Random Close Packing Of Spheres Research Articles

Descriptor-free unsupervised learning method for local structure identification in particle packings.

Local structure identification is of great importance in many scientific and engineering fields. However, mathematical and supervised learning methods mostly rely on specific descriptors of local structures and can only be applied to particular packing configurations. In this work, we propose an improved unsupervised learning method, which is descriptor-free, for local structure identification in particle packing. The point cloud is used as the input of the improved method, which directly comes from spatial positions of particles and does not rely on specific descriptors. The improved method constructs an autoencoder based on the point cloud network combined with Gaussian mixture models for dimension reduction and clustering. Numerical examples show that the improved method performs well in local structure identification of quasicrystal disk and sphere packings, achieving comparable accuracy with previous methods. For disordered packings, which have been considered having nearly no local structures, the improved method identifies a nontrivial seven-neighbor motif in the maximally dense random packing of disks and finds acentric structural motifs in the random close packing of spheres, which demonstrate the ability on identification of new and unknown local structures. The improved unsupervised learning method would help obtain information from massive simulation and experimental results as well as devising new order parameters for particle packings.

Determining random packing density and equivalent packing size of superballs via binary mixtures with spheres

We propose a new approach to determining the random packing densities of superballs via binary mixtures with spheres. The main idea of the approach is to suppress order formations in non-spherical particle packings via the polydispersity of particle shapes, which avoids using order metrics. The packing density of superballs in a mixture can be segregated using a linear fitting method with the concept of equivalent packing size (or size ratio with unit spheres) which represents the effective size (or volume) of a non-spherical particle in a binary mixture with spheres. We systemically study the packing properties of binary mixtures consisting of spheres and superballs and obtain the equivalent packing sizes of superballs. Our results show that the equivalent packing size ratio always corresponds to the minimal packing density or specific volume (reciprocal of packing densities) variation, and is independent of the solid volume fraction. The specific volumes of mixtures with the equivalent packing size ratio are always the upper bound for all the solid volume fractions. The linear relationship between the specific volume and solid volume fraction is only observed in the mixtures with superballs of small surface shape parameters (shapes close to a sphere), which results from the highly disordered nature in the mixtures. Moreover, the ideal random packing densities of mono-sized superballs obtained via the linear fitting method are surprisingly close to those of the MDRPs (maximally dense random packings), further verifying that the MDRPs of non-spherical particles correspond to the ideal random packings whose degrees of order are at the same level with that of the random close packing of spheres. Our work leads to a better understanding towards the random and binary packings and sheds new light on the essence of the MDRP. Our work also guides the optimal particle size distributions of powders in chemical engineering process.

Influence of particle size distribution on random close packing of spheres.

The densest amorphous packing of rigid particles is known as random close packing. It has long been appreciated that higher densities are achieved by using collections of particles with a variety of sizes. For spheres, the variety of sizes is often quantified by the polydispersity of the particle size distribution: the standard deviation of the radius divided by the mean radius. Several prior studies quantified the increase of the packing density as a function of polydispersity. A particle size distribution is also characterized by its skewness, kurtosis, and higher moments, but the influence of these parameters has not been carefully quantified before. In this work, we numerically generate many sphere packings with different particle radii distributions, varying polydispersity and skewness independently of one another. We find that the packing density can increase significantly with increasing skewness and in some cases skewness can have a larger effect than polydispersity. However, the packing fraction is relatively insensitive to the higher moment value of the kurtosis. We present a simple empirical formula for the value of the random close packing density as a function of polydispersity and skewness.

Packing fraction of trimodal spheres with small size ratio: An analytical expression

In previous papers analytical expressions were derived and validated for the packing fraction of bimodal hard spheres with small size ratio, applicable to ordered (crystalline) [H. J. H. Brouwers, Phys. Rev. E 76, 041304 (2007);H. J. H. Brouwers, Phys. Rev. E 78, 011303 (2008)] and disordered (random) packings [H. J. H. Brouwers, Phys. Rev. E 87, 032202 (2013)]. In the present paper the underlying statistical approach, based on counting the occurrences of uneven pairs, i.e., the fraction of contacts between unequal spheres, is applied to trimodal discretely sized spheres. The packing of such ternary packings can be described by the same type of closed-form equation as the bimodal case. This equation contains the mean volume of the spheres and of the elementary cluster formed by these spheres; for crystalline arrangements this corresponds to the unit cell volume. The obtained compact analytical expression is compared with empirical packing data concerning random close packing of spheres, taken from the literature, comprising ternary binomial and geometric packings; good agreement is obtained. The presented approach is generalized to ordered and disordered packings of multimodal mixes.

Thermodynamic status of random close packing

An amorphous ground state reminiscent of random close packing (RCP) of spheres terminates a liquid phase that spans all temperatures. On the Gibbs density surface, the liquid phase has bounded by a supercritical percolation line, two-phase liquid–gas coexistence line, and below the triple point, the metastable liquid branch terminating at T = 0 K with the RCP state. There is no “continuity of gas and liquid phases”; they are separated by a supercritical mesophase bounded by percolation transitions. As a consequence, the gas phase cannot be a starting point for liquid-state theory. RCP has a thermodynamic status. For square-well model fluids, evidence from computer simulations shows that the amorphous ground state is the same RCP state of the hard-sphere model. The RCP limiting density state of the hard-sphere fluid, with its reproducible and well-characterized structure, can be obtained by well-defined irreversible and reversible processes which establish a thermodynamic status. Empirical results, within margins of numerical uncertainty, are that the amorphous ground state density corresponds to a packing fraction = 0.6366 ± 0.0005 (Buffon’s constant 2/π) and a residual entropy per sphere ΔS(0) ∼ kB (Boltzmann constant). We conclude that RCP of spheres is a well-defined state with a thermodynamic status, and a central role in the description of liquid phase equilibria, as originally suggested, 50 years ago, by J.D. Bernal. We postulate that the metastable RCP state is the starting point for a modern theory of the liquid state.

Modeling and characterization of two-phase composites by Voronoi diagram in the Laguerre geometry based on random close packing of spheres

Abstract An effective method is developed for modeling and characterization of two-phase composites using Voronoi diagram in the Laguerre geometry based on random close packing of spheres (RCP-LV). First, RCP is generated with two groups of spheres and each group has its own volume distribution. Then LV diagram is performed basing on the sphere packing to generate the grains of two phases, thus the model of two-phase composite is obtained. Because the model is based on the RCP, the grain volume distributions of the two phases are easily controlled by changing the volume distributions of two groups of spheres in RCP. Various geometrical and topological characterizations are conducted, yielding useful information of this kind of composite. This modeling method can be utilized in deep investigation of structure–property relations in two-phase composites.

Random close packing of disks and spheres in confined geometries

Studies of random close packing of spheres have advanced our knowledge about the structure of systems such as liquids, glasses, emulsions, granular media, and amorphous solids. In confined geometries, the structural properties of random-packed systems will change. To understand these changes, we study random close packing in finite-sized confined systems, in both two and three dimensions. Each packing consists of a 50-50 binary mixture with particle size ratio of 1.4. The presence of confining walls significantly lowers the overall maximum area fraction (or volume fraction in three dimensions). A simple model is presented, which quantifies the reduction in packing due to wall-induced structure. This wall-induced structure decays rapidly away from the wall, with characteristic length scales comparable to the small particle diameter.

Expansion evolution during foaming of amorphous metals

Amorphous Pd 43Ni 10Cu 27P 20 foam of 38, 49, and 70% porosity is produced by expanding a 25% porosity amorphous precursor in the supercooled liquid state in three isothermal stages of varying durations. Metallographic examination suggests that expansion evolves by bubble growth towards a limit at which bubbles become critically packed. This limit is found to be close to the limit of random close packing of spheres of 63.7%. Beyond this critical limit, bubbles tend to impinge and coalesce and expansion progresses by growth of bubble clusters via stretching of intracellular membranes. The expansion evolution is modeled by means of a dynamic treatment of over-damped growth of individual bubbles. The model captures the expansion evolution reasonably well up to porosities between 50–70%, hence verifying that a transition from a bubble growth mechanism to a cluster growth mechanism takes place at some intermediate porosity.

Spectral properties of contact matrix: Application to proteins

A protein can be modelled by a set of points representing its amino acids. Topologically, this set of points is entirely defined by its contact matrix (adjacency matrix in graph theory). The contact matrix characterizing the relation between neighboring amino acids is deduced from Voronoi or Laguerre decomposition. This method allows contact matrices to be defined without any arbitrary cut-off that could induce arbitrary effects. Eigenvalues of these matrices are related with elementary excitations in proteins. We present some spectral properties of these matrices that reflect global properties of proteins. The eigenvectors indicate participation of each amino acids to the excitation modes of the proteins. It is interesting to compare the protein modelled as a close packing of amino acids, with a random close packing of spheres. The main features of the protein are those of a packing, a result that confirms the importance of the dense packing model for proteins. Nevertheless there are some properties, specific to the hierarchical organization of the protein: the primary chain order, the secondary structures and the domain structures.

Chord length distribution of Voronoi diagram in Laguerre geometry with lognormal-like volume distribution

The Voronoi diagram in the Laguerre geometry based on random close packing of spheres (RCP-LV diagram) has been found to be a better representation of polycrystalline structure than the conventional Poisson–Voronoi diagram. Stereology of the RCP-LV diagram with lognormal-like volume distribution has been investigated by the classical intercept count method. An improved five-parameter gamma distribution function is proposed to integrate the fact that probability density of the chord length remains non-zero as the chord length approaches zero. The proportional coefficient between the average grain size and the average chord length varies from 1.60 to 1.14 as the coefficient of variation of grain volume increases from 0.4 to 2.2. It is shown that there exists the possibility that not only the average grain size but also the grain size distribution can be estimated if the chord distribution characterization is fully explored with the aid of RCP-LV diagram simulation.

Scaling behaviors of colloidal aggregates under uniform pressure

We present a theoretical model for the compaction of a colloidal sediment under uniaxial mechanical pressure in the continuous three-dimensional space. The initial system is formed with aggregated particles dispersed in a fluid, and softly sedimented in a vessel. When a uniform pressure is applied, it evolves irreversibly through successive creation and destruction of bonds between the particles. The rules governing the bonds depend on both geometrical constraints and current stresses. Numerical simulations of such systems exhibit three different scenarios, corresponding, respectively, to the fragile, elastic, and plastic behaviors. In the elastic regime, where most bonds are permanent, the pressure scales as a power law of the volume fraction of particles, with a numerical exponent equal to 4.4. In the plastic regime, where many bonds are broken and many others created, the pressure also scales with volume fraction, but the exponent is much lower, equal to 1.7. These scaling behaviors agree remarkably well with recent experiments realized on the compaction of systems with aggregated silica particles in the oedometer cell. They also can be explained with simple theoretical arguments using a plausible morphology of the resistant paths acting throughout the system. Finally, at very large applied pressures, all these regimes converge to the random close packing of spheres.

Random close packing of spheres in a round cell

The paper describes the method of construction of cells containing large numbers (10 5–10 6) of close packed spheres, ranging in packing fractions from 0.2 to 0.60. The cells are constructed in a round form, utilizing varying surface curvature to prevent ordered packing. The properties of the cells are described by a series of distributions: (i) polar angle of contact points, (ii) maximum displacement of loose spheres, (iii) and Voronoi volumes for individual spheres. The cells of varying density appear to show the liquid-to-solid transition. All cells described here contain loose spheres. It is proposed that cells with zero loose spheres are ideal amorphous solids. Ideal amorphous solids are characterized by a number of elements, described herein.

Is random close packing of spheres well defined?

Despite its long history, there are many fundamental issues concerning random packings of spheres that remain elusive, including a precise definition of random close packing (RCP). We argue that the current picture of RCP cannot be made mathematically precise and support this conclusion via a molecular dynamics study of hard spheres using the Lubachevsky-Stillinger compression algorithm. We suggest that this impasse can be broken by introducing the new concept of a maximally random jammed state, which can be made precise.

Vanishing elasticity for wet foams: Equivalence with emulsions and role of polydispersity

We present an experimental study of the rheology of polydisperse aqueous foams of different gas volume fractions φ. With oscillatory deformation at fixed frequency, we determine the behavior of the maximum stress as a function of the strain amplitude. At low strain, the maximum stress increases linearly, defining a shear modulus G. At progressively higher strains, the response eventually becomes nonlinear, defining the yield strain and the yield stress. While φ decreases toward φc=0.635±0.01, G goes to zero, and the yield stress decreases by many orders of magnitude with a quadratic behavior. The yield strain, which can be extrapolated to 0.18±0.02 at φ=1, has a minimum value of 0.045±0.010 at φc. This behavior shows the occurrence of a melting transition located at φc, which can be correlated to the random close packing of spheres. We compare these results to similar ones obtained previously for monodisperse and polydisperse emulsions. Our new experiments clarify the rheological similarities between emulsions and foams, as well as the role of polydispersity. We find that as long as polydispersity is moderate, it does not play a crucial role in the elastic response of foams and emulsions.

Algorithm for Random Close Packing of Spheres with Periodic Boundary Conditions

The isotropic algorithm is constructed for random close packing of equisized spheres with triply periodic boundary conditions. All previously published packing methods with periodic boundaries were kinetics-determined; i.e., they contained a densification rate as an arbitrary parameter. In contrast, the present algorithm is kinetic-independent and demonstrates an unambiguous convergence to the experimental results. To suppress crystallization, the main principles of our algorithm are (1) to form a contact network at an early stage and (2) retain contacts throughout the densification, as far as possible. The particles are allowed to swell by the numerical solution of the differential equations of densification. The RHS of these equations is calculated efficiently from a linear system by a combination of conjugate gradient iterations and exact sparse matrix technology. When an excessive contact occurs and one of the existing bonds should be broken to continue the densification, an efficient criterion based on multidimensional simplex geometry is used for searching the separating bond. The algorithm has a well-defined termination point resulting in a perfect contact network with the average coordination number six (for particle number N>1) and a system of normal reactions between the spheres maintaining the structure. These forces are the counterpart of the algorithm and can be used to calculate small elastic particle deformations in a granular medium. Extensive calculations are presented for 50 @? N 5 400 and demonstrate very good agreement with the experimental packing density (about 0.637 ) and structure.

Algorithm for Random Close Packing of Spheres with Periodic Boundary Conditions

The isotropic algorithm is constructed for random close packing of equisized spheres with triply periodic boundary conditions. All previously published packing methods with periodic boundaries were...

A small-angle x-ray scattering study of microstructure evolution during sintering of sol-gel-derived porous nanophase titania

The evolution of microstructure as a function of firing temperature in sol-gel derived porous titania xerogels was investigated by small-angle x-ray scattering (SAXS). SAXS curves for xerogels fired below 550 °C exhibit a well-defined structure peak. This peak indicates the presence of a high degree of order in the electron density correlations associated with the interparticle structure factor. Results from electron microscopy and SAXS give a primary particle mean diameter of 50 Å, while scaling analysis of the scattered intensity at large momentum transfer values yields the Porod exponent 4, indicating a sharp transition zone between the solid and void phases. The pore volume fraction of the unfired xerogel is consistent with random close-packing of spheres. The internal surface area decreases almost linearly with increasing firing temperature. Rapid grain growth and pore coarsening begin near 90% of theoretical density, and lead to a breakdown in pore interconnectedness and the development of isolated pores. Observed enhanced sintering properties may be attributed primarily to the large surface-to-mass ratio of the sol-gel particles. The SAXS curves were adequately fit using a bicontinuous phase model developed for late-stage spinodal decomposition structures. Alternatively, the microstructure can be described by a hierarchical close-packing of spheres model. SAXS results are compared with data from gas adsorption and electron microscopy.

Numerical modeling of polymolecular adsorption of argon on a model amorphous oxide

Using the Monte Carlo method for the grand canonical ensemble, we have calculated the Ar adsorption isotherms in the range 77–107 K on an inhomogeneous surface with random close packing of spheres (the Bemal model), representing oxygen ions in a model oxide. The isotherm at 77 K is very close to the standard Ar adsorption isotherm on nonporous silica. The difference involves the weak step character of the calculated isotherm. The adsorbate on the model oxide at 77 K has a layered structure, reminiscent of the structure of Ar on the basal face of graphite, but significantly more extended. The atomic area of Ar depends on the method for determination of the capacity of the monolayer and lies within the range 0.1–0.13 nm2.

Effects of three-body interactions on the structure of clusters

The structure of 54 and 147 atom clusters was studied using molecular dynamics at constant temperature and incorporating three-body interactions to the usual pairwise additive potentials among atoms. The effects of three-body interactions of the triple-dipole and exchange overlap type on the aggregation of clusters are: (1) to diminish the coordination number resulting in a global expansion of the clusters; (2) to increase internal disorder in such a way as to lower the crystallization temperature; (3) to favor certain features characteristic of random close packing of spheres at the expenses of destroying locally paired tetrahedra and octahedra, in the temperature range 0.25 ⩽ T ⩽ 0.4.

Hard-core square-well fermions

Based on the well-known low-density expansion for the ground-state energy per particle of a many-fermion system interacting pairwise via a central potential we develop a perturbation scheme in which the zero-order state is that of the purely-repulsive-spheres fluid. For the hard-core plus square-well pair-interaction case we have deduced, for two-species fermion matter, perturbation formulas up to sixth order in the attractive coupling parameter. We explicitly report results up to fourth order here. Pad\'e-approximant analyses in the density are carried out in each order with the intent of providing the basis necessary for extensive equation-of-state calculations of various many-fermion systems. In particular, for zero order (the fermion hard-sphere fluid) we predict an uncertainty-principle divergence in the energy, and hence pressure, corresponding to the random close packing of spheres, at a density of about 0.5 times the recently obtained value for boson hard spheres and about 0.2 times the well-known empirical value for classical hard spheres.