Jamming III: Characterizing randomness via the entropy of jammed matter

The nature of randomness in disordered packings of frictional and frictionless spheres is investigated using theory and simulations of identical spherical grains. The entropy of the packings is defined through the force and volume ensemble of jammed matter and this is shown to be difficult to calculate analytically. A mesoscopic ensemble of isostatic states is then utilized in an effort to predict the entropy through the definition of a volume function that is dependent on the coordination number. Equations of state are obtained relating entropy, volume fraction and compactivity characterizing the different states of jammed matter, and elucidating the phase diagram for jammed granular matter. Analytical calculations are compared to numerical simulations using volume fluctuation analysis and graph theoretical methods, with reasonable agreement. The entropy of the jammed system reveals that random loose packings are more disordered than random close packings, allowing for an unambiguous interpretation of both limits. Ensemble calculations show that the entropy vanishes at random close packing (RCP), while numerical simulations show that a finite entropy remains in the microscopic states at RCP. The notion of a negative compactivity, which explores states with volume fractions below those achievable by existing simulation protocols, is also explored, expanding the equations of state. The mesoscopic theory reproduces the simulations results in shape well, though a difference in magnitude implies that the entire entropy of the packing may not be captured by the methods presented herein. We discuss possible extensions to the present mesoscopic approach describing packings from random loose packing (RLP) to RCP to the ordered branch of the equation of state in an effort to understand the entropy of jammed matter in the full range of densities from RLP to face-centered cubic (FCC) packing.     

Polytetrahedral nature of the dense disordered packings of hard spheres

We study the structure of numerically simulated hard sphere packings at different densities by investigating local tetrahedral configurations of the spheres. Clusters of tetrahedra adjacent by faces present relatively dense aggregates of spheres atypical for crystals. The number of spheres participating in such polytetrahedral configurations increases with densification of the packing, and at the Bernal's limiting density (the packing fraction around 0.64) all spheres of the packing become involved in such tetrahedra. Thus the polytetrahedral packing cannot provide further increase in the density, and alternative structural change (formation of crystalline nuclei) begins henceforth.     

Description of the geometry of crystals with a hexagonal close-packed structure based on pair interaction potentials

The pair force interaction potential that allows one to describe a deviation from spherical symmetry, which is typical for hexagonal close-packed structures, is constructed using the ??spherically symmetric?? Mie potential that depends only on the interatomic distance. The parameters of the considered potential, which ensure the stability of hexagonal close-packed lattices, are obtained for a wide range of metals, namely, beryllium, gadolinium, hafnium, holmium, dysprosium, yttrium, cobalt, lutetium, magnesium, osmium, rhenium, ruthenium, scandium, thallium, terbium, technetium, titanium, thulium, cerium, zirconium, and erbium. It is shown that for this pair interaction potential the hexagonal close-packed structure is energetically more favorable than the face-centered cubic structure. The proposed potential can be used to perform computational experiments and analytical investigations.     

基于原子晶体构型的高压固氩中的多体相互作用

X射线衍射实验显示固氩是面心立方(fcc)晶格结构,目前对晶体氩的研究只限于两体,三体以及四体相互作用势.本文利用多体展开方法和超分子单、双(三)重激发耦合簇理论(CCSD(T))对固氩fcc晶格结构的三体和四体的几何构型、几何参数、不同体积下所有三体和四体构型的势能以及各构型所占比例等几个方面进行了准确的量子化学计算.结果表明:所有三体构型中对总的三体势能贡献最大的是构型1、构型6、构型12和构型23;三体势及其交换部分和色散部分的计算结果与现有解析经验势在长程部分符合得非常好,但在短程部分有较小差异.所有的四体构形中对总的四体势能贡献最大的是构型1,构型2,构型4,构型5,构型7和构型8;四体势及其交换势部分和色散部分的计算结果尚无解析经验势可比较.利用这些特殊构型的相关数据并结合其它构型,可拟合出更准确的三体经验势函数及其参数,也为拟合四体经验势函数及其参数提供了重要的参考价值.     

基于原子晶体构型的高压固氩中的多体相互作用

X射线衍射实验显示固氩是面心立方(fcc)晶格结构,目前对晶体氩的研究只限于两体,三体以及四体相互作用势。本文利用多体展开方法和超分子单、双 (三)重激发耦合簇理论(CCSD(T))对固氩fcc晶格结构的三体和四体的几何构型、几何参数、不同体积下所有三体和四体构型的势能以及各构型所占比例等几个方面进行了准确的量子化学计算。结果表明:所有三体构型中对总的三体势能贡献最大的是构型1、构型6、构型12和构型23;对三体势及其交换部分和色散部分的计算结果与现有解析经验势在长程部分符合得非常好,但在短程部分有较小差异。所有的四体构形中对总的四体势能贡献最大的是构型1,构型2,构型4,构型5,构型7和构型8;对四体势及其交换势部分和色散部分的计算结果尚无解析经验势可比较。利用这些特殊构型的相关数据并结合其它构型,可拟合出更准确的三体经验势函数及其参数,也为拟合四体经验势函数及其参数提供了重要的参考价值。     

Statistical Analysis of Simulated Random Packings of Spheres

This paper describes two algorithms for the generation of random packings of spheres with arbitrary diameter distribution. The first algorithm is the force‐biased algorithm of Mościński and Bargieł. It produces isotropic packings of very high density. The second algorithm is the Jodrey‐Tory sedimentation algorithm, which simulates successive packing of a container with spheres following gravitation. It yields packings of a lower density and of weak anisotropy. The results obtained with these algorithms for the cases of log‐normal and two‐point sphere diameter distributions are analysed statistically, i. e. standard characteristics of spatial statistics such as porosity (or volume fraction), pair correlation function of the system of sphere centres and spherical contact distribution function of the set‐theoretical union of all spheres are determined. Furthermore, the mean coordination numbers are analysed. These results are compared for both algorithms and with data from the literature based on other numerical simulations or from experiments with real spheres.     

Numerical Simulation of Random Close Packings in Particle Deformation from Spheres to Cubes

Variation of packing density in particle deforming from spheres to cubes is studied. A new model is presented to describe particle deformation between different particle shapes. Deformation is simulated by relative motion of component spheres in the sphere assembly model of a particle. Random close packings of particles in deformation form spheres to cubes are simulated with an improved relaxation algorithm. Packings in both 2D and 3D cases are simulated. With the simulations, we find that the packing density increases while the particle sphericity decreases in the deformation. Spheres and cubes give the minimum (0.6404) and maximum (0.7755) of packing density in the deformation respectively. In each deforming step, packings starting from a random configuration and from the final packing of last deforming step are both simulated. The packing density in the latter case is larger than the former in two dimensions, but is smaller in three dimensions. The deformation model can be applied to other particle shapes as well.     

Tuning oxygen packing in silica by nonhydrostatic pressure

The transformation of SiO2 from low pressure tetrahedral phases into denser octahedral phases takes place via the collapse of the oxygen sublattice into a close-packed arrangement. The transition paths and the resulting products are known to be affected by the presence of anisotropic stresses, which are difficult to control, so interpretation of the experimental results is problematic. Based on nonhydrostatic molecular dynamics simulations, we show that the collapse of the oxygen sublattice in the specific case of cristobalite is concomitant with the disappearance of tetrahedral units and that non hydrostatic stresses can be tuned to yield phases with different oxygen close-packed sublattices, including the alpha-PbO2-like phase, for which we provide a microscopic formation path, and phases with a cubic close packing, like anatase, not seen in experiments yet.     

Numerical Simulation of Random Close Packing with Tetrahedra

The densest packing of tetrahedra is still an unsolved problem. Numerical simulations of random close packing of tetrahedra are carried out with a sphere assembly model and improved relaxation algorithm. The packing density and average contact number obtained for random close packing of regular tetrahedra is 0.6817 and 7.21 respectively, while the values of spheres are 0.6435 and 5.95. The simulation demonstrates that tetrahedra can be randomly packed denser than spheres. Random close packings of tetrahedra with a range of height are simulated as well. We find that the regular tetrahedron might be the optimal shape which gives the highest packing density of tetrahedra.     

Self-imaging in photonic crystals in a subwavelength range

Self-imaging was observed in simulations and in microwave experiments that were performed on a face-centered cubic photonic crystal structure. The structure was composed of dielectric spheres that generally were smaller than the incident wavelength. Such self-imaging phenomena occurred along the direction of propagation and at distances smaller than the propagating wavelength.     

Experiments on random packings of ellipsoids

Recent simulations indicate that ellipsoids can pack randomly more densely than spheres and, remarkably, for axes ratios near 1.25:1:0.8 can approach the densest crystal packing (fcc) of spheres, with a packing fraction of 74%. We demonstrate that such dense packings are realizable. We introduce a novel way of determining packing density for a finite sample that minimizes surface effects. We have fabricated ellipsoids and show that, in a sphere, the radial packing fraction phi(r) can be obtained from V(h), the volume of added fluid to fill the sphere to height h. We also obtain phi(r) from a magnetic resonance imaging scan. The measurements of the overall density phi(avr), phi(r) and the core density phi(0) = 0.74 +/- 0.005 agree with simulations.     

Structural transitions in granular packs: statistical mechanics and statistical geometry investigations

We investigate equal spheres packings generated from several experiments and from a large number of different numerical simulations. The structural organization of these disordered packings is studied in terms of the network of common neighbours. This geometrical analysis reveals sharp changes in the network’s clustering occurring at the packing fractions (fraction of volume occupied by the spheres respect to the total volume, ρ) corresponding to the so called Random Loose Packing limit (RLP, ρ ～ 0.555) and Random Close Packing limit (RCP, ρ ～ 0.645). At these packing fractions we also observe abrupt changes in the fluctuations of the portion of free volume around each sphere. We analyze such fluctuations by means of a statistical mechanics approach and we show that these anomalies are associated to sharp variations in a generalized thermodynamical variable which is the analogous for these a-thermal systems to the specific heat in thermal systems.     

Properties of Moving Discrete Breathers in Beryllium

Physics of the Solid State - Discrete breathers (DBs) have been described among pure metals with face-centered cubic (FCC) and body-centered cubic (BCC) lattice, but for hexagonal close-packed...     

Crystal nucleation in the hard sphere system

The structure and growth of crystal nuclei that spontaneously form during computer simulations of the simplest nontrivial model of a liquid, the hard sphere system, is described in this work. Compact crystal nuclei are observed to form at densities within the coexistence region of the phase diagram. The nuclei possess a range of morphologies with a predominance of multiply twinned particles possessing in some cases a significant decahedral character. However the multiply twinned particles do not form from an initial decahedral core but appear to nucleate as blocks of a face-centered cubic crystal partially bounded by stacking faults.     

Dense regular packings of irregular nonconvex particles

We present a new numerical scheme to study systems of nonconvex, irregular, and punctured particles in an efficient manner. We employ this method to analyze regular packings of odd-shaped bodies, both from a nanoparticle and from a computational geometry perspective. Besides determining close-packed structures for 17 irregular shapes, we confirm several conjectures for the packings of a large set of 142 convex polyhedra and extend upon these. We also prove that we have obtained the densest packing for both rhombicuboctahedra and rhombic enneacontrahedra and we have improved upon the packing of enneagons and truncated tetrahedra.     

Crystalline—amorphous interface packings for disks and spheres

We have employed a computer simulation method for uniaxial compression to create random, but spatially inhomogeneous, disk and sphere packings in contact with exposed faces of their own close-packed crystals. The disk calculations involved 7920 movable particles, while the sphere cases utilized over 4000 particles. Rates of compression to the jamming limit were varied over two orders of magnitude, and in three dimensions this produced a clear distinction between the cases of jamming against (001) and (111) faces of the sphere crystal. Specifically, epitaxial order next to the (001) face was markedly enhanced by slowing the compression; for the (111) face the epitaxial order was quite insensitive to the compression rate.     

Packing of elastic wires in spherical cavities

We investigate the morphologies and maximum packing density of thin wires packed into spherical cavities. Using simulations and experiments, we find that ordered as well as disordered structures emerge, depending on the amount of internal torsion. We find that the highest packing densities are achieved in low torsion packings for large systems, but in high torsion packings for small systems. An analysis of both situations is given in terms of energetics and comparison is made to analytical models of DNA packing in viral capsids.     

Ground state of a dipolar crystal

We provide some of the strongest evidence to date that the ground state structure of an infinite collection of point dipoles with hardcore sphere interactions is body-centered tetragonal. The structure with the next highest binding energy is not face-centered cubic; a particular honeycomb structure has lower energy.     

Electronic structure of the Zr-He system

Ab initio calculations of the electronic structure of hexagonal close-packed and face-centered cubic zirconium with the impurity of helium atoms of about 6 at % have been performed. It has been established that the presence of helium significantly changes the electronic structure of zirconium and leads to a considerable redistribution of its electron density. Calculated values of chemical shifts of the core states of Zr atoms due to the presence of helium atoms in its lattice have been discussed.     

Applied electric field to fabricate colloidal crystals with the photonic band-gap in communication waveband

The macropore silica colloidal crystal templates were assembled orderly in a capillary glass tube by an applied electric field method to control silica deposition. In order to achieve the photonic band gap (PBG) of colloidal crystal in optical communication waveband, the diameter of silica microspheres is selected by Bragg diffraction formula. An experiment was designed to test the bandgap of the silica crystal templates. This paper discusses the formation process and the close-packed fashion of the silica colloidal crystal templates was discussed. The surface morphology of the templates was also analyzed. The results showed that the close-packed fashion of silica array templates was face-centered cubic (FCC) structure. The agreement is very good between the experimental data and the theoretical calculation.