Dense and nearly jammed random packings of freely jointed chains of tangent hard spheres

Dense packings of freely jointed chains of tangent hard spheres are produced by a novel Monte Carlo method. Within statistical uncertainty, chains reach a maximally random jammed (MRJ) state at the same volume fraction as packings of single hard spheres. A structural analysis shows that as the MRJ state is approached (i) the radial distribution function for chains remains distinct from but approaches that of single hard sphere packings quite closely, (ii) chains undergo progressive collapse, and (iii) a small but increasing fraction of sites possess highly ordered first coordination shells.     

Force indeterminacy in the jammed state of hard disks

Granular packings of hard disks are investigated by means of contact dynamics which is an appropriate technique to explore the allowed force realizations in the space of contact forces. Configurations are generated for given friction coefficients, and then an ensemble of equilibrium forces is found for fixed contacts. We study the force fluctuations within this ensemble. In the limit of zero friction, the fluctuations vanish in accordance with the isostaticity of the packing. The magnitude of the fluctuations has a nonmonotonous friction dependence. The increase for small friction can be attributed to the opening of the angle of the Coulomb cone, while the decrease as friction increases is due to the reduction of connectivity of the contact network, leading to local, independent clusters of indeterminacy. We discuss the relevance of indeterminacy to packings of deformable particles and to the mechanical response properties.     

Microscopic mean-field theory of the jamming transition

Dense particle packings acquire rigidity through a nonequilibrium jamming transition commonly observed in materials from emulsions to sandpiles. We describe athermal packings and their observed geometric phase transitions by using equilibrium statistical mechanics and develop a fully microscopic, mean-field theory of the jamming transition for soft repulsive spherical particles. We derive analytically some of the scaling laws and exponents characterizing the transition and obtain new predictions for microscopic correlation functions of jammed states that are amenable to experimental verifications and whose accuracy we confirm by using computer simulations.     

Jamming in two-dimensional packings

We investigate the existence of random close and random loose packing limits in two-dimensional packings of monodisperse hard disks. A statistical mechanics approach-based on several approximations to predict the probability distribution of volumes-suggests the existence of the limiting densities of the jammed packings according to their coordination number and compactivity. This result has implications for the understanding of disordered states in the disk packing problem as well as the existence of a putative glass transition in two-dimensional systems.     

Elegance of disordered granular packings: a validation of Edward's hypothesis

We have found a way to analyze Edwards' density of states for static granular packings in the special case of round, rigid, frictionless grains assuming a constant coordination number. It obtains the most entropic density of single grain states, which predicts several observables including the distribution of contact forces. We compare these results against empirical data obtained in dynamic simulations of granular packings. The agreement is quite good, helping validate the use of statistical mechanics methods in granular physics. The differences between theory and empirics are mainly related to the coordination number, and when the empirical data are sorted by that number we obtain several insights that suggest an underlying elegance in the density of states.     

Numerical study of the stress response of two-dimensional dense granular packings

We investigate the Green function of two-dimensional dense random packings of grains in order to discriminate between the different theories of stress transmission in granular materials. Our computer simulations allow for a detailed quantitative investigation of the dynamics which is difficult to obtain experimentally. We show that both hyperbolic and parabolic models of stress transmission fail to predict the correct stress distribution in the studied region of the parameters space. We demonstrate that the compressional and shear components of the stress compare very well with the predictions of isotropic elasticity for a wide range of pressures and porosities and for both frictional and frictionless packings. However, the states used in this study do not include the critical isostatic point for frictional particles, so that our results do not preclude the fact that corrections to elasticity may appear at the critical point of jamming, or for other sample preparation protocols, as discussed in the main text. We show that the agreement holds in the bulk of the packings as well as at the boundaries and we validate the linear dependence of the stress profile width with depth.     

Unexpected density fluctuations in jammed disordered sphere packings

We computationally study jammed disordered hard-sphere packings as large as a million particles. We show that the packings are saturated and hyperuniform, i.e., that local density fluctuations grow only as a logarithmically augmented surface area rather than the volume of the window. The structure factor shows an unusual nonanalytic linear dependence near the origin, S(k) approximately |k|. In addition to exponentially damped oscillations seen in liquids, this implies a weak power-law tail in the total correlation function, h(r) approximately -r(-4), and a long-ranged direct correlation function c(r).     

Protocol dependence of mechanical properties in granular systems

We study the protocol dependence of the mechanical properties of granular media by means of computer simulations. We control a protocol of realizing disk packings in a systematic manner. In 2D, by keeping material properties of the constituents identical, we carry out compaction with various strain rates. The disk packings exhibit the strain rate dependence of the critical packing fraction above which the pressure becomes non-zero. The observed behavior contrasts with the well-studied jamming transitions for frictionless disk packings. We also observe that the elastic moduli of the disk packings depend on the strain rate logarithmically. Our results suggest that there exists a time-dependent state variable to describe macroscopic material properties of disk packings, which depend on its protocol.     

Simulation and Statistical Analysis of Random Packings of Ellipsoids

The paper describes an algorithm for the generation of random packings of ellipsoids of revolution, which is a natural generalization of an older algorithm for the case of spherical particles and belongs to the class of collective rearrangement algorithms. It yields packings with a broad spectrum of densities and geometrical properties, aspect ratio which depend on and the various parameters controlling the motions of ellipsoids during the simulation. The ellipsoid systems obtained are characterized by second‐order characteristics, namely pair correlation and orientation correlation functions. Furthermore, the constructed packings are described in a phase diagram taken from statistical physics.     

双压电变形反射镜的优化设计

利用Ｋｏｋｏｒｏｗｓｋｉ导出的自由边界条件下双压电变形镜表面位移－电压所满足的偏微分方程,运用高斯迭代法求解该方程,得出变形镜的静态响应函数矩阵ｆ（ｒ,０）,考虑到由大气湍流造丰的光学波前畸变中各泽尼克基元模式的统计权重分布,以及变形的实际装夹定位方式,对双压电变形反射镜的优化设计进行了分析。结果表明：变形的校正能力随被校正的各项泽尼克模式像差在大气湍流造成的波前畸变中所占统计权重的不同而有很大的     

The statistical mechanics of granular systems composed of spheres and elongated grains

In this paper we will apply a statistical mechanical theory of granular materials to a system composed of a mixture of elongated rods and spheres (both mono- and polydisperse). We will show for a system of rods and spheres that the system's constituents do not completely phase segregate even at minimum close packing. The generalisation of this to packings of rods and bi-disperse spheres produces a similar result except that the two species of sphere do phase segregate at some particular packing.     

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.     

Does gravity cause load-bearing bridges in colloidal and granular systems?

We study structures which can bear loads, "bridges", in particulate packings. To investigate the relationship between bridges and gravity, we experimentally determine bridge statistics in colloidal packings. We vary the effective magnitude and direction of gravity, volume fraction, and interactions, and find that the bridge size distributions depend only on the mean number of neighbors. We identify a universal distribution, in agreement with simulation results for granulars, suggesting that applied loads merely exploit preexisting bridges, which are inherent in dense packings.     

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.     

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.     

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.     

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.