattice structures. Additionally, we explore the use of this method to characterize jamming in 2D granular cubic systems, focusing on the orientation of oblique square grains in tilted vibrated layers under cyclic shear.

Coupled Dynamical Phase Transitions in Driven Disk Packings

Under the influence of oscillatory shear, a monolayer of frictional granular disks exhibits two dynamical phase transitions: a transition from an initially disordered state to an ordered crystalline state and a dynamic active-absorbing phase transition. Although there is no reason a priori for these to be at the same critical point, they are. The transitions may also be characterized by the disk trajectories, which are nontrivial loops breaking time-reversal invariance.

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.

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.

Flow-induced surface crystallization of granular particles in cylindrical confinement

An interesting phenomenon that a layer of crystallized shell formed at the container wall during an orifice flow in a cylinder is observed experimentally and is investigated in DEM simulation. Different from shear or vibration driven granular crystallization, our simulation shows during the flow the shell layer is formed spontaneously from stagnant zone at the base and grows at a constant rate to the top with no external drive. Roughness of the shell surface is defined as a standard deviation of the surface height and its development is found to disobey existed growth models. The growth rate of the shell is found linearly proportional to the flow rate. This shell is static and served as a rough wall in an orifice flow with frictionless sidewall, which changes the flow profiles and its stress properties, and in turn guarantees a constant flow rate.

Jammed packings of 3D superellipsoids with tunable packing fraction, coordination number, and ordering

We carry out numerical studies of static packings of frictionless superellipsoidal particles in three spatial dimensions. We consider more than 200 different particle shapes by varying the three shape parameters that define superellipsoids. We characterize the structural and mechanical properties of both disordered and ordered packings using two packing-generation protocols. We perform athermal quasi-static compression simulations starting from either random, dilute configurations (Protocol 1) or thermalized, dense configurations (Protocol 2), which allows us to tune the orientational order of the packings. In general, we find that superellipsoid packings are hypostatic, with coordination number zJ < ziso, where ziso = 2df and df = 5 or 6 depending on whether the particles are axi-symmetric or not. Over the full range of orientational order, we find that the number of quartic modes of the dynamical matrix for the packings always matches the number of missing contacts relative to the isostatic value. This result suggests that there are no mechanically redundant contacts for ordered, yet hypostatic packings of superellipsoidal particles. Additionally, we find that the packing fraction at jamming onset for disordered packings of superellipsoidal depends on at least two particle shape parameters, e.g. the asphericity A and reduced aspect ratio β of the particles.

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.

Self-assembly of granular spheres under one-dimensional vibration

The self-assembly of uniform granular spheres is related to the fundamentals of granular matter such as the transitions of phases, order/disorder and jamming states. This paper presents a DEM (discrete element method) study of the continuous self-assembly of uniform granular spheres from random close packing (RCP) to partially and nearly fully ordered packings under one-dimensional (1D) sinusoidal vibration without other interventions. The effects of the vibration amplitude and frequency are investigated in a wide range. The structures of the packings are characterized in terms of packing fraction and other microscopic structural parameters, including the coordination number, bond-orientational orders, and, in particular, ordered clusters, by adaptive common neighbor analysis (a-CNA). It is shown that 1D vibrations can also lead to the self-assembly of uniform granular spheres with packing fractions exceeding the RCP limit, and FCC (face centered cubic) and HCP (hexagonal close packed) structures coexist in the self-assembled packings while their total fraction can reach nearly 100%. The structures of these packings can be better correlated with the vibration velocity amplitude rather than the commonly used vibration intensity. The dynamics of such self-assembly is also preliminarily analyzed. Our study not only presents the conditions for the self-assembly of uniform granular spheres under 1D vibration, but also characterizes the order-disorder transitions during the process, which can improve our understanding of the fundamentals of granular materials and jamming states.

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.

Phase diagram of hard squares in slit confinement

This work shows a complete phase diagram of hard squares of side length σ in slit confinement for H < 4.5, H being the wall to wall distance measured in σ units, including the maximal packing fraction limit. The phase diagram exhibits a transition between a single-row parallel 1-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ and a zigzag 2-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\diamond }$$\end{document}◇ˆ structures for H_c(2) = (2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\bf{2}}}$$\end{document}2 − 1) < H < 2, and also another one involving the 1-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ and 2-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ structures (two parallel rows) for 2 < H < H_c(3) (H_c(n) = n − 1 + \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\bf{2}}{\boldsymbol{n}}-{\bf{1}}}$$\end{document}2n−1/n is the critical wall-to-wall distance for a (n − 1)-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ to n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond }$$\end{document}◇ transition and where n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond }$$\end{document}◇ represents a structure formed by tilted rectangles, each one clustering n stacked squares), and a triple point for H_t \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\simeq }}$$\end{document}≃ 2.005. In this triple point there coexists the 1-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□, 2-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□, and 2-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\diamond }$$\end{document}◇ˆ structures. For regions H_c(3) < H < H_c(4) and H_c(4) < H < H_c(5), very similar pictures arise. There is a (n − 1)-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ to a n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond }$$\end{document}◇ strong transition for H_c(n) < H < n, followed by a softer (n − 1)-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ to n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ transition for n < H < H_c(n + 1). Again, at H \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\gtrsim }}$$\end{document}≳ n there appears a triple point, involving the (n − 1)-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□, n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□, and n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond }$$\end{document}◇ structures. The similarities found for n = 2, 3 and 4 lead us to propose a tentative phase diagram for H_c(n) < H < H_c(n + 1) (n ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{{\mathbb{N}}}}$$\end{document}ℕ, n > 2), where structures (n − 1)-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□, n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□, and n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond }$$\end{document}◇ fill the phase diagram. Simulation and Onsager theory results are qualitatively consistent.