Structural and entropic insights into the nature of the random-close-packing limit

Disordered packings of equal sized spheres cannot be generated above the limiting density (fraction of volume occupied by the spheres) of ??0.64 without introducing some partial crystallization. The nature of this “random-close-packing” limit (RCP) is investigated by using both geometrical and statistical mechanics tools applied to a large set of experiments and numerical simulations of equal-sized sphere packings. The study of the Delaunay simplexes decomposition reveals that the fraction of “quasiperfect tetrahedra” grows with the density up to a saturation fraction of ?30% reached at the RCP limit. At this limit the fraction of aggregate “polytetrahedral” structures (made of quasiperfect tetrahedra which share a common triangular face) reaches it maximal extension involving all the spheres. Above the RCP limit the polytetrahedral structure gets rapidly disassembled. The entropy of the disordered packings, calculated from the study of the local volume fluctuations, decreases uniformly and vanishes at the (extrapolated) limit ?K?0.66. Before such limit, and precisely in the range of densities between 0.646 and 0.66, a phase separated mixture of disordered and crystalline phases is observed

Structure characterization of hard sphere packings in amorphous and crystalline states

The channel size distribution in hard sphere systems, based on the local neighbor correlation of four particle positions, is investigated for all volume fractions up to jamming. For each particle, all three particle combinations of neighbors define channels, which are relevant for the concept of caging. The analysis of the channel size distribution is shown to be very useful in distinguishing between gaseous, liquid, partially and fully crystallized, and glassy (random) jammed states. A common microstructural feature of four coplanar particles is observed in crystalline and glassy jammed states, suggesting the presence of "hidden" two-dimensional order in three-dimensional random close packings.Comment: 5 pages, 5 figure

Principal component component analysis of persistent homology rank functions with case studies of spatial point patterns, sphere packing and colloids

Persistent homology, while ostensibly measuring changes in topology, captures multiscale geometrical information. It is a natural tool for the analysis of point patterns. In this paper we explore the statistical power of the persistent homology rank functions. For a point pattern X we construct a filtration of spaces by taking the union of balls of radius a centred on points in X, X-a = U-x is an element of x B(x, a). The rank function beta(k)(X) : {(a, b) is an element of R-2 : a R is then defined by beta(k)(X)(a, b) = rank (l* : H-k(X-a) -> H-k(X-b)) where l* is the induced map on homology from the inclusion map on spaces. We consider the rank functions as lying in a Hilbert space and show that under reasonable conditions the rank functions from multiple simulations or experiments will lie in an affine subspace. This enables us to perform functional principal component analysis which we apply to experimental data from colloids at different effective temperatures and to sphere packings with different volume fractions. We also investigate the potential of rank functions in providing a test of complete spatial randomness of 2D point patterns using the distances to an empirically computed mean rank function of binomial point patterns in the unit square. (C) 2016 Elsevier B.V. All rights reserved

Dichteoptimierung und Strukturanalyse von Hartkugelpackungen

Bei der Verwendung von Hartkugelpackungen als Modelle für verschiedene Systeme in Physik, Chemie und den Ingenieurwissenschaften kommen einige Fragen auf, z.B. nach dem Zusammenhang zwischen der Packungsdichte und der Radienverteilung der Kugeln bzw. der Packungsstruktur. Der erste Teil dieser Arbeit beschäftigt sich mit dem Problem der optimalen Packungsdichte von zufällig dichten Packungen. Es wird ein Optimierungsalgorithmus vorgestellt, der aus einer vorgegebenen Klasse von Radienverteilungen diejenige bestimmt, für die die Packungsdichte maximal wird. Die Packungsstruktur kann man durch verschiedene statistische Größen charakterisieren, die im zweiten Teil dieser Arbeit beschrieben werden. Dabei wird die Abhängigkeit dieser Größen von der Packungsdichte und der Radienverteilung untersucht und gezeigt, dass in monodispersen Packungen mit zunehmender Dichte erhebliche strukturelle Veränderungen auftreten: Im Dichteintervall zwischen 0,64 und 0,66 erfolgt offenbar ein Übergang von ungeordneten zu kristallinen Packungen, bei weiterer Verdichtung entwickelt sich schließlich eine FCC-Struktur

Edwards statistical mechanics for jammed granular matter

International audienc

Modeling of the Structure of Disordered Metallic Alloys and Its Transformation Under Thermal Forcing

The morphology of disordered binary metallic alloys is investigated. The structure of disordered binary metallic alloys is modeled as a randomly close packed (RCP) assembly of atoms. It was observed through a 2-D binary hard sphere experiment that RCP structure can be modeled as a mixture of nano-crystallites and glassy matter. We define the degree of crystallinity as the fraction of atoms contained in nano-crystallites in an RCP medium. Nano-crystallites by size in a crystallite size distribution were determined experimentally to define the morphology of the RCP medium. Both the degree of crystallinity and the crystallite size distribution have been found to be determined by the composition of a given binary mixture. A 2-D Monte Carlo simulation was developed in order to replicate the RCP structure observed in the experiment which is then extended to cases of arbitrary composition. Crystallites were assumed to be spherical with isotropic cross sections. The number of atoms in an individual crystallite in 2-D is simply transformed into the number of atoms in 3-D; we then obtain the crystallite size distribution in 3-D. This experiment accounts for the contribution from the repulsive core of the inter-atomic potential. The attractive part of the potential is recovered by constructing spherical nano-crystallites of a given radius from a crystalline specimen of each given alloy. A structural model of a disordered alloy is thus obtained.With the basic structure of the RCP medium defined, the response to heating would be in the form of changes to the crystallite size distribution. This was first investigated in a hard sphere mechanical oven experiment. The experimental setup consists of a 2-D cell which is driven by two independent stepper motors. The motors drive a binary RCP bed of spheres on a slightly tilted plane according to a chaotic algorithmm. The motors are driven at four different speed settings. The RCP medium was analyzed using a sequence of digital images taken of the beds. The bursts of images provide a Gaussian distribution of particle speeds in x and y directions thus giving rise to the notion of temperature. This temperature scales with the motor speed settings. The measured average degree of crystallinity is found to decrease as the effective temperature was raised suggesting that nano-crystallites dissociate under thermal forcing. The evolution of a specimen\u27s structure is calculated rigorously by means of the law of mass action formalism. A system of thermal dissociation reaction equations is written out for the set of nano-crystallites according to the 3-D crystallite size distribution. The equilibrium treatment is justified because the energy differences between metastable RCP structures fall within kT. Thermal dissociation of one surface atom at a time is assumed because the energy cost in dissociation of a surface atom on a nano-crystallite is significantly less than that of a multi atom cluster. The full set of reaction equations cover all possible dissociation steps, which may amount to several thousand for a disordered alloy specimen. The primary determining factor in each of these dissociation equations is the dissociation potential or the amount of attractive energy needed to remove a surface atom on a nano-crystallite of a given size. The attractive potential between atoms is calculated using a Lennard-Jones potential between a pair of atoms for which quantum chemistry calculations exist in the literature. All interactions impinged on the surface atom by all other atoms in a crystallite are summed. As the nano-crystallites dissociate due to heating, the structure of the alloy changes, and this leads to modifications of alloy\u27s transport properties. The model is found to predict the melting temperature of various disordered binary alloys as well as refractory metals in good agreement with known data. The structure model for disordered binary alloys gives an excellent characterization of the alloy morphology. It therefore provides fruitful avenues for making predictions about how thermophysical properties of disordered binary alloys change as the alloy temperature is raised by heating