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Organizing principles for dense packings of nonspherical hard particles: Not all shapes are created equal

Physical Review E 2012年第1期86卷 011102-011102页

作者： Salvatore Torquato Yang Jiao []Department of Chemistry Department of Physics Princeton Center for Theoretical Science Program of Applied and Computational Mathematics Princeton Institute of the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA

We have recently devised organizing principles to obtain maximally dense packings of the Platonic and Archimedean solids and certain smoothly shaped convex nonspherical particles [Torquato and Jiao, Phys. Rev. E 81, 041310 (2010)]. Here we generalize them in order to guide one to ascertain the densest packings of other convex nonspherical particles as well as concave shapes. Our generalized organizing principles are explicitly stated as four distinct propositions. All of our organizing principles are applied to and tested against the most comprehensive set of both convex and concave particle shapes examined to date, including Catalan solids, prisms, antiprisms, cylinders, dimers of spheres, and various concave polyhedra. We demonstrate that all of the densest known packings associated with this wide spectrum of nonspherical particles are consistent with our propositions. Among other applications, our general organizing principles enable us to construct analytically the densest known packings of certain convex nonspherical particles, including spherocylinders, “lens-shaped” particles, square pyramids, and rhombic pyramids. Moreover, we show how to apply these principles to infer the high-density equilibrium crystalline phases of hard convex and concave particles. We also discuss the unique packing attributes of maximally random jammed packings of nonspherical particles.

关键词： PARTICLES (Nuclear physics) PACKINGS (Chromatography) CONCAVE functions CATALANS POLYHEDRA

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Maximally dense packings of two-dimensional convex and concave noncircular particles

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Physical Review E 2012年第3期86卷 031302-031302页

作者： Steven Atkinson Yang Jiao Salvatore Torquato Department of Chemistry Department of Physics Princeton Center for Theoretical Science Program of Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA

Dense packings of hard particles have important applications in many fields, including condensed matter physics, discrete geometry, and cell biology. In this paper, we employ a stochastic search implementation of the Torquato-Jiao adaptive-shrinking-cell (ASC) optimization scheme [Nature (London) 460, 876 (2009)] to find maximally dense particle packings in d-dimensional Euclidean space Rd. While the original implementation was designed to study spheres and convex polyhedra in d≥3, our implementation focuses on d=2 and extends the algorithm to include both concave polygons and certain complex convex or concave nonpolygonal particle shapes. We verify the robustness of this packing protocol by successfully reproducing the known putative optimal packings of congruent copies of regular pentagons and octagons, then employ it to suggest dense packing arrangements of congruent copies of certain families of concave crosses, convex and concave curved triangles (incorporating shapes resembling the Mercedes-Benz logo), and “moonlike” shapes. Analytical constructions are determined subsequently to obtain the densest known packings of these particle shapes. For the examples considered, we find that the densest packings of both convex and concave particles with central symmetry are achieved by their corresponding optimal Bravais lattice packings; for particles lacking central symmetry, the densest packings obtained are nonlattice periodic packings, which are consistent with recently-proposed general organizing principles for hard particles. Moreover, we find that the densest known packings of certain curved triangles are periodic with a four-particle basis, and we find that the densest known periodic packings of certain moonlike shapes possess no inherent symmetries. Our work adds to the growing evidence that particle shape can be used as a tuning parameter to achieve a diversity of packing structures.

关键词： MAXIMAL functions DIMENSIONAL analysis CONVEX geometry CONCAVE functions PARTICLES (Nuclear physics) DISCRETE geometry POLYGONS

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Densest binary sphere packings

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Physical Review E 2012年第2期85卷 021130-021130页

作者： Adam B. Hopkins Frank H. Stillinger Salvatore Torquato Department of Chemistry Department of Physics Princeton Center for Theoretical Science Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The densest binary sphere packings in the α-x plane of small to large sphere radius ratio α and small sphere relative concentration x have historically been very difficult to determine. Previous research had led to the prediction that these packings were composed of a few known “alloy” phases including, for example, the AlB2 (hexagonal ω), HgBr2, and AuTe2 structures, and to XYn structures composed of close-packed large spheres with small spheres (in a number ratio of n to 1) in the interstices, e.g., the NaCl packing for n=1. However, utilizing an implementation of the Torquato-Jiao sphere-packing algorithm [Torquato and Jiao, Phys. Rev. E 82, 061302 (2010)], we have discovered that many more structures appear in the densest packings. For example, while all previously known densest structures were composed of spheres in small to large number ratios of one to one, two to one, and very recently three to one, we have identified densest structures with number ratios of seven to three and five to two. In a recent work [Hopkins et al., Phys. Rev. Lett. 107, 125501 (2011)], we summarized these findings. In this work, we present the structures of the densest-known packings and provide details about their characteristics. Our findings demonstrate that a broad array of different densest mechanically stable structures consisting of only two types of components can form without any consideration of attractive or anisotropic interactions. In addition, the structures that we have identified may correspond to currently unidentified stable phases of certain binary atomic and molecular systems, particularly at high temperatures and pressures.

关键词： SPHERE packings RADIUS (Geometry) PHASE transformations (Physics) ALGORITHMS TEMPERATURE effect ANISOTROPY PRESSURE

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Density of States for a Specified Correlation Function and the Energy Landscape

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Physical Review Letters 2012年第8期108卷 080601-080601页

作者： C. J. Gommes Y. Jiao S. Torquato Department of Chemical Engineering University of Liège B-4000 Liège Belgium Department of Chemistry Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics and Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA

The degeneracy of two-phase disordered microstructures consistent with a specified correlation function is analyzed by mapping it to a ground-state degeneracy. We determine for the first time the associated density of states via a Monte Carlo algorithm. Our results are explained in terms of the roughness of an energy landscape, defined on a hypercubic configuration space. The use of a Hamming distance in this space enables us to define a roughness metric, which is calculated from the correlation function alone and related quantitatively to the structural degeneracy. This relation is validated for a wide variety of disordered structures.

关键词： ENERGY levels (Quantum mechanics) CORRELATION (Statistics) MICROSTRUCTURE GROUND state (Quantum mechanics) MONTE Carlo method CONFIGURATION space QUANTITATIVE research

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Nonequilibrium static diverging length scales on approaching a prototypical model glassy state

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Physical Review E 2012年第2期86卷 021505-021505页

作者： Adam B. Hopkins Frank H. Stillinger Salvatore Torquato Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Maximally random jammed states of hard spheres are prototypical glasses. We study the small wavenumber k behavior of the structure factor S(k) of overcompressed million-sphere packings as a function of density up to the jammed state. We find both a precursor to the glassy jammed state evident long before the jamming density is reached and two associated growing length scales, one extracted from the volume integral of the direct correlation function c(r) and the other from the small-k behavior of the structure factor S(k), that can diverge at the “critical” jammed state. We also define a nonequilibrium index X and use it to demonstrate that the packings studied are intrinsically nonequilibrium in nature well before the critical state is reached. The extension of the results reported in the present work to supercooled atomic-liquid models in which the atoms interact with both repulsive and attractive forces is also discussed.

关键词： PROTOTYPES GLASS FACTOR structure DENSITY functionals NONEQUILIBRIUM thermodynamics MATHEMATICAL models CORRELATION (Statistics) CRITICAL point

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Origin of the Energy Barrier to Chemical Reactions of O2 on Al(111): Evidence for Charge Transfer, Not Spin Selection

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Physical Review Letters 2012年第19期109卷 198303-198303页

作者： Florian Libisch Chen Huang Peilin Liao Michele Pavone Emily A. Carter Departments of Mechanical and Aerospace Engineering and Chemistry Program in Applied and Computational Mathematics and Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544 USA Theoretical Division Los Alamos National Laboratory New Mexico 87544 USA

Dissociative adsorption of molecular oxygen on the Al(111) surface exhibits mechanistic complexity that remains surprisingly poorly understood in terms of the underlying physics. Experiments clearly indicate substantial energy barriers and a mysteriously large number of adsorbed single oxygen atoms instead of pairs. Conventional first principles quantum mechanics (density functional theory) predicts no energy barrier at all; instead, spin selection rules have been invoked to explain the barrier. In this Letter, we show that correct barriers arise naturally when embedded correlated electron wave functions are used to capture the physics of the interaction of O2 with the metal surface. The barrier originates from an abrupt charge transfer (from metal to oxygen), which is properly treated within correlated wave function theory but not within conventional density functional theory. Our potential energy surfaces also identify oxygen atom abstraction as the dominant reaction pathway at low incident energies, consistent with measurements, and show that charge transfer occurs in a stepwise fashion.

关键词： CHEMICAL reactions CHARGE transfer ADSORPTION SURFACE chemistry WAVE mechanics QUANTUM theory

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Microstructural degeneracy associated with a two-point correlation function and its information content

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Physical Review E 2012年第5期85卷 051140-051140页

作者： C. J. Gommes Y. Jiao S. Torquato Department of Chemical Engineering University of Liège Liège 4000 Belgium Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA

A two-point correlation function provides a crucial yet an incomplete characterization of a microstructure because distinctly different microstructures may have the same correlation function. In an earlier Letter [Gommes, Jiao, and Torquato, Phys. Rev. Lett. 108, 080601 (2012)], we addressed the microstructural degeneracy question: What is the number of microstructures compatible with a specified correlation function? We computed this degeneracy, i.e., configurational entropy, in the framework of reconstruction methods, which enabled us to map the problem to the determination of ground-state degeneracies. Here, we provide a more comprehensive presentation of the methodology and analyses, as well as additional results. Since the configuration space of a reconstruction problem is a hypercube on which a Hamming distance is defined, we can calculate analytically the energy profile of any reconstruction problem, corresponding to the average energy of all microstructures at a given Hamming distance from a ground state. The steepness of the energy profile is a measure of the roughness of the energy landscape associated with the reconstruction problem, which can be used as a proxy for the ground-state degeneracy. The relationship between this roughness metric and the ground-state degeneracy is calibrated using a Monte Carlo algorithm for determining the ground-state degeneracy of a variety of microstructures, including realizations of hard disks and Poisson point processes at various densities as well as those with known degeneracies (e.g., single disks of various sizes and a particular crystalline microstructure). We show that our results can be expressed in terms of the information content of the two-point correlation functions. From this perspective, the a priori condition for a reconstruction to be accurate is that the information content, expressed in bits, should be comparable to the number of pixels in the unknown microstructure. We provide a formula to calculate the

关键词： MICROSTRUCTURE CORRELATION (Statistics) INFORMATION theory ENTROPY (Information theory) GROUND state (Quantum mechanics) MONTE Carlo method POISSON processes

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Singularities of quantum control landscapes

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Physical Review A 2012年第1期86卷 013405-013405页

作者： Re-Bing Wu Ruixing Long Jason Dominy Tak-San Ho Herschel Rabitz Department of Automation Tsinghua University Beijing 100084 China and Center for Quantum Information Science and Technology TNList Beijing 100084 China Department of Chemistry Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Quantum control landscape theory was formulated to assess the ease of finding optimal control fields in simulations and in the laboratory. The landscape is the observable as a function of the controls, and a primary goal of the theory is the analysis of landscape features. In what is referred to as the kinematic picture of the landscape, prior work showed that the landscapes are generally free of traps that could halt the search for an optimal control at a suboptimal observable value. The present paper considers the dynamical picture of the landscape, seeking the existence of singular controls, especially of a nonkinematic nature along with an assessment of whether they correspond to traps. We analyze the necessary and sufficient conditions for singular controls to be kinematic or nonkinematic critical solutions and the likelihood of their being encountered while maximizing an observable. An algorithm is introduced to seek singular controls on the landscape in simulations along with an associated Hessian landscape analysis. Simulations are performed for a large number of model finite-level quantum systems, showing that all the numerically identified kinematic and nonkinematic singular critical controls are not traps, in support of the prior empirical observations on the ease of finding high-quality optimal control fields.

关键词： QUANTUM theory LANDSCAPES COMPUTER simulation OPTIMAL control theory ALGORITHMS

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Densest local sphere-packing diversity. II. Application to three dimensions

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Physical Review E 2011年第1期83卷 011304-011304页

作者： Adam B. Hopkins Frank H. Stillinger Salvatore Torquato Department of Chemistry and Department of Physics Princeton Center for Theoretical Science Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained for selected values of N up to N=1054. In the predecessor to this paper [A. B. Hopkins, F. H. Stillinger, and S. Torquato, Phys. Rev. E 81, 041305 (2010)], we described our method for finding the putative densest packings of N spheres in d-dimensional Euclidean space Rd and presented those packings in R2 for values of N up to N=348. Here we analyze the properties and characteristics of the densest local packings in R3 and employ knowledge of the Rmin(N), using methods applicable in any d, to construct both a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings. In R3, we find wide variability in the densest local packings, including a multitude of packing symmetries such as perfect tetrahedral and imperfect icosahedral symmetry. We compare the densest local packings of N spheres near a central sphere to minimal-energy configurations of N+1 points interacting with short-range repulsive and long-range attractive pair potentials, e.g., 12–6 Lennard-Jones, and find that they are in general completely different, a result that has possible implications for nucleation theory. We also compare the densest local packings to finite subsets of stacking variants of the densest infinite packings in R3 (the Barlow packings) and find that the densest local packings are almost always most similar as measured by a similarity metric, to the subsets of Barlow packings with the smallest number of coordination shells measured about a single central sphere, e.g., a subset of the fcc Barlow packing. Additionally, we observe that the densest local packings are dominated by the dense arrangement of spheres with centers at distance Rmin(N). In particular, we find two “maracas” packings at N=77 and N=93, each consisting of a few unjammed sphere

关键词： DENSITY functionals SPHERES MATHEMATICAL models EUCLIDEAN metric CORRELATION (Statistics) SYMMETRY

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Phase Diagram and Structural Diversity of the Densest Binary Sphere Packings

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Physical Review Letters 2011年第12期107卷 125501-125501页

作者： Adam B. Hopkins Yang Jiao Frank H. Stillinger Salvatore Torquato Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The densest binary sphere packings have historically been very difficult to determine. The only rigorously known packings in the α−x plane of sphere radius ratio α and relative concentration x are at the Kepler limit α=1, where packings are monodisperse. Utilizing an implementation of the Torquato-Jiao sphere-packing algorithm [S. Torquato and Y. Jiao, Phys. Rev. E 82, 061302 (2010)], we present the most comprehensive determination to date of the phase diagram in (α,x) for the densest binary sphere packings. Unexpectedly, we find many distinct new densest packings.

关键词： SPHERE packings PHASE diagrams SOLID geometry KEPLER'S laws CRYSTALLOGRAPHY CELLS

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