Novel Features Arising in the Maximally Random Jammed Packings of Superballs

Dense random packings of hard particles are useful models of granular media and are closely related to the structure of nonequilibrium low-temperature amorphous phases of matter. Most work has been done for random jammed packings of spheres, and it is only recently that corresponding packings of nonspherical particles (e.g., ellipsoids) have received attention. Here we report a study of the maximally random jammed (MRJ) packings of binary superdisks and monodispersed superballs whose shapes are defined by |x_1|^2p+...+|x_2|^2p<=1 with d = 2 and 3, respectively, where p is the deformation parameter with values in the interval (0, infinity). We find that the MRJ densities of such packings increase dramatically and nonanalytically as one moves away from the circular-disk and sphere point. Moreover, the disordered packings are hypostatic and the local arrangements of particles are necessarily nontrivially correlated to achieve jamming. We term such correlated structures "nongeneric". The degree of "nongenericity" of the packings is quantitatively characterized by determining the fraction of local coordination structures in which the central particles have fewer contacting neighbors than average. We also show that such seemingly special packing configurations are counterintuitively not rare. As the anisotropy of the particles increases, the fraction of rattlers decreases while the minimal orientational order increases. These novel characteristics result from the unique rotational symmetry breaking manner of the particles.Comment: 20 pages, 8 figure

Controlling the Short-Range Order and Packing Densities of Many-Particle Systems

Questions surrounding the spatial disposition of particles in various condensed-matter systems continue to pose many theoretical challenges. This paper explores the geometric availability of amorphous many-particle configurations that conform to a given pair correlation function g(r). Such a study is required to observe the basic constraints of non-negativity for g(r) as well as for its structure factor S(k). The hard sphere case receives special attention, to help identify what qualitative features play significant roles in determining upper limits to maximum amorphous packing densities. For that purpose, a five-parameter test family of g's has been considered, which incorporates the known features of core exclusion, contact pairs, and damped oscillatory short-range order beyond contact. Numerical optimization over this five-parameter set produces a maximum-packing value for the fraction of covered volume, and about 5.8 for the mean contact number, both of which are within the range of previous experimental and simulational packing results. However, the corresponding maximum-density g(r) and S(k) display some unexpected characteristics. A byproduct of our investigation is a lower bound on the maximum density for random sphere packings in $d$ dimensions, which is sharper than a well-known lower bound for regular lattice packings for d >= 3.Comment: Appeared in Journal of Physical Chemistry B, vol. 106, 8354 (2002). Note Errata for the journal article concerning typographical errors in Eq. (11) can be found at http://cherrypit.princeton.edu/papers.html However, the current draft on Cond-Mat (posted on August 8, 2002) is correct

Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. II. Anisotropy in particle shape

We extend the results from the first part of this series of two papers by examining hyperuniformity in heterogeneous media composed of impenetrable anisotropic inclusions. Specifically, we consider maximally random jammed packings of hard ellipses and superdisks and show that these systems both possess vanishing infinite-wavelength local-volume-fraction fluctuations and quasi-long-range pair correlations. Our results suggest a strong generalization of a conjecture by Torquato and Stillinger [Phys. Rev. E. 68, 041113 (2003)], namely that all strictly jammed saturated packings of hard particles, including those with size- and shape-distributions, are hyperuniform with signature quasi-long-range correlations. We show that our arguments concerning the constrained distribution of the void space in MRJ packings directly extend to hard ellipse and superdisk packings, thereby providing a direct structural explanation for the appearance of hyperuniformity and quasi-long-range correlations in these systems. Additionally, we examine general heterogeneous media with anisotropic inclusions and show for the first time that one can decorate a periodic point pattern to obtain a hard-particle system that is not hyperuniform with respect to local-volume-fraction fluctuations. This apparent discrepancy can also be rationalized by appealing to the irregular distribution of the void space arising from the anisotropic shapes of the particles. Our work suggests the intriguing possibility that the MRJ states of hard particles share certain universal features independent of the local properties of the packings, including the packing fraction and average contact number per particle.Comment: 29 pages, 9 figure

Hyperuniformity and its Generalizations

Disordered many-particle hyperuniform systems are exotic amorphous states characterized by anomalous suppression of large-scale density fluctuations. Here we substantially broaden the hyperuniformity concept along four different directions. This includes generalizations to treat fluctuations in the interfacial area in heterogeneous media and surface-area driven evolving microstructures, random scalar fields, divergence-free random vector fields, as well as statistically anisotropic many-particle systems and two-phase media. Interfacial-area fluctuations play a major role in characterizing the microstructure of two-phase systems , physical properties that intimately depend on the geometry of the interface, and evolving two-phase microstructures that depend on interfacial energies (e.g., spinodal decomposition). In the instances of divergence-free random vector fields and statistically anisotropic structures, we show that the standard definition of hyperuniformity must be generalized such that it accounts for the dependence of the relevant spectral functions on the direction in which the origin in Fourier space (nonanalyticities at the origin). Using this analysis, we place some well-known energy spectra from the theory of isotropic turbulence in the context of this generalization of hyperuniformity. We show that there exist many-particle ground-state configurations in which directional hyperuniformity imparts exotic anisotropic physical properties (e.g., elastic, optical and acoustic characteristics) to these states of matter. Such tunablity could have technological relevance for manipulating light and sound waves in ways heretofore not thought possible. We show that disordered many-particle systems that respond to external fields (e.g., magnetic and electric fields) are a natural class of materials to look for directional hyperuniformity.Comment: In pres

Robust Algorithm to Generate a Diverse Class of Dense Disordered and Ordered Sphere Packings via Linear Programming

We have formulated the problem of generating periodic dense paritcle packings as an optimization problem called the Adaptive Shrinking Cell (ASC) formulation [S. Torquato and Y. Jiao, Phys. Rev. E {\bf 80}, 041104 (2009)]. Because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in $d$-dimensional Euclidean space $\mathbb{R}^d$, it is most suitable and natural to solve the corresponding ASC optimization problem using sequential linear programming (SLP) techniques. We implement an SLP solution to produce robustly a wide spectrum of jammed sphere packings in $\mathbb{R}^d$ for $d=2,3,4,5$ and $6$ with a diversity of disorder and densities up to the maximally densities. This deterministic algorithm can produce a broad range of inherent structures besides the usual disordered ones with very small computational cost by tuning the radius of the {\it influence sphere}. In three dimensions, we show that it can produce with high probability a variety of strictly jammed packings with a packing density anywhere in the wide range $[0.6, 0.7408...]$. We also apply the algorithm to generate various disordered packings as well as the maximally dense packings for $d=2,3, 4,5$ and 6. Compared to the LS procedure, our SLP protocol is able to ensure that the final packings are truly jammed, produces disordered jammed packings with anomalously low densities, and is appreciably more robust and computationally faster at generating maximally dense packings, especially as the space dimension increases.Comment: 34 pages, 6 figure

Basic Understanding of Condensed Phases of Matter via Packing Models

Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure and bulk properties of condensed phases of matter, including low-temperature states (e.g., molecular and colloidal liquids, crystals and glasses), multiphase heterogeneous media, granular media, and biological systems. The densest packings are of great interest in pure mathematics, including discrete geometry and number theory. This perspective reviews pertinent theoretical and computational literature concerning the equilibrium, metastable and nonequilibrium packings of hard-particle packings in various Euclidean space dimensions. In the case of jammed packings, emphasis will be placed on the "geometric-structure" approach, which provides a powerful and unified means to quantitatively characterize individual packings via jamming categories and "order" maps. It incorporates extremal jammed states, including the densest packings, maximally random jammed states, and lowest-density jammed structures. Packings of identical spheres, spheres with a size distribution, and nonspherical particles are also surveyed. We close this review by identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298