Jamming and crystallization in athermal polymer packings

Dense packings of chains of hard spheres possess characteristic features that do not have a counterpart in corresponding packings of monomeric spheres especially near the maximally random jammed (MRJ) state. From the modelling perspective the additional requirement that spheres keep their connectivity while maximizing the occupied volume fraction imposes severe constraints on generation algorithms of dense chain configurations. The extremely sluggish dynamics imposed by the uncrossability of chains precludes the use of deterministic or stochastic dynamics to generate all but dilute polymer packings. As a viable alternative, especially tailored chain-connectivity-altering Monte Carlo (MC) algorithms have been developed that bypass this kinetic hindrance and have actually been able to produce packings of hard-sphere chains in a volume fraction range spanning from infinite dilution up to the MRJ state. Such very dense athermal polymer packings share a number of structural features with packings of monomeric hard spheres, but also display unique characteristics due to the constraints imposed by connectivity. We give an overview of the most relevant results of our recent modeling work on packings of freely-jointed chains of tangent hard spheres about the MRJ state, local structure, chain dimensions and their scaling with density, topological constraints in the form of entanglements and knots, contact network at jamming, and entropically driven crystallization.     

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.     

Dense packing in the monodisperse hard-sphere system: A numerical study

We report a numerical study of the close packing of monodisperse hard spheres. The close packings of hard spheres are produced by the Lubachesky-Stillinger (LS) compression algorithm and span the range from the disordered states to the ordered states. We provide quantitative evidence for the claim that the density and structural order of the arrested close packing can be determined by the compression rate, i.e., with slower rates producing denser and more ordered structures. Through deeply analyzing the structure of the resulting arrested close packings, a transition region has been identified in the plane of density and reciprocal compression rate, in between what have been historically thought of as amorphous and crystalline packings. We also find clear system size dependences in studying the structural properties of the packings from the disordered ones to the ordered ones. These detailed investigations, on the structure of the arrested close packings, may provide a link between the glassy states and the crystalline states in the hard spheres.     

Phyllotactic description of hard sphere packing in cylindrical channels

We develop a simple analytical theory that relates dense sphere packings in a cylinder to corresponding disk packings on its surface. It applies for ratios R=D/d (where d and D are the diameters of the hard spheres and the bounding cylinder, respectively) up to R=1+1/sin(π/5). Within this range the densest packings are such that all spheres are in contact with the cylindrical boundary. The detailed results elucidate extensive numerical simulations by ourselves and others by identifying the nature of various competing phases.     

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.     

Entropy of jammed matter

We investigate the nature of randomness in disordered packings of frictional spheres. We calculate the entropy of 3D packings through the force and volume ensemble of jammed matter, a mesoscopic ensemble and numerical simulations using volume fluctuation analysis and graph theoretical methods. Equations of state are obtained relating entropy, volume fraction and compactivity characterizing the different states of jammed matter. At the mesoscopic level the entropy vanishes at random close packing, while the microscopic states contribute to a finite entropy. The entropy of the jammed system reveals that the random loose packings are more disordered than random close packings, allowing for an unambiguous interpretation of both limits.     

Rheology of nanosized hairy grain suspensions

Suspensions of nanosized hairy grains have been prepared by grafting long polydimethylsiloxane chains (molecular weight ) onto silica particles (radius ), dispersed into a good solvent of PDMS. Depending on the particle volume fraction, different rheological behaviors are observed. In the very dilute regime, the suspensions are perfectly stable and the particles behave almost as hard spheres: flow penetration inside the corona is then very weak. When the particle volume fraction goes to the close packing volume fraction, the suspension viscosity does not diverge as for hard spheres due to the increase of flow penetration inside the corona and to corona entanglements. The particles have then the same behavior as polymer stars having an intermediate number of arms (). Finally, in the concentrated regime (), the suspensions form irreversible gels. We shown that this unexpected gelation phenomenon is related to the presence of the silica cores: grafted PDMS chains can adsorb onto different particles and form irreversible bonds between the cores. The viscosity and elastic modulus evolutions during gelation are well described by the scalar percolation model of sol-gel transition. Received 23 March 1998     

A hybrid helical structure of hard-sphere packing from sequential deposition

Abstract

A hybrid helical structure of equal-sized hard spheres in cylindrical confinement was discovered as a ‘by-product’ of the recently developed sequential deposition approach [Physical Review E 84, 050302(R) (2011)] for constructing the densest possible packings of such systems. Unlike the conventional triple-helix structure where its three strands of spheres are packed densely to form triads of close-packed, mutually touching spheres, in this novel helical phase, only two of its three strands of spheres are packed in this densest arrangement and the overall structure resembles a hybrid of the single and the double helix. This article explains how this previously unknown structure can be constructed via the abovementioned sequential deposition of spheres, which involves manipulating the positions of a few spheres to create a template for the deposition process. The findings show that it is possible to discover new structures through varying only the configuration of the few spheres that form the template, where this approach relies on a sensitive dependence of the deposition-generated structures on the template.     

Nucleation and crystal growth in sheared granular sphere packings

We investigate the nucleation of ordered phases, their symmetries, and distributions in dense frictional hard sphere packings as a function of particle volume fraction ?, by imposing cyclic shear and constant applied pressure conditions. We show, with internal imaging, that the nucleating crystallites in the bulk consist of 10-60 spheres with hexagonal close packed (hcp) order and nonspherical shape, that are oriented preferentially along the shear axis. Above ?=0.62±0.005, crystallites with face centered cubic (fcc) order are observed with increasing probability, and ordered domains grow rapidly. A polycrystalline phase with domains of fcc and hcp order is observed after hundreds of thousands of shear cycles.     

Modified thermodynamic perturbation theory for fused–sphere dimer fluids

Wertheim's thermodynamic perturbation theory of first order (TPT1) is based on the approximation that the monomer–monomer distribution functions can be approximated by the reference fluid distribution functions regardless of the amount of bonding. This is remarkably accurate for chains formed by tangent spheres, but no longer valid for chains of fused spheres. This constitutes the reason for the inadequacy of TPT1 for fused sphere chains. We present a systematic modification of TPT1, the path integral perturbation method, that takes into account the variations of the distribution functions with extent of bonding. We demonstrate the accuracy of the theory for mixtures of hard spheres and diatomics over a range of extent of bonding (pure monomers to pure dimers) and degree of fusion (bond length 0–1). We found that the choice of reference fluid was decisive for the accuracy of the model's predictions. The proposed theory can accurately predict the properties of mixtures of hard spheres and diatomics, and of the pure fused diatomic fluids. The results from the path integral theory are in excellent agreement with simulation results, and compare favourably with the results from the Tildesley–Streett and the Boublík–Nezbeda equations of state.     

Suppressed compressibility at large scale in jammed packings of size-disperse spheres

We analyze the large-scale structure and fluctuations of jammed packings of size-disperse spheres, produced in a granular experiment as well as numerically. While the structure factor of the packings reveals no unusual behavior for small wave vectors, the compressibility displays an anomalous linear dependence at low wave vectors and vanishes when q→0. We show that such behavior occurs because jammed packings of size-disperse spheres have no bulk fluctuations of the volume fraction and are thus hyperuniform, a property not observed experimentally before. Our results apply to arbitrary particle size distributions. For continuous distributions, we derive a perturbative expression for the compressibility that is accurate for polydispersity up to about 30%.     

Do binary hard disks exhibit an ideal glass transition?

We demonstrate that there is no ideal glass transition in a binary hard-disk mixture by explicitly constructing an exponential number of jammed packings with densities spanning the spectrum from the accepted amorphous glassy state to the phase-separated crystal. Thus the configurational entropy cannot be zero for an ideal amorphous glass, presumed distinct from the crystal in numerous theoretical and numerical estimates in the literature. This objection parallels our previous critique of the idea that there is a most-dense random (close) packing for hard spheres [Torquato, Phys. Rev. Lett. 84, 2064 (2000)10.1103/PhysRevLett.84.2064].     

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.     

Non-crystalline structures in a Lennard-Jones potential

It is sometimes proposed that cunning packings of spheres could account for the structure of a metallic amorphous material. For such packings, relaxation is thought to result only in a small improvement of local distortions or stresses due to the building of the model, without changing much the radial distribution functions of the atoms.From this issue, it seems rather that any initial packing of spheres do reach (by relaxation in a Lennard-Jones potential) the f.c.c. structure or a reproducible “amorphous state”.     

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.     

Statistical verification of crystallization in hard sphere packings under densification

The performance of various structure characteristics in the task of indicating structural peculiarities in packings of hard spheres is investigated. Various characteristics based on Voronoi polyhedra, spherical harmonics, and Delaunay simplices are considered together with the pair correlation function and the mean number of r-close triples. They are applied to a set of hard sphere packings of density φ from 0.62 to 0.72. It turns out that all used structure characteristics are able to indicate changes of order from non-crystalline to crystalline packings. However, not all of them are sensitive enough to indicate different stages of structure transformation under densification. The characteristics based on Delaunay simplices turn out to be the most sensitive for this purpose. For the models considered three principal structure classes are found: packings of densities lower than the known critical value 0.64 showing a non-crystalline behavior; packings with considerable crystalline regions for φ up to 0.66–0.67; rather complete crystals although with numerous defects for φ above 0.67.     

Voronoï tessellation reveals the condensed matter character of folded proteins

The packing geometry of amino acids in folded proteins is analyzed via a modified Vorono? tessellation method which distinguishes bulk and surface. From a statistical analysis of the Vorono? cells over 40 representative proteins, it appears that the packings are in average similar to random packings of hard spheres encountered in condensed matter physics, with a quite strong fivefold local symmetry. Moreover, the statistics permits one to establish a classification of amino acids in terms of increasing propensity to be buried in agreement with what is known from chemical considerations.     

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.     

A model of binary assemblies of spheres

We propose a theoretical model of random binary assemblies of spheres at any packing fraction. We use the notion of geometrical neighborhood between grains that is defined through two generalizations of the Vorono? tessellation: the radical (or Laguerre) tessellation and the navigation map. The model is tested on different numerical packings. We find a weak local segregation for high packing fraction. We also find that the higher the size ratio of the particles, the more important the segregation. Received 19 February 2001 and Received in final form 27 June 2001